ANNALES
Universitatis Scientiarum
Budapestinensis
de Rolando Eotvos nominatae
SECTIO MATHEMATICA TOMUS XLIX. REDIGIT A.´ CSASZ´ AR´ ADIUVANTIBUS L. BABAI, A. BENCZUR,´ K. BEZDEK., M. BOGNAR,K.B´ OR¨ OCZKY,¨ I. CSISZAR,´ J. DEMETROVICS, GY. ELEKES, A. FRANK, J. FRITZ, E. FRIED, A. HAJNAL, G. HALASZ,´ A. IVANYI,´ A. JARAI,´ P. KACSUK, I. KATAI,´ E. KISS, P. KOMJATH,´ M. LACZKOVICH, L. LOVASZ,´ GY. MICHALETZKY, J. MOLNAR,´ P. P. PALFY,´ GY. PETRUSKA, A. PREKOPA,´ A. RECSKI, A. SARK´ OZY,¨ F. SCHIPP, Z. SEBESTYEN,´ L. SIMON, P. SIMON, GY. SOOS,´ J. SURANYI,´ L. SZEIDL, T. SZONYI,˝ G. STOYAN, J. SZENTHE, G. SZEKELY,´ A. SZUCS,˝ L. VARGA, F. WEISZ
2006
2008. februar´ 10. –13:41
ANNALES
Universitatis Scientiarum
Budapestinensis
de Rolando Eotvos nominatae
SECTIO BIOLOGICA incepit anno MCMLVII SECTIO CHIMICA incepit anno MCMLIX SECTIO CLASSICA incepit anno MCMXXIV SECTIO COMPUTATORICA incepit anno MCMLXXVIII SECTIO GEOGRAPHICA incepit anno MCMLXVI SECTIO GEOLOGICA incepit anno MCMLVII SECTIO GEOPHYSICA ET METEOROLOGICA incepit anno MCMLXXV SECTIO HISTORICA incepit anno MCMLVII SECTIO IURIDICA incepit anno MCMLIX SECTIO LINGUISTICA incepit anno MCMLXX SECTIO MATHEMATICA incepit anno MCMLVIII SECTIO PAEDAGOGICA ET PSYCHOLOGICA incepit anno MCMLXX SECTIO PHILOLOGICA incepit anno MCMLVII SECTIO PHILOLOGICA HUNGARICA incepit anno MCMLXX SECTIO PHILOLOGICA MODERNA incepit anno MCMLXX SECTIO PHILOSOPHICA ET SOCIOLOGICA incepit anno MCMLXII
2008. februar´ 10. –13:41 ANNALES UNIV. SCI. BUDAPEST., 49 (2006), 3–14
TESSELLATION-LIKE ROD-JOINT FRAMEWORKS
By
GYULA NAGY Received July
1. Introduction
One of the simplest structures in statics are the tessellation-like rod-joint frameworks including grid frameworks. Consider a rod-joint framework in the form of a rectangular grid. This framework is not rigid. How can one add diagonal braces to some of the squares in such a way as to make the framework rigid? This problem was solved generally by Maxwell [5], but in his result the time complexity of
3
n n
deciding the rigidity is O ( )where is the number of the joints. We are n interested in an algorithm with time complexity O ( ) or less. Such a result for grid bracing problem was given by Bolker and Crapo [1, 2] and also by Gasp´ ar,´ Radics, and Recski [4]. In this paper we give a new proof of Bolker–Crapo’s theorem and we consider the eight semiregular (Archimedean) tessellations, and the special hexagon bracing problem in the plane. The word “special” means that we assume the opposite edges of the regular polygon in the tessellation remain parallel during any motion of the vertices. These result are useful from an
2 n
algorithmic point of view, for instance the number of the joints is O ( )in
n case of the n grid and the size of the auxiliary graph of the framework,
2 n and hence the time complexity of the proposed algorithm is O ( ) while
6 n according to Maxwell the time complexity would be O ( ) if we use Gaussian elimination for deciding the rank of the rigidity matrix.
2008. februar´ 10. –13:24 4 GYULA NAGY
2. Tessellation-like rod-joint framework
2.1. c-graph
Define the c-graph (corresponding graph) of the framework F as follows:
the vertices of the c-graph correspond to the joints of the framework F and there is an edge between two points of c if and only if there is a rod between the corresponding two joints of the framework. Consider a rod-joint
framework in one dimension (on a line or on an arc).
Lemma A framework is rigid in one dimension if and only if its
c graph is connected
Proof The connectivity of the c-graph means we can get from every point to every other point along edges of the graph, that means, the joints of the framework can move together to the same direction with the same velocity. If the c-graph is not connected then the frameworks corresponding to its components can move independently of each other.
2.2. Square tessellation
Consider the square tessellation on the plane with unit edges. Its squares are in rows and in columns. A finite set of unit squares from the tessellation is semi convex if every line parallel with the row or the column intersects the set in an interval. Let us correspond joints to the vertices of the semi convex tessellation and correspond rods to the sides of the semi convex tessellation. Hence we get to the square tessellation-like rod-joint framework. Inserting braces in the diagonals of some squares we want to make the square grid rigid. We can see a square tessellation-like rod-joint framework
with some braces in Fig. 1. The horizontal rods of the i -th column are parallel
with each other during any motion of the joints so they can be denoted by a
x j
vector i . Similarly, the vertical rods of the -th row are parallel with each
y other during any motion of the joints so they can be denoted by a vector j . Thus we can describe the move of the square tessellation-like rod-joint framework with some vectors disregarding the translation of the framework. These vectors form an auxiliary gadget. The vectors of the auxiliary gadget can rotate independently of each other around the origin if there is no diagonal brace in the framework. We can see the auxiliary gadget of the framework of
2008. februar´ 10. –13:24
TESSELLATION-LIKE ROD-JOINT FRAMEWORKS 5 Fig
Fig. 1 on top of the right hand side in Fig. 2. We translate the vectors to top of the right direction because of the visibility.
The auxiliary graph of the braced square tessellation-like rod-joint frame-
x
work is a bipartite graph. The point i in the first point class corresponds to
x y y
j j
vector i and the point in the second point class corresponds to ,and
x y
j
an edge i exists if and only if there is a diagonal brace in the square
i j x
determined by the -th column and the -th row. In this case the vector i
y is perpendicular to vector j in the auxiliary gadget. If every square of the square tessellation-like rod-joint framework re-
mains square during any motions of the joints then the square tessellation-like
x
rod-joint framework is rigid in the plane. In this case every vector i is
y
perpendicular to every vector j in the auxiliary gadget.
Theorem The square tessellation like rod joint framework with some diagonal braces is rigid if and only if its auxiliary graph is connected
Proof The head of the vectors in the auxiliary gadget are on a unit circle and some of them are of distance 2 from each other. These vectors are originally perpendicular in the gadget. Let us construct a framework on the circle. Its joints are the head of the vectors and its rods exist if there is a diagonal brace in the corresponding square, that means these two vectors must be perpendicular to each other. The square tessellation-like rod-joint framework is rigid in the plane if and only if the former framework is rigid on the circle. This framework lies on a one dimensional circular arc. Using Lemma 2.1 this framework is rigid if and only if its c-graph is connected.
2008. februar´ 10. –13:24 6 GYULA NAGY
But this c-graph is isomorphic to the auxiliary graph, because their points correspond to the columns and to the rows of the square tessellation-like framework, and their edges correspond to the diagonal braces. If the bracing graph of the square grid framework is not connected then the square grid framework is not rigid, see the graph in the bottom of the right hand side on
Fig. 2, which is the bracing graph of the square grid framework on Fig. 1.
Fig y We can see a possible motion of the joints on Fig. 2. Since edge x2 3 is an independent component of the bracing graph, the corresponding motion of the gadget can be seen at the top of the right hand side of Fig. 2. The motion of the framework (the joints and the rods) is shown on the left hand side of Fig. 2.
This result can be generalized in several directions.
Corollary Theorem is true for non degenerated parallelogram
this consists of parallelograms and its graph is isomorphic
tessellations c to the
c graph of the square tessellation like rod joint framework instead of
square tessellation but not true for degenerated parallelogram tessellations
Corollary We regard the regular triangle tessellation as a non
degenerated parallelogram tessellation with each diagonal brace Hence the regular triangle tessellation like rod joint framework is rigid
The result of Theorem 2.1 is useful from an algorithmic point of view.
2008. februar´ 10. –13:24
TESSELLATION-LIKE ROD-JOINT FRAMEWORKS 7
The number of the joints is O mn in case of the m n
Corollary ( )
square tessellation and the size of the auxiliary graph of the framework and
O mn because the
hence the time complexity of the proposed algorithm is ( )
complexity of checking connectivity is O N where N is the number
time ( ) of the vertices and edges in the graph while according to Maxwell the time
3
mn O complexity would be (( ) )
3. Hexagon tessellation and semiregular tessellations
While studying the rigidity of the hexagon tessellation Gabor´ Fejes Toth´ suggested to consider the rigidity of all the semiregular (Archimedean) tes- sellations. The semiregular (Archimedean) tessellations have incongruent regular polygons and equivalent surrounding of the vertices. Let us denote such a tessellation by a symbol giving the number of sides of the polygons sur- rounding a vertex in their proper cyclic order. There are eight semiregular (Archimedean) tessellation in the plane as follows (3,12,12), (4,8,8), (4,6,12), (3,4,6,4), (3,3,3,4,4), (3,3,4,3,4), (3,6,3,6), (3,3,3,3,6) [3].
3.1. The semiregular (Archimedean) tessellation-like rod-joint frameworks
The polygons of a semiregular (Archimedean) tessellation are in rows. In case of the square tessellation the columns will also be called rows. Fig. 1 shows four horizontal rows and three vertical rows. Among each set of parallel rows, one will be denoted by thin line in the figure of the tessellation. A finite set of polygons from the tessellation is “row semi convex” if every line “parallel” with the rows intersects the set of the polygon in an interval. Consider a “row semi convex” tessellation. Let us correspond joints to the vertices of the polygons and rods to the sides of the polygons. Hence we obtain a semiregular (Archimedean) tessellation-like rod-joint framework.
2008. februar´ 10. –13:24 8 GYULA NAGY
3.2. The (3,3,3,3,6)-, (3,3,3,4,4)- and the (3,3,4,3,4) tessellation-like rod-joint frameworks
3.2.1. (3,3,3,3,6) tessellation
Firstly we shall consider the (3,3,3,3,6) tessellation with unit edges. Fig
It is easy to see that the (3,3,3,3,6) tessellation-like rod-joint framework is almost rigid, because there are many rigid triangles between the polygons of the tessellation (Fig. 3). Some hexagons on the boundary of the framework could be not rigid. These hexagons must be made rigid with short diagonal braces.
3.2.2. (3,3,3,4,4)- and the (3,3,4,3,4) tessellation
The rigidity of the (3,3,3,4,4)- and the (3,3,4,3,4) tessellation-like rod- joint frameworks is implied by Corollary 2.1. In Figures 4 and 5 we denote the rows by thin lines. We can see the auxiliary gadget at the top of the right and the auxiliary graph at the bottom of the right. In the auxiliary graph some edges are originally included because the common edges of the neighboring triangles form diagonal braces in the parallelogram tessellation. In the auxiliary graphs there will be new edges (denoted by thick line) if we put some diagonal braces into the square of the (3,3,3,4,4)- and the (3,3,4,3,4) tessellation-like rod-joint framework.
2008. februar´ 10. –13:24
TESSELLATION-LIKE ROD-JOINT FRAMEWORKS 9
Fig
Fig
Theorem The and the tessellation like rod joint frameworks are rigid if and only if their auxiliary graphs are connected
Proof The statement of the theorem is implied by Corollary 2.1.
Corollary In case of the framework we have to insert
into every square row one diagonal brace this result is obvious In case of
the framework only one diagonal brace is enough to connect the auxiliary graph
2008. februar´ 10. –13:24 10 GYULA NAGY
3.3. The special tessellation frameworks
3.3.1. The special (4,6,12) framework Let us consider the rest of the special semiregular (Archimedean) tes- sellation, and the special hexagon tessellation bracing problem in the plane. The word “special” means that we assume the opposite edges of the regular polygon in the tessellation remain parallel during any motion of the vertices. The special assumption is unnecessary in the former cases, but it is necessary for the rest of the special semiregular tessellations. We can construct this kind of “special” framework using some new rods or joints. In case of the hexagon we add a new joint into the center of the hexagon, and three new rods from the center joint to every second one of the hexagon joints. To make these tessellations rigid we use bracing elements along the shortest diagonals of the polygons because the two neighboring sides and the shortest diagonals form triangles. Hence the two neighboring sides cannot rotate around the common joint. The rigidity problems of these tessellations are similar to each other, hence for simplicity we shall restrict our consideration to one of the former frameworks, for example the (4,6,12) framework. The drawing of the frame- work and auxiliary graph of the (4,6,12) tessellation framework are shown in Fig. 6. Given a (4,6,12) tessellation with unit edges. Its polygons are in parallel rows that are in six kinds of different direction. The six directions are denoted by thin lines. The special assumption implies the next statement. The rods of a row that are perpendicular to the direction of the row are parallel with each other during any motion of the joints so they can be denoted by a vector. Thus we can describe the move of the special (4,6,12) tessellation-like rod-joint framework with some vectors disregarding the translation of the framework only. These vectors form an auxiliary gadget. The vectors of the auxiliary gadget can rotate independently each other around the origin if there is no diagonal brace in the framework. The auxiliary graph of the braced special
(4,6,12) tessellation-like rod-joint framework is a sixpartite graph. The point
a a b
i j
i in the first point class corresponds to vector and the point in the
b a b
i j
second point class corresponds to j , and an edge exists if and only if
a b
j
there is a short diagonal brace between the i -th row and -throw.Inthis
a b
j case the head of vector i is braced to vector in the auxiliary gadget. The
2008. februar´ 10. –13:24
TESSELLATION-LIKE ROD-JOINT FRAMEWORKS 11 Fig auxiliary gadget is rigid if and only if the special (4,6,12) tessellation-like
rod-joint framework is rigid in the plane.
Theorem The special regular hexagon the and the
tessellation like rod joint framework with some diagonal braces is rigid if and only if its auxiliary graph is connected
Proof This proof is similar to that of Theorem 1.1. The head of the vectors in the auxiliary gadget are on a unit circle. Let a framework be on this circle. Its joints are the heads of the vectors and its rods exist if there is a diagonal brace in the corresponding rows, that means these two vectors can move together. The (4,6,12) tessellation-like rod-joint framework is rigid in the plane if and only if the former framework is rigid on the circle. This framework lies on a one dimensional circular arc. Using Lemma 1.1 this framework is rigid if and only if its c-graph is connected. But the c-graph is isomorphic to the auxiliary graph of the framework, because their points correspond to the rows of the framework, and their edges correspond to the diagonal bracing. If the bracing graph of the square grid framework is not connected then the square grid framework is not rigid.
2008. februar´ 10. –13:24 12 GYULA NAGY
3.3.2. The (3,6,3,6), (3,12,12) and (3,4,6,4) special framework Let us consider the (3,6,3,6), (3,12,12) and (3,4,6,4) special semiregular (Archimedean) tessellation bracing problem in the plane. The rigidity problems of these tessellations are similar to each other, hence for simplicity we shall restrict our consideration to one of the former frameworks for instance the (3,4,6,4) framework. The illustration of the framework, auxiliary gadget and auxiliary graph of tessellation (3,4,6,4) framework are shown on Fig. 7. The three different
directions of the row are denoted by thin lines. Fig
We can describe the move of the special (3,4,6,4) tessellation-like rod- joint framework with some vectors disregarding the translation of the frame- work only. These vectors form the auxiliary gadget of the framework. The vectors of the squares and the hexagons can rotate independently each other around the origin if there is no diagonal brace in the framework, but the vec- tors of the triangles denoted by capital letter can move together only, because
they form a rigid triangle. The auxiliary graph is a sixpartite graph. The
a a b c
i j k
point i in the first point class corresponds to vector and the points ,
b c
k
in the second and the third point classes correspond to vector j and respec-
a b
j
tively, and an edge i exists if and only if there is a short diagonal brace
a b a
j i
between the i -th row and the -th row. In this case the vector is braced to
b A
vector j in the auxiliary gadget. The vectors of the triangles denoted by , C B and are braced with each other, hence between the corresponding points of them there are edges in the auxiliary graph. The auxiliary gadget is rigid if
2008. februar´ 10. –13:24 TESSELLATION-LIKE ROD-JOINT FRAMEWORKS 13 and only if the special (3,4,6,4) tessellation-like rod-joint framework is rigid
in the plane.
Theorem The special and tessellation
like rod joint framework with some diagonal braces is rigid if and only if its auxiliary graph is connected
Proof The proof is similar to that of Theorem 3.2.
Corollary Theorems and are true for non degenerated
special a ne regular hexagon tessellation like rod joint frameworks and the
a ne semiregular tessellation like rod joint frameworks respectively these graphs are isomorphic to the
consist of a ne regular polygons and their c c
graphs of the special regular hexagon tessellation like rod joint frameworks and the semiregular tessellation like rod joint frameworks respectively
The result of Theorems 3.1, 3.2, 3.3 are useful from an algorithmic point
of view:
Corollary Let n be the sum of the number of the rows in the
di erent directions Hence n equals the number of the points of the auxiliary
graph of the special tessellation like rod joint framework The time complex
ity of the rigidity algorithm of the former framework braced with diagonals
2
is linear in the size of the graph hence O because the time complexity
( n )
O N where N is the number of the vertices and
of checking connectivity is ( ) edges in the graph
Acknowledgment I thank Andras´ Recski for his help in developing this work. Research partially supported by the Hungarian National Research Fund and by the National Office for Research and Technology (Grant Number OTKA 67651).
References Env Plan B
[1] E D Bolker Bracing grids of cubes, 4 (1977), 157–172.
H Crapo SIAM J Appl [2] E D Bolker and Bracing rectangular frameworks I,
Math 36(3) (1979), 473–490.
oth Regular Figures [3] L Fejes T , Pergamon Press, Oxford (1964).
2008. februar´ 10. –13:24
14 GYULA NAGY: TESSELLATION-LIKE ROD-JOINT FRAMEWORKS
aspar N Radics A Recski [4] Zs G and Square Grids with Long “Diagonals”,
Optimizations Methods and Software 10 (1998), 217–231. Phil Maga [5] J C Maxwell On Reciprocal Figures and Diagrams of Forces,
sine 27 (1864), 250–261. Acta Technica Acad Sci [6] Gy Nagy Diagonal bracing of special cube grids,
Hung 106(3–4) (1994), 265–273.
t Proc st Japanese Hungarian [7] N Radics Rigidity of -story Buildings, in:
Symposium on Discrete Mathematics and Its Applications (1999), 181–187. Discrete Mathemat [8] A Recski Bracing Cubic Grids – A Necessary Condition,
ics 73 (1988/89), 199–206.
Matroid Theory and its Applications in Electric Network Theory [9] A Recski
and in Statics,Akademiai´ Kiado,´ Budapest and Springer, Berlin (1989).
Gyula Nagy Szent Istvan´ University Ybl Miklos´ Faculty of Architecture and Civil Engineering Budapest
Hungary nagygyu ymmf hu
2008. februar´ 10. –13:24 ANNALES UNIV. SCI. BUDAPEST., 49 (2006), 15–24
IMPROVING SIZE-BOUNDS FOR SUBCASES OF SQUARE-SHAPED SWITCHBOX ROUTING*
By
ANDRAS´ RECSKI, GABOR´ SALAMON and DAVID´ SZESZLER´ Received June
1. Introduction
Several different models are known for the detailed routing phase of VLSI-design [6], namely, for the case when the position of pins to be in- terconnected is given already, and the connections must be established using a minimal size cubic grid. Regarding either the position of the pins, or the routing model of the layers, we can distinguish between different problems and models. Although these problems are under intensive research, the exact number of layers needed is still not known for most cases. In this paper we focus on routing problems having square-shaped layers. We summarize the earlier results, upper and lower bounds given for the differ- ent subcases of the Switchbox Routing problem. Some results are improved to decrease the gap between the bounds.
2. Basic definitions
G tracks
A switchbox is a rectangular grid consisting of horizontal (num-
columns h
bered from 0 to w + 1) and vertical (numbered from 0 to +1),
width h height
where w is the and is the of the switchbox. In this paper, we
w h G assume square shaped layers, namely = . The boundary points of are
called terminals. Depending on which boundary of the switchbox they are
* This work was supported by the Grant No. OTKA T042559 of the Hungarian National Science Foundation.
2008. februar´ 10. –13:24
16 ANDRAS´ RECSKI, GABOR´ SALAMON, DAVID´ SZESZLER´
Southern Eastern Western situated on, the terminals are called Northern, , or . The “corners” of the switchbox are not regarded as terminals, routings must
not use them either. switchbox routing problem
A net is a collection of terminals. A is a set
N fN N g
= 1 n of pairwise disjoint nets.
solution H fH H g The of a routing problem is a set = 1 n of pairwise
vertex disjoint subgraphs (also called wires, these are usually Steiner trees)
k H N
i
of the -layer rectangular grid such that i connects the terminals of for n i =1 . The wires can access the terminals on any layer. Edges of the
wires that join adjacent vertices of two consecutive layers are called vias.
A specific routing problem can be solved either in the unconstrained
layer model Manhattan model
k ,orinthe .
k layer model
In the unconstrained no further restrictions are applied to
H Manhattan model layers or to subgraphs i . Contrarily, in the multilayer consecutive layers must contain wire segments of different directions. Thus, in this model, layers with horizontal (East–West) and with vertical (North– South) wire segments alternate. According to the position of terminals, several subcases of the switchbox routing problem are considered, as follows:
Single Row Routing problem, if all terminals are Northern;
Channel Routing problem, if all terminals are either Southern or North- ern;
Γ-Routing problem, if all terminals are either Northern or Western;
C-Routing problem, if all terminals are either Southern or Western or Northern;
General Switchbox problem, if terminals are situated on all four bound- aries of the grid.
3. Square-shaped switchbox routing
In this paper, a survey is given on the results concerning the number of layers needed in both models. Table 1 shows the different routing problems and gives the most important references. Our new results on C-shape routing are printed in bold face letters. Note that the figures in the table show the minimum number of layers that
enables to solve any routing problem. Hence, some specific problems can be solved using fewer layers than the corresponding universal bound. On the
2008. februar´ 10. –13:24 IMPROVING SIZE-BOUNDS FOR SUBCASES OF SQUARE-SHAPED SWITCHBOX ROUTING 17 other hand, obviously, any upper bound in the Manhattan model is an upper bound in the unconstrained model, and any lower bound in the unconstrained model is a lower bound in the Manhattan model.
Table Known bounds for the minimum number of layers in different square-shaped routing problems Manhattan model Unconstrained model
=2[2] =2 [2,8,12]
2 =3[2,5]
3 [7, 8, 10]
3 [1, 12] 2
4 3 [12]
4 3 5 [11] 5 [11]
4[1] = 6 [3, 11]
6 [11]
Before giving the details of these results, we recall a theorem of Hambr-
h
usch [4]. There exists a switchbox routing problem of the w grid which
w h
m m
cannot be solved using k layers, where =max .
w h
This theorem implies the restriction of any further analysis to the special m
case where m has a bounded value. This paper focuses on the case of =1, w namely h = . A simple algorithm for Single Row Routing in the 2-layer Manhattan model is based on the fundamental studies of Gallai [2]. Further results on the effect of using more layers are given in [9], while finding the minimum wire-length solution turned out to be NP-hard [12]. Channel Routing problems cannot always be wired on two layers in the Manhattan model, as Fig. 1(a) shows an example. Note that this example can be extended to an arbitrary sized one. However, three layers are always enough in the Manhattan model [2, 5], and even two layers can be enough in the unconstrained model [7, 8, 10]. Wiring a Gamma Routing problem in the Manhattan model can require at least three layers [12], and can be trivially solved on four layers in the
2008. februar´ 10. –13:24 18 ANDRAS´ RECSKI, GABOR´ SALAMON, DAVID´ SZESZLER´
12 12
4 3
3 4
2 1 2 1 (a) 3 layers are needed for (b) 6 layers are needed for Channel Routing Switchbox Routing
Fig Examples where lower bounds are sharp in the Manhattan model
Manhattan model. For the sake of completeness we mention here a short
proof of this latter claim.
Proposition Every square shaped Gamma Routing problem can be wired on four layers in the Manhattan model
Proof Let us have four layers, the first and the third are used for vertical wire segments only, while the second and fourth are used for horizontal segments only. For all terminals of the Western side introduce a horizontal wire segment alongside the whole height of the grid on the second layer. Similarly, for all terminals of the Northern side introduce a vertical wire segment alongside the whole width of the grid on the third layer. If a net has at least one terminal on the Western side and at least one on the Northern side, connect their corresponding wire segments with a via hole between the second and third layers.
If a net has terminals only on the Western side, we can use the first layer w to connect its corresponding wire segments on the second layer. As at most 2 nets can have this property, the Gallai-algorithm can solve this using a single vertical layer. Similarly, the fourth layer is sufficient to accomodate all horizontal seg- ments for nets having terminals on the Northern side only. Gamma Routing problems are always solvable on three layers in the unconstrained model. Furthermore, the following theorem can be stated as
a straightforward consequence of the method in [12].
Theorem Every Gamma Routing problem can be solved in the un
constrained model using three layers Moreover if there is no net having
2008. februar´ 10. –13:24
IMPROVING SIZE-BOUNDS FOR SUBCASES OF SQUARE-SHAPED SWITCHBOX ROUTING 19
at least two Northern and at least two Western terminals then such a wiring exists that uses the third layer for a single net only
Szeszler´ [11] gave an efficient linear time algorithm to solve the general
m e Switchbox problem in the Manhattan model using at most 2d +4layers. This yields a 6-layer solution for every square-shaped switchbox routing
problem. Moreover, the construction includes a 5-layer solution for all C- h shaped routing problem where w = . As the problem of Fig. 1(b) cannot be solved on five layers [3], this bound is sharp. Note that for this case, only a small size example is known, unlike in the case of Gamma Routing, where we have examples of arbitrary size. In the unconstrained model, no such example is known, but a lower bound of four layers is already proved. As mentioned above, Szeszler’s´ algorithm gives a trivial way to solve C-shaped routing problems on five layers. Our new results give a lower bound of four layers in the Manhattan model and a lower bound of three layers in
the unconstrained model as stated in the following theorems:
For all w there exists a C shaped routing problem
Theorem 3
on the w w grid which cannot be solved on three layers in the Manhattan model
Proof Suppose that the routing problem of Fig. 2 can be solved on 3
l l
v layers in the Manhattan model. Let h ,and denote the number of horizontal,
and vertical layers, respectively. Fig 4 layers are needed in the Manhattan model
2008. februar´ 10. –13:24
20 ANDRAS´ RECSKI, GABOR´ SALAMON, DAVID´ SZESZLER´ f
Consider a vertical cut e and a horizontal cut ,asshowninFig.2.Each
w wl
of them crosses grid edges on each layer. Thus, wires of at most h nets
e wl f
can cross ,andatmost v nets can cross .
3 w
However, in this example, all 2 nets must be wired through f ,and
w k w k
through e , as well. Note that if =4 +1or =4 + 2, then wires of net
w 4 may not cross e .
Since
3 w
wl
v and
2
3 w
w
1 if 3
wl 2
h w
4 if =3
l l
v we obtain h 2and 2, proving that at least two horizontal and at least two vertical layers are needed for wiring this problem.
In the following, we show that C-shaped routing problems cannot always
be solved on 2 layers, even in the unconstrained model.
w w Consider a C-shaped routing problem of size w (Fig. 3), where is
an arbitrary non-negative even integer. Fig 3 layers are needed in the unconstrained model
To prove the unsolvability of this problem on 2 layers we introduce the
SW NW
k concepts of k -corner and -corner.
2008. februar´ 10. –13:24
IMPROVING SIZE-BOUNDS FOR SUBCASES OF SQUARE-SHAPED SWITCHBOX ROUTING 21
w
Definition k SW A South-Western corner of size 1 2 ( k -
corner) is a set of grid-points containing a point ( i j ) if and only if
i j k
= 0 AND (Southern part of Western terminals) OR
i k j
AND = 0 (Western part of Southern terminals) OR
i j k i j k
1 AND + + 1 (internal points).
w
Definition k NW A North-Western corner of size 1 2 ( k -
corner) is a set of grid-points containing a point ( i j ) if and only if
i j w k
= 0 AND (Northern part of Western terminals) OR
i k j w
AND = + 1 (Western part of Northern terminals) OR
i k w k j w j i w k 1 AND AND (internal points).
Note that each of these corners defines a cut in the grid. Fig. 4 shows an
NW w SW4-corner and a 3-corner of a grid for =8.
Fig Example of corners
Definition border edge SW NW
k An edge is a of the k -corner ( -
corner), if it has exactly one end-point in the corner.
SW NW border point
k
A point of the k -corner ( -corner) is a of the corner,
SW NW
k
if there is a border edge of the k -corner ( -corner) incident to it.
leaves SW NW
k (Wiring of) a net the k -corner ( -corner), if it has at least one terminal inside and at least one terminal outside the corner.
2008. februar´ 10. –13:24
22 ANDRAS´ RECSKI, GABOR´ SALAMON, DAVID´ SZESZLER´
SW NW
k At first, some basic propositions are proved on the k -corners ( -
corners) of any feasible layer solution of our example. From this point we
focus on the properties of such a solution.
Proposition An SW corner NW corner is left by exactly k
k
k 2
nets and has exactly border edges on both layers Furthermore each
2 k
border point of an SW corner NW corner is used by the wiring of exactly
k k two nets in any feasible solution
Proof On one hand, the first part of the proposition is a trivial conse-
quence of the construction of our example. On the other hand, the facts that
k SW NW
k 2 nets must leave the k -corner ( -corner), and that the number of
border points is k imply the second part.
Proposition Wires leaving terminals k and w k in the SW
+1 k
corner NW corner corner NW corner must not enter the SW
k
k
k 1 1
SW NW
Proof The -corner ( -corner) must be left by wires of k
k 1 1
2( k 1) other nets, which occupy all of its border points, and so wires
w k SW NW
realizing nets k and +1 cannot even enter the -corner ( - k k 1 1
corner).
Proposition If a solution exists then on each border edge of the
SW corner NW corner this solution uses exactly one layer
k k
Proof k k SW
At first, consider the edges (1 )–(1 +1)ofthe k -corner. k
As wires of nets k and + 1 must both enter the grid, the only possibility k to use both layers on this edge would be to route wires of nets k and +1
through it. However, as it was shown above, wires of net k + 1 cannot enter
SW
the k -corner.
k k SW
The same logic can be applied to the edge ( 1)–( +1 1) of the k -
k w w k w k
corner, and edges ( k w)–( +1 ), and (1 )–(1 +1)ofthe
NW
k -corner.
SW Let us consider the pairs of border edges of the k -corner leaving it from (or to) a common end-point. On these pairs of edges one can route the wires of at most two nets, due to their common point. Hence, it is easy to
see that if a solution uses two layers on any of them, then it should use two
k k k
layers on one of the edges (1 k)–(1 +1),or( 1)–( +1 1) to enable all
SW
nets to leave the k -corner. As this is not allowed, the only way to let all
SW
nets leave the k -corner is to use exactly one layer on each edge of it.
NW Note that the same logic can be applied for the k -corner.
2008. februar´ 10. –13:24 IMPROVING SIZE-BOUNDS FOR SUBCASES OF SQUARE-SHAPED SWITCHBOX ROUTING 23
As a result, one can see that there is a bijection between the border edges
w w NW of the SW -corner ( -corner) and the wires of nets leaving it.
2 2
w
As there is a common border edge of the two corners, namely (1 2 )
w
w
m SW (1 + 1), a single net must leave here both the -corner, and the
2 2 w NW -corner. 2
Let us define a cut C by taking the union of the points of these corners.
w SW All nets except m leave this cut twice, namely both from the -corner,
2 w and the NW -corner.
2 C Net m should leave cut in any feasible solution, but there is no room to do this, which contradicts the fact that a feasible 2-layer solution exists.
As the value of w can be arbitrary large in the above example, a C-shaped routing of any size can be unsolvable in the 2-layer unconstrained model.
Note that the same considerations can be taken when w has an odd
SW NW w value. In this case, w -corner ( -corner) is defined by adding
2 2
w
SW NW w point (1 )tothe w -corner ( -corner). The union of the
2 2 2
SW NW w w w -corner and the -corner should be left by the wires of all
2 2 nets but it has only w 1 available border edges.
This implies the following theorem:
For all w there exists a C shaped routing problem
Theorem 3
on the w w grid which cannot be solved on two layers in the unconstrained model
4. Summary
Our new results improve lower bounds for the number of layers needed to solve any C-shaped routing problem. Namely, we gave two problem instances that cannot be solved using less than 4 or 3 layers in the Manhattan or in the unconstrained model, respectively. Both can be extended to an arbitrary large size. However, for 6 problems out of 10 mentioned here (see Table 1) the exact number of needed layers is still not known; this can be a subject of further research.
2008. februar´ 10. –13:24 24 RECSKI, SALAMON, SZESZLER:´ SIZE-BOUNDS FOR SQUARE-SHAPED SWITCHBOX ROUTING
References F Wettl [1] E Boros A Recski and Unconstrained multilayer switchbox rout-
ing, Annals of Operations Research 58 (1995), 481–491.
Unpublished results A Hajnal J Suranyi [2] T Gallai , announced in: and
Uber¨ die Auflosung¨ von Graphen in vollstandige¨ Teilgraphen, Ann Univ
otvos Sect Math
Sci Budapest E 1 (1958), 115–123. A Recski [3] B Golda B Laczay Z Mann Cs Megyeri and Implementation
of VLSI routing algorithms, in Proc of INES International Confer
ence on Intelligent Engineering Systems, pp. 383–388, 2002. IEEE Trans Computer [4] S E Hambrusch Channel routing in overlap models,
Aided Design of Integrated Circ Syst CAD-4 (1985), 23–30. J Stevens [5] A Hashimoto and Wire routing by optimizing channel assignment,
in Proc of the Eighth Design Automation Conference, pp. 214–224, 1971. Combinatorial Algorithms for Integrated Circuit Layout [6] T Lengauer , Teub-
ner, Stuttgart and Wiley, Chicester, 1990.
Marek Sadowska E S Kuh [7] M and General channel-routing algorithm,
Proc of the IEE GB , 130 (1983), 83–88. Com [8] A Recski Minimax results and polynomial algorithms in VLSI routing,
binatorics Graphs Complexity Ann Discrete Math pp. 261–273, 1992.
[9] A Recski Some polinomially solvable subcases of detailed routing problem
in VLSI design, Discrete Applied Mathematics 115 (2001), 199–208. F Strzyzewski
[10] A Recski and Vertex-disjoint channel routing on two lay-
programming and combinatorial optimization R Kannan
ers, in Integer eds and W R Pulleyblank , pp. 397–405, University of Waterloo Press,
Waterloo, Ontario, 1990.
er Ann Univ
[11] D Szeszl Switchbox routing in multilayer manhattan model,
otvos Sect Math
Sci Budapest E 40 (1997), 155–164. Polynomial algorithms in VLSI routing [12] T Szkaliczki , PhD thesis, Technical University of Budapest, 1998.
Andras´ Recski, Gabor´ Salamon, David´ Szeszler´ Department of Computer Science and Information Theory Budapest University of Technology and Economics
Hungary
gsala cs bme hu szeszler cs bme hu recski cs bme hu, ,
2008. februar´ 10. –13:24 ANNALES UNIV. SCI. BUDAPEST., 49 (2006), 25–42
SEMI-SEPARATED SETS IN GENERALIZED CLOSURE SPACES
By
MARCELINA MOCANU Received August
1. Introduction X
In what follows, we denote by P( ) the power set of a non-empty
u P X P X
set X . An arbitrary set-valued set function : ( ) ( ) is called
X u generalized
generalized closure operator and the pair ( )issaidtobea
space u GCO X u GCS
closure (briefly, is a and ( )isa ). The dual of the
u interior operator u P X P X
operator u is the , -Int : ( ) ( )definedby
A X n u X n A
u -Int( )= ( ).
X u closed closed in X u
AsetA is said to be or ( ) (respectively,
u closed almost closed in X u u A A u A
almost or ( )) if ( )= (respectively, ( )
A A X u open open in X u
). A set is said to be or ( ) (respectively,
u open almost open in X u A u A A
almost or ( )) if = -Int( ) (respectively,
u A -Int( )). Due to its generality, the notion of generalized closure operator, viewed as a link between topology and algebra, has an increasing importance for applications [8].
In a topological space (X ), with the usual closure operator Cl, two
B A B A B subsets A and are said to be semi-separated if Cl( ) = Cl( )= . A subset of a topological space is connected if and only if it is not the union of two non-empty semi-separated sets. In the study of various forms of connectedness, the notion of pair of
semi-separated sets is an important prerequisite. The notion of pair of - separated sets [3], which generalizes and unifies several modifications of the
Mathematics Subject Classi cation : 54A05, 54A10
2008. februar´ 10. –13:24 26 MARCELINA MOCANU semi-separation property in topological spaces [10], [15], [13], is a particular case of the notion of pair of semi-separated sets in a GCS. The aim of the present paper is to prove the basic properties of semi-separated sets in generalized closure spaces, under minimal assumptions on the corresponding generalized closure operators. The fact that our assumptions can not be weakened is proven by examples, for almost every result. We give general sufficient conditions for semi-separation, and, in some special cases, necessary conditions for semi-separation. Characterizations in terms of semi-separation of complete graphs and connected graphs, as well as of continuous functions on graphs, are given. We study the invariance of semi-separation property to the relativization to a subspace. We prove that the preimages ot two semi-separated sets, under a continuous function between GCS’s, are semi- separated. A similar invariance result is proven for images under an almost open injective function between GCS’s.
2. Preliminaries
2.1. Generalized closure operators
P X P X
Let u v : ( ) ( ).
v u u v
Remark (i) = -Int if and only if = -Int. u
(ii) A set is u -closed (respectively, almost -closed) if and only if its u
complement is u -open (respectively, almost -open). u
(iii) The family of u -closed (respectively, -open) sets is contained in u the family of almost u -closed (respectively, almost -open) sets. The corre-
sponding families agree if u is expansive.
u u
(iv) X is almost -closed, the empty set is almost -open.
u isotone A B X u A u B
Definition is called if implies ( ) ( ),
A u A A X idempotent u u A u A
expansive if ( ) for every , if ( ( )) = ( )for
X sub linear u A B u A u B A B X
every A , if ( ) ( ) ( ) whenever .
P X P X
Note that u : ( ) ( ) is isotone (respectively, idempotent) if and u
only if u -Int is isotone (respectively, idempotent), while is expansive if and
A A A X only if u -Int( ) for every . The properties of isotony and expansivity are independent, as the follow- ing examples show.
2008. februar´ 10. –13:24
SEMI-SEPARATED SETS IN GENERALIZED CLOSURE SPACES 27
X x X
Example Let be a set with at least two elements and 0 .
P X P X u A A nfx g A X u
Define u : ( ) ( )by ( )= 0 for all .Then is
u A x A u
isotone and A ( ) if and only if 0 , consequently is not expansive.
X fa b cg u P X P X
Example Let = and : ( ) ( ) be such that
A A A X u fc g fa cg
u ( )= for every , with the following exceptions: ( )=
fb g fb cg u u u fc g
and u ( )= .Then is expansive. is not isotone, since ( )is
fb cg
not included in u ( ).
If u P X P X is isotone the following proper
Proposition : ( ) ( )
ties are equivalent
u is expansive
u A A for every non empty subset A of X
( )
u
Proof (1) (2): Obviously, even if is not isotone.
u A X
(2) (1): Assume that is not expansive. Then there exists
A n u A u B u A u B B such that B := ( ) . By isotony, ( ) ( ), hence ( ) = ,
i.e., (2) is false.
u u
Remark If is isotone, then every union of almost -open sets u
(every intersection of almost u -closed sets) is almost -open (respectively,
u u
almost u -closed). If is isotone and expansive, then every union of -open
u u
sets (every intersection of u -closed sets) is -open (respectively, -closed). u
If u is isotone, expansive and idempotent, then the union of all -open sets
X u A
contained in an arbitrary set A is -Int( ) and the intersection of all
A u A
u -closed sets containing is ( ).
C P X
Example If ( ) is a closure system, i.e., a non-empty family
P X P X
closed to arbitrary intersections, one defines c : ( ) ( )by
A fB B C A B g
c ( )= : , C The set operator c , called the GCO generated by , is expansive, isotone
and idempotent. The given closure system C agrees with the family of all
X u P X P X c -closed subsets of . Conversely, if : ( ) ( ) is expansive and
isotone, then the family of u -closed sets is a closure system, that generates u
the GCO u provided that is, in addition, idempotent.
X A X u A
Example Let be a vector space. For every let ( )be u the linear hull (respectively, the convex hull) of A. Then the -closed sets are
2008. februar´ 10. –13:24
28 MARCELINA MOCANU X exactly the linear subspaces of X (respectively, the convex subsets of ) and,
in both cases, u is expansive, isotone and idempotent, but it is not sub-linear.
Next we discuss the connections between some closure operators intro-
duced in [9], [3], [13].
Example m X f Xg
Let X be a minimal structure on , i.e.,
m P X A X m A
X
( ) [7]. For every subset of ,the X -closure of and the
m A m A fB A B X n B m g
X X
X -interior of are defined by -Cl( )= :
m A fC C A C m g m
X X
and X -Int( )= : , respectively. Then -Cl
m
is expansive, isotone and idempotent, and its dual X -Int is isotone and
u m
idempotent [6], [9]. If = X -Cl, then the family of all open sets in the
X u m
GCS ( ) is the family of all unions of sets that belong to X [7].
P X
Remark Consider, as in [2] and [3], an isotone operator : ( )
X A X A A open
P( )andthesets with ( ), called . The family of all generalized topology -open sets is a [2], i.e., contains the empty set and is
closed to arbitrary unions. The complements of all -open sets form a closure c
system, that generates a GCO denoted in [3] by , which is expansive,
X
isotone and idempotent. A subset A of is -open in the sense of [2], [3] if
A X c X X
and only if is open in the GCS ( ). If ( )= , then the family of all
m c m
X
-open sets is a minimal structure X and = -Cl. X Remark Assume that ( ) is a topological space, with the closure
operator and the interior operator denoted by Cl and Int, respectively. For
= Int (respectively, =Cl Int, =Int Cl, =Int Cl Int, =
A X A
=Cl Int Cl) a set is -open if and only if is open (respectively,
semi-open [5], preopen [11], -open [12], -open [1]).
P X P X associated with
In [13], a set operator : ( ) ( )issaidtobe
topology U U U A X
the if ( )forall ,andaset is said to be
U U A U
-semi open if there is an open set such that ( ). Assume in the following that the set operator , associated with the topology ,is
isotone. Then the family of all -semi open sets is a generalized topology
containing , in particular is a minimal structure, which will be denoted in
m A X
the following by X . The union of all -semi open sets contained in
A m
is an -semi open set, denoted by -sInt( ). Note that -sInt = X -Int. If
m in Remark 2.3 we take = -sInt, then the family of all -open sets is X .
2008. februar´ 10. –13:24 SEMI-SEPARATED SETS IN GENERALIZED CLOSURE SPACES 29
2.2. Neighborhoods
u P X P X neighborhood
Definition Let : ( ) ( ). The
function N X P P X x X N x u
u : ( ( )) assigns to each the family ( )=
N P X x u N g u neighborhoods N P X
= f ( ): -Int( ) of its . ( )issaidto
u neighborhood of a set A N N A N N x u
be an (write u ( )) if ( ) for each
A
x .
u A A u N N is a -neighborhood of if and only if -Int( ). If no confusion
can occur, we will use the term neighborhood instead u -neighborhood.
u
Remark Each almost -open set is a neighborhood of itself. If
P X P X
u : ( ) ( ) is expansive, then every set which is a neighborhood u of itself is u -open; the assumption “ expansive” cannot be removed from
this statement, as the following example shows.
u P X P X
Example Let : ( ) ( ) be as in Example 2.1 and assume
u A A fx g A
that X has at least two elements. Then -Int( )= 0 for every subset A
of X , hence every subset is a neighborhood of itself. On the other hand, is
x A u -open if and only if 0 .
The following characterizations in terms of neighborhoods are known
x X
(see [14]). u is isotone if and only if for each every superset of a
x u
neighborhood of x is a neighborhood of . is expansive if and only if each
X x u
neighborhood of an arbitrary x contains . is idempotent if and only
x X N N x u N N x u
if, for each , u ( ) is equivalent to -Int( ) ( ).
Let u P X P X be isotone and expansive The
Proposition : ( ) ( )
following properties are equivalent
u is idempotent
For each x X every neighborhood of x contains a u open neighborhood
of x
Proof x X N N x u
(1) (2): Let and u ( ). Since is idempo-
N u x u
tent, u -Int( )isan -open neighborhood of and, since is expansive,
N N
u -Int( ) . u
(2) (1): Assume that (2) holds and is not idempotent. Then there
X u A u u A u
exists A such that -Int( ) -Int( -Int( )). Since is expansive,
u A u A x u A n u u A A
u -Int( -Int( )) -Int( ) Let -Int( ) -Int( -Int( )). We have
N x u U x U A u
u ( ). By (2), there exists an -open set such that .Since
u u U u u A x u u A is isotone, U = -Int( -Int( )) -Int( -Int( )), hence -Int( -Int( )), a contradiction.
2008. februar´ 10. –13:24
30 MARCELINA MOCANU Remark To prove the implication (1) (2) it suffices to assume
that u is expansive.
Let u P X P X A X and x X Consider
Proposition : ( ) ( )
the following properties
x u
( A)
V A for every u neighborhood V of x
Then The converse holds provided that u is isotone
Proof X n A N x
Note that (1) is false if and only if u ( ). If (1) is
X n A u
false, then taking V := we see that (2) is false. Suppose that is
A u V x
isotone. If we assume that V = for some -neighborhood of ,we
x u V u X n A X n A N x
get -Int( ) -Int( ), hence u ( ). If u is not isotone, implication (1) (2) from Proposition 2.3 does not
hold in general, as the following example shows.
X u x c A
Example Let ( ) be the GCS from Example 2.2. Let = , =
fb g V fc g x N x u V X n u X n V fc g
= and = .Then u ( ), since -Int( )= ( )= ,
u A V A and x ( ), but = .
We recall the comparison between two closure spaces on the same set.
u v X v
Definition Let and be GCO’s on . We say that is
u u coarser v v A u A A X
ner than ,or is than ,if ( ) ( )forall and denote
v
this by u .
Let u v P X P X The following state
Proposition : ( ) ( )
ments are equivalent
u v
A v A for all A X u
Int( ) Int( )
N x N x for every x X u
( ) v ( )
P X P X then u v if and only if If u v
Corollary : ( ) ( ) =
N x N x for every x X u ( )= v ( )
2008. februar´ 10. –13:24 SEMI-SEPARATED SETS IN GENERALIZED CLOSURE SPACES 31
2.3. Subspaces of generalized closure spaces
P X P X
The relativization of an operator u : ( ) ( ) to a non-empty
Y X u P Y P Y u A u A Y Y v
Y
subset of is Y : ( ) ( ), ( )= ( ) .AGCS( )
Y X v u is said to be a subspace of (X u)if and is the relativization of
to Y .
Let X u and Y v be GCS s such that Y X The
Lemma ( ) ( )
following are equivalent
Y v is a subspace of X u
( ) ( )
v A Y u A X n Y for all A Y
Int( )= Int( ( ))
Lemma Let u P X P X Y X and v u Then
: ( ) ( ) = Y
x fY U U N x g for every x Y N u
v ( ) : ( )
Proof x Y N x
Let .If v ( )= there is nothing to prove. Let
N N x x v N Y u N X n Y
v ( ). Then -Int( )= -Int( ( )), by Lemma 2.8,
P N X n Y N x N Y P
hence := ( ) u ( )and = .
Let u P X P X Y a non empty subset of X and
Theorem : ( ) ( )
P Y P Y Ifu is isotone the following properties are equivalent
v : ( ) ( )
v is the relativization of u to Y
N x fY N N N x g for every x Y v
( )= : u ( )
N x fY N N N x g Proof u
(1) (2): By Lemma 2.2, v ( ) : ( ) for
x Y x Y N N x x Y u N Y
every .Let and u ( ). Then -Int( )
u X n Y Y u Y N X n Y v Y N
-Int( N ( )) = -Int(( ) ( )) = -Int( ),
Y N N x N x v
by isotony and Lemma 2.1, hence v ( ). It follows that ( )
fY N N N x g
: u ( ) .
N x N x u
(2) (1): Using the preceding implication we see that v ( )= ( )
Y
x Y v u for every , hence = Y by Corollary 2.1. The following proposition shows the importance of the isotony condition
in Theorem 2.1.
Proposition Let u P X P X be expansive If N x
: ( ) ( ) u ( )=
Y
fY N N N x g for every nonempty subset Y of X and every
= : u ( )
x Y thenu is isotone
N x fY N N N x g Proof u
Assume by contrary that u ( )= : ( )
Y
X x Y u
for every nonempty subset Y of and every , but is not isotone.
B X u A n u B u Then there exists A such that -Int( ) -Int( ) .Since
2008. februar´ 10. –13:24
32 MARCELINA MOCANU
A A U A Y A X n B
is expansive, u -Int( ) . Consider := , := ( )and
x u A n u B x U Y U N x x u U
-Int( ) -Int( ). Then , u ( )and -Int(
X n Y u B x Y u U X n Y u U
( )) = -Int( ), hence -Int( ( )) = Y -Int( ).
U Y U fY N N N x gn N x u
Then = : u ( ) ( ), a contradiction. Y
3. Semi-separated sets
X u A B X
Definition Let ( )beaGCS.Twosubsets and of
X u U V
are said to be semi separated in ( ) if there are neighborhoods and
B U B V A
of A and , respectively, such that = = .
A B
Remark (1) Semi-separation is hereditary: if and are semi-
A A B B A B separatedin(X u), 1 and 1 ,then 1 and 1 are semi-separated
in (X u). The empty set is semi-separated from itself. A
(2) Every two disjoint almost u -open subsets are semi-separated: if
u X U A V B and B are disjoint almost -open subsets of ,then := and := satisfy all the conditions of Definition 3.1.
(3) Every two disjoint almost u -closed subsets are semi-separated. In-
B u X B X n A
deed, if A and are disjoint almost -closed subsets of ,then
X n A A X n B u X n B U X n B u
-Int( )and -Int( ), hence := and
X n A
V := satisfy all the conditions of Definition 3.1.
P X P X u v
(4) If u v : ( ) ( ) satisfy then every two sets which are X v
semi-separated in (X u) are also semi-separated in ( ), by Proposition 2.4.
Let X u be a GCS and A B be subsets of X
Theorem ( )
u A B A u B thenA and B are semi separated in X u
a If ( ) = ( )= ( )
b The converse of the implication in a holds provided that u is isotone
u A B A u B A X n u B
Proof (a) Assume that ( ) = ( )= ,thatis, ( )=
u X n B B X n u A u X n A U X n B N A
= -Int( )and ( )= -Int( ). Then := u ( ),
V X n A N B U B V A A B
:= u ( )and = = , hence and are
semi-separated in (X u).
A B X u
(b) Assume that u is isotone and , are semi-separated in ( ).
U N A V N B U B V A u
There exist u ( ), ( ) such that = = .
u U u X n B X n u B B u V
Then A -Int( ) -Int( )= ( )and -Int( )
u X n A X n u A A u B B u A -Int( )= ( ), hence ( )= and ( )= .
2008. februar´ 10. –13:24
SEMI-SEPARATED SETS IN GENERALIZED CLOSURE SPACES 33
Let X u be a GCS with u isotone and let A
Corollary ( )
X u if and only if be subsets of X Then A and B are semi separated in
B ( )
u B A u B
( A) = ( )=
X u be a GCS with u isotone If the subsets A Let
Corollary ( ) 1
A of X are both almost u open or almost u closed then the sets A n A
and 2 1 2
A n A are semi separated
and 2 1
Proof i j f g u u A n A
j
Let ( ) (1 2); (2 1) .Since is isotone, ( i )
A n A u A u X n A A X n A A u
i i j j i i
( j ) ( ) ( ) ( ). If is almost -closed or
A u u A X n A u X n A A
i i j j
j is almost -open then ( ) ( )= or ( ) = , hence
u A n A A n A
j j i
( i ) ( )= .
u Remark The assumption “ is isotone” cannot be removed from
Theorem 3.1 (b), as the following example shows.
X u M fa bg
Example Let ( ) be as in Example 2.2. For := we have
u M u M u M X n u fc g fb g u u M
u ( -Int( )) ( ), since -Int( )= ( )= , ( -Int( )) =
b cg u M fa bg A X n u M B u M U X n M
= f and ( )= .Set := ( ), := -Int( ), :=
M A u U B u V u
and V := .Wehave = -Int( ), = -Int( ) and, since is expansive,
B V A A B X u
U = = .Then and are semi-separated in ( ), by
B A u u M X n u M fc g Definition 3.1. But u ( ) = ( -Int( )) ( ( )) = .
Remark Using the notations of Remark 2.3, we point out that two
X U V
sets U V are -separated in the sense of [3] if and only if
X c X
are semi-separated in the GCS ( ). If ( ) is a topological space and
P X P X A B
: ( ) ( ) is an isotone operator associated to , two subsets
A B
of X are said to be -semi-separated if and only if are -separated
X
for = -sInt [13] (see Remark 2.4). If ( ) is a topological space and
A B X =Cl Int (respectively, =Int Cl), then two sets which are
-separated are termed semiseparated [10] (respectively, preseparated [15]). It is natural to think that every two semi-separated sets are disjoint, but
this is not true in general, as the following proposition shows.
Let u P X P X The following properties are
Proposition : ( ) ( )
equivalent
u is not expansive
The set fx g is semi separated from itself for some x X
There is a non empty subset of X which is semi separated from itself
There exist two non empty semi separated subsets of X one of which
2008. februar´ 10. –13:24
34 MARCELINA MOCANU
containing the other
There exist two non empty semi separated subsets of X which are not
disjoint
Proof u M X
(1) (2): Since is not expansive, there is such
M n M A u M n M A
u -Int( ) .Let -Int( ) be non-empty. Then is semi-
u A M A
separated by itself, because M is a -neighborhood of and = .We
fx g x u M n M
can choose A = ,where -Int( ) .
Implications (2) (3), (3) (4) and (4) (5) are obvious.
A B
(5) (1): Assume that (5) is true and (1) is false. Let , be semi-
X u U N A
separated in ( ) which are not disjoint. There exist u ( )and
V N B U B V A u
u ( ) such that = = .Since is expansive, we
u U U X n B A B
have A -Int( ) , a contradiction with .
u A X
Remark If is isotone, a subset of is semi-separated from
A A itself if and only if u ( ) = . See also Proposition 3.1 and Proposition 2.1.
The following proposition is useful in the study of connectedness in
GCS’s.
Let X u be a GCS Assume that X A B Con
Proposition ( ) =
sider the following statements
A and B are semi separated in X u
The sets ( )
A and B are disjoint and almost u open
Then
a implies
b implies provided that u is expansive
Proof (a) See Remark 3.1 (2).
B X u u
(b) Suppose that A and are semi-separated in ( ). Since is
B B X n A
expansive, A and are disjoint. Then = . For every generalized
A X A X n A
closure space ( X u)andany ,thesets and are semi-separated
U V X
in (X u) if and only if there exist some subsets and of such that
A u U V X n A u V
U -Int( )and -Int( )
A u U U
Since u is expansive, the inclusions above imply = -Int( )= and
v V V A B u
B = -Int( )= , hence and are -open.
X u A X A X n A
Remark Let ( )beaGCSand .Then and u are semi-separated in (X u) if and only if there exist some (almost -open)
2008. februar´ 10. –13:24
SEMI-SEPARATED SETS IN GENERALIZED CLOSURE SPACES 35
V X U A u U V X n A
subsets U and of such that -Int( )and
u V -Int( ).
We study the relativization of the semi-separation property:
Theorem Let X u be a GCS and let u be the relativization of u
( ) Y
to Y X Then
Y u are semi a Every subsets A B of Y which are semi separated in
( Y )
X u
separated in ( )
X u u is isotone every subsets A B of Y which are semi separated in
b If ( )
are semi separated in Y u
( Y )
Proof u v A B Y
Denote Y by .Let be subsets of .
B Y v
(a) Assume that A and are semi-separated in ( ). There exist some
V Y A v U B v V U B
subsets U and of such that -Int( ), -Int( )and =
A A u U X n Y B u V
= V = . By Lemma 2.1, -Int( ( )) and -Int(
X n Y U U X n Y N A V V X n Y N B u
( )). Then := ( ) u ( ), := ( ) ( )
B V A A B X u
and U = = , hence and are semi-separated in ( ).
A B X u
(b) Assume that u is isotone and and are semi-separated in ( ).
A B u B A v A B
Then u ( ) = = ( ) , by Theorem 3.1 (b). Obviously ( ) = =
B A A B Y v = v ( ) , hence and are semi-separated in ( ) by Theorem 3.1 (a).
The assumption “u is isotone” cannot be removed from Theorem 3.2 (b),
according to the following example.
X u A B A fc g
Example Let , , and be as in Example 3.1. Then =
fb g X u
and B = are semiseparated in ( ), but they are not semi-separated
Y u Y fb cg u u Y
Y
in ( Y ), where := and is the relativization of to . Indeed,
A B Y u
if we suppose that and are semi-separated in ( Y ), then there exist
M N Y A u Y n M B u Y n N
Y
and subsets of such that Y -Int( ), -Int( ),
N B M u
A and . Taking into account that is expansive, this implies
c g N u N Y fc g fb g M u M Y fb g
f ( ) and ( ) .But
b g M Y M ffb g fb cgg u M Y fb cg fb g
f implies , hence ( ) = .
A B Y u
This contradiction shows that and are not semi-separated in ( Y ). Remark Taking into account Remark 3.3 we can see that Re- mark 3.1 (2), Theorem 3.2 and Corollary 3.2 furnish Theorem 2, Theorem 4 and Theorem 5 in [10], as well as Theorem 2, Theorem 4 and Theorem 5
in [15].
G V E V
Example Let =( ) be an undirected graph, where is the
A V u A V vertex set and E is the set of edges. For every ,wedefine 1( )
2008. februar´ 10. –13:24
36 MARCELINA MOCANU
A V x V u A
(respectively, u2( ) ) such that a vertex belongs to 1( ) (respec-
A x A x A
tively, u2( )) if and only if, either ,or has a neighbor belonging to
G A
(respectively, x can be joined by a path in to a vertex belonging to ).
m
u u A fu A m g
Clearly, u2 1. Moreover, 2( )= 1 ( ): =12 , for every
m
A V u u u u P V P V k
,where 1 = 1 1. The set operators k : ( ) ( ), =
z
m
u u
=1 2, are expansive and isotone, 2 is idempotent, while 1 is not idempotent
A u A
in general. Note that u1-Int( ) (respectively, 2-Int( )) is the set of vertices
X n A
belonging to A which have no neighbors in (respectively, which cannot
G X n A A u B
be joined by paths in to vertices in ). We see that k -Int( )if
u A B k A B V
and only if k ( ) ,for =1 2, whenever .
V V u V u
The sets A B are semi-separated in ( 1) (respectively, in ( 2))
A y B if and only if any two vertices x and are not neighbors (re-
spectively, cannot be joined by a path in G ). Note the following symmetry:
u A B u B A k A B V
k k ( ) = implies ( ) = ,for =1 2, whenever .
In graph theory, a graph G is called complete if any two distinct vertices are neighbors, and connected if any two distinct vertices can be joined by a
path in G . We get the following characterizations.
Let G V E be a graph having more than one vertex
Theorem =( )
a G is not complete if and only if there exist two non empty subsets
V which are semi separated in V u
of ( 1)
b G is not connected if and only if there exist two non empty subsets
of V which are semi separated in
( V u2)
A B V V u
Proof If two non-empty sets are semi-separated in ( 1)
x A y B x y (respectively, in ( V u2)), taking and we see that vertices and
are not neighbors (respectively, cannot be joined by a path in G ), hence the G
graph G is not complete (respectively, is not connected). Conversely, if the G
graph G is not complete (respectively, is not connected), then there exist
V x y
x y such that and are not neighbors (respectively, cannot be joined
fx g fy g
by a path in G ) and it follows that the sets and are semi-separated
V u x y
in (V u1) (respectively, in ( 2)). Moreover, if and cannot be joined by
u fx g u fy g apathinG ,then ( ) ( )= .
Example 3.3 suggests the following result.
2008. februar´ 10. –13:24
SEMI-SEPARATED SETS IN GENERALIZED CLOSURE SPACES 37
Let u P X P X such that u B X n B
Proposition : ( ) ( ) ( ) ( )=
u X n B B whenever B X Then a set A X is almost
implies ( ) =
u closed if and only if A is almost u open
A X u A A
Proof For all the following relations are equivalent: ( ) ,
A X n A u X n A A A X n u X n A A u A
u ( ) ( )= , ( ) = , ( ), -Int( ).
Corollary With the notations of Example the following are
equivalent for a set A V
V u A is closed in
( 2)
A is open in
( V u2)
V u A and V n A are semi separated in
( 2)
Moreover if A is a non empty proper subset of V theneachoftheabove
statements is equivalent to the following
Any vertices x A and y V n A cannot be joined by a path in G
4. Images and preimages of semi-separated sets
Recall the generalization of continuity to generalized closure spaces.
X u Y v
Definition Let ( )and( ) be generalized closure
X u Y v continuous at x X spaces. A function f :( ) ( )issaidtobe if
1
V N f x f V N x f X u Y v u
for all v ( ( )) implies ( ) ( ). :( ) ( )issaid
x X to be continuous if it is continuous at every point .
The following characterizations of continuity in GCS’s are known.
For f X u Y v the following conditions are
Lemma :( ) ( )
equivalent
f is continuous
1 1
u f B f v B for every subset B of Y
( ( )) ( ( ))
1 1
f v B u f B for every subset B of Y
( Int( )) Int( ( ))
X u Y v Consider the following Let f
Proposition :( ) ( )
statements
f is continuous
The preimage under f of each almost v open subset of Y is an almost
u open subset of X
2008. februar´ 10. –13:24
38 MARCELINA MOCANU
Then
a
b provided that v is idempotent and expansive and u is isotone
f B Y v
Proof (a) Assume that is continuous and let be almost -open,
v B
i.e., B -Int( ). By Lemma 4.1,
1 1 1
B f v B u f B f ( ) ( -Int( )) -Int( ( ))
1
B u
hence f ( )isalmost -open. u
(b) Let v be idempotent and expansive and let be isotone. Assume that
V N f x f x v V
(2) is true. Let v ( ( )), i.e., ( ) -Int( ), which is equivalent
1
f v V v v V v to x ( -Int( )). Since is idempotent, the set -Int( )is -open, in
particular almost v -open. According to our assumption,
1 1
v V u f v V
f ( -Int( )) -Int( ( -Int( ))) u
Since v is expansive and is isotone, we have
1 1
f v V u f V
u -Int( ( -Int( ))) -Int( ( ))
1 1
x u f V f V N x
It follows that -Int( ( )), i.e., ( ) u ( ).
Let f X u Y v be a continuous function If
Theorem :( ) ( )
1 1
A and B are semi separated in Y v thenf A and f B are
the sets ( ) ( ) ( )
X u
semi separated in ( )
A B Y v U
Proof Assume that and are semi-separated in ( ). Let
N A V N B U B V A v
v ( )and ( ) such that = = . Applying
1 1 1 1
A f v U u f U f B
Lemma 4.1, we get f ( ) ( -Int( )) -Int( ( )) and ( )
1 1 1 1
v V u f V f U N f A f
( -Int( )) -Int( ( )). It follows that ( ) u ( ( )),
1 1 1 1 1 1
f V N f B f U f B f V f A
( ) u ( ( )) and ( ) ( )= ( ) ( )= , hence
1 1
A f B X u
f ( )and ( ) are semi-separated in ( ).
Remark In [14] Theorem 4.1 was proven under the assumption v that u and are isotone, expansive and preserve the empty set. There the proof makes use of the characterization of semi-separated sets in isotonic
spaces, given by Corollary 3.1.
V E G V E Let G =( )and =( ) be two undirected graphs.
2008. februar´ 10. –13:24
SEMI-SEPARATED SETS IN GENERALIZED CLOSURE SPACES 39
f V V connectivity
Definition We say that a function : is
G G f x f y
preserving, with respect to the graphs and ,if ( )and ( ) can be
x y V G joined by a path in G whenever can be joined by a path in .
Note that, if the graph G has only isolated vertices, then every function
V V
f : is connectivity-preserving.
u
Denote by u2 and 2 the set operators associated as in Example 3.3 to
G
the graphs G and , respectively.
A function f V V is connectivity preserving with
Theorem :
V u V u is to the graphs G and G if and only if f
respect :( 2) ( 2) continuous
Proof Using Definition 4.1 and the characterizations of neigborhoods
given in Example 3.3, we see that the following are equivalent:
u V u V
(1) f :( 2) ( 2) is continuous;
V W V u ff x g W
(2) For every x and all such that 2( ( ) ) ,wehave
1
fx g f W
u2( ) ( ).
f V V
Necessity. Assume that : is connectivity-preserving, with
G x V W V
respect to the graphs G and .Let and such that
ff x g W y u fx g x y
u2( ( ) ) .If 2( ), then and can be joined by a path
f f y u ff x g W
in G .Since is connectivity-preserving, ( ) 2( ( ) ) , hence
1
f W
y ( ). We proved that (2) holds. f
Su ciency. Assume that (2) holds, but is not connectivity-preserving.
V x y
Then there exist x y such that and can be joined by a path
f x f y G
in G , but ( )and ( ) cannot be joined by a path in . It follows that
1
ff x g V nff y g u fx g f V nff y g
u2( ( ) ) ( ) . We deduce that 2( ) ( ( ) )
y u fx g V nfy g
, hence 2( ), a contradiction.
1 1
Let f V V such that f A and f B are
Corollary : ( ) ( )
V u whenever A and B are semi separated in V u
semi separated in ( 2) ( 2)
Then f V u is continuous
:(V u2) ( 2)
f V u V u
Proof Assume, by contrary, that :( 2) ( 2) is not contin-
V x y
uous. Then there exist x y such that and can be joined by a path
f x f y G ff x g
in G , but ( )and ( ) cannot be joined by a path in . Then the sets ( )
1 1
f y g V u f ff x g f ff y g and f ( ) are semi-separated in ( 2), but ( ( ) )and ( ( ) ) are not semi-separated in ( V u2), which is a contradiction.
2008. februar´ 10. –13:24
40 MARCELINA MOCANU
f V u V u is continuous if and only if f
Corollary :( 2) ( 2) pulls back every pair of semi separated sets into a pair of semi separated sets
Proof Apply Theorem 4.1 and Corollary 4.1.
f X u Y v almost open
Definition We say that :( ) ( )is if
u A v f A A X
f ( -Int( )) -Int( ( )) for all .
f X u Y v f Remark Let :( ) ( ) be a bijection. By Lemma 4.1, is
1 almost open if and only if f is continuous.
We prove a counterpart of Proposition 4.1 for almost open functions.
Let f X u Y v Consider the following
Proposition :( ) ( )
statements
f is almost open
The image under f of each almost u open subset of X is an almost v open
subset of Y
Then
a
b provided that u is idempotent and expansive and v is isotone
f U X u
Proof (a) Assume that is continuous and let be almost -open,
u U
i.e., U -Int( ). According to Definition 4.3,
U f u U v f U
f ( ) ( -Int( )) -Int( ( ))
U v
hence f ( )isalmost -open. v (b) Let u be idempotent and expansive and let be isotone. Assume that
(2) holds.
u u A
Since u is idempotent, -Int is also idempotent, hence -Int( )is(al-
A X A X f u A most) u -open for all .Let .Then ( -Int( )) is almost
v -open, i.e.,
u A v f u A
f ( -Int( )) -Int( ( -Int( ))) v
Since u is expansive and is isotone, we have
f u A v f A
v -Int( ( -Int( ))) -Int( ( ))
u A v f A The latter two inclusions imply f ( -Int( )) -Int( ( )).
2008. februar´ 10. –13:24
SEMI-SEPARATED SETS IN GENERALIZED CLOSURE SPACES 41
If f X u Y v is almost open and injective u
Theorem :( ) ( )
X u then f U f V are v are isotone and U V are semi separated in
and ( ) ( ) ( )
semi separated in
( Y v )
f X u Y v u
Proof Assume that :( ) ( ) is almost open and injective,
U V X u U u X n
and v are isotone and are semi-separated in ( )(i.e., -Int(
V u X n U
V )and -Int( )). Then
U v f X n V v Y n f V
f ( ) -Int( ( )) -Int( ( )) and
V v f X n U v Y n f U
f ( ) -Int( ( )) -Int( ( ))
U f V Y v
hence f ( ) ( ) are semi-separated in ( ).
P X P X P Y P Y
Remark Let : ( ) ( )and : ( ) ( ) be isotone
X Y continuous
operators. In [3] a function f : is said to be ( )
1
V V Y f X Y
if f ( )is -open whenever is -open. In other words, :
1
f V X c V
is ( )-continuous if ( ) is open in the GCS ( ) whenever is
Y c f X Y
open in the GCS ( ). Similarly, : is said to be ( )-open
U U X f U
if f ( )is -open whenever is -open [3], i.e., ( ) is open in the
Y c U X c
GCS ( ) whenever is open in the GCS ( ).
c c
Since the operators and are isotone, expansive and idempotent,
we may apply Proposition 4.1 (respectively, Proposition 4.2) to show that
X Y f
f : is ( )-continuous (respectively, ( )-open) if and only of :
X c Y c
( ) ( ) is continuous in the sense of Definition 4.1 (respectively, almost open in the sense of Definition 4.3). It follows that Theorem 4.1 and Theorem 4.3 furnish Lemma 2.1 and Lemma 2.3 in [3], respectively.
References
R A Mahmoud
[1] M E Abd El Monsef S N El Deeb and -open sets and Bull Fac Sci Assiut Univ
-continuous mapping, 12 (1983), 77–90.
Acta Math Hungar aszar [2] A Cs Generalized topology, generalized continuity;
96 (2002), 351–357.
aszar Acta Math Hungar
[3] A Cs -connected sets, 101 (2003), 273–279.
R Ghosh H Dasgupta Bull [4] S and Connectedness in monotone spaces,
Malays Math Sci Soc 27(2) (2004), 129–148. Amer [5] N Levine Semi-open sets and semi-continuity in topological spaces,
Math Monthly 70 (1963), 36–41.
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[6] H Maki On generalizing semi-open and preopen sets, in:
Topological Spaces Theory and its Applications on , August 1996, Yat-
sushiro College of Technology, 13–18.
M Annales Univ [7] M Mocanu On -continuous functions and product spaces,
Sci Budapest 47 (2004), 65–89.
R E Jamison Informa [8] J L Pfaltz and Closure systems and their structure,
tion Sciences 139 (2001), 275–286. T Noiri [9] V Popa and On the definitions of some generalized forms of con-
tinuity under minimal conditions, Mem Fac Sci Kochi Univ Math 22
(2001), 9–18. U Tapi [10] S N Maheswari and Connectedness of a stronger type in topological
spaces, Nanta Math 12(1) (1979), 102–109. S N El Deeb [11] A S Mashhour M E Abd El Monsef and On precontinuous
and weak precontinuous mappings, Proc Math Phys Soc Egypt 53 (1982),
47–53.
astad Paci c J Math [12] O Nj On some classes of nearly open sets, 15 (1965),
961–970.
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connected properties in topological spaces, Sci Math Jap Online 6 (2002),
465–472.
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Technical report, Institute for Theoretical Chemistry, University of Vienna.
http www tbi univie ac at pap ers wpl ist
P Paik P J Karadniz Univ Fac Art [15] S S Thakur and On -connected spaces,
Sci Ser Math Phys 10 (1987), 55–62.
Marcelina Mocanu Department of Mathematics, University of Bacau˘
Spiru Haret 8, 600114 Bacau,˘ Romania tiberium easynet ro mmocanu ub ro,
2008. februar´ 10. –13:24 ANNALES UNIV. SCI. BUDAPEST., 49 (2006), 43–52
THE FOURIER INTEGRALS OF FUNCTIONS OF BOUNDED VARIATION
By
SHIMSHON BARON, ELIJAH LIFLYAND and ULRICH STADTMULLER¨ Received March
1. Introduction
Various summability properties of number series as well as their appli-
cations to Fourier series are well studied. In his papers [11, 12], G. Goes developed the theory of C -complementary spaces of Fourier coefficients, that is with respect to Cesaro` methods of summability. In [13] he used this
as a bridge between known theorems on summability factors for the Cesaro` summability method C and effective sufficient conditions for the multipliers of Fourier series. These results were extended to general Toeplitz methods by
M. Tynnov [17, 18]. A comprehensive exposition may be found in [1, x 28]. This concept allows to study summability and multiplier problems for Fourier series by means of summability methods for numerical series. Similar results for integrals are less investigated – some references may be found in [2] where an attempt to build a general theory was made. Of course, this is impossible without understanding, generally speaking, the
Fourier transform in the distributional sense. The main object of this topic b
are various spaces X of the Fourier integrals
Z
ix t
t x e dx f ( ) ( ) R
ˆ f f X
such that is the Fourier transform of
Z
ix t
ˆ 1 x f x f t e dt ( )= ( )=(2 ) ( ) R
Mathematics Subject Classi cation : 42A38, 42A45; 42A24, 40C15
2008. februar´ 10. –13:57
44 SHIMSHON BARON, ELIJAH LIFLYAND, ULRICH STADTMULLER¨
X kf k kf k X Given a normed space, we define X ˆ = . In a sense, here we attempt to do for Fourier integrals what we did in [3] for double Fourier series. In particular, our scope is such that we avoid the distributional approach. The paper is organized as follows. After the present introduction we give in the next section the formulations of our main results as well as examples and corollaries. We prove the results in the last section.
2. Problems and results
We first observe that the Fourier transform of a function of bounded
V variation vanishing at infinity, written F 0, exists as an improper integral
everywhere except maybe the origin (see, e.g., [4, Ch. 1, x 2]). For functions of bounded variation the notion of the Fourier–Stieltjes transform proved to be more natural in many respects as the usual Fourier
transform:
Z
ix t
c 1
x e dF t dF ( )=(2 ) ( )
R
b
d V We study conditions under which f belongs to the space , 0,
that is, when
1 c
x dF x (1) (ix ) ( )= ( )
where F is a function of bounded variation. For this we need a notion of fractional derivative. The one corresponding to our scope is naturally defined
( )
g via the Fourier transform: g ,the th derivative of , is the function for
which
d
( )
x ix g x g ( )=( ) ˆ ( ) We shall study these spaces in connection with summability. Let the
summability method be defined by a single function , a multiplier, as follows
Z
t
ix t
f x t e dt (2) ( N )( )= ( )
R N
It is clear that should be defined at each point, so let be continuous.
L The following representation is useful in many cases (when f (R)itis
2008. februar´ 10. –13:57 THE FOURIER INTEGRALS OF FUNCTIONS OF BOUNDED VARIATION 45 merely equivalent to (2), see, e.g., [16, Ch. I, Th. 1.16]; moreover, this is true
under any assumptions which provide the validity of the Parseval identity)
Z
b
f x N N t x f t dt (3) ( N )( )= ( ( )) ( ) R In what follows we assume that
ˆ (4) both and are integrable on R
and
(5) (0) = 1
We shall make use of either (2) or (3) as a definition of the linear means N just according to which of the two formulas is valid.
When a function is already represented by a Fourier integral, to take
it
its th derivative means to multiply the integrand by ( ) . Therefore, by
(1 )
f
( N ) we understand
Z
t
ix t
(1 ) (1 )
f x t it e dt ( N ) ( )= ( )( )
R N
Our first result is the following:
(1 )
Let and be such that t N t it are integrable
Theorem ( ) ( )( )
b
for all N In order that f belong to d V it is necessary and su cient that
(1 )
f O
(6) ( N ) = (1)
L(R)
Remark Clearly, =0and = 1 are two main cases. Slightly less general versions of these cases are known from [10] and [9], respectively.
For more information, see [7] or [15]. multiplier
Here and for the sequel we recall that a function s is called a
b b
X Y f X sf Y
of class ( ) if for each we have .
1 Thus this theorem is a criterion for ( ix ) to be a multiplier of class
( L L) (see, e.g., [7, Th. 6.5.6]).
R
jt j
1
t Si jt j u udu
Let ( )= ( )= 0 sin , which is bounded and non-
t jt j f
negative, and ( )=(1 )+. We wish to check that ( N ) are uniformly in-
R N
1
j f x j dx tegrable for this choice. The value 0 ( N )( ) is uniformly bounded,
2008. februar´ 10. –13:57
46 SHIMSHON BARON, ELIJAH LIFLYAND, ULRICH STADTMULLER¨
x N y Nx as for any bounded .Given 1 , we obtain for = 1by
changing the order of integration
f y ( N )( )=
1
Z
h i
1 2 1 2
N u Nu y y y u yu y yu = sin cos (1 )sin( )+ cos( ) du
0
R R
1
j f x j dx N j f y j dy N
The main term for R ( N )( ) = R ( )( ) is
1
Z Z
1 1
u Nu yu du dy
y sin( )sin( )
1 0
N
the uniform integrability of the rest is rather obvious. For y 2 and
y N
1 2, we obtain the needed bounds by estimating the inner integral as N
y +
Z
y
1 1 + N
1
u udu
cos log
N
2 2 y
jy N j
N j y N y
and straightforward calculations, e.g. for y 2 use that log ( + ) (
N j N N y N N N N
) 2 ( ). For the integrals from 2to 1and +1to2
R R
N
1 2 1
udu y dy use that u cos exists and = log 4. The estimate over
1 N 2
y N Si jt j L L
N 1 + 1 is simple. Hence ( ) is a multiplier of class ( ).
t jt j t Taking the same and ( )=1for 1and ( )=0otherwise, the partial Fourier integrals, we easily derive that this function cannot be a
multiplier of class ( L L). Indeed, we immediately arrive at
1
Z
t 1
1 1 2
xt dt x x N x x
1 cos = 1 sin + (1 cos )
N N
0
which is non-integrable for any N 1.
t
The other negative example is delivered, again with the same ,by ( )=
t t
=1for t 0and ( )= 1for 0. The calculations are simple:
N
Z
t
1 1 2
xt dt x N x Nx
1 sin = sin N 0
2008. februar´ 10. –13:57 THE FOURIER INTEGRALS OF FUNCTIONS OF BOUNDED VARIATION 47
and the right-hand side is non-integrable for any N . This gives a different
proof that the Hilbert transform is not a bounded operator on L.
Further, we introduce the following (cf. [2])
b
d V V
Definition We define the complementary space to , written ,
as the space of all Fourier integrals
Z
ix t
t g t e dt g ( ) ˆ ( ) R
for which the integral Z
ˆ x g x dx f ( )ˆ( )
R
b
f d V
is -summable for all .
V Theorem In order that g it is necessary and su cient that the
( 1)
sequence g is boundedly convergent
( N )
X
For f g we define the inner product by the integral
Z
f gi f t g t dt h := ( ) ( )
R
e t t
Further, define the function e by the formula ( ):=1forall R.
X a summability factor of type A function s is called ( ) if for every
ˆ ˆ b hh ei h X -summable functional with the values for the functional with
ˆ ˆ s h ei hh si
the values h = is -summable.
If a function s X is a summability factor of type
Theorem ( )
then s is a multiplier of class V
( V )
( ) Let s mean the well-known fractional derivative in the Riemann– Liouville sense.
( )
Let If s is even s is absolutely continuous s
Corollary 0
has a nite limit at in nity and
Z
( +1)
t s t dt ( ) +
R
s is a multiplier of the class V V
then ( )
2008. februar´ 10. –13:57 48 SHIMSHON BARON, ELIJAH LIFLYAND, ULRICH STADTMULLER¨
3. Proofs
b
Necessity f d V
Proof of Theorem Let .Then
Z
t
(1 ) ix t
f x it it t e dt
( N ) ( )= ( )( ) ( )
R N
Z
t
ix t
c
dF t e dt = ( )
R N
By the Stieltjes version of (3) (see, e.g., [7, Th. 6.5.6]) we have
Z
(1 ) ˆ
f x N N t x dF t ( N ) ( )= ( ( )) ( ) R
Hence
Z Z
(1 ) ˆ
f N j N t x j d jF t j dx
( N ) ( ( )) ( )
L(R) R R
ˆ F Since is integrable on R,and is of bounded variation we may use here
and in what follows the Fubini theorem freely. This yields
Z Z
(1 ) ˆ
f d jF t j N j N t x j dx
( N ) ( ) ( ( ))
L(R) R R
ˆ
kF k k k L = BV (R) and we are done.
Su ciency Denote
x
Z
(1 )
x f t dt
N
ΦN ( )= ( ) ( )
This sequence possesses the following properties. First, (6) provides that
f g
both the family ΦN and the family of variations of these functions are
(1 )
k f k L
uniformly bounded (by the same constant ( N ) (R)). In virtue of the
f g
boundedness of ΦN along with their variations, the first Helly’s theorem
fN g
(see, e.g., [14, Th. 9.1.1]) ensures the existence of a subsequence k ,
N k
k as , such that
x
lim ΦN =Φ( )
k
k
2008. februar´ 10. –13:57 THE FOURIER INTEGRALS OF FUNCTIONS OF BOUNDED VARIATION 49
at any point x on the whole R, where Φ is a function of bounded variation
k f k (bounded by N along with its total variation). Our next (and final) L(R)
step is to show that
Z Z
ix t ix t
c 1 1
d x e d t e d t
(7) Φ( )=(2 ) Φ( ) = lim (2 ) ΦN ( )
k
k R R Generally speaking, the second Helly’s theorem is true only under additional assumptions (see, e.g., [15, Th. 9.1.3]). For example, it holds true if the last limit is uniform on every finite interval (see [5, Lemma 1]). As in [9], we
have
Z
t
ix t
1 1
it t e d x
( ) ( )=(2 ) ΦN ( )
k
N k
R
1
t N it t
The right-hand side is continuous as well as ( k ), hence ( ) ( ) t
almost everywhere coincides with a continuous function ( ). By (4) and (5),
t N t N t ( k ) ( ) converges, as ,to ( ) uniformly on every finite interval. Thus, (7) is true, which completes the proof.
Proof of Theorem We are checking the conditions for convergence
of
Z
x ˆ g x f x dx (8) ˆ( ) ( )
R N
b
d V
Since f ,wehave
Z Z
x x
1 1 ˆ 1 c dF g x ix ix f x dx g x ix x dx
ˆ ( )( ) ( ) ( ) = ˆ( )( ) ( ) N R N R
for some F of bounded variation. By Parseval’s identity this is
Z
1
g t dF t ( N ) ( ) ( ) R
Taking
n
t x
1 t F ( )=
0 otherwise
yields
Z
x
ˆ ( 1)
g x f x dx g x
ˆ ( ) ( ) =( N ) ( )
R N
V
and the right-hand side is boundedly convergent provided g .
2008. februar´ 10. –13:57
50 SHIMSHON BARON, ELIJAH LIFLYAND, ULRICH STADTMULLER¨
( 1)
g Conversely, let ( N ) be boundedly convergent. Then the opera-
tion similar to that above and Parseval’s identity yield
Z Z
x x
ˆ 1 1 ˆ g x f x dx g x ix ix f x dx
ˆ ( ) ( ) = ˆ( )( ) ( ) ( ) N
R N R
Z
( 1)
g t dF t = ( N ) ( ) ( )
R
( 1)
g x F Since ( N ) ( ) is boundedly convergent and is of bounded varia-
tion, Lebesgue’s dominated convergence theorem shows that the integral (8)
converges as N , which completes the proof.
f V
Proof of Theorem Let . It means that
ˆ ˆ h f g i h f g ei
N
N ˇ = ˇ
V s
converge for every g . This yields, since is a summability factor of
type ( ), that
ˆ ˆ h s f g ei h s f g i
N
N ˇ = ˇ
V s
converge for every g . This just means that acts as a multiplier taking
the space V into itself.
C
Proof of Corollary We recall, that -summability with 0
t jt j
is the case when ( )=(1 )+ . We apply a result of J. Cossar [8] on the Cesaro` C -summability of integrals (see [6] for the complete version of
this result; the partial case of integer is due to G. H. Hardy [14]) due to
which if and satisfy the conditions of Corollary 4, then is a summability
C s factor of type ( C ) for integrals. In view of Theorem 3 the function is a multiplier of the indicated class.
References Introduction to the theory of summability of series [1] S Baron , 2nd ed., Valgus,
Tallinn, 1977 (in Russian). E Liflyand [2] S Baron and Complementary spaces and multipliers for Fourier
transforms, Acta Sci Math Szeged 60 (1995), 49–57.
Baron E Liflyand U Stadtmuller [3] S and Complementary spaces and multipliers of double Fourier series for functions of bounded variation,
J Math Anal Appl 250 (2000), 706–721.
2008. februar´ 10. –13:57
THE FOURIER INTEGRALS OF FUNCTIONS OF BOUNDED VARIATION 51 Lectures on Fourier integrals [4] S Bochner , Princeton Univ. Press, Princeton,
N.J., 1959. Bull Amer Math Soc [5] S Bochner A theorem on Fourier–Stirltjrs integrals,
40 (1934), 271–278. J London Math Soc [6] D Borwein A summability factor theorem, 25 (1950),
302–315.
R J Nessel Fourier analysis and approximation [7] P L Butzer and ,Birkhauser¨
Verlag, Basel, 1971. J London Math Soc [8] J Cossar A theorem on Cesaro` summability, 16 (1941),
56–68.
er [9] H Cram On the representation of functions by certain Fourier integrals,
Trans Amer Math Soc 46 (1939), 190–201.
nguez Duke [ ] A G Dom 10 The representation of functions by Fourier integrals,
Math J 6 (1940), 246–255.
Goes BK [11] G -Raume¨ und Matrixtransformationen fur¨ Fourierkoeffizienten,
Math Z 70 (1959), 345–371. Goes [12] G Komplementare¨ Fourierkoeffizientenraume¨ und Multiplikatoren,
Math Ann 137 (1959), 371–384.
[13] G Goes Charakterisierung von Fourierkoeffizienten mit einem Summierbar- Math keitsfaktorentheorem und Multiplikatoren, Studia 19 (1960), 133–
148. Messenger Math [14] G H Hardy Notes on some points in the integral calculus,
40 (1911), 87–91, 108–112. Fourier analysis in probability theory [15] T Kawata , Academic Press, Inc., New
York, 1972.
G Weiss Introduction to Fourier analysis on Euclidean spaces [16] E Stein and ,
Princeton Univ. Press, Princeton, N.J., 1971.
Tynnov T Tartu Ulik [17] M -complementary spaces of Fourier coefficients,
Toimetised 192 (1966), 65–81 (in Russian).
2008. februar´ 10. –13:57
52 SHIMSHON BARON, ELIJAH LIFLYAND, ULRICH STADTMULLER¨ Tartu
[18] M Tynnov Summability factors, Fourier coefficients and multipliers;
Ulik Toimetised 192 (1966), 82–97 (in Russian).
Shimshon Baron Elijah Liflyand Department of Mathematics Department of Mathematics Bar-Ilan University Bar-Ilan University Ramat-Gan 52900 Ramat-Gan 52900
Israel Israel
baron math biu ac il liflyand math biu ac il
Ulrich Stadtmuller¨ Department of Mathematics University of Ulm 89069 Ulm
Germany ulrich stadtmueller uni ulm de
2008. februar´ 10. –13:57 ANNALES UNIV. SCI. BUDAPEST., 49 (2006), 53–57
INTERSECTIONS OF OPEN SETS IN GENERALIZED TOPOLOGICAL SPACES
By
AKOS´ CSASZ´ AR*´ Received November
0. Introduction
X
Let X be a (non-empty) set and a topology on . In the literature, a lot
X kA
of papers use the following construction: for a subset A ,let denote
A kernel A
the intersection of all -open sets containing (usually called of )
A kA and then they consider the class of all sets A such that = (see e.g. [1],
[12] or [17]). Another often used construction is the following: one considers
X A G A G all sets A such that, if , the closure of is contained in . In these constructions, the role of open sets is often given to sets taken from another class (see e.g. [9], [14], [15], [16]). The purpose of the present paper is to formulate a rather general form of
the above constructions and, in particular, to look for conditions under which
the result is a generalized topology (briefly GT), i.e., a subset of the power
X set exp X of such that and every union of elements of belongs
to (see [6]).
1. Preliminaries
X A X
For a set X , let us consider a subset exp and .Let
open closed
us call the elements of and their complements; let be
i A
the family of all -closed sets. According to [9], denotes the union of
A i A
all -open sets contained in ,orlet = if there is no such set in .
Mathematics Subject Classi cation : 54A05 * Research supported by Hungarian Foundation for Scientific Research, grant No. T 49786.
2008. februar´ 10. –13:24
54 AKOS´ CSASZ´ AR´
A A
Similarly, let c denote the intersection of all elements of containing
A X
and let c = if there is no such set in .By[9],
i i A i A A c A c c A
(1 1) = =
and
A B X i A i B c A c B
(1 2) implies and
A i A A i A
so that, by [5], the sets A such that = (that is ) constitute a
GT . Moreover
i X A X c A c X A X i A
(1 3) ( )= and ( )=
Againby[9],if is a GT then
A c c A
k
(1 4) = k
k K
k K ultratopology
so that is an (i.e., a topology such that every intersection of a
non-empty family of -open sets is -open). Therefore is an ultratopology
as well.
X base
A family exp is said to be a for a GT if is composed
of and all unions of subfamilies of (see [10] or [11]).
2. The case of a GT
X
Consider a family exp and the GT defined as above.
Proposition The family is a base for the GT
N N i N
Proof If then = is either empty or a union of some
N N N
elements of . Conversely, if = then as is a GT. If
i N N N
then clearly N = and . Hence if is the union of some elements
N of then belongs to the GT .
Cf. [9], 2.8.
Corollary If is a GT then the family of the closed sets is
a base for the ultratopology
2008. februar´ 10. –13:24 INTERSECTIONS OF OPEN SETS IN GENERALIZED TOPOLOGICAL SPACES 55
3. Generalized closed sets
X
Let us now consider exp . Let us denote by ( ) the family of
X A L c A L
the sets A having the following property: if then .
closed The elements of this family are called ( ) .
The terminology is justified by
Every closed set belongs to
Lemma ( )
Proof F F i X F F c F
implies X = ( ) so that, by (1.3), = .
With the help of the operations introduced above, it is easy to characterize
the family ( ):
Proposition The following statements are equivalent
a A
( )
b c A c A
C c A C X c A C Proof
a) b): Define = .If = then clearly .If
X C L A L
C then is the intersection of all sets such that ( ) = .
L c A L c A C c A
For every such set , , consequently = .
A L c A L c A L
b) a): If then so that by b). Hence
A ( ).
If is a GT then is a GT as well
Theorem ( )
Proof c A
As ,wehave = and ( ). If k ( )for
S
A k K A A L A L k
k
and = , choose .Then k for each ,
k K
c A L c A L
hence k , so that (1.4) implies .
X
If is a GT on , we can choose = and then the families , and ( ) are all GT’s. The following simple example shows that they can
be distinct.
X fa b cg f fa bg fa cg Xg
Example Let = , = . Then for
f fb g fc g Xg i A A
= ,wehave = .Now is equal to for =
fa g fb g A fb g fa bg fc g A fc g fa cg
= or ,to for = or ,to for = or ,
b cg A fb cg X A X A i A
to f for = and to for = . Therefore = holds
fb g fc g fb cg X f fb g fc g fb cg Xg for A = so that = .Further
2008. februar´ 10. –13:24
56 AKOS´ CSASZ´ AR´
f fa g fa bg fa cg Xg c c
= . Finally, by = , and are defined as
follows:
fa g fb g fc g fa bg fa cg fb cg X
A = ;
A fa g fa bg fa cg fa bg fa cg X X
c = ;
c A X fb g fc g X X X X
=
A c A c A
Therefore ( ) is composed of the sets with , i.e., equals
fa g fa bg fa cg fb cg Xg
f .
m
4. The case = , =
In the paper [12], the following particular case of ( ) is considered:
X minimality m m X
one chooses a GT on and a (i.e., a subset exp
m m A X
satisfying ). Then ( ) is composed of all sets such that
A M m c A M A mg closed
implies ; these sets are called in [12]. Different particular cases are studied in [4] and [18] so that the results in [12] produce generalizations of the results in [4] or [18].
Here we formulate a slight sharpening of [12], 4.2:
Theorem If A is mg closed then c A A does not contain any
m closed set F Conversely if m is a GT such that m and the above
condition is ful lled then A is mg closed
Proof A mg F c A A
Suppose that is -closed and assume that
M X F m F X A A M
is m-closed. Then = and implies .By
c A M X F c A F
hypothesis, = so that = in contradiction to
F c A F
, except if = .
m c A A
Conversely (cf. the proof of [12], 4.2), if is a GT and does
m F A M m F c A X M
not contain any -closed set ,let .Then = ( )
m X M X A F c A A F
is -closed and implies . Hence = and
c A M A mg
: is -closed.
The above theorem generalizes [4], Theorems 2.6 and 2.9.
References
M Ganster [1] F G Arenas J Dontchev and On -sets and dual of generalized
continuity, Questions Answers Gen Topology 15 (1997), 3–13.
2008. februar´ 10. –13:24
INTERSECTIONS OF OPEN SETS IN GENERALIZED TOPOLOGICAL SPACES 57 B K Lahiri [2] P Bhattacharayya and Semi-generalized closed sets in topol-
ogy, Indian J Math 29 (1987), 375–382.
Caldas E Ekici S Jafari T Noiri [3] M and On the class of contra -
continuous functions, Annales Univ Sci Budapest Sectio Math 49 (2006),
???–???.
T Noiri
[4] M Caldas S Jafari and On -generalized closed sets in topologi- http dx doi org s cal spaces, Acta Math Hungar ,
.
aszar Acta Math Hungar
[5] A Cs Generalized open sets, 75 (1997), 65–87.
Acta Math Hun aszar [6] A Cs Generalized topology, generalized continuity,
gar 96 (2002), 351–357.
Annales Univ aszar [7] A Cs Extremally disconnected generalized topologies,
Sci Budapest Sectio Math 47 (2004), 91–96.
Acta Math Hun aszar [8] A Cs Generalized open sets in generalized topologies,
gar 106 (2005), 53–66.
aszar Acta Math [9] A Cs Ultratopologies generated by generalized topologies,
Hungar 110 (2006), 153–157.
aszar Acta Math Hungar [10] A Cs Normal generalized topologies, 115(4)
(2007), 309–313.
aszar Acta [11] A Cs Modification of generalized topologies via hereditary classes,
Math Hungar 115(1–2) (2007), 29–36.
aszar Acta Math Hungar
[12] A Cs Generalized closed sets, , (to appear????).
Csaszar Annales Univ [13] A -open sets in generalized topological spaces,
Budapest Sectio Math 49? (2006?), ????–????.
T Noiri Acta Math [14] M Ganster S Jafari and On pre- -sets and pre- -sets,
Hungar 95(4) (2002), 337–343.
uldurdek O B Ozbak r Acta Math Hungar [15] A G and On -semi-open sets,
109 (2005), 347–355.
T Noiri s p Acta Math [16] E Hatir and -sets and some weak separation axioms,
Hungar 103(3) (2004), 225–232. Rend Circ Mat Palermo [17] N Levine Generalized closed sets in topology, 19
(1970), 89–96.
V Popa g Rend Circ Mat [18] T Noiri and Between closed sets and -closed sets,
Palermo 55(2) (2006), 175–184.
Akos´ Csasz´ ar´ Department of Analysis, Lorand´ Eotv¨ os¨ University, Pazm´ any´ P. set´ any´ 1./c
H-1117 Budapest, Hungary
csaszar ludens elte hu
2008. februar´ 10. –13:24 2008. februar´ 10. –15:13 ANNALES UNIV. SCI. BUDAPEST., 49 (2006), 59–63
-OPEN SETS IN GENERALIZED TOPOLOGICAL SPACES
By
AKOS´ CSASZ´ AR´ Received November
The concept of a -open set is generalized in the way that the given topology is replaced by a generalized topology.
0. INTRODUCTION
L X
Let ( X ) be a topological space. According to [1], a set is said
L L A X
to be a -set iff is the intersection of all open supersets of .Aset
A L D L D
is said to be -closed iff = where is a -set and is closed. A
X X A set A is said to be -open iff is -closed. The purpose of the present paper is to show that these constructions work
in the more general case when the topology is replaced by a generalized topology in the sense of [3] and to show some simple applications.
1. PRELIMINARIES
generalized topology
Let X be a (non-empty) set. According to [3], a
X X
(briefly GT) on X is a subset of the power set exp of such that
and every union of elements of belongs to . The elements of are called
open closed , their complements . According to e.g. [5], the union of all
AMS Subject Classification 54A05.
Key words and phrases. -set, -closed set, -open set, generalized topology. * Research supported by Hungarian Foundation for Scientific Research, grant No. T 49786.
2008. februar´ 10. –13:24
60 AKOS´ CSASZ´ AR´
A X i A
-open subsets of defines and the intersection of all -closed
A c A A B X
supersets of defines . These operations fulfil, for , ,
i i A i A A
(1 1) =
c c A c A A
(1 2) =
i A i B c A c B A B
(1 3) and if
c A X i X A i A X c X A
(1 4) = ( )and = ( )
i A c A
is always -open and is -closed.
X X
If is a GT on , then, in general, need not hold. If, all the
same, X then is said to be a strong GT (see [4]). In the general case
S
i X M fM M g c X M
= = : , hence = .
base
A family B is said to be a for a GT (see[6]or[7])iffevery B
element of is union of elements of .
If is a GT, we define, according to [5], the GT ( )tobe
A X A i c i A A c i A A
composed of all sets satisfying ( ,
i c A A c i A i c A A c i c A
, , ). For these GT’s, the following
inclusions always hold:
(1 5)
and
(1 6)
A X A i c A A
As a consequence of (1.4) and [2], 1.8, is -closed iff ,
A c i A A A c i c A A
is -closed iff , is -closed iff , is -closed iff
A i c i A A A i c A c i A
, finally is -closed iff .
2. -OPENSETSINAGT
X L X
Consider a GT on .Let be composed of all sets equal to L
the intersection of all -open supersets of .
Lemma L i L is the intersection of a family of open sets
closed A L D L L X
Let us say that a set A is - iff = where or =
and D is -closed.
Lemma Asetis closed i it is the intersection of a family of sets
each of them being open or closed
A L D L D L
Proof If = , , is -closed, then is the intersection of
A a family of -open sets by 2.1, hence is the intersection of these -open
2008. februar´ 10. –13:24
-OPEN SETS IN GENERALIZED TOPOLOGICAL SPACES 61
D L X A
sets and the -closed set . In the case = , is the intersection of the
X D
-closed sets and .
Conversely, if A is the intersection of a family of partly -open and
A L D L
partly -closed sets, then = where is the intersection of the -open
D
factors, hence L by 2.1, and is the intersection of the -closed factors,
D X
hence -closed as well. If all factors are -open, then we can put = ,
L X
and if all factors are -closed then we put = .
X open X A
Let us say that A is iff is -closed. Let us denote by
the family of all -open sets. We deduce from 2.2:
Lemma is a GT and the family B of all open and all closed
sets is a base for the GT
More precisely:
Theorem is a strong GT
X X
Proof is -open since = is -closed.
If it is necessary, we write ( ) instead of .
If and are GT s on X such that then
Lemma ( )
( )
Proof 2.3.
3. SOME APPLICATIONS
In the following, we examine the GT’s ( ), ( ), ( ), ( ), ( )fora
given GT and the GT’s , , , , defined with the help of the given
GT .
If is an arbitrary GT on X then is the discrete
Theorem ( )
topology on X
x X fx g
Proof We show that, for ,theset is either -open or -closed,
fx g fx g
hence ( )-open. In fact, if then by (1.5). On the other hand,
fx g i fx g c i fx g X M i c fx g
if then = , hence = , while
M c i fx g i c fx g fx g fx g
is contained in so that = so that is
-closed.
2008. februar´ 10. –13:24
62 AKOS´ CSASZ´ AR´
If is an arbitrary GT on X then is the discrete
Corollary ( )
topology on X
Proof (1.5) and 2.5.
For ( ), we have a weaker statement:
If is a strong GT on X then is the discrete topology
Theorem ( )
on X
x X fx g fx g fx g
Proof If and then by (1.5). If then
i fx g c i fx g X fx g
= , hence = as is -closed by , consequently
is -closed.
The statement of 3.3 need not hold if the GT is not strong:
X fa b cg f fa gg
Example Let = and = . Now we have
c fb g fb cg i fb cg fb g fb g
= , = does not contain so that .On
i fb g c fb cg fb g fb g
the other hand, = , = is not contained in so that is
fb g not -closed and ( ). Therefore ( ) cannot be equal to the discrete
topology.
X It can happen that is a topology on and neither ( ) nor ( ) equals
to the discrete topology:
Example X fa bg f Xg i fa g c
Let = , = .Now = , = so
fa g c fa g X i X X fa g
that , while = , = and is not -closed. Therefore
( ) does not be equal to the discrete topology. By (1.5) and 2.5 the same is true for ( ).
REFERENCES [1] F G Arenas J Dontchev and M Ganster On -sets and dual of general-
ized continuity, Questions Answers Gen Topology 15 (1997), 3–13.
aszar Acta Math Hungar
[2] A Cs Generalized open sets, 75 (1997), 65–87.
Acta Math Hun Csaszar [3] A Generalized topology, generalized continuity,
gar 96 (2002), 351–357.
aszar Ann Univ Sci [4] A Cs Extremally disconnected generalized topologies,
Budapest Sectio Math 47 (2004), 91–96.
Csaszar Acta Math [5] A Generalized open sets in generalized topologies,
Hungar 106 (2005), 53–66.
2008. februar´ 10. –13:24
-OPEN SETS IN GENERALIZED TOPOLOGICAL SPACES 63
aszar Acta
[6] A Cs Modification of generalized topologies via hereditary classes,
Math Hungar
Acta Math Hungar aszar [7] A Cs Normal generalized topologies,
Akos´ Csasz´ ar´ Department of Analysis, Lorand´ Eotv¨ os¨ University, Pazm´ any´ P. set´ any´ 1./c
H-1117 Budapest, Hungary
csaszar ludens elte hu
2008. februar´ 10. –13:24 2008. februar´ 10. –15:13
ANNALES UNIV. SCI. BUDAPEST., 49 (2006), 65–73 F
ON E-SETS AND -SETS AND DECOMPOSITIONS OF A -CONTINUITY, QUASI-CONTINUITY AND -CONTINUITY
By
ERDAL EKICI and SAEID JAFARI Received October
1. Introduction
In general topology, mathematicians have introduced in several papers different and interesting new decompositions of continuous functions as well as of some other weak forms of continuous functions. In two recent papers,
Al-Nashef [2] has introduced a decomposition of -continuity and quasi-
B
continuity via LC -sets and -sets in 2002. Aslim and Ayhan [4] have
A LC
introduced an other decomposition of -continuity via -sets and -sets
F E in 2004. In this paper, we introduce the notions of E-sets, -sets, -continu-
ity and F-continuity. Also, the relationships between these classes of sets and other related sets are investigated. Together with some other forms of
continuities, we obtain some new decompositions of -continuity, quasi-con-
tinuity and A-continuity. Y
In this paper (X )and( ) represent topological spaces. For a sub-
X K K K
set K of a space ,cl( )andint( ) denote the closure of and the interior
K X
of K , respectively. A subset of a space ( ) is called regular open (resp.
K K K K
regular closed) [16] if K =int(cl( )) (resp. = cl(int( ))). A subset
x K
is said to be -open [18] if for each , there exists a regular open
x G K x X
set G such that . A point is called a -cluster point
K U U x
of K [18] if int(cl( )) for each open set containing .Thesetofall
K K K
-cluster points of is called the -closure of and is denoted by -cl( ).
K K K fx X x U K
If -cl( )= ,then is said to be -closed. The set :
X g K
for some regular open set U of is called the -interior of and is
K K X denoted by -int( ). A subset of a space ( ) is called semiopen [10]
Mathematics Subject Classi cation : 54C05, 54C08
2008. februar´ 10. –13:57
66 ERDAL EKICI, SAEID JAFARI
b
(resp. -open [13], -open [1], -open [3] or -open [8] or sp-open [6],
K K
preopen [11], -preopen [15], -semiopen [14]) if cl(int( )) (resp.
K K K K K K
K int(cl(int( ))), cl(int(cl( ))), int(cl( )) cl(int( )),
K K K K K
K int(cl( )), int( -cl( )), cl( -int( ))). The complement
of a -semiopen set is called a -semiclosed set. The intersection of all
K X
-semiclosed sets, each containing a set in a topological space is called
K K
the -semiclosure of and it is denoted by -scl( ) [14].
K X
Definition A subset of a space ( ) is called
K D X fV T V T g
(1) a D-set [7] if ( )= : and is -closed ;
K V T V T
(2) a DS-set [7] if = ,where is open and is -semiclosed;
K A X fV T V T T g
(3) an A-set [17] if ( )= : , = cl(int( )) ;
K LC X fV T V T (4) an LC -set [2] if ( )= : is -open and is
closedg;
B K B X fV T V T (5) an -set [2] if ( )= : is -open and is
semiclosedg;
K A X fV T V T T g
(6) an A-set [4] if ( )= : is -open and cl(int( )) = ;
K B X fV T V T X g
(7) a B2-set [4] if 2( )= : is -open and cl( )= ;
K LC X fV T V T T g
(8) an LC -set [5] if ( )= : ,cl( )= .
f X Y
Definition A function :( ) ( ) is called -continuous [1] (resp. -continuous [12], -continuous [8], quasi-continuous [9], precontin-
1
f V
uous [11], -almost continuous [15]) if ( )is -open (resp. -open,
V
-open, semiopen, preopen, -preopen) for each . F
2. E-sets and -sets
K X E
Definition A subset of a topological space is called an -set if
V T V T
K = ,where is -open and is -closed.
X E X
The family of all E-sets of a space will be denoted by ( ).
Lemma Let X be a topological space and A X Then A is
open if and only if A T where V is open and T is dense
= V int( )
Theorem The following are equivalent for a subset K of a space X
K E X
( )
V T where V is a D set and T is dense K
= int( )
K E X K U F U
Proof ( ): Let ( ). This implies = where is -open
U W T W and F is -closed. By Lemma 4, we have = where is open and
2008. februar´ 10. –13:57
F A
ON E - AND -SETS, AND DECOMPOSITIONS OF -, QUASI- AND -CONTINUITY 67
K U F W T F W F T
int( T ) is dense. Moreover, we have = = =( )
F D T
such that W is a -set and int( )isdense.
K V T V D T V
( ): Let = where is a -set and int( ) is dense. Since is
M R V M R
a D-set, there exist an open set and a -closed set such that = .
V T M R T M T R M T
We have K = = =( ) where is an -open
E X
by Lemma 4. Thus, K ( ).
K X F
Definition A subset of a topological space is called an -set if
V T V T
K = ,where is -open and is -semiclosed.
X F X
The family of all F-sets of a space will be denoted by ( ).
Theorem The following are equivalent for a subset K of a space X
K F X
( )
V T where V is a DS set and T is dense K = int( )
Proof It is similar to that of Theorem 5.
K X
Remark (1) The following diagram holds for a subset of a space :
B
LC -set -set
F
E-set -set
A-set
E
(2) Every -open and every -closed set is an -set and every -open F and every -semiclosed set is an -set. None of these implications is reversible as shown in the following exam-
ples.
X fx y w zg f X fx g fw g fx yg
Example Let = and let =
x wg fx y wg fx w zgg
f .Then
w z g E F
(1) the set f is an -set and so an -set but it is not -open;
x zg E F (2) the set f is an -set and so an -set but it is neither -closed nor
-semiclosed;
z g E A
(3) the set f is an -set but it is not an -set;
y w z g LC B E (4) the set f is an -set and an -set but it is neither an -set
nor an F-set.
X fa b c d g f X fb g fd g fb d gg
Example Let = and let = .
c d g F LC E The set f is an -set and it is neither an -set nor an -set.
2008. februar´ 10. –13:57
68 ERDAL EKICI, SAEID JAFARI
Theorem The following are equivalent for a subset K of a space X
X K E
( )
K K for some open set V
= V cl( )
K E X K V F V
Proof ( ): Let ( ). We have = where is -open
K F K F F
and F is -closed. Since , -cl( ) -cl( )= . This implies
K V F K V K K V K
V -cl( ) = -cl( ) and hence = -cl( ).
K V K V K
( ): Let = -cl( )forsome -open set .Since -cl( )is
K E X
-closed, ( ).
Theorem The following are equivalent for a subset K of a space X
K F X
( )
V K for some open set V K = scl( )
Proof Similar to that of Theorem 11.
Theorem The following are equivalent for a subset K of a space X
K is open
K is preopen and an E set
K is preopen and an F set
K is preopen and an E set
K is preopen and an F set
K is a B E set
2 set and an
K is a B F set
2 set and an
K is preopen and an A set
K is preopen and an A set
K is a B LC set
2 set and an
E Proof (1) (2): Since every -open set is both preopen and -set, the
proof is obvious.
(2) (3) (5): Obvious.
(2) (4) (5): Obvious.
K F K
(5) (1): Let be -preopen and an -set. Since is -preopen, then
K K F X K U K
K int( -cl( )). From ( )wehave = -scl( )forsome
U K K K K
-open set . On the other hand -scl( )= int( -cl( )) = int( -cl( ))
K K U K U K
since K int( -cl( )). Hence, = -scl( )= int( -cl( )). Thus,
K is -open.
(1) (6): It follows from the fact that every -open set is both E a B2-set and an -set.
(6) (7): Obvious.
2008. februar´ 10. –13:57
F A
ON E - AND -SETS, AND DECOMPOSITIONS OF -, QUASI- AND -CONTINUITY 69
K B F K B
(7) (1): Let be a 2-set and an -set. Since is a 2-set, by F
Proposition 3.11 [4] then it is preopen. Hence, K is preopen and an -set.
Thus, by (3) K is -open. (1) (8) (9): Obvious.
(9) (4): It follows from Remark 8.
(6) (10): Obvious.
K B LC K B
(10) (1): Let be a 2-set and an -set. Since is a 2-set, by LC
Proposition 3.11 [4] then it is preopen. Hence, K is preopen and an -set.
Thus, by Theorem 3.13 [4] K is -open.
Theorem The following are equivalent for a subset K of a space X
K is an A set
K is an A set and a D set
A A Proof (1) (2): It follows from the fact that every -set is an -set
and an D-set.
K A D K LC
(2) (1): Let be an -set and a -set. Then is an -set. By A
Theorem 3.9 [4], K is an -set.
Theorem The following are equivalent for a subset K of a space X
K is semiopen
K is open and an E set
K is open and an E set
K K Proof (1) (2): Let be semiopen. This implies that is -open.
Also, by Proposition 3.2 [2], there exists a regular closed set A such that
A B B A D K K = where int( ) is dense. Since is a -set, by Theorem 5 is an
E-set.
(2) (3): Obvious.
K E
(3) (1): Let be -open and an -set. By Theorem 5, there exists a
A K A B B D
D-set such that = where int( ) is dense. Since every -set is an
K LC LC -set, by Proposition 3.5 [2] is an -set. Thus, by Theorem 3.7 [2],
K is semiopen.
Theorem The following are equivalent for a subset K of a space X
K is open
K is a B A set and a D set
2 and
K is a preopen A set and a D set
K is a preopen A set and a D set Proof (1) (2): Obvious.
2008. februar´ 10. –13:57
70 ERDAL EKICI, SAEID JAFARI B (2) (3): Since every 2-set is preopen by Proposition 3.11 [4], the proof is completed.
(3) (4): Obvious.
K A D K
(4) (1): Let be a -preopen -set and a -set. Since is
D E
-preopen and a -set, it is -preopen and an -set. So, by Theorem 13
K A D
K is -open. Since is an -set and a -set, by Theorem 14 it is an K
A-set. Thus, by Theorem 3.2 [17], is open. A
3. Decompositions of -continuity, quasi-continuity and -continuity
f X Y E
Definition A function :( ) ( ) is called -continuous (resp.
1 1
f V E X f V F X V
F-continuous) if ( ) ( ) (resp. ( ) ( )) for each .
f X Y B
Definition A function :( ) ( ) is called -continu-
LC B
ous [2] (resp. A-continuous [4], -continuous [2], 2-continuous [4],
1 1 1
f V B X f V A X f V
D-continuous [7]) if ( ) ( ) (resp. ( ) ( ), ( )
1 1
X f V B X f V D X V LC
( ), ( ) 2( ), ( ) ( )) for each .
f X Y
Remark The following diagram holds for a function : :
B
LC -continuous -continuous
E F A-continuous -continuous -continuous
None of these implications is reversible as shown in the following exam-
ples:
X fx y w zg f fx g fw g fx yg fx wg
Example Let = and =
x y wg fx w zg Xg
f .Then
X X f x z f y z f w x
(1) the function f :( ) ( ), defined as ( )= , ( )= , ( )= ,
z w E A
f ( )= ,is -continuous but it is not -continuous;
X X h x z h y w h w y
(2) the function h :( ) ( ), defined as ( )= , ( )= , ( )= ,
z x LC B E h ( )= ,is -continuous and -continuous but it is not -continuous
and F-continuous.
X fa b c d g f X fb g fd g fb d gg
Example Let = and = .
X X f a c f b a
Define a function f :( ) ( ) as follows: ( )= , ( )= ,
c b f d b f F E f ( )= , ( )= .Then is -continuous but it is neither -continuous
nor LC -continuous.
2008. februar´ 10. –13:57
F A
ON E - AND -SETS, AND DECOMPOSITIONS OF -, QUASI- AND -CONTINUITY 71
For a function f X Y the following are equivalent
Theorem :
f is continuous
f is precontinuous and E continuous
f is precontinuous and F continuous
f is almost continuous and E continuous
f is almost continuous and F continuous
f is B continuous and E continuous
2
f is B continuous and F continuous
2
f is precontinuous and A continuous
f is almost continuous and A continuous
f is B LC continuous 2 continuous and
Proof It follows from Theorem 13.
For a function f X Y the following are equivalent
Theorem :
f is A continuous
f is A continuous and D continuous
Proof The proof is an immediate consequence of Theorem 14.
For a function f X Y the following are equivalent
Theorem :
f is quasi continuous
f is continuous and E continuous
f is continuous and E continuous
Proof It follows from Theorem 15.
For a function f X Y the following are equivalent
Theorem :
f is continuous
continuous A continuous and D continuous f is B
2
f is precontinuous A continuous and D continuous
f is almost continuous A continuous and D continuous
Proof The proof is an immediate consequence of Theorem 16.
2008. februar´ 10. –13:57 72 ERDAL EKICI, SAEID JAFARI
References
R A Mahmoud
[1] M E Abd El Monsef S N El Deeb and -open sets and Bull Fac Sci Assiut Univ
-continuous mappings, 12 (1983), 77–90.
Acta [2] B Al Nashef A decomposition of -continuity and semi-continuity,
Math Hungar 97(1–2) (2002), 115–120.
c b Mat Bech
[3] D Andrijevi On -open sets, 48 (1996), 59–64. Y Ayhan [4] G Aslim and Decompositions of some weaker forms of continuity,
Acta Math Hungar 105(1–2) (2004), 85–91.
General Topology [5] N Bourbaki , Part I, Addison Wesley, Reading, Mass.,
1996. M Przemski [6] J Dontchev and On the various decompositions of continu-
ous and some weakly continuous functions, Acta Math Hungar 71(1–2)
(1996), 109–120.
S Jafari D DS
[7] E Ekici and On -sets, -sets and decompositions of continu-
AB Italian J Pure and Appl ous, A-continuous and -continuous functions;
Math , (2007), (to appear).
A El Atik A study of some types of mappings on topological spaces [8] A ,
Master’s Thesis, Faculty of Science, Tanta Univ., Egypt, 1997. Fund Math [9] S Kempisty Sur les functions quasicontinues, 19 (1932), 184–
197. Amer [10] N Levine Semi-open sets and semi-continuity in topological spaces,
Math Monthly 70 (1963), 36–41. S N El Deeb [11] A S Mashhour M E Abd El Monsef and On precontinuous
and weak precontinuous mappings, Proc Math Phys Soc Egypt 53 (1982),
47–53.
S N El Deeb [12] A S Mashhour I A Hasanein and -continuous and -open
mappings, Acta Math Hungar 41 (1983), 213–218.
astad Paci c J Math [13] O Nj On some classes of nearly open sets, 15 (1965),
961–970.
M J Son [14] J H Park B Y Lee and On -semiopen sets in topological space,
J Indian Acad Math 19(1) (1997), 59–67.
Raychaudhuri M N Mukherjee [15] S and On -almost continuity and -
preopen sets, Bull Inst Math Acad Sinica 21 (1993), 357–366.
[16] M Stone Application of the theory of Boolean rings to general topology,
Trans Amer Math Soc 41 (1937), 374–481. Acta Math Hungar [17] J Tong A decomposition of continuity, 48(1–2) (1986), 11–15.
2008. februar´ 10. –13:57
F A
ON E - AND -SETS, AND DECOMPOSITIONS OF -, QUASI- AND -CONTINUITY 73
cko H Amer Math Soc Transl [18] N V Veli -closed topological spaces, 78 (1968), 103–118.
Erdal Ekici Saeid Jafari Department of Mathematics College of Vestsjælland South Canakkale Onsekiz Mart University Herrestræde 11 Terzioglu Campus 4200 Slagelse 17020 Canakkale Denmark
Turkey jafari stofanet dk eekici comu edu tr
2008. februar´ 10. –13:57 2008. februar´ 10. –15:13 ANNALES UNIV. SCI. BUDAPEST., 49 (2006), 75–86
ON THE CLASS OF CONTRA -CONTINUOUS FUNCTIONS
By
MIGUEL CALDAS, ERDAL EKICI, SAEID JAFARI AND TAKASHI NOIRI Received October
1. Introduction
Maki [18] in 1986 introduced the notion of -sets in topological spaces. A A -set is a set if it is equal to its kernel (= saturated set), i.e., to the
intersection of all open supersets of A. Arenas et al. [1] introduced and investigated the notion of -closed sets by involving -sets and closed sets.
This enabled them to obtain some nice results. Quite recently, Caldas et al. [3] introduced the notion of -closure of a set by utilizing the notion of -open sets defined in [1]. Recently, Jafari and Noiri introduced and investigated
the notions of contra-precontinuity [14], contra- -continuity [12] and con- tra-super-continuity [13] as a continuation of research done by Dontchev [8] and Dontchev and Noiri [9] on the interesting notions of contra-continuity and contra-semi-continuity, respectively. Caldas and Jafari [6] introduced and
investigated the notion of contra- -continuous functions in topological spaces.
The present paper has as its purpose to investigate some properties of contra
-continuous functions, by using -open sets.
Y X Y
Throughout this paper, by (X )and( )(or and )wealways X
mean topological spaces. Let A be a subset of . We denote the interior, the
c
A A X nA A
closure and the complement of a set A by Int( ), Cl( )and or ,re- X
spectively. A subset A of is said to be regular open (resp. regular closed) if
A A A A X
A =Int(Cl( )) (resp. = Cl(Int( ))). A subset of a space is called pre-
A A
open [17] (resp. semi-open [16], -open [21], -open [2]) if Int(Cl( ))
A A A A A
(resp. A Cl(Int( )), Int(Cl(Int( ))), Cl(Int(Cl( )))). The complement of a preopen (resp. semi-open, -open, -open) set is said to be
Mathematics Subject Classi cation : 54C10, 54D10
2008. februar´ 10. –13:25
76 MIGUEL CALDAS, ERDAL EKICI, SAEID JAFARI, TAKASHI NOIRI
preclosed (resp. semi-closed, -closed, -closed). The collection of all closed
X X
(resp. clopen) subsets of X will be denoted by C( ) (resp. CO( )). We set
fV X x V g x X X x
C( X x )= C( ): for . We define similarly CO( ).
X g X
A subset B of a topological space is called -closed in [15] if
G B G G X A X
Cl( B ) whenever and is open in . A subset of ( )is
closed A L D L D
called [1] if = ,where is a -set and is a closed set.
open
The complement of a -closed set is called . We denote the collection
O X X
of all -open sets (resp. -closed sets) by ( ) (resp. C( )). We set
X x fU x U O X g X x fU x U
O ( )= : ( ) and C( )= :
X g x X cluster
C( ) . A point in a topological space ( ) is called a
point U X x A U
of A [3] if every -open set of containing fulfils .The
closure A A
set of all -cluster points is called the of and is denoted by Cl ( )
B X neighborhood
([1, 3]). A subset x of a topological space is said to be
x X U x U B ofapoint
[3] if there exists a -open set such that x .
Lemma Let A B and A i I be subsets of a topological
1 3 i
X The following properties hold
space ( )
If A is closed for each i I then A is closed
i i I i
If A is open for each i I then A is open
i i I i
closed if and only if A A
Ais =Cl ( )
A fF X A F g Cl ( )= ( ):
C
A A
Cl ( )
A B then A B
If Cl ( ) Cl ( )
A is closed
Cl ( )
Recall that a topological space (X )issaidtobe:
(i) T ([3, 4]) if every singleton is -open or -closed.
1 2
T x y X
(ii) - 2 ([3, 4]) if for any pair of distinct points and in ,thereexist
O X x V O X y U V
U ( )and ( ) such that = .
y X
(iii) Ultra Hausdorff [22] if for each pair of distinct points x and in there
X x V X y U V exist U CO( )and CO( ) such that = .
A space X is called locally indiscrete [20] if every open set is closed.
f X Y
Definition function : is said to be:
Y
(i) -continuous [1] if the inverse image of every closed set in is -closed
Y X
in X , equivalently if the inverse image of every open set in is -open in . Y (ii) LC -continuous [11] if the inverse image of every open set in is locally
closed in X .
2008. februar´ 10. –13:25 ON THE CLASS OF CONTRA -CONTINUOUS FUNCTIONS 77
2. Contra -continuous functions
f X Y Definition A function : is said to be contra -continuous
1
V X V Y
if f ( )is -closed in for each open set of .
T
fU X
Definition Let A be a subset of a space ( ). The set
A U g A A
: is called the kernel of [19] and is denoted by ker( ).
The following properties hold for
Lemma Jafari and Noiri [13]
the subsets A B of a space X
x if and only if A F for any F X x
ker(A) C( )
A A and A A if A is open in X
ker( ) =ker( )
A B then A B
If ker( ) ker( )
X Y be a function from a topological space X Let f
Theorem :
into a topological space Y The following statements are equivalent
f is contra continuous
The inverse image of each closed set in Y is open in X
x V there x in X and each closed set V in Y with f
For each point ( )
exists a open set U in X such that x U f V
( U )
For every subset A of X f f A
(Cl ( A)) ker( ( ))
1 1
B of Y f B f B
For each subset Cl ( ( )) (ker( ))
Proof (1) (2): see Definition 2.
x X V f x (2) (3): Let and be a closed set containing ( ). By (2),
1
f V x f U V
U = ( )isa -open set containing such that ( ) .
A X y f A
(3) (4): Let be any subset of . Suppose that ker( ( )). Then,
Y y f A V by Lemma 2.1 there exists V C( ) such that ( ) = .Forany
1
x f V U O X x f U V x
( ), by (3) there exists x ( ) such that ( ) . Hence
f A U f A f U f A V A U
x x
( x ) ( ) ( ) ( ) = and = . This shows
1 1
A x f V f V A
that x Cl ( )forany ( ). Therefore, ( ) Cl ( )= and
f A y f A
hence V (Cl ( )) = . Thus, (Cl ( )). Consequently, we obtain
A f A
f (Cl ( )) ker( ( )).
Y
(4) (5): Let B be any subset of . By (4) and Lemma 2.1, we have
1 1 1 1
f B f f B B f B f B
f (Cl ( ( ))) ker( ( ( ))) ker( )andCl ( ( )) (ker( )).
V Y
(5) (1): Let be any open set of . Then, by Lemma 2.1(2) we have
1 1 1 1 1
V f V f V f V f V
Cl ( f ( )) (ker( )) = ( )andCl( ( )) = ( ). This
1
V X shows that f ( )is -closed in . This completes the proof.
2008. februar´ 10. –13:25
78 MIGUEL CALDAS, ERDAL EKICI, SAEID JAFARI, TAKASHI NOIRI
f X Y Definition A function : is said to be contra-continuous [8]
(resp. contra- -continuous [12], contra-pre-continuous [14], contra-semi-con- g tinuous [9], contra- -continuous [6] and contra- -continuous [5]) if for each
1
Y f V
open set V of , ( ) is closed (resp. -closed, preclosed, semi-closed,
g X -closed and -closed) in .
For the functions defined above, we have the following implications:
contra continuity contra continuity contra pre continuity
contra continuity contra semi continuity contra continuity Remark (1) The following Examples 2.4 and 2.5 show that -con-
tinuous and contra- -continuous are independent concepts.
(2) The following Examples 2.5 and 2.9 show that contra- -continuous and
contra- g -continuous are independent concepts.
Example The identity function on the real line (with the usual topol- ogy) is continuous and hence -continuous but not contra -continuous, since
the preimage of each singleton fails to be -open.
X X fa b cg
Example Let ( ) be a topological space such that, =
f fa g fa bg Xg O X f fa g fc g fa bg fa cg
and = . Clearly ( )=
b cg Xg f X X f b a f c b
f .Let : be a function defined by ( )= , ( )= and
a c f f f ( )= .Then is contra -continuous, but is not -continuous and also it
is not contra- g -continuous.
Remark It should be mentioned that every contra-continuous func-
tion is contra- g -continuous and none of the implications in the above diagram
is reversible as shown by the following examples.
X Y fa b cg f fa g fc g fa cg Xg
Example Let = = = and
f fa g Y g f X X = . Then the identity function :( ) ( ) is contra- -
continuous but not contra-continuous.
X fa b cg f fa g Xg f fb g
Example [12] Let = = and =
fc g fb cg Xg X X . Then the identity function f :( ) ( ) is contra- -
continuous but not contra-continuous.
X fa b cg f fa g Xg f fb g
Example [5] Let = = and =
c g fb cg Xg f X X g f . Then the identity function :( ) ( ) is contra- -
continuous but not contra-continuous and also it is not contra- -continuous.
2008. februar´ 10. –13:25
ON THE CLASS OF CONTRA -CONTINUOUS FUNCTIONS 79
Example [9] A contra-semi-continuous function need not be con-
R R f x x x tra-pre-continuous. Let f : be the function ( )=[ ], where [ ]is
the Gaussian symbol. If V is a closed subset of the real line, its preimage
1
f V n n n Z U U = ( ) is the union of intervals of the form [ +1), ; hence is
semi-open being union of semi-open sets. But f is not contra-pre-continuous,
1
R
since f (0 5 1 5) = [1 2) is not preclosed in .
Example [9] A contra-pre-continuous function need not be con-
fa bg f Xg f fa g Xg
tra-semi-continuous. Let X = , = and = .The
X Y identity function f :( ) ( ) is contra-pre-continuous as only the
1
X f fa g fa g
trivial subsets of X are open in ( ). However, ( )= is not f
semi-closed in (X ); hence is not contra-semi-continuous. T Recall that a topological space (X ) is called a -space [15] if every 1 2
generalized closed subset of X is closed or equivalently if every singleton is
open or closed [10].
Let X be a T space and f X Y be a func Lemma ( ) :
1 2
tion If f is contra continuous resp contra semi continuous contra pre
continuous contra continuous contra g continuous then f is contra continuous
Proof It follows directly from Theorem 2.6 of [1].
X Y RC Recall that a function f : is said to be -continuous [9] if for
1
Y f V X
each open set V of , ( ) is regular-closed in .
Theorem The following statements are equivalent for a func
tion f Y
: X
i f is RC continuous
ii f is contra pre continuous and semi continuous
iii f is contra continuous and continuous
iv f is contra continuous and continuous
f X Y is RC continuous then f is contra continuous If :
Proof (1) It follows directly from Theorem 3.9 of [14] and Theo- rem 3.11 of [9].
(2) Every RC -continuous function is contra-continuous and hence contra
-continuous. We present a new decomposition of contra-continuity.
2008. februar´ 10. –13:25
80 MIGUEL CALDAS, ERDAL EKICI, SAEID JAFARI, TAKASHI NOIRI
For a function f X Y the following conditions are
Theorem :
equivalent
f is contra continuous
f is contra g continuous and LC continuous
f is contra g continuous and contra continuous
Proof It follows directly from [1, Theorem 2.4 and Lemma 2.2(i)].
If a function f X Y is contra continuous and Y
Theorem :
is regular then f is continuous
x X V Y
Proof Let be an arbitrary point of and an open set of
x Y W Y
containing f ( ). Since is regular, there exists an open set in
x W V f
containing f ( ) such that Cl( ) .Since is contra -continuous, so
O X x f U W
by Theorem 2.2 there exists U ( ) such that ( ) Cl( ). Then
U W V f
f ( ) Cl( ) . Hence, is -continuous.
f X Y
Remark By Example 2.4, a -continuous function : is Y
not always contra -continuous even if is regular.
A A
Recall that Caldas et al. [3] define the -frontier of denoted by Fr ( ),
A n A A A X nA
as Fr ( A)=Cl ( ) Int ( ), equivalently Fr ( )=Cl ( ) Cl ( ).
The set of points x X such that f X Y is
Theorem :( ) ( )
not contra continuous at x is identical with the union of the frontiers of
Y containing f x
the inverse images of closed sets of ( )
Necessity f x
Proof . Suppose that is not contra -continuous at a point
F Y f x f U
of X . Then there exists a closed set containing ( ) such that ( )
U O X x U X n
is not a subset of F for every ( ).Hencewehave (
1 1
f U O X x x X n f F
( F )) for every ( ). It follows that Cl ( ( )).
1 1 1
f F f F x f F
We also have x ( ) Cl ( ( )). This means that Fr ( ( )).
1
x f F F Y f x Su ciency. Suppose that Fr ( ( )) for some ( ( ))
C
x X Now, we assume that f is contra -continuous at . Then there exists
1
O X x f U F x U f F
U ( ) such that ( ) . Therefore, we have ( )
1 1
f F X n f F
and hence x Int ( ( )) Fr ( ( )). This is a contradiction. This
x
means that f is not contra -continuous at .
X S
Definition A space ( )issaidtobe -space if every -open X subset of X is semi-open in .
2008. februar´ 10. –13:25
ON THE CLASS OF CONTRA -CONTINUOUS FUNCTIONS 81
Lemma A space X is locally indiscrete if and only if every open
set of X is open in X
X Y is contra continuous and X is If a function f
Theorem :
a S space resp locally indiscrete then f is contra semi continuous resp contra continuous continuous
Proof It follows immediately from definitions.
Theorem If X is a topological space and for each pair of distinct
points x x in X there exists a function f of X into a Urysohn topolog
1 and 2
Y such that f x f x and f is contra continuous at x and x
ical space ( 1) ( 2) 1 2
X is T
then 2
x x X
Proof Let 1 and 2 be any distinct points in . Then by hypothesis,
f X Y
there is a Urysohn space Y and a function : , which satisfies
y f x i y y i
the conditions of the theorem. Let i = ( )for =12. Then 1 2.
Y U U y y
Since is Urysohn, there exist open neighborhoods y1 and 2 of 1
y Y U U f y
and 2, respectively, in such that Cl( y1 ) Cl( 2 )= .Since is contra
x W x X
x
i
-continuous at i , there exists a -open neighborhood of in such
i
f W U i W W
y x x
that ( x ) Cl( )for =12. Hence we get = since i
i 1 2
U U X T y
Cl( y1 ) Cl( 2 )= . Hence is - 2.
Corollary If f isacontra continuous injection of a topological
space X into a Urysohn space Y then X is T
2
x x X f
Proof For each pair of distinct points 1 and 2 in , is a contra
X Y f x f x
-continuous function of into a Urysohn space such that ( 1) ( 2)
X T
since f is injective. Hence by Theorem 2.20, is - 2.
Corollary If f isacontra continuous injection of a topological
X into an Ultra Hausdor space Y thenX is T
space 2
x x X f
Proof Let 1 and 2 be any distinct points in .Since is injective
f x f x V V Y and Y is Ultra Hausdorff, ( 1) ( 2), and there exist 1 2 CO( )such
1
f x V f x V V V x f V O X i
that ( 1) 1, ( 2) 2 and 1 2 = .Then i ( ) ( )
1 1
f V f V X T
for i =1 2and ( 1) ( 2)= . Thus is - 2.
X X X P
We say that the product space = n has property if,
1
A X i n
i
whenever i is a -open set in a topological space ,for =12 ,
A A X X X n then 1 n is also -open in the product space = 1 .
2008. februar´ 10. –13:25
82 MIGUEL CALDAS, ERDAL EKICI, SAEID JAFARI, TAKASHI NOIRI
Let f X Y and f X Y be two functions
Theorem 1 : 1 2 : 2
where
X X has the property P X =
1 2
Y is a Urysohn space
and f are contra continuous f
1 2
Then f x f x f x g is closed in the product space X X X
( x1 2): 1( 1)= 2( 2) = 1 2
A f x x f x f x g
Proof Let denote the set ( 1 2): 1( 1)= 2( 2) . In order to show
X X A x x A
that A is -closed, we show that ( 1 2) is -open. Let ( 1 2) .
x f x Y
Then f1( 1) 2( 2). Since is Urysohn, there exist open neighborhoods
V V f x f x V V f
1 and 2 of 1( 1)and 2( 2) such that Cl( 1) Cl( 2)= .Since i
1
i f V
( =12) is contra -continuous, (Cl( i )) is a -open set containing
i
1 1
x X i f V f V i
i in ( =12). Hence by (1), 1 (Cl( 1)) 2 (Cl( 2)) is -open.
1 1
x f V f V X X A
Furthermore (x1 2) 1 (Cl( 1)) 2 (Cl( 2)) 1 2 .It
X A A
follows that X1 2 is -open. Thus is -closed in the product space
X X
X = 1 2.
Corollary Assume that the product space X X has the prop
erty P Iff Y is contra continuous and Y is a Urysohn space then
: X
f x x f x f x g is closed in the product space X X
A = ( 1 2): ( 1)= ( 2)
Theorem Let f Y be a function and g X X Y the
: X :
g x x f x for every x X Then f is contra
graph function given by ( )=( ( ))
continuous if and only if g is contra continuous
Proof X W X Y
Let x and let be a closed subset of contain-
x W fx g Y fx g Y g x
ing g ( ). Then ( ) is a closed set in containing ( ).
x g Y Y fy Y x y W g Also f is homeomorphic to . Hence :( ) is a
1
f ff y x y W g
closed subset of Y .Since is contra -continuous, ( ):( )
1 1
X x ff y x y W g g W is a -open subset of .Further ( ):( ) ( ).
1
W g
Hence g ( )is -open. Then is contra -continuous.
Y X F Conversely, let F be a closed subset of .Then is a closed subset
1
Y g g X F
of X .Since is contra -continuous, ( )isa -open subset
1 1
g X F f F f
of X .Also ( )= ( ). Hence is contra -continuous.
Theorem Let X and Z be any topological spaces and Y be locally
X Z of a contra continuous Then the composition g f
indiscrete :
f X Y and a continuous function g Y Z is contra
function : :
continuous
2008. februar´ 10. –13:25 ON THE CLASS OF CONTRA -CONTINUOUS FUNCTIONS 83
Proof It follows from the definitions.
Let fX g be any family of topological spaces
Theorem : Ω
Q
If f X is a contra continuous function then f X X
: X Pr :
Q
continuous for each where is the projection of X
is contra Ω Pr
onto X
U
Proof We shall consider a fixed Ω. Suppose is an arbitrary open
Q
1
U X f
set in X .ThenPr ( ) is open in .Since is contra -continuous,
1 1 1
U f U X
we have by definition f (Pr ( )) = (Pr ) ( )is -closed in .
f
Therefore Pr is contra -continuous.
If f X Y is a contra continuous function and
Theorem :
Y Z is a continuous function then g f X Z is contra
g : :
continuous X
Definition A topological space is said to be:
X
(1) -compact [3] if every -open cover of has a finite subcover (resp.
X X X X A is -compact relative to if every cover of by -open sets of
has a finite subcover); X
(2) strongly S -closed [8] if every closed cover of has a finite subcover
X S A S
(resp. A is strongly -closed if the subspace is strongly -closed [8]);
X
(3) mildly -compact if every -clopen cover of has a finite subcover.
X Y f x f x x X g
Recall that for a function f : , the subset ( ( )) :
f G f X Y
is called the graph of and is denoted by ( ).
G f f X Y
Definition Agraph ( ) of a function : is said to be
x y X Y nG f U O X
contra -closed if for each ( ) ( ) ( ), there exist ( )
V Y y U V G f
containing x and C( ) containing such that ( ) ( )= .
G f is contra closed in X Y if and only if for each
Lemma ( )
X Y nG f there exist U O X containing x and V Y
( x y) ( ) ( ) ( ) C( )
y such that f U V
containing ( ) =
Theorem If f Y is contra continuous and Y is Urysohn
: X
f is contra closed in X Y G
then ( )
Proof X Y nG f f x y
Let ( x y) ( ) ( ), then ( ) and there exist open
W f x V y W V W f
sets V , such that ( ) , and Cl( ) Cl( )= .Since
U O X x f U V
is contra -continuous, there exists ( ) such that ( ) Cl( ).
U W G f
Therefore, we obtain f ( ) Cl( )= . This shows that ( ) is contra
X Y -closed in .
2008. februar´ 10. –13:25
84 MIGUEL CALDAS, ERDAL EKICI, SAEID JAFARI, TAKASHI NOIRI
Let X be locally indiscrete If f X Y has a contra
Theorem :
closed graph then the inverse image of a strongly S closed set K of Y is
closed in X
1
K S Y x f K
Proof Assume that is a strongly -closed set of and ( ).
k K x k G f U O X x
For each ,( ) ( ). By Lemma 2.29, there exist k ( )
T T
V Y k f U fK V k K g V
k k k
and C( ) such that ( ) k = .Since : is a
K K
closed cover of the subspace K , there exists a finite subset 0 such that
S T
K fV k K g U fU k K g U X k
k : 0 .Set = : 0 ,then is open since is
T T
1
K U U f K locally indiscrete. Therefore f ( ) = and ( )= . This shows
1
K X
that f ( )isclosedin .
If f X Y is contra continuous and K is compact
Theorem :
X then f K is strongly S closed in Y
relative to ( )
Proof fH I g f K
Let : be any cover of ( )byclosedsetsof
f K I K Y
the subspace ( ). For each , there exists a closed set of
T
f H K K x K x I
such that = ( ). For each ,thereexist ( ) such
f x K U O X x
that ( ) . By Theorem 2.2, there exists x ( ) such that
(x )
f U K fU x K g K x
( x ) . Since the family : is a -open cover of ,there x
( )
S
K K K fU x K g
exists a finite subset 0 of such that x : 0 . Therefore, we
S S
f K ff U x K g x K g fK
obtain ( ) ( x ): which is a subset of : . x
0 ( ) 0
S
f K x K g f K S Y H
Thus ( )= f : and hence ( ) is strongly -closed in . x
( ) 0
If f X Y is a contra continuous and continuous
Theorem :
surjection and X is mildly compact then Y is compact
Proof fV I g Y f
Let : be an open cover of .Since is contra
1
ff V I g
-continuous and -continuous, we have that ( ): is a -clopen
X I I
cover of X .Since is mildly -compact, there exists a finite subset 0 of
S S
1
X V I g f Y ff fV
such that = ( ): 0 .Since is surjective, = :
I Y 0 g and therefore is compact.
Recall that a function is called preclosed [7] if the image of every closed Y
subset of X is preclosed in .
Let f X Y be a surjective preclosed contra
Theorem :
continuous function If X is locally indiscrete then Y is locally indiscrete
V Y f Proof Let be any open set of . By hypothesis, is contra -
1
V U X X continuous and therefore f ( )= is -closed in .Since is locally
2008. februar´ 10. –13:25
ON THE CLASS OF CONTRA -CONTINUOUS FUNCTIONS 85
X f V
indiscrete, the set U is closed in .Since is preclosed, then is also
V V V V preclosed in Y . Now we have Cl( ) = Cl(Int( )) .Thismeansthat
is closed and hence Y is locally indiscrete.
Let f X Y be a contra continuous function If X
Theorem :
is locally indiscrete then f is contra continuous
Proof Obvious.
References
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continuity, Questions Answers Gen Topology 15 (1997), 3–13.
R A Mahmoud
[2] M E Abd El Monsef S N El Deeb and -open sets and Bull Fac Assiut Univ
-continuous mappings, 12 (1983), 77–90.
G Navalagi [3] M Caldas S Jafari and More on -closed sets in topological
spaces, Revista Colombiana 41(2) (2007), (to appear).
S Jafari
[4] M Caldas and On some low separation axioms via -open and Rend Circ Mat Palermo
-closure operator, 54 (2005), 195–208. M Simoes
[5] M Caldas S Jafari T Noiri and A new generalization of contra- Chaos Solitons and Fractals continuity via Levine’s g -closed sets, 32(4)
(2007), 1597–1603.
S Jafari [6] M Caldas and Some properties of contra- -continuous functions,
Mem Fac Sci Kochi Univ Ser A Math 22 (2001), 19–28.
T Noiri p [7] S N El Deeb I A Hasanein A S Mashhour and On -regular
spaces, Bull Math Soc Sci Math R S Roumanie 27(75) (1983), 311–315.
S In [8] J Dontchev Contra-continuous functions and strongly -closed spaces,
ternat J Math Math Sci 19 (1996), 303–310.
T Noiri Math Pannonica [9] J Dontchev and Contra-semicontinuous functions,
10 (1999), 159–168.
T Kyungpook Math J [10] W Dunham -spaces, 17 (1977), 161–169.
1 2
I L Reilly LC [11] M Ganster and Locally closed sets and -continuous functions,
Internat J Math Math Sci 3 (1989), 417–424.
Jafari T Noiri [12] S and Contra- -continuous functions between topological
spaces, Iran Int J Sci 2 (2001), 153–167.
T Noiri Annal Univ Sci [13] S Jafari and Contra-super-continuous functions,
Budapest 42 (1999), 27–34.
T Noiri Bull Malaysian [14] S Jafari and On contra-precontinuous functions,
Math Sci Soc 25 (2002), 115–128.
2008. februar´ 10. –13:25
86 CALDAS, EKICI, JAFARI, NOIRI: ON THE CLASS OF CONTRA -CONTINUOUS FUNCTIONS Rend Circ Mat Palermo [15] N Levine Generalized closed sets in topology,
19 (1970), 89–96. Amer [16] N Levine Semi-open sets and semi-continuity in topological spaces,
Math Monthly 68 (1961), 44–46. S N El Deeb [17] A S Mashhour M E Abd El Monsef and On precontinuous
and weak precontinuous mappings, Proc Math Phys Soc Egypt 53 (1982),
47–53.
Generalized sets and the associated closure operator [18] H Maki , The Special Issue in Commemoration of Prof. Kazusada IKEDA’ Retirement, 1. Oct.
(1986), 139–146.
sevic R R Bull Math [19] M Mr On pairwise 0 and pairwise 1 bitopological spaces,
Soc Sci Math R S Roumanie 30(78) (1986), 141–148. Ann Acad Sci Fenn [20] T Nieminen On ultrapseudocompact and related topics,
Ser A I Math 3 (1977), 185–205.
astad Paci c J Math [21] O Nj On some classes of nearly open sets, 15 (1965), 961–970.
[22] R Staum The algebra of bounded continuous functions into a nonarchimedean
field, Paci c J Math 50 (1974), 169–185.
Miguel Caldas Erdal Ekici Departamento de Matematica´ Aplicada Department of Mathematics Universidade Federal Fluminense Canakkale Onsekiz Mart University Rua Mario´ Santos Braga, s/n Terzioglu Campus 24020-140, Niteroi,´ RJ 17020 Canakkale
Brasil Turkey
gmamccs vm uff br eekici comu edu tr
Saeid Jafari Takashi Noiri College of Vestsjælland South Department of Mathematics Herrestræde 11 Yatsushiro College of Technology 4200 Slagelse Yatsushiro, Kumamoto, 866-8501
Denmark Japan
jafari stofanet dk t noiri nifty com
2008. februar´ 10. –13:25 ANNALES UNIV. SCI. BUDAPEST., 49 (2006), 87–93
A NOTE TO THE PAPER “ON A FAST VERSION OF A PSEUDORANDOM GENERATOR”
By
KATALIN GYARMATI* Received April
1. Introduction
C. Mauduit and A. Sark´ ozy¨ [5, pp. 367–370] introduced the following
measures of pseudorandomness of binary sequences:
N
E fe e e g f g N
For a finite binary sequence N = 1 2 1 +1 write
t 1
X
U E t a b e
N a ( )= +jb
j =0
D d d d d k
and, for =( 1 k ) with non-negative integers 1 ,
M
X
V E M D e e e
d d d n n n ( N )= + + + 1 2 k
n =1
well distribution measure E
Then the of N is defined as
t
1
X
W E E t a b j e j U
N a jb
( N )=max ( ( ) =max +
a b t a b t
j =0
a b t N a
where the maximum is taken over all a b t such that and 1
a t b N correlation measure of order k E +( 1) .The of N is defined
Mathematics Subject Classi cation : 11K45 * Research partially supported by Hungarian Scientific Research Grants OTKA T043623, T043631 and T049693.
2008. februar´ 10. –13:25 88 KATALIN GYARMATI
as
M
X
C E jV E M D j e e e
d d d N N n n n k ( )=max ( ) =max + + +
1 2 k
M D M D
n =1
D d d d M
where the maximum is taken over all =( 1 2 k )and such that
d d d M d N k
1 2 k + .
E
A sequence N is considered as a “good” pseudorandom sequence if
W E C E k
k N
each of these measures ( N ), ( ) (at least for small ) is “small” in
o N N
terms of N (in particular, all are ( )as ). Indeed, it was proved
N
E f g
in [2, Theorem 1, 2] that for a truly random sequence N 1 +1 each
p p
N N N of these measures is log and . Later Alon, Kohayakawa, Mauduit, Moreira and Rodl¨ [1] improved on these bounds. Numerous binary sequences have been tested for pseudorandomness by J. Cassaigne, S. Ferenczi, C. Mauduit, J. Rivat and A. Sark´ ozy,¨ and large families of pseudorandom binary sequences have been constructed first by L. Goubin, C. Mauduit, A. Sark´ ozy¨ [4]. I gave a further construction of this
type in [6], [7].
g p
Let p be an odd prime and be a fixed primitive root modulo ,andfor
n n p ( n p)=1ind denotes the index or discrete logarithm of modulo . Thus
ind n is defined as the unique integer with
ind n
n p
(1) g (mod )
n p and 1 ind 1. Using the discrete logarithm I introduced a new large family of pseudorandom sequences with strong pseudorandom proper- ties in [6]. However the sequences in this family can be generated very slowly, so in [7] I slightly modified the construction so that the new family can be generated much faster. In this paper I will improve on results in [7]. Recently the discrete logarithm is used more and more frequently in cryptography. Z. Chen, S. Li and G. Xiao [3] generalized the pseudorandom constructions based on the notion of index using elliptic curves. Throughout this paper we
will use the following
p g p
Notation Let be an odd prime, be a fixed primitive root modulo .
n f X F X k
Define ind by (1). Let ( ) p [ ] be a polynomial of degree 1
which has no multiple roots. Moreover, let
j p
m 1
N x m x m with m ,andlet be coprime with :( )=1.
2008. februar´ 10. –13:25 A NOTE TO THE PAPER “ON A FAST VERSION OF A PSEUDORANDOM GENERATOR” 89
The crucial idea of the new, faster construction defined in [7] was to m
reduce ind n modulo :
n
Construction Let ind denote the following function: For all 1
n p
1, let
n n m x m
ind n ind (mod )and1 ind
x m E fe e g
(ind n exists since ( ) = 1). Define the sequence = p p 1 1 1
by
m
f n
+1 if 1 ind ( ) 2 , e
(2) n =
m
n m p j f n 1if2 ind f ( ) or ( ).
Note that this construction also generalizes the Legendre symbol con-
x
struction described in [4] and [5]. Indeed, in the special case m =2 =1 e
the sequence n defined in (2) becomes
f
(n )
+1 if = 1,
p
e
n =
f n
( )
p j f n
1if =1or ( ). p
In [7], I proved that this construction has good pseudorandom properties: c
1 2
W E C E p p
each of the measures ( ), k ( )islessthan (log ) under p p 1 1
certain conditions on the polynomial f . However in Theorem 3 in [7] for the correlation measure we obtained an upper bound which is optimal only for
large m . Indeed, there the following was proved:
Suppose that m is even or m is odd with m j p and
Theorem A 2 1
at least one of the following conditions holds
a f is irreducible
1 2 u
b If f has the factorization f over F where N and
p u
= 1 2 i
the s are irreducible over F then there exists a such that exactly
p
i
one or two of s have the degree
i
c
=2
k
k p or p
d (4 ) (4 )
Then
( +1) k
!
1 2 +1
p E k p p
(3) C ( ) 9 4 (log ) + p
1
m
2008. februar´ 10. –13:25 90 KATALIN GYARMATI
If m is odd, then it is proved in section 4 in [7] that
p
C E
( ) p
1
m
thus the second term in (3) can not be dropped completely. If m is even, x I will improve on Theorem A under the further assumption that f ( )hasno
multiple roots.
Theorem Suppose that m is even and at least one of the conditions a
b c and d holds in Theorem A Moreover we suppose that all conditions
assumed in the Notation hold in particular f X F x has no multiple
( ) p [ ]
roots Then
1 2 +1
p E k p
(4) C ( ) 9 4 (log ) p 1
(4) is considerably sharper than (3) if
2 +
m p
Then the second term is much larger than the first term in (3). In particular,
O p for m = (1) the second term is , so (3) becomes trivial, while our Theorem 1 still gives good upper bound.
2. Proof of Theorem 1
fd d d g d d Consider any D = with non-negative integers
1 2 1 2
d M M d p
and positive integers with + 1. Then arguing as
in the proof of Theorem 2 in [7] (formulas (11) and (12)) we have
1 2 +1
p j V E M D j k p
( N ) 9 4 (log )
m
2
Y X X
x
2
j j
p g
+ ( 1)
m
j l
m
=1 =1 1 j
1
n d f n d f 1 ( + ) ( + ) 1
is a perfect m -th power
X
1 2 +1
p p (5) =9k 4 (log ) +
2 P We will prove that the sum 2 is empty, which follows from the following lemma:
2008. februar´ 10. –13:25
A NOTE TO THE PAPER “ON A FAST VERSION OF A PSEUDORANDOM GENERATOR” 91
Suppose that the conditions of Theorem hold Then if
Lemma 1
m andf n d f n d is a perfect m th power
1 1( + ) ( + )
1 1
then
m j
1 2
Indeed, this is a a sharpened version of Lemma 4 in [7]. Assuming the
f x F
further condition that ( ) has no multiple roots in p we will be able to
prove this stronger result. P
If 2 is empty then by (5) we have
1 2 +1
jV E M D j k p p ( N ) 9 4 (log ) which proves Theorem 1. Thus it remains to prove Lemma 1. We will need the following definition
and lemma:
Definition A B Z A B
Let and be multi-sets of the elements of p .If +
Z m
represents every element of p with multiplicity divisible by , i.e., for
c Z
all p , the number of solutions of
b c a A b B a + =
(the a’s and b’s are counted with their multiplicities) is divisible by m ,then
B P
the sum A + is said to have property .
Lemma Let A fa a a g D fd d d g Z If one p = r =
1 2 1 2
of the following two conditions holds
i r g and fr g p
minf 2 max 1
r
p or p r
ii (4 ) (4 )
then there exist c c Z and a permutation q q of d d
p ( ) ( )
1 1 1
i
such that for all 1
a d c a A d D
+ = i
has at least one solution and the number of solutions is less than or equal
to i Moreover for all solutions a A d D we have d fq q q g
1 2 i
q a c q is always a solution and d
i i = i =
Proof This is Lemma 5 in [7]. Now we return to the proof of Lemma 1. The following equivalence
relation was defined in [4] and also used in [6] and [7]: We will say that
X X F X
the polynomials ( ) ( ) p [ ] are equivalent, ,ifthereisan
a F X X a
p such that ( )= ( + ). Clearly, this is an equivalence relation.
2008. februar´ 10. –13:25
92 KATALIN GYARMATI
f X F
Write ( ) as the product of irreducible polynomials over p .Letus
group these factors so that in each group the equivalent irreducible factors
X a X a f X
are collected. Consider a typical group ( + 1) ( + r ). Then ( )
f X X a X a g X g X
is of the form ( )= ( + 1) ( + r ) ( )where ( )hasno
X a i r
irreducible factors equivalent with any ( + i )(1 ).
X f X d f X d m Let h ( )= 1( + ) ( + ) be a perfect -th power where
1
m h x 1 . Then writing ( ) as the product of irreducible polyno-
1
F X a d i r j
p
j mials over , all the polynomials ( + i + ) with 1 ,1 occur amongst the factors. All these polynomials are equivalent, and no other
irreducible factor belonging to this equivalence class will occur amongst the X irreducible factors of h ( ).
Since distinct irreducible polynomials cannot have a common zero, each
X m
of the zeros of h ( ) is of multiplicity divisible by , if and only if in each
X a d h X
j
group, formed by equivalent irreducible factors ( + i + )of ( ), every
X c m
polynomial of form ( + ) occurs with multiplicity divisible by .Inother
A fa a g D fd d d d g d
r
words, writing = , = where i
1 1 1
D h X f X d f X d
1 has the multiplicity i in ( ( )= ( + ) ( + ) is a perfect
1
A D P
m -th power), then for each group + must possess property .
D D fd d g Let D be the simple set version of , more exactly, let = .
1
D A
A and satisfy the conditions of Lemma 2. So by Lemma 2 for the sets
D c c Z
and we have the following: There exist p and a permutation
1
q q d d d d
j
( )=( j )of( ) such that if 1 1
1
a d c a A d D + = i
then we have
d fq q g fd d g
j j 1 i =
1 i
d q a c q j j
i i and = i , = is a solution. Here ( ) is a permutation
1
i j j j of (1 ). Define the i ’s by = (so ( )=( )is
1 i
1
q q d
the same permutation of ( ) as the permutation ( )=( j
1 1 1
d d d j )of( )). Returning to the multi-set case, using this notation
1
we get that the number of the solutions
a d c a A d D + = i
is of the form
i i i i
i 1 1 + 2 2 + +
f g j i
i i where i j 0 1 for 1 and = 1. (We study the number of
solutions by multiplicity since D is a multi-set.)
2008. februar´ 10. –13:25
A NOTE TO THE PAPER “ON A FAST VERSION OF A PSEUDORANDOM GENERATOR” 93
D P i
Since A + possesses property for all 1 we have
m j
i
(6) + + + i i i i 1 1 2 2
By induction on i we will prove that
m j
(7) i
m j Indeed, for i = 1 by (6) and =1weget . We will prove that if (7)
1 1 1
j i j holds for i 1, then it also holds for = .
By the induction hypothesis we have
m j m j m j
1 2 j 1 j
Using this and (6) for i = we get:
m j j which was to be proved.
References
V Rodl [1] N Alon Y Kohayakawa C Mauduit C G Moreira and Mea-
sures of pseudorandomness for finite sequences: typical values, Proc Lon
don Math Soc 95(3) (2007), 778–812.
A Sarkozy [2] J Cassaigne C Mauduit and On finite pseudorandom binary
sequences VII: The measures of pseudorandomness, Acta Arith 103 (2002),
97–118. G Xiao [3] Z Chen S Li and Construction of pseudo-random binary sequences
from elliptic curves by using discrete logarithm, in: Sequences and Their
SETA Applications , Lecture Notes in Computer Science, 4086,
Springer, (2006), 285–294.
A Sarkozy [4] L Goubin C Mauduit and Construction of large families of
pseudorandom binary sequences, J Number Theory 106 (2004), 56–69.
A Sarkozy
[5] C Mauduit and On finite pseudorandom binary sequences I: Arith Measures of pseudorandomness, the Legendre symbol; Acta 82
(1997), 365–377.
Gyarmati Periodica [6] K On a family of pseudorandom binary sequences,
Math Hungar 49 (2004), 45–63. General The [7] K Gyarmati On a fast version of a pseudorandom generator, in:
ory of Information Transfer and Combinatorics, Lecture Notes in Computer Science 4123, Springer, (2006), 326–342.
Katalin Gyarmati Alfred´ Renyi´ Institute of Mathematics Realtanoda´ utca 13–15.
H-1053 Budapest, Hungary gykati cs elte hu
2008. februar´ 10. –13:25 2008. februar´ 10. –15:13 ANNALES UNIV. SCI. BUDAPEST., 49 (2006), 95–101
CORRECTIONS TO MY PAPER “DISCRIMINATOR OR DUAL DISCRIMINATOR?”
By
ERVIN FRIED Received February
0. Preliminaries
dual discriminator
In one part of my original paper Discriminator or , Universalis which appeared in Algebra , there were several inaccuracies, for various reasons. Here we give a corrected version, trying to make it as selfcontained as possible. To be able to keep in track with the original paper we follow the original numbering1. Firstly, we need some definitions:
3
f A A
Definitions A ternary function : will be called a semidis-
c A fa cg fa c g
criminator if for all a for every choice = and for every
a b A
b ( ) we have either one of the following cases:
a a c c f a b c a
case(d): f ( )= and ( )=
a a c a f a b c c case(dd): f ( )= and ( )=
An algebra is called a semidiscriminator algebra if a term p yields a
semidiscriminator in it. A ternary term p in a variety is a semidiscriminator term if it yields a semidiscriminator function in every subdirectly irreducible member of the variety. A variety is called a semidiscriminator variety if it possesses a semidis- criminator term. A variety of type 3 is called a pure semidiscriminator variety, if the operation itself is a semidiscriminator term.
Mathematics Subject Classi cation : 08A50; 03C05, 05C99 1
This manuscript was sent to Algebra Universalis in 2003, but it went astray and was lost.
2008. februar´ 10. –13:57
96 ERVIN FRIED p Remarks (1) If in the definition always case(d) holds then is called
a discriminator, and if always case(dd) then it is a dual discriminator. For
q x y z p x p x y z z a semidiscriminator term p ,theterm ( )= ( ( ) ) is a dual discriminator term, hence the variety is a dual discriminator variety. Thus, we
can apply the results of [4] or [2]. In particular, in any algebra of the variety
d a b
c (Θ( )) holds if and only if the equations
q c d q a b c c d q c d q a b d c = ( ( )) = and = ( ( )) are satisfied.
(2) The two cases in the definition do not exclude each other. It is trivial,
c that both cases are satisfied if and only if a = .
(3) There is a one-to-one correspondence between semidiscriminator al- c gebras and “two-colored” graphs: Let the edge, connecting a and be red in case(d) and blue in case(dd). Every edge in the graph has exactly one color except the loops which have both colors.
3. Pure (Semi)discriminator varieties
We need a result of K. Baker ([1]) which is very simple for dual discrim-
inator varieties:
Proposition In any dual discriminator variety the intersection of nitely many principal congruences is a principal congruence
Proof Clearly, it is enough to prove our statement for two congruences. We shall give principal intersection polynomials in the sense of [1] (see,
also, [4]). We claim, that
c d q a b c q a b d
Θ( a b) Θ( )=Θ( ( ) ( )) x y z
where q ( ) is a dual discriminator term. The right hand side is contained
b c d q a b c q a b d in the left hand side, since both a = and = yields ( )= ( )in every subdirectly irreducible algebra of the variety. Conversely, factorizing by
the right hand side and consider the given congruence identity in a subdirectly
c d
irreducible algebra. We have Θ( a b) Θ( ) = 0, the least congruence. If
a b q a b c c
Θ( a b) = 0, we are done. Otherwise, , therefore, ( )= and
a b d d c d q ( )= ,i.e.Θ( ) = 0, by assumption.
Next, we are going to describe the pure semidiscriminator varieties by a set of identities.
2008. februar´ 10. –13:57
CORRECTIONS TO MY PAPER “DISCRIMINATOR OR DUAL DISCRIMINATOR?” 97
Theorem Let S be a variety of type with the ternary operational
x y z S is a pure semidiscriminator variety with the semidis p
symbol ( )
x y z in every subdirectly irreducible algebra if and only if p
criminator ( )
the following hold
x y x x
(1) p ( )= ;
q x y z p x p x y z z is the dual discriminator i e
the term ( )= ( ( ) )
x z z z q x y x x q x x z x
1) q ( )= ( ( )= ) ( )=
x y q x y z q x y z
(2) 2) q ( ( )) = ( )
x y p u v z p q x y u q x y v q x y z
3) q ( ( )) = ( ( ) ( ) ( ));
u p x y z and for every v we have
for the term = ( )
u x q u z v u
(3) q ( ( )) = ;
q x y q p x x y p x y y x
(4) x = ( ( ( ) ( ) ))
q x y q p x x y p x y y y
y = ( ( ( ) ( ) ));
x x y p y x x
(5) p ( )= ( )
Proof x y z Suppose, p ( ) is a semidiscriminator. (1) is obvious. (2) fol-
lows from the fact that q is the dual discriminator, therefore it satisfies the
identities in [2]. Hence, S is a dual discriminator variety with the dual
discriminator q . Therefore, it is enough to verify the other identities for
subdirectly irreducible algebras.
fx zg For the semidiscriminator, we have u , and (3) is satisfied in
both cases.
y a b
For x = , (4) holds, trivially. Suppose, . Then, the definition of
a a b p a b b
the semidiscriminator yields p ( ) ( ) in both cases (d) and (dd).
p a a b p a b b c c c Hence, we have the equality q ( ( ) ( ) )= for every , yield-
ing (4).
y a b p a b b p a a b (5) holds in case x = , clearly. Suppose, .Then ( ) ( )
follows from the definition of the semidiscriminator. Since in the definition
u y v p v y u p a a b a
p ( )and ( ) belong to the same case, ( )= implies
b b a b p b a a a p ( )= , which yields ( )= . The other case is “dual”.
Conversely, assume all the identities above hold. (2) implies, that S is a
dual discriminator variety. Therefore, it is enough to deal with algebras, where
x y z a c p a b c
q ( ) is the dual discriminator. Choose any in the algebra. ( )
a c a c a cg
f follows from (1) if = . Next, let . If the algebra has only two
p a b c fa cg b elements, then d = ( ) for every in the algebra. Otherwise,
2008. februar´ 10. –13:57
98 ERVIN FRIED
a q d c b q d a q d c b d
suppose d . Then, by (3), we have ( )= ( ( )) = for
d c q p a b c fa cg
each b , i.e., = ,since is the dual discriminator. Hence, ( ) . b
Suppose, a . Then, by (4), we have
p a a c p a c c a a q p a a c p a c c c c
q ( ( ) ( ) )= and ( ( ) ( ) )=
a a c p a c c d q d d a q d d c
This means, that if p ( )= ( )= ,then ( )= ( ),
c a c p a a c
yielding a = . In other words, for distinct and , the elements ( ) a c c
and p ( ) are distinct, as well. a y c
Finally, (5) implies, that if p ( ) satisfies the condition of the first c y a case in the definition, then so does p ( ).
4. Subvarieties of pure semidiscriminator varieties
Now, we turn to the description of some subvarieties of pure semidis-
criminator varieties. Let S be the semidiscriminator variety, defined by the
identities in the previous section. Of course, every subvariety of S is uniquely given by listing all the subdirectly irreducible members. Since we are in a
congruence distributive variety, a class S of subdirectly irreducible algebras
of S consists exactly of the subdirectly irreducible members of a subvariety,
if and only if S is closed under ultraproducts, subalgebras and homomorphic images. Since we are in a dual discriminator variety, subdirectly irreducibles are simple, hence, homomorphic images are not needed. We make use of the one-to-one correspondence between the subdirectly
irreducible members of S and the two-colored graphs discussed in the pre- vious section. It is clear, that the algebra-mappings are exactly the graph-
mappings. Notice, however, that the image of an edge is red (blue) only if the
original edge is red (blue). It will be easier, to use graph-theoretical concepts.
An n element n two colored graph will be called
Definition 2
a red cycle of length n if for its vertices a a the edges
1 n
a a a a a a n
( ) ( n ) ( )
1 2 n 1 1
are red and all the other edges are blue Blue cycles of length n are de ned
dually A two colored graph contains a red blue cycle of length n if it
contains a subgraph which is a red blue cycle of length n
Proposition Let S be a set of subdirectly irreducible elements of
the variety S A nite subdirectly irreducible algebra of S belongs to the
2008. februar´ 10. –13:57
CORRECTIONS TO MY PAPER “DISCRIMINATOR OR DUAL DISCRIMINATOR?” 99
subvariety generated by the elements of S if and only if it is isomorphic to a
subalgebra of some element of S
Proof The “if” part is trivial. Suppose the subdirectly irreducible alge-
bra S belongs to the generated subvariety. As we mentioned at the beginning S
of this section, S must be a subalgebra of an ultraproduct of elements of .
s S
s s s k Let 1 k be the elements of ,and 1 elements of the direct prod-
uct, which are mapped to the corresponding elements of S, when factoring by
s s the ultrafilter. The subalgebra of the direct product, generated by 1 k is
contained, obviously, in a direct product of subalgebras of elements of S,such
that S is a factor of this direct product by the original ultrafilter. List all the
p s s s s S
u v
j equalities ( i )= in .Let be the natural projection of the direct
product to some components. Since we are in an ultraproduct, the relation
p v
s s s u s
j ( ( i ) ( ) ( )) = ( ) holds in almost all component, i.e., almost all
components are isomorphic to S.
Let N f n n g be any in nite sequence of
Theorem = 2 1 2
S N denote the class of subdirectly irreducible members of S
integers Let ( )
which do not contain a say red cycle of length n N Then the subvariety
S generated by S N is not nitely based
( N ) ( )
The number of these varieties is continuum S Proof If the subdirectly irreducible algebra does not contain a red
cycle, no subalgebra of S contains a red cycle. Then, by proposition 4.2, the
N S N
subdirectly irreducible members of S( ) are exactly the elements of ( ).
N k
Now, consider a finite set Σ of identities, valid in S( ). Let be
N
the number of variables in the members of Σ, and choose an n such
n k C n S N
that .Let n be a red cycle of length . By the definition of ( ),
C S N k C n n ( ). Take any elements of . Observe, that the subalgebra they
generate does not contain other elements. Since this subalgebra contains no
S N C
cycle, at all, it belongs to ( ). Thus, n satisfies all the identities in Σ.
N S N Therefore, Σ is not a basis for the identities of S( ), i.e., ( ) is not finitely based. By proposition 4.2, distinct sequences define distinct varieties. Since the number of the given sequences is continuum, so is the number of constructed varieties. Observe, that we not only proved that there is a continuum of non-finitely based subvarieties, but we, in some sense, also constructed them.
We finish this section by giving an equational basis for every vari-
N ety S( ).
2008. februar´ 10. –13:57
100 ERVIN FRIED
Theorem For every n there is a nite system of identities such
that these identities hold in a subdirectly irreducible member of the pure
semidiscriminator variety de ned in Theorem if and only if it contains
as a two colored graph no say red cycle of length n
N nite if N is nite and S
This yields an equational base for every ( )
in nite otherwise
p x y z n Proof Let ( ) denote the semidiscriminator term, and let 2be fixed. As the first element of desired finite system of identities we list all the identities of Theorem 3.4.
Suppose, the subdirectly irreducible element S of the variety contains
a a n a red cycle ( 1 n ) of length . Define the following operation on the
integers:
n
j i j j n
if j 1( ),
j i =
i otherwise.
p a a a a i j n
i j i j
Using this operation, we have ( i )= (1 , ). n
If S does not contain a red cycle of length , then whenever these
a a
identities hold, the subalgebra consisting of 1 n can not be proper,
a a i j n
j
i.e., we must have i = for some 1 . (As the cycles are
i j n
simple, this equality holds for each 1 .) In other words, the
V
p x x x x j i j n g f
i j i j
congruence = Θ( ( i )= 1 , implies, say, the
x
congruence Θ( x1 2). By Baker’s theorem, is a principal congruence of the
f g f g
n n n
form =Θ( n )where are well-defined elements of the free algebra
x x generated by 1 n . Thus, a two-colored graph, containing no red cycle
of length n , satisfies the identities
x q x x q f g x x q x x q f g x
n n n
1 = ( 1 2 ( n 1)) and 2 = ( 1 2 ( 2))
a a
Conversely, if 1 n is a red cycle in a colored graph, then the mapping
x a i n f g b
n n
i
i ,for1 sends and to the same element , yielding
a q a a b b i f g
i = ( 1 2 )= for 1 2 , a contradiction.
References
[1] K A Baker Primitive satisfaction and equational problems for lattices and
other algebras, Trans Amer Math Soc 190 (1974), 125–150.
Acta Sci Math Szeged [2] E Fried Dual discriminator, revisited; 64 (1998),
581–597. Algebra Universalis [3] E Fried Discriminator or dual discriminator?, 46 (2001), 25–41.
2008. februar´ 10. –13:57
CORRECTIONS TO MY PAPER “DISCRIMINATOR OR DUAL DISCRIMINATOR?” 101 A F Pixley [4] E Fried and The dual discriminator function in universal algebra,
Acta Sci Math Szeged 41 (1979), 83–100.
onsson Math Scand [5] B J Algebras whose congruence lattices are distributive, 21 (1967), 110–121.
[6] R McKenzie Para-primal varieties: a study of finite axiomatizability and de-
finable principal congruences in locally finite varieties; Algebra Universalis 8 (1978), 336–348.
Fried Ervin Department of Algebra and Number Theory, Lorand´ Eotv¨ os¨ University, Pazm´ any´ P. set´ any´ 1./c
H-1117 Budapest, Hungary fried cs elte hu
2008. februar´ 10. –13:57 2008. februar´ 10. –15:13 ANNALES UNIV. SCI. BUDAPEST., 49 (2006), 103–108
A WEAKER FORM OF PRE-R0
By
MIGUEL CALDAS, SAEID JAFARI and TAKASHI NOIRI Received February
1. Introduction
The notion of R0 topological spaces is introduced by Shanin [13] in 1943.
By definition, a topological space is R0 if every open set contains the closure of each of its singletons. Later, Davis [4] rediscovered it and studied some properties of this weak separation axiom. Many researchers further investi- gated properties of R0 topological spaces and many interesting results have been obtained in various contexts (see: [5], [7], [8], [11], [12]). In 1982, Mashhour et al. [10] introduced the notion of preopen sets which are also known under the name of locally dense sets [3] in the literature. Since then, this notion received wide usage in general topology. In 2000, Caldas et al. [1]
introduced and investigated the fundamental properties of the separation ax- R
iom pre-R0. In this paper, we offer another characterization of pre- 0.We
R R
T also introduce a new separation axiom called pre- T . It turns out that pre-
is weaker than pre-R0. X
Throughout the paper ( X )(orsimply ) will always denote a topolog- X
ical space. For a subset A of , the closure, the interior and the complement
X A A X A X of A in are denoted by Cl( ) Int( )and , respectively. By PO( )
and PC( X ) we denote the collection of all preopen sets and the collection
of all preclosed sets of ( X ), respectively.
Since we shall require the following known notions, notations and some properties, we recall them in this section.
Mathematics Subject Classi cation : 54D10
2008. februar´ 10. –13:25
104 MIGUEL CALDAS, SAEID JAFARI, TAKASHI NOIRI
A X
Definition Let be a subset of a space ( ). Then:
A A
(1) A is said to be preopen [10] if Int(Cl( )).
X A
(2) A is said to be preclosed [10] if is preopen or equivalently if
A Cl(Int(A)) .
(3) The intersection of all preclosed sets containing A is called the preclosure A
of A [6] and is denoted by pCl( ).
Let X be a space and A B subsets
Lemma El Deeb et al ( )
of X Then the following hold
A if and only if A V for every V X such that x
pCl( ) PO( )
x V
A is preclosed in X if and only if A A
( ) =pCl( )
A B if A B
pCl( ) pCl( )
A pCl(pCl(A)) = pCl( )
2. Another Characterization of pre- R0 spaces
X A X
Definition Let ( ) be a space and . Then the pre-kernel
A A fG
of A [9], denoted by pKer( ), is defined to be the set pKer( )=
X j A G g
PO( ) .
Let X be a space A X and x y X Then the
Lemma ( )
following hold
y x g if and only if x fy g
pKer( f ) pCl( )
A fx X j fx g A g
pKer( )= pCl( )
x g fy g if and only if fx g fy g
pKer( f ) pKer( ) pCl( ) pCl( )
X R Definition A topological space ( ) is called a pre- 0 space [1] if
every preopen set contains the preclosure of each of its singletons equivalently
fy g y fx g x pCl( ) if and only if pCl( ). The following Lemma 2.2 is Theorem 3.8 in [1] and Lemma 2.3 is a
special case of Corollary 3.9 in [1].
A space X is pre R if and only if for any x X
Lemma ( ) 0
x g fx g
pCl(f ) pKer( )
A space X is pre R if and only if for any x X
Lemma ( ) 0
x g fx g
pCl(f )=pKer( )
x g x Since pCl( f ) is the intersection of all preclosed sets containing ,
Lemma 2.3 suggests a natural definition of pre-R0.
2008. februar´ 10. –13:25
A WEAKER FORM OF PRE-R0 105
X R Definition A topological space ( )ispre- 0 if the intersection of
all preopen sets containing x coincides with the intersection of all preclosed
sets containing x .
For a topological space X the following properties
Theorem ( )
are equivalent
X is a pre R space
( ) 0
fx g fx g for each x X
pKer( ) pCl( )
X R y
Proof (1) (2): Suppose that ( )isapre- 0 space. Let
fx g x fy g
pKer( ), then by Lemma 2.1(1) pCl( ) and by Lemma 2.2
fy g y fx g
x pKer( ). Therefore, by Lemma 2.1(1) pCl( ) and hence
x g fx g
pKer( f ) pCl( ).
x fy g y fx g
(2) (1): Let pCl( ). Then by Lemma 2.1(1) pKer( ).
fx g fx g x fy g
By (2) we obtain y pKer( ) pCl( ). Therefore pCl( ) implies
fx g
y pCl( ). The converse is obvious. Now it follows from Definition 3 that R ( X )ispre- 0.
Observe that Lemma 2.2 and Theorem 2.4 show the symmetry of pre- R0 in another sense.
3. A weaker form of pre- R0
We begin with the following definitions. X
Definition A topological space ( )issaidtobe
R x y X fx g fy g
(1) pre- H if for any points and in ,pCl( ) pCl( ) implies
x g fy g fx g fy g
pCl(f ) pCl( )= , or .
R x X fx g fx g fx g fx g
(2) pre- D if for ,pCl( ) pKer( )= implies that pD( )=
x g nfx g fx g fx g
=pCl(f ) is preclosed, where pD( ) is the pre-derivate of . R
It follows from Lemma 2.2 and Theorem 2.4 that if X is not pre- 0,
fx g n fx g
then there is some x such that pKer( ) pCl( ) , and there is some
fx g n fx g x such that pCl( ) pKer( ) . This suggests a new separation axiom. In the following definition, a set that contains at most one point is said to be
degenerate.
Definition X R
A topological space ( ) is called a pre- T space if for
fx g n fx g fx g n fx g
any x , both pKer( ) pCl( ) and pCl( ) pKer( ) are degenerate.
R R
Obviously, every pre- 0 space is pre- T . In general the converse may not be true.
2008. februar´ 10. –13:25
106 MIGUEL CALDAS, SAEID JAFARI, TAKASHI NOIRI
X fa b c d g
Example Let = and
f fa g fa bg fc d g fa c dg X g
=
X R R fa g n fa g fb g
Then ( )ispre- T but it is not pre- 0 since pCl( ) pKer( )= .
X fa b cg f fa g fa bg fa cg Xg
Example Let = and = .
X R R R fa g X H
Then ( )ispre- D and pre- but it is not pre- 0 since pCl( )=
a g fa g
and pKer(f )= .
X fa b cg f fa g fa bg Xg X
Example Let = and = .Then( )
T R fa g X fa g fa g
is pre- 0 but is not pre- T since pCl( )= and pKer( )= .
Theorem If X is pre R then X is pre R
D
( ) T ( )
Proof X R
Suppose that is pre- T and denote
x fx g fx g
( ) p =pCl( ) pKer( )
fx g x D fx g x E D E p
Then pCl( )=( ) p and pKer( )=( ) ,where and are
D fx g E fx g x fx g
degenerate sets such that pKer( )and pCl( ). If ( ) p = ,
x g fx g D fx g fx g E fx g
then pCl( f )= and pKer( )= . We prove that pD( )=
x g nfx g D U
=pCl(f ) = is a preclosed set. Let be a preopen set containing
x g X n U X n U fx g D
pKer( f ). Then is a preclosed set, and ( ) pCl( )= or .
n U fx g D D
If (X ) pCl( )= ,then is the intersection of two preclosed sets
X n U fx g fx g U
and hence D is also preclosed. If ( ) pCl( )= ,thenpCl( )
U D fx g V x V
and D .Since pKer( ), there is a preopen set such that
V fx g X n V D
and D .ThenpCl( ) ( )= is a preclosed set. Therefore
fx g x fx g X R
p
pD( ) is preclosed whenever ( ) = . Hence is pre- D .
Theorem If X is pre R then X is pre R
H
( ) T ( )
Proof X R x y X fx g fy g
Let be pre- T and . Suppose pCl( ) pCl( )
X a x a y a fx g fy g
and there is an a such that , but pCl( ) pCl( ). Then
fx g a fy g x fa g y fa g
a pCl( )and pCl( ). Hence pKer( )and pKer( ).
fa g a E E E fa g
Since pKer( )=( ) p ,where is a degenerate set and pCl( ),
fa g
there are following four possible cases for x y pKer( ):
x a y a x fa g y fa g p
(i) ( ) p and ( ) .Wehave pCl( )and pCl( )and
fx g a fy g fx g fy g fa g also a pCl( )and pCl( ). Hence pCl( )=pCl( )=pCl( )
which is a contradiction.
fx g E y a x fa g y fa g
(ii) = and ( ) p .Wehave pCl( )and pCl( ). Since
fy g fy g fa g
a pCl( ), we have pCl( )=pCl( ). We have two cases to consider
fx g y fx g fy g fa g
between y and pCl( ). Case (1): pCl( ). Then pCl( )=pCl( ).
fa g x X n fa g X n fa g Since x pCl( ), pCl( ). Since pCl( ) is preopen,
2008. februar´ 10. –13:25
A WEAKER FORM OF PRE-R0 107
x g X n fa g fx g n fx g fa g
pKer( f ) pCl( ) and hence pCl( ) pKer( ) pCl( )
fx g n fx g y ag
f . Hence pCl( ) pKer( ) is not a degenerate set, a contradiction
X R y fx g y fa g
to the fact that is pre- T .Case(2): pCl( ). Since pCl( )and
fx g y fx g
a pCl( ), we have pCl( ), a contradiction.
x a fy g E
(iii) ( ) p and = . Similar to case (ii).
x g fy g E fx g fy g
(iv) f = = .WehavepCl( )=pCl( ). But this is impossible.
x g fy g fx g fy g fx g
Therefore if pCl( f ) pCl( ), we have pCl( ) pCl( )= ,
fy g X R or . It follows that is pre- H .
References
T Noiri R R [1] M Caldas S Jafari and Characterizations of pre- 0 and pre- 1
topological spaces, Topology Proccedings 25 (2000), 17–30. T Noiri [2] M Caldas S Jafari and On some weaker separation axioms than
pre- R0, (under preparation).
E Michael Illinois [3] H Corson and Metrizability of certain countable unions,
J Math 8 (1964), 351–360.
[4] A S Davis Indexed systems of neighborhoods for general topological spaces,
Amer Math Monthly 68 (1961), 886–893.
R Mat Vesnik [5] K K Dube A note on 0 topological spaces, 11 (1974), 203–
208.
T Noiri p
[6] S N El Deeb I A Hasanein A S Mashhour and On -regular
Math Soc Sci Math R S Roumanie spaces, Bull 27(75) (1983), 311–
315.
J Rozycki T [7] D W Hall S K Murphy and On spaces which are essentially 1,
J Austr Math Soc 12 (1971), 451–455. Gen Topol Appl
[8] H Herrlich A concept of nearness, 4 (1974), 191–212.
T Noiri Univ Bacau Stud [9] S Jafari and Functions with preclosed graphs,
Cerc St Ser Mat 8 (1998), 53–56. S N El Deeb [10] A S Mashhour M E Abd El Monsef and On precontinuous
and weak precontinuous mapping, Proc Math Phys Soc Egypt 53 (1982),
47–53.
X R
[11] M Mrshevich Some properties of the space 2 for a topological 0-space,
Uspekhi Mat Nauk Russian Math Surveys Gen 34(6) (1979), 166-170;
34(6) (1979), 188–193.
R Ann Univ Sci Budapest Eotvos [12] S A Naimpally On 0-topological spaces,
Sect Math 10 (1967), 53–54.
2008. februar´ 10. –13:25
108 MIGUEL CALDAS, SAEID JAFARI, TAKASHI NOIRI: A WEAKER FORM OF PRE-R0 Dokl Akad Nauk SSSR [13] N A Shanin On separation in topological spaces, 38 (1943), 110–113.
Miguel Caldas Saeid Jafari Departamento de Matematica´ Aplicada College of Vestsjælland South Universidade Federal Fluminense Herrestræde 11 Rua Mario´ Santos Braga, s/n 4200 Slagelse 24020-140, Niteroi,´ RJ Denmark
Brasil jafari stofanet dk gmamccs vm uff br
Takashi Noiri Department of Mathematics Yatsushiro College of Technology Yatsushiro, Kumamoto, 866
Japan t noiri nifty com
2008. februar´ 10. –13:25 ANNALES UNIV. SCI. BUDAPEST., 49 (2006), 109–114
ON TRIPLES OF CONSECUTIVE LEGENDRE SYMBOLS
By
KATALIN KOVACS´
Received February
p p n
In [1] Davenport proved that for any prime p large enough ( ( ))
n i
b +
f g b N
and =( n ) 1 1 there exists a such that =
1 p
i n = i , =1 . In [2] A. Weil improved on Davenport’s theorem. These
existence proofs estimate some character-sums and get a result for the number n of solutions depending of p and and are based on the statistical behaviour of the quadratic residues. The purpose of my paper is the determine explicit
triples satisfying Davenport’s theorem in case n = 3 or describe an algorithm how the get these triples from other ones. (The values of the triples may
depend on the prime p .) The proof is based on combinatorial ideas and results
concerning the quadratic residues.
For an arbitrary prime p and Theorem 17 =(1 2 3)
3
g there exists a well de ned set B of elements such that either
f 1 1
i
b +
i
(1) = i =1 2 3
p
for at least one element of B or there is a short algorithm how to get B
from an other one
a
i
Notation L a a i
n n ( )=( ) means that = i , =
1 1 p
=1 n.
Mathematics Subject Classi cation : 11A15, 11A25 * Research partially supported by the Hungarian National Foundation for Scientific Re- search Grants No. T043623 and T049693.
2008. februar´ 10. –13:25 110 KATALIN KOVACS´
The following proof proves Davenport’s theorem on a constructive way
in the case n = 3. First we prove some lemmas. Lemma 3 almost proves
Davenport’s theorem in the case n = 3 based on lemmas 1 and 2. Lemma 2
N
is not necessary to prove the theorem. Instead of lemma 2 we find c d
c c c L d d d such that L( +1 +2) = (1 1 1) and ( +1 +2)= ( 1 1 1) using
lemma 4.
If p k and L a a a then
Lemma =4 +1 ( 1 2 3)=( 1 2 3)
L a p a p a
( p 3 2 1)=( 3 2 1)
k L a a a
If p =4 1and ( 1 2 3)=( 1 2 3)then
p a p a p a
L( 3 2 1)=( 3 2 1)
There exists a c N such that L c c c or
Lemma ( +1 +2)=(1 1 1)
c c c if p L( +1 +2)=( 1 1 1) 7
3
Lemma For the primes p f g
11 all the triples ( 1 2 3) 1 1
p k and at least triples occur if p k
occur if =4 1 =4 +1
ai d
Lemma If p at d and i then
= + = i =1 2 3
p
i a
t +
i
(2) = i =1 2 3
p p
Proof of the Lemmas
p a a
i
1 i
1. It is true as = , i =1 2 3.
p p p
P P
P 2
x x x x
( +1) 2 (2 +2) z 1
2. U = = = = 1
p p p
x p
f p g 1 1 z 0 2 3 2 as
2 2
u p
(3) z 1 0mod
1 1 1
z u z u c c p z c c u c c
( )( + ) 1mod ( + ) 2, ( ) 2,
c 1.
1 2
c c s zs z uniquely determines the pair ( )as 2 +1 0 has at most two
solutions for any fixed z (using the Viete´ formulas) therefore (3) has exactly p 3
2 solutions.
U L p
= 1 means that if (1 2) = ( )then i = i
1 p 2 +1
p
p 3 1
i i i i 2 times and i +1 2 times. If = +1 = +2 is excluded then the
2008. februar´ 10. –13:25
ON TRIPLES OF CONSECUTIVE LEGENDRE SYMBOLS 111 p 3 2 neighbouring residue pairs are disjunct and only 2 elements are missing.
As there cannot be more identical neighbouring residue pairs these 2 elements
z
are single as and or are between 2 identical pairs beginning at and
1 p 2
1 4
z z L
p for some (by lemma 1). As = = 1 we have (1 2 3 4 )=
p p
p =(1 1 1 1 1 1 )or(11 1 1 1 1 1 1 1 1 )for 19
depending on the place of the first single element. We get the contradiction
6 10
by counting and , resp. It can be verified that p [11 23] either.
p p
b i i
2 +2 2 b +
i
3. By lemma 2 there exist a b such that = = , =1 2 3
p p p
i
2 b +2 1
f g
with =1or = 1. If 0 we multiply by 2 again. Sooner
p
or later (multiplying by 2 repeatedly) we get a triple of residues (1 1 1)
or ( 1 1 1) as there does not exist a sequence of neighbouring identical
k
residues of arbitrary length. For p =4 1 we have both triples by lemma 1.
If we continue the triple (1 1 1) or ( 1 1 1) we get at least one
p k
of the triples (1 1 1), (1 1 1), ( 1 1 1), ( 1 1 1). If =4 +1
k
then by lemma 1 we get an other triple also. If p =4 1then(11 1)
L p
and ( 1 1 1) both occur. Therefore (1 1) = (1 1 1
1 1 1 1 1) or (1 1 1 1 1 1) or (1 1 1 1 1
L p
1 1 1) and the triple (-1,1,-1) is a triple in (1 1). It is
trivial that (1 1 1) occurs. We prove that ( 1 1 1) occurs also and so by
Lemma 1 we are ready. If ( 1 1 1) and ( 1 1 1) (its symmetric partner in
p
Lemma 1) are missing then L(1 1) has a special form: block of 1’s of
k
length k ( 3), 1 1 1 1 (alternating block), block of 1’s of length .
p L p p L p p
If k ( 1) 2then ( 3 1) = (( 3) 2 ( 1) 2) = (1 1) or
k p L p L L p ( 1 1) is a contradiction. If =( 1) 2then ( 5) = (2) (( 5) 2) = 1
is a contradiction for p 11.
at ai p
t i a a ai d
a 2
4. It is true as + = + = .
p p p p p p
2008. februar´ 10. –13:25 112 KATALIN KOVACS´
Proof of the Theorem In the following table we consider the possible
2 3 5
8 cases depending on the values of .
p p p
1 2 3 4 5 6 7 8 9 10
p p p p p p p p p p
1 +1 +1 +1 +1 +1 +1 1+1+1+1
2 +1 +1 +1 +1 1+1 1+1+1 1
3 +1 +1 1+1+1 1 1+1+1+1
4 +1 +1 1+1 1 1 1+1+1 1
5 +1 1+1+1+1 1 1 1+1 1
6 +1 1+1+1 1 1 1 1+1+1
7 +1 1 1+1+1+1 1 1+1 1
8 +1 1 1+1 1+1 1 1+1+1
The triple (1 1 1) does not occur in rows 4, 6, 8 only. Suitaible triples
which guarantee (1 1 1) are in these cases as follows:
L L L
Row 4. If L(17) = 1 then (15 16 17) = ( (3) (5) 1 1) = (1 1 1) and if
L L L L
L(17) = 1then (49 50 51) = (1 (2) (3) (17)) = (1 1 1).
3
L L L L
Row 6. If L(13) = 1then (25 26 27) = (1 (2) (13) (3) =
=(1 1 1).
L L L L L
If L(7 13) = ( 1 1) then (12 13 14) = ( (3) (4) 1 (2) (7)) =
L p k k
=(1 1 1). As (2 3) = ( 1 1) must have the form 24 +11or24 + 13.
k k a t k d
If p =24 +11=3 8 + 11 then by the choice =3, =8 , =11in
k k k L L
Lemma 4 we have L(8 +1 8 +2 8 +3) = (3) ( 8 5 2) = (1 1 1). If
k k L a t k
p =24 +13= 4(6 +2)+5and (7) = 1 then by the choice =4, =6 +2,
L k k k L L
d = 5 in Lemma 4 we have (6 +3 6 +4 6 +5) = (4) ( 1 3 7) = (1 1 1).
L L
Row 8. If L(7) = 1then (14 15 16) = (1 1 1). If (13) = 1then
L p
L(24 25 26) = (1 1 1). As (2 3) = ( 1 1) then must have the form
k p k k L
24k + 5 or 24 5. If =24 +5=6 4 +5and (7 13) = (1 1) then by
t k d L k k k
the choice a =6, =4 , = 5 in Lemma 4 we have (4 +1 4 +2 4 +
L p k k
+3)= L(6) (1 7 13) = (1 1 1). If =24 5=3(8 2) + 1 then by the
t k d L k k k
choice a =3, =8 2, = 1 in Lemma 4 we have (8 1 8 8 +1)=
L
= L(3) (2 5 8) = (1 1 1).
k L a a a
If p =4 1then ( +1 +2)=(1 1 1) implies
p a p a p a L( 2 1 )=( 1 1 1)
by Lemma 1.
k p L p
If p =4 + 1 then let 1 denote the least prime for which ( 1)= 1
f p d p d g L
and let s denote the element in 2 1 1 .If (2) = 1 we have
p z d L p L t z L( 1 )= ( 1) ( + )= 1.
2008. februar´ 10. –13:25
ON TRIPLES OF CONSECUTIVE LEGENDRE SYMBOLS 113
k L
If p =4 +1and (2) = 1 then the following triples guarantee
( 1 1 1):
k p k L
Row 5. p must have the form 24 +13 as =4 +1 and (2 3) = ( 1 1).
t k d
By the choice a =2, =12 +6, = 1 in Lemma 4 we have
k k k L L
L(12 +7 12 +8 12 +9)= (2) (1 3 5) = ( 1 1 1)
L L
Row 6. If L(7) = 1then (5 6 7) = ( 1 1 1). If (7 13) = (1 1)
then L(13 14 15) = ( 1 1 1).
k p k L
p must have the form 24 +13as =4 +1and (2 3) = ( 1 1). If
p k a t k d L(7 13) = (1 1) then by the form =6(4 1)+19 ( =6, =4 1, =19
in Lemma 4) we have
k k k L L
L(4 4 +1 4 +2)= (6) ( 13 7 1) = ( 1 1 1)
L p
Row 7. If L(11) = 1then (10 11 12) = ( 1 1 1). must have
p k L L
the form 24k +5as =4 +1and (2 3) = ( 1 1). If (7)=1thenby
k L k k k
the form p =3(8 +1)+2inLemma4wehave (8 +2 8 +3 8 +4)=
L L p m
= L(3) (1 4 7) = ( 1 1 1). Considering (5) = 1 we have = 120 +29
m L p m
or p = 120 19. If (7) = 1thenbytheform =15 8 +29inLemma4
m m m L L
we have L(8 +1 8 +2 8 +3)= (15) ( 14 1 16) = ( 1 1 1). If
p m
L(11) = 1 then by the form = 10(12 4) + 21 in Lemma 4
m m m L L
L(12 3 12 2 12 1) = (10) ( 11 1 9) = ( 1 1 1)
Row 8. The contstruction is similar to the case 7. If L(19) = 1then
p k p k
L(18 19 20) = ( 1 1 1). must have the form 24 +5(as =4 +1
L p m
and L(2 3) = ( 1 1)) futhermore considering (5) = 1 = 120 +53
m L p k
or p = 120 43. If (7)=1thenbytheform =3(8 +1)+2in
k k k L L
Lemma 4 we have L(8 +2 8 +3 8 +4)= (3) (1 4 7) = ( 1 1 1).
p k
If L(17 19) = (1 1) in the case =2(12 +9) 13 by Lemma 4 we
k k k L L
have L(12 +1012 +1112 + 12) = (2) (15 17 19) = ( 1 1 1).
p m
If L(7 17) = ( 1 1) then in the case = 10(12 + 4) + 13 by Lemma 4
m m m L L L
L(12 +5 12 +6 12 +7) = (10) ( 3 7 17) = ( 1 1 1). If (7) = 1
m
then in the case p = 15(8 6) 47 by Lemma 4 we have
m m m L L
L(8 5 8 4 8 3) = (15) ( 32 17 2) = ( 1 1 1)
k B
For p =4 1 we use the ideas of lemma 3 to derive the further ’s
B B p k
from (1 1 1) and ( 1 1 1). The proof works for =4 + 1 also with
f g L
the exceptional cases (1 1 1) ( 1 1 1) if (2) = 1. Considering
L
the table we need to construct triples for ( 1 1 1) in case 6 if (7) = 1
L L and for (1 1 1) in case 7 if (7) = 1 only. If (7 13) = ( 1 1) then
2008. februar´ 10. –13:25
114 KATALIN KOVACS´
L L
L(13 14 16) = ( 1 1 1) and if (7 13) = ( 1 1) then (24 25 26) =
L L
=( 1 1 1) in row 6. If (7) = 1then (14 15 16) = (1 1 1) in row 7.
L p
Remark A further construction for (1 1 1) if (2) = 1: If 1 is the
p L p p p
minimal prime such that L( 1)= 1then ( 1 1 1 1 +1)=(1 1 1).
The proof works for all p 19. The theorem is valid for the primes 11 and 19 either. For 13 and 17, e.g. (1 1 1) is missing.
References Acta Mathematica [1] H Davenport On character sums of finite fields, 71 (1939), 99–121.
[2] A Weil Sur les courbes algebriques´ et les variet´ es´ qui s’en deduisent,´ in
es Sci Indust Actualit , vol 1041, Hermann, Paris, 1948.
Katalin Kovacs´ Mathematical Institute, Eotv¨ os¨ University
Pazm´ any´ Peter´ set´ any´ 1/c, Budapest, H-1117, Hungary pkovacsk cs elte hu
2008. februar´ 10. –13:25 ANNALES UNIV. SCI. BUDAPEST., 49 (2006), 115–136
STABILIZATION OF SOLUTIONS TO A NONLINEAR SYSTEM MODELLING FLUID FLOW IN POROUS MEDIA
By
AD´ AM´ BESENYEI* Received July
1. Introduction
In this paper we investigate a system of nonlinear equations that models fluid flow through porous medium. A porous medium, roughly speaking, is a solid medium with lots of tiny holes, for example limestone. Such medium consits of two parts, the solid matrix and the holes. If the fluid carries dissolved chemical species, a variety of chemical reactions can occur that can change the porosity. This process was modelled by J. Logan, M. R. Petersen,
T. S. Shores in [10] by the following system of equations in one dimension:
t x u t x
(1) ( ) t ( )=
jv t x ju t x K t x p t x u t x ku t x g t x
x x x
= ( ( ) x ( )) + ( ( )) ( ) ( ) ( ) ( ( ))
t x bu t x g t x
(2) t ( )= ( ) ( ( ))
K t x p t x bu t x g t x x
(3) ( ( ( )) x ( )) = ( ) ( ( ))
v t x K t x p t x t x
(4) ( )= ( ( )) x ( ) 0 (0 1)
with initial and boundary conditions
x u x x x x
u (0 )= 0( ) (0 )= 0( ) (0 1)
u t u t u t t
( 0) = 1( ) x ( 1) = 0 0
t p t t p ( 0) = 1 ( 1) = 0 0
Mathematics Subject Classi cation : 35K60, 35J60 * This work was supported by the Hungarian National Foundation for Scientific Research under grant OTKA T 049819.
2008. februar´ 10. –14:16
116 AD´ AM´ BESENYEI
u p
where is the porosity, is the concentration of the dissolved solute, is the
k b K g pressure, v is the velocity, further, , , are given constants, and are given real functions. For the details of making this model and on flow in such media see [7, 10] and the references there. In what follows, we investigate the
following system of nonlinear differential equations:
D t x f t x t x u t x x x
(5) t ( )= ( ( ) ( )) (0 )= 0( )
n
X
D u t x D t x t x u t x Du t x t x D t x a
t p p
i ( ) i ( ( ) ( ) ( ) ( ) ( )) +
(6) i =1
t x t x u t x Du t x t x D t x
+ a0( ( ) ( ) ( ) p( ) p( )) =
t x t x u x
= g ( ( )) (0 )=0
n
X
b t x t x u t x t x D t x
D p p
i (7) i [ ( ( ) ( ) ( ) ( ))] +
i =1
t x t x u t x t x D t x h t x t x u t x + b0( ( ) ( ) p( ) p( )) = ( ( ) ( ))
with boundary conditions homogenous Dirichlet or Neumann, for example
n
X
t x t x u t x Du t x t x D t x
a p p
i i ( ( ) ( ) ( ) ( ) ( )) =0
i =1
x t p( t x )=0 Ω 0
where is the unit normal along the boundary. (The variable p is written p
by boldface letter for the purpose of distinguishing it from exponents p1 2).
S S S S Moreover, if Ω= 1 2 where 1 2 = , then we can pose different boundary conditions on the elements of the partition. That is the case in the
model (1)–(4) where the partitions are the endpoints of the interval [0 1]. Clearly, we can assume the boundary conditions to be homogeneous by sub- stracting a suitable function from the unknown function. The above system was investigated in [4] and it was shown there that
due to some assumptions (that will be introduced later in this paper) the solution of (2) is positive thus we can divide equation (1) by . Hence the above system is a generalization of the model (1)–(4). Observe that for
fixed u equation (5) is an ordinary differential equation with respect to the
function ;forfixed and p equation (6) is a parabolic equation with
u respect to the function u ; and for fixed and equation (7) is an elliptic equation with respect to the function p. This shows that the above system is a hybrid evolutionary/elliptic problem, thus theorems of „usual” systems on
2008. februar´ 10. –14:16 STABILIZATION OF SOLUTIONS TO A NONLINEAR SYSTEM 117
partial differential equations do not work. In [4] existence of weak solutions
T to the above system were proved in (0 T) Ω(where0 )by
using the theory of operators of monotone type. In the following we consider
the equations in (0 ) Ω. First we define the weak forms then we prove
(under suitable conditions) existence of weak solutions, the boundedness of the solutions in appropriate norms in the time interval (0 ), further, we
show the stabilization of the solutions, in the sense that they converge to
the stationary solutions as t . We use some results and arguments of [11, 12, 13]. Finally, we give some examples.
2. Notation
n 1
R C
Let Ω be a bounded domain with the uniform regularity
p p T property (see [1]), further, 2 1 2 be real numbers. For 0
1 p
i
Q T Q W
let T := (0 ) Ω, and let := (0 ) Ω. Denote by (Ω) the
usual Sobolev space with the norm
1
p
Z
i n
X
p p
A
i i
kv k jv j jD v j p
1 = ( + j ) i W (Ω) Ω
j =1
D j
where j denotes the distributional derivative with respect to the -th variable
D D D V n (later we use the notation =(1 )). In addition, let i be a
1 p p
1 i
i W
closed linear subspace of the space W (Ω) which contains 0 (Ω) (the
p p
1
i i
C W L T V
closure of 0 (Ω) in (Ω)), and let (0 ; i ) be the Banach space
p
i
u T V ku k
of measurable functions :(0 ) i such that is integrable and the
V i
norm is given by
1
Z
p
T
i
p
i
p
dt u k ku t k
k = ( )
i
T V L
(0 ; ) V i
0 i
q
p 1 1
i i
L T V L T V
The dual space of (0 ; i )is (0 ; )where + =1and
p q
i
i i
T
V V X
is the dual space of i . In what follows, we use the notation :=
i i
p
i
L T V V V h i
i
= (0 ; i ). The pairing between and is denoted by ,the
i
q p
i i
L T V L T V
pairing between (0 ; )and (0 ; i ) is denoted by [ ], further,
i
D u t
t stands for the distributional derivative (with respect to the variable )
2008. februar´ 10. –14:16
118 AD´ AM´ BESENYEI
p
i
u L T V u
of a function (0 ; i ). It is well known (see [14]) that if
q
p 2
i i
L T V D u L T V u C T L u
t
(0 i ) (0 ; )then ([0 ] (Ω)) so that (0)
i
p
i
L V
makes sense. Denote by loc(0 ; i ) the space of measurable functions
p
i
u V T u j L T V
i :(0 ) i such that for every 0 , (0 ; ).
(0 T )
q
i
T D u j L T V
t If for every 0 there exists ( T ) (0 ; )then
(0 ) i
q
i
D u X L V L Q
t = (0 ; ). Finally, let ( ) denote the space of i
i loc loc
Q R j L Q T
T
functions : such that Q ( ) for every 0 .
T 3. The problem in (0 )
3.1. Assumptions
We formulate some assumptions in order to prove existence of weak
u D u solutions. In the following ,(0 ), ( 0 ) refer for variables ,( ),
(p Dp), respectively. n
n +1 +1
a Q R R R R i n
A1. Functions i : ( =0 )havethe
t x Q
Caratheodory´ property, i.e., they are measurable in ( ) for every n
n +1 +1
R R R
( 0 0 ) and continuous in ( 0 0 ) n
+1 n +1
R R R t x Q
for a.a. ( ) .
+ q
R R k L A2. There exist a continuous function c1: and a function 1 1(Ω)
such that
ja t x j
i ( 0 0 )
p
p2 2
p p q
1 1 1 1 q
j j j j j j j j k x c
1( ) 0 + + 0 1 + 1 + 1( ) n
n +1 +1
t x Q R R R
for a.a. ( ) and every ( 0 0 ) n
( i =0 ).
C t x Q
A3. There exists a constant 0 such that for a.a. ( ) and every n
˜ n +1 +1
R R R
( 0 0 ) ( 0 0 )
n
X ˜ ˜
a t x a t x
i i i i ( 0 0 ) ( 0 0 ) ( )
i =1
˜ p1
C j j
2008. februar´ 10. –14:16
STABILIZATION OF SOLUTIONS TO A NONLINEAR SYSTEM 119
R R A4. There exist a constant c2 0, a continuous function : and a
1
L
function k2 (Ω) such that
n
X
p
p1 1
j a t x c j j j k x i i ( 0 0 ) 2 0 + ( ) 2( )
i =0 n
n +1 +1
t x Q R R R for a.a. ( ) and every ( 0 0 ) .
n +1
b Q R R R R i n
B1. Functions i : ( =0 )havethe
t x Q
Caratheodory´ property, i.e., they are measurable in ( ) for every
n +1
R R R R R ( 0 0 ) and continuous in ( 0 0 )
n +1
R t x Q
for a.a. ( ) .
+ ˆ q
R R k L B2. There exist a continuous functionc ˆ1: and a function 1 2 (Ω)
such that
p1
p p
2 1 2 1 q ˆ jb t x j c j j j j j k x j 2 i ( 0 0 ) ˆ1( ) 0 + + 0 + 1( )
n +1
t x Q R R R i n for a.a. ( ) and every ( 0 0 ) ( =0 ).
ˆ C t x Q
B3. There exists a constant 0 such that for a.a. ( ) and every
n +1
R R R
( 0 0 ) ( 0 ˜0 ˜)
n
X
b t x b t x
i i i ( 0 0 ) i ( 0 ˜0 ˜) ( ˜ )
i =0
p
ˆ p2 2 C j j j j
0 ˜0 + ˜
R R B4. There exist a constantc ˆ2 0, a continuous function ˆ: and a
ˆ 1 L
function k2 (Ω) such that
n
X
p p
p2 2 1 ˆ b t x c j j j j j j k x i i ( 0 0 ) ˆ2 0 + ˆ( ) 0 + 2( )
i =0
n +1
t x Q R R R for a.a. ( ) and every ( 0 0 ) .
2
f Q R R
F1. Function : is of Caratheodory´ type, i.e., it is measurable
2 2
t x Q R R
in ( ) for every fixed ( 0) and continuous in ( 0)
t x Q I R
for a.a. ( ) . Further, for every bounded set there exists a
+
R R continuous function K1: such that
p1
q
K j d j j d R
(i) j 1( 0) 1 0 2 + 2 for every 0 , with some nonnegative
d I constants d1 2 (depending on ),
2008. februar´ 10. –14:16 120 AD´ AM´ BESENYEI
˜
t x Q I R (ii) for a.a. ( ) and every ( 0) ( 0) ,
˜ ˜
f t x f t x j K j j j ( 0) ( 0) 1( 0)
+
R R t x F2. There exists a continuous function K2: such that for a.a. ( )
˜ 2
Q R and all ( 0) ( 0)
˜ ˜
f t x f t x j K j j
j ( 0) ( 0) 2( ) 0 0
L t x Q
F3. There exists (Ω) such that for a.a. ( ) and every
2
R x f t x
( 0) ,( ( )) ( 0) 0.
g Q R R
G1. Function : is of Caratheodory´ type, i.e., it is measurable
t x Q R R
in ( ) for every and continuous in for a.a.
t x Q
( ) .
+ q
R R k L G2. There exist a continuous function c3: and a function 3 1(Ω)
such that
g t x j c k x
j ( ) 3( ) 3( )
t x Q R
for a.a. ( ) and every .
2
h Q R R
H1. Function : is of Caratheodory´ type, i.e., it is measurable
2 2
t x Q R R
in ( ) for every ( 0) and continuous in ( 0) for
t x Q
a.a. ( ) .
+ q
R R k L H2. There exist a continuous function c4: and a function 4 2(Ω)
such that
p1
q
jh t x j c j k x ( 0) 4( ) j 0 2 + 4( )
2
t x Q R for a.a. ( ) and every ( 0) .
3.2. Weak form
If the above assumptions are satisfied, we may define the weak form of Q
the system (5)–(7) in . First for arbitrary
p p
1 2
L Q u L V L V t loc( ) loc(0 ; 1) p loc(0 ; 2)and (0 )
define functionals
u p t V B u t V
[ A( )]( ) 1 [ ( p)]( ) 2
t V H u t V [ G ( )]( ) 1 [ ( )]( ) 2
2008. februar´ 10. –14:16 STABILIZATION OF SOLUTIONS TO A NONLINEAR SYSTEM 121
by
A u p t v i
h[ ( )]( ) 1 :=
Z
n
X
t x t x u t x Du t x t x D t x D v x dx
a p p i = i ( ( ) ( ) ( ) ( ) ( )) 1( ) + Ω
i =1
Z
t x t x u t x Du t x t x D t x v x dx + a0( ( ) ( ) ( ) p( ) p( )) 1( )
Ω
B u t v i
h[ ( p)]( ) 2 :=
Z
n
X
t x t x u t x t x D t x D v x dx
b p p i = i ( ( ) ( ) ( ) ( )) 2( ) + Ω
i =1
Z
t x t x u t x t x D t x v x dx + b0( ( ) ( ) p( ) p( )) 2( )
Ω
Z
h t v i g t x t x v x dx [ G ( )]( ) 1 := ( ( )) 1( )
Ω
Z
H u t v i h t x t x u t x v x dx (8) h[ ( )]( ) 2 := ( ( ) ( )) 2( )
Ω
V v V T for all v1 1 and 2 2. Further, for every 0 we introduce
operators
T T T T T T
A L Q X X X B L Q X X X
T T T
T : ( ) 1 2 ( 1 ) : ( ) 1 2 ( 2 )
T T T
G L Q X H L Q X X
T T T
T : ( ) ( 1 ) : ( ) 1 ( 2 )
T T
X w X
such that for all w1 1 and 2 2 ,
Z
T
u w h A u t w t i dt A p p [ T ( ) 1]= [ ( )]( ) 1( )
0
Z
T
u w h B u t w t i dt B p p [ T ( ) 2]= [ ( )]( ) 2( )
(9) 0
Z
T
G w h G t w t i dt
[ T ( ) 1]= [ ( )]( ) 1( )
0
Z
T
H u w h H u t w t i dt
[ T ( ) 2]= [ ( )]( ) 2( ) 0
2008. februar´ 10. –14:16
122 AD´ AM´ BESENYEI
T
L D L X T In addition, let us introduce the linear operators T : ( ) ( 1 ) by the
formula
T T
D L fu X D u X g L u D u u
t t T ( T )= 1 : ( 1 ) (0) = 0 =
Now we are ready to define the weak form of (5)–(7), namely, we say that
p p
1 2
L Q u L V L V
loc( ) loc(0 ; 1) p loc(0 ; 2) are weak solutions of
T
the system (5)–(7) if for every 0 ,
Z
t
t x x f s x s x u s x ds Q
(10) ( )= 0( )+ ( ( ) ( )) a.e. in T
0
uj u j A j j G j
L p
Q Q T Q Q T Q
(11) T ( )+ ( )= ( )
T T T T T
j uj j H j uj
B p
Q Q Q T Q Q
(12) T ( )= ( )
T T T T T It is well known (see e.g. [9]) that one obtains the above weak forms by
considering sufficiently smooth solutions and then using Green’s theorem,
X after that one considers the equations in the spaces i . It is clear that if
1 p
i
V W the boundary condition is homogenous Neumann than i = (Ω) (since the boundary term vanishes in Green’s theorem) and if we have homogeneous
1 p
i
V W Dirichlet boundary condition then i = 0 (Ω) (in order to eliminate the boundary terms in Green’s theorem). Further, if we have a partition then for example in our one dimensional equation (1) with homogenous bound-
1 p
fv W v t g ary conditions V1 = 1(0 1) : ( 0) = 0 , and in addition
1 p2
W V2 = 0 (0 1). In the next section we prove that the earlier introduced assumptions imply existence of solutions of the above system.
3.3. Existence of weak solutions
Theorem Suppose that conditions A A B B F F G G
are satis ed Then for all L there exist solutions
H H 0 (Ω)
p p
1 2
L Q u L V L V of problem
( ) loc(0 ; 1) p loc(0 ; 2) (10) (12)
T
Proof The main idea is the following. By [4] for every 0
Q there exist solutions in T . By using a diagonal process we can choose
weakly convergent subsequences from the solutions and then prove that the Q
limit functions are solutions in .
2008. februar´ 10. –14:16
STABILIZATION OF SOLUTIONS TO A NONLINEAR SYSTEM 123
T
Let ( k ) be a monotone increasing sequence of positive numbers such
T T
k k
that k + . Then by Theorem 2.1 in [4], for every we have
p p
Q u L T V T V L L
1 p 2 T k k k
( ), (0 ; 1), k (0 ; 2), such that
k
Z
t
t x x f s x s x u s x ds t x Q
k k T (13) k ( )= 0( )+ ( ( ) ( )) ( )
0 k
u A u G
L p
k T k k k T k
(14) T + ( )= ( )
k k k
u H u
B p
k k k T k k
(15) T ( )= ( )
k k
j
(We omit the notation Q if it is not confusing, since the operators and the
T m norms contain the information about the space). By similar arguments as in
Proposition 2.2 of the cited paper, it is easy to see that
k k k k k k
(16) k + Q L L
L ( ) 0 (Ω) (Ω)
T m
Now by following the proof of Theorem 2.1 in [4] word for word and by
L Q m
T
using the boundedness of the sequence ( k )in ( )forall one obtains
m
j m N u j L j
u p
k Q k Q T Q
that for fixed ,( ), ( k )and( ) are bounded in
m
T T T
m m m
T T T
m m m
X X
X1 ,( 1 ) and 2 , respectively.
u L u
m p k T k
Now let =1.Since( ) ( 1 ) ( k ) are bounded sequences in reflex-
T T T
1 1 1
X X
ive Banach spaces 1 ( 1 ) X2 , respectively, there exist subsequences T
T1 1
u u u X X
p p p
k
( ) ( ) ( ) ( k ) and functions , such that k 1 k 1 1 1 1 2
T1
u X
u weakly in
1 k 1 1 T
1
L u L u X
T
T weakly in ( ) 1 1 k 1 1 1
T1
p p weakly in X 1 k 1 2
If (u ) is given then k k m
m 1 1
L u
u p
( ) ( T ) ( ) k k m m k k m m k k m
m 1 1 1 1 1 1
m 1
T T T
m m m
1 1 1
X X
are bounded in reflexive spaces X1 ( 1 ) 2 , thus there exist
u u u
p p m
m k
subsequences ( m k ) ( ) ( ) ( ) and functions k m k
m 1 1
T T
m m
X X
1 pm 2 such that
T
m
u u X
m
m k weakly in 1
T
m
L u L u X
m
m k T
T weakly in ( ) m
m 1
T
m
X p pm m k weakly in 2
2008. februar´ 10. –14:16
124 AD´ AM´ BESENYEI
m
It is easy to see that for each fixed l the above weak convergences hold in
T T T
l l l
X X u j u j X
m
pm
Q l
( ) , respectively, which yields = and Q = T
1 1 2 T
l l
u l m u
= l for . Consequently there exist unique functions :(0 )
V V u j u j u
m m
p m p p Q
1 :(0 ) 2 such that Q = = and
T T
m m
p1
D L m N u L V ( T ) for every . This means that (0 ; )and m loc 1
p2
L V u u
p p p k k k k k
loc(0 ; 2). Consider the sequences ( k )=( ) ( )=( )and
T
m
u u X
p p k k
the appropriate sequence ( k ). Observe that weakly in 1 ,
T m
weakly in X2 . Thus by the well known embedding theorem (see [9]) we
p
u L Q
u 1
T
may assume that k in ( ). Then Propositions 2.2, 2.3 in [4] yield
m
m L Q
m m
k
that for every there exists ( T ) such that ( ) a.e.
m
Q
in T and
m
Z
t
t x x f s x s x u s x ds t x Q
m m m ( )= ( )+ ( ( ) ( )) ( ) T 0 m
0
u
Since for every fixed m the solution of the above equation is unique,
u u Q
m
further, functions ( ) are the restrictions of the function to T ,it
m
L Q j
m
follows that there exists a unique ( ) such that = Q T
loc m
for every m and
Z
t
t x x f s x s x u s x ds t x Q ( )= 0( )+ ( ( ) ( )) ( )
0
By similar arguments as in Proposition 2.2 in [4] it is easy to see that
L Q m N
( ). Now fix . We conclude from the above arguments that
Q
T
k a.e. in
m
T p
m 1
X L Q Q u u
T T
k weakly in strongly in ( ) a.e. in ; m
1 m
T
m
u u L L X
k T
T weakly in ( ) ; m
m 1
T
m
X p p k weakly in 2 By applying word for word step 3 of the proof of Theorem 2.1 in [4], by the
above convergences it is not difficult to show that (for a suitable subsequence)
T T
m m
u u X X
p pm k
k strongly in 1 , strongly in 2 and
u j A j uj j G j
L p
Q T Q Q Q T Q
T + ( )= ( )
m m m
T T T T T
m m m m m
B j j H j uj uj
p
T Q Q T Q Q
( Q )= ( )
m m
T T T T T
m m m m m
2008. februar´ 10. –14:16
STABILIZATION OF SOLUTIONS TO A NONLINEAR SYSTEM 125
This means that u p are solutions in (0 ), so the proof of the theorem is complete.
4. Boundedness of solutions
In this section we show that under some further assumptions, the so-
lutions of (10)–(12) are bounded in appropriate norms in the time interval
(0 ). First suppose the following.
R R B4 . There exist a constantc ˆ2 0, a continuous function ˆ: and a
ˆ 1 L
function k2 (Ω) such that
n
X
p
p2 2 2 ˆ b t x c j j j j j j k x i i ( 0 0 ) ˆ2 0 + ˆ( ) 0 + 2( )
i =0
n +1
t x Q R R R
for a.a. ( ) and every ( 0 0 ) .
+ q
R R k L H2 . There exist a continuous function c4: and a function 4 2 (Ω)
such that
2
q
jh t x j c j k x ( 0) 4( ) j 0 2 + 4( )
2
t x Q R
for a.a. ( ) and every ( 0) .
Theorem Assume that conditions A A B B B F F G
G H H are ful lled and let u solutions of Then
p be (10) (12)
2
Q u L L L V L
( ) (0 ; (Ω)) p (0 ; 2)
Proof L Q
In Theorem 1 we have verified that ( )(whichwasa
2
t ku t k trivial consequence of (16)). In the following let y ( )= ( ) 2 .First L (Ω)
2
C T L y T
note that u ([0 ] (Ω)) thus is continuous in [0 ] (see, e.g., [14]).
u
We show that y is bounded in (0 ). Since is a solution of (11) for all
T T T
0 , thus for arbitrary 0 1 2 we have
T T Z
Z 2 2
hD u t u t i dt h A u t u t i dt
t ( ) ( ) + [ ( p)]( ) ( ) = T
T1 1 T
Z 2
G t u t i dt (17) = h[ ( )]( ) ( )
T1
2008. februar´ 10. –14:16
126 AD´ AM´ BESENYEI
L Q k G t k
By ( )andG2, [ ( )]( ) is bounded, thus by Young’s
V1
inequality we obtain for 0that T
Z 2
G t u t i dt h[ ( )]( ) ( )
T1
T
Z 2
p 1
p1 1
dt ku t k k G t k
( ) + [ ( )]( )
V
q
V
q p1 1 1 1 1
T1
T
Z 2 p
1
p1
u t k c dt
k ( ) + ( ) V p1 1
T1 T
Z 2
hD u t u t i dt y T y T By using the relation t ( ) ( ) = ( 2) ( 1) (see [14]) and
T1 condition A4 on the left hand side of (17), further, by applying the above
estimates with sufficiently small on the right hand side it follows T
Z 2
1 1 p1
T y T c ku t k dt (y ( ) ( )) + ( )
1 2 2 V 2 2 1
T1
T Z
Z 2
j t x k x j dx dt const ( ( )) 2( ) +1
T1 Ω
L Q
Since ( )and is continuous, the integrand (with respect to
T T variable t ) is bounded, thus const ( 2 1) is an upper bound of the right
2
V
hand side. By the continuous embedding L (Ω) 1,
ku t k ku t k
( ) 2 const ( ) V L (Ω) 1
hence T
Z 2
p1
T y T y t dt T T (18) y ( 2) ( 1) + const ( ) 2 const ( 2 1)
T1
2008. februar´ 10. –14:16 STABILIZATION OF SOLUTIONS TO A NONLINEAR SYSTEM 127
Now we show that the above inequality implies the boundedness of y . Sup-
T T
pose the contrary. Then for every M 0thereexist0 1 2 such
T M M y t M T t T
that y ( 2)= and 1 ( ) if 1 2. By choosing these T
T1 2, from (18) we obtain
T T M T T
const ( 2 1)( 1) const ( 2 1)
y
for every M 0. This is impossible, thus is bounded.
L V It remains to show that p (0 ; 2). The proof goes the same way
as the previous part (moreover it is simpler since there is no derivative with
B u H u respect to t ), from ( p)= ( ), by using conditions B4 ,H2 and
the boundedness of we obtain
p2 2
p t k ku t k k ( ) const ( ) +1
V 2 2 L (Ω)
2
t ku t k In the previous part of the proof we have shown that y ( )= ( ) 2 is
L (Ω)
L V bounded thus the above inequality implies the desired p (0 ; 2). The proof of Theorem 2 is complete.
5. Stabilization of solutions
In this section we consider a special case of the system (10)–(12), namely,
p p q q q V V V h f
let p1 = 2 = (thus 1 = 2 = , 1 = 2 = ), = and consider the
T
following problem: for every 0
t
Z
t x x f s x s x u s x ds t x Q
(19) ( )= 0( )+ ( ( ) ( )) ( ) T
0
uj u j A j j G j
L p
Q Q T Q Q T Q
(20) T ( )+ ( )= ( )
T T T T T
j uj j F j uj
B p
Q Q Q T Q Q
(21) T ( )= ( )
T T T T T
F H F
The above operator T is defined through formula (9) with = ,where
h f
operator F is given by (8) with = . We note that in the original model f (1)–(4) h = thus the above sytem is also the generalization of it. In what
follows, we show stabilization of the solutions of the above sytem. That is, we prove the convergence (in some sense) of solutions as t to the solutions
of the stationary system. We need some further assumptions: n
n +1 +1
R R R R i
´ a
A5. There exist Caratheodory functions i :Ω ( =
x
=0 n), i.e., they are measurable in Ω for every ( 0 0 )
n n n
n +1 +1 +1 +1
R R R R R R and continuous in ( 0 0 )
2008. februar´ 10. –14:16
128 AD´ AM´ BESENYEI
x
for a.a. x Ω. Further, for a.a. Ω and every ( 0 0 )
n n
R R R R R
, ,
t x a x a
i
lim i ( 0 0 )= ( 0 0 )
t
n +1
R R R R i
´ b
B5. There exist Caratheodory functions i :Ω ( =
x
=0 n), i.e., they are measurable in Ω for every ( 0 0 ) n
n +1 +1
R R R R x
and continuous in ( 0 0 ) for a.a. Ω.
n +1
R R R
Further, for a.a. x Ω and every ( 0 0 ) , ,
t x b x b
i
lim i ( 0 0 )= ( 0 0 )
t
C t x Q
AB There exists a positive constant such that for a.a. ( ) ,every n
˜ ˜ n +1 +1
R R R
and ( 0 0 ), ( 0 ˜0 ˜) ,
n
X ˜ ˜ ˜
a t x a t x
i i i i ( 0 0 ) ( 0 ˜0 ˜) ( )+
i =0
n
X ˜
b t x b t x
i i i + i ( 0 0 ) ( 0 ˜0 ˜) ( ˜ )
i =0
p p p
˜ p ˜
j j j j j j j C j 0 0 + + 0 ˜0 + ˜
F1/(i ) Suppose (instead of condition F1/(i)) that there exist nonnegative con-
2
q
d jK j d j j d R
stants d1 2 such that 1( 0) 1 0 + 2 for every 0 .
x f t x F4. There exists a positive constant m such that ( ( )) ( 0)
2 2
x t x Q R m
( ( )) for a.a. ( ) and every ( 0) .
g R R
G3. There exists a Caratheodory´ function :Ω such that for a.a.
R
x Ω and every ,
t x g x g
lim ( )= ( )
t
Now introduce operators
A L V V V
: (Ω)
B L V V V
: (Ω)
G L V
: (Ω)
2008. februar´ 10. –14:16 STABILIZATION OF SOLUTIONS TO A NONLINEAR SYSTEM 129
by
hA u vi
( p) :=
Z
n
X
x x u x Du x x D x D v x dx
a p p
i = i ( ( ) ( ) ( ) ( ) ( )) ( ) +
Ω i =1
Z
x x u x Du x x D x v x dx
+ a ( ( ) ( ) ( ) p( ) p( )) ( ) 0
Ω
hB u vi
( p) :=
Z
n
X
x x u x x D x D v x dx
b p p
i = i ( ( ) ( ) ( ) ( )) ( ) +
Ω i =1
Z
x x u x x D x v x dx
+ b ( ( ) ( ) p( ) p( )) ( ) 0
Ω
Z
hG vi g x x v x dx ( ) := ( ( )) ( )
Ω
V
where v .
Theorem Assume that conditions A A B B B B AB F
F i G G are satis ed with p p p Then there exist
F ) = 1 = 2
u V V such that for the solutions u of problem p p (19) (21)
t +1
Z
p
2
t in L u t u in L ku s u k ds
( ) (Ω) ( ) (Ω) ( ) 0
V t 1
t +1
Z
p
k s k F u t in V G t G ds
p( ) p 0 [ ( )]( ) 0 [ ( )]( ) ( )
V
t 1
in V as t further
A u G
( p )= ( )
B u
( p )=0
s t t Remark Precisely, by the convergence ( ) 0as where
+
R M M
s : is a measurable function, and is a normed space, we mean
t ks t k t t
that for all 0 there exists 0 such that ( ) M for a.a. 0.
2008. februar´ 10. –14:16
130 AD´ AM´ BESENYEI
t L t x
Proof First we show that ( ) in (Ω) as .Fix Ω
x x
and assume that 0( ) ( ). By using similar arguments as in the proof
t x x t
of Proposition 2.5 in [4] we obtain that ( ) ( )for 0. Then
t x t x u t x m t x x conditions F3 and F4 yield f ( ( ) ( )) ( ( ) ( )).
Since is absolutely continuous, it is a.e. differentiable, so
t x f t x t x u t x m t x x
( )= ( ( ) ( )) ( ( ) ( ))
By the positivity of we obtain
t x
( )
m
t x x ( ) ( )
hence
mt
t x x x e
( ) ( ) 0( )
mt
x x x e t x x
When 0( ) ( ) one has the estimate 0( ) ( ) ( ),
mt
t k k k e t
thus k ( ) ( ) . This means that ( ) L
L (Ω) 0 (Ω)
L t t x x
( )in (Ω) as . Note, that this implies that ( ) ( )
t
for a.a. x Ωas .
u t V t Now we prove that [ F ( )]( ) 0in as . Clearly, by
conditions F1, F1/(i ), F3,
Z
q
q
k F u t k jf t x t x u t x j dx
[ ( )]( ) ( ( ) ( )) V
Ω
Z
q q
jK u t x j j t x x j dx 1( ( )) ( ) ( )
Ω
q
2
ku t k k t k
const ( ) + const ( )
2 L
L (Ω) (Ω)
t k The multiplicator of the term k ( ) is bounded by the previous
L (Ω)
section, thus the right hand side of the above inequality tends to 0 as t .
G t G V
Now we verify that [ ( )]( ) ( )in . Observe that
Z
q
q
kG t G k jg t x t x g x j dx
( )]( ) ( ) ( ( )) ( ) V2
Ω
and the integrand is a.e. convergent to 0 in Ω as t by G3. Further,
q
jg x j c k x jg t x t x g x j
by G2, G3, ( ) 3( ) 3( ), hence ( ( )) ( )
q
kc k kc k jk x j
( ) + ( ) ( ) and the right hand side is Q L L
3 ( ) 3 (Ω) 3
integrable, thus Lebesgue’s theorem implies that the above integral converges to0ast .
2008. februar´ 10. –14:16 STABILIZATION OF SOLUTIONS TO A NONLINEAR SYSTEM 131
Now we show that problem
A u G
(22) ( p )= ( )
B u
(23) ( p )=0
u V V
has a unique solution p (for fixed ). First observe
a b g
i
that functions i satisfy conditions A1–A4, B1–B4, G1–G2, H1– t x H2 (with variables x instead of ( )). Then by the using the idea of the
main theorem of [4] (successive approximation) it is easy to see that there
u V V exist solutions p of problem (22)–(23) (the proof goes
almost word for word, but it is simpler since there is no in the above
u problem). Uniqueness follows from condition AB, indeed if u1 p1 and 2 p2
are solutions then
hA u A u u u i
0= ( 1 p1) ( 2 p2) 1 2 +
hB u B u i
+ ( 1 p1) ( 2 p2) p1 p2
p
1 p
C ku u k k k
+ p p V
1 2 1 2 V u which implies u1 = 2 and p1 = p2.
In order to show the desired convergences we prove inequality for u
and p. From equations (19)–(21) and (22)–(23) we obtain
hD u t u u t u i h A u t A u u t u i
t ( ( ) ) ( ) + [ ( p)]( ) ( p ) ( ) +
h B u t B u t i
+ [ ( p)]( ) ( p ) p( ) p =
h G t G u t u i h F u t t i
(24) = [ ( )]( ) ( ) ( ) + [ ( )]( ) p( ) p
Observe that the first term on the left hand side of the above equation equals Z
1 2
y t y t u t u y to 2 ( )where ( )= ( ( ) ) (note that is bounded in [0 )by Ω Theorem 2). Further, for the second and third terms of the above equation we
have by condition AB and Young’s inequality
h A u t A u u t u i
[ ( p)]( ) ( p ) ( ) +
h B u t B u t i
+ [ ( p)]( ) ( p ) p( ) p =
h A u t A u t u t u i
= [ ( p)]( ) [ ( p )]( ) ( ) +
i h B u t B u t t
+ [ ( p)]( ) [ ( p )]( ) p( ) p +
h A u t A u u t u i
+ [ ( p )]( ) ( p ) ( ) +
i h B u t B u t
+ [ ( p )]( ) ( p ) p( ) p
p
p p p
C ku t u k k t k ku t u k
( ) + p( ) p ( )
V V V p
2008. februar´ 10. –14:16
132 AD´ AM´ BESENYEI
p
q
p 1
k k t A u t A u k k
p( ) p [ ( p )]( ) ( p )
V
V
q
p q
1 q
k B u t B u k
(25) [ ( p )]( ) ( p )
V q
q
with 0. On the right hand side of (24) by Young’s inequality we obtain
jh G t G u t u i h F u t t ij
[ ( )]( ) ( ) ( ) + [ ( )]( ) p( ) p
p p
p p
ku k t u k t k
(26) ( ) + p( ) p +
V V
p p q
1 q 1
k k G t G k F u t k
+ [ ( )]( ) ( ) + [ ( )]( )
V V
q q
q q
where the last two terms (as we have verified earlier) tend to 0 as t .
We show that last two terms of the right hand side on (25) converges to 0 as
t . Clearly,
q
k A u t A u k
[ ( p )]( ) ( p )
V
Z
n
X
ja t t u Du D
p p i ( ( ) )
i =0 Ω
q
a u Du D j
p p
i ( )
The integrand on the right hand side is a.e. convergent in Ω as t
t x x x by condition A5 and since ( ) ( ) for a.a. Ω. Further, by
conditions A2, A5,
q
ja t t u Du D a u Du D j
p p p p
i
i ( ( ) ) ( )
kc k kc k
const ( ) + ( ) L Q L Q
1 ( ) 1 ( )
p p p p
q
ju j jDu j j j jD j kk k
+ + p + p + 1 L (Ω) 1
where the right hand side is integrable in L (Ω), so by Lebesgue’s theorem
q
k A u t A u k t
we obtain [ ( p )]( ) ( p ) 0as .The convergence of the last term in (25) can be provedV similarly.
Now, by choosing sufficiently small in (24) and by using (25), (26) and
the above convergences we obtain
p p
y t ku t u k k t k t
(27) ( ) + const ( ) + const p( ) p ( )
V V
t t where ( ) 0as and the constants are positive. By the continuous
p 2
L
embedding L (Ω) (Ω)
p
p
t y t k t k t y 2
( ) + const ( ) + const p( ) p ( ) V
2008. februar´ 10. –14:16
STABILIZATION OF SOLUTIONS TO A NONLINEAR SYSTEM 133 t
It is not difficult to show that this inequality implies lim y ( )=0(see
t
the proof of Theorem 2 in [13]).
T By integrating (27) over ( T 1 + 1) we obtain
T +1
Z
p
y T y T ku t u k dt
( +1) ( 1) + const ( ) +
V
T 1 T
T +1 +1
Z Z
p
dt k t t dt k
+const p( ) p ( )
V
T
T 1 1 t Clearly the right hand side tends to 0, and by the convergence of y ( )the first difference on the left hand side tends to 0, too, which yields the desired convergences. The proof of stabilization is complete.
6. Examples
Now we give some examples of functions which fulfil conditions of the
previous theorems. Consider the following functions
a t x
i ( 0 0 )=
r
p1 2 ˜ ˜ ˜ 1 1 P t x P Q j j P t x P Q j j i
(28) = ( ) ( ) ( 0 ) i + ( ) ( ) ( 0 ) n
for i =1
t x a ( )=
(29) 0 0 0
r
p1 2 ˜ ˜ ˜ 1 1
t x P Q j j P t x P Q j j
= P( ) ( ) ( 0 ) 0 0 + 0( ) 0( ) ( 0 ) 0 0
t x b
i ( 0 0 )=
p r
2 2 ˜ ˜ ˜ 2 1 R t x R S j j R t x R S j j i
(30) = ( ) ( ) ( 0) i + ( ) ( ) ( 0) n
for i =1
t x b ( )=
(31) 0 0 0
r
p2 2 ˜ ˜ ˜ 2 1 R t x R S j j R t x R S j j
= ( ) ( ) ( 0) i + ( ) ( ) ( 0) 0 0
where n
˜ ˜ ˜ ˜ ˜ +1 P P P L Q P P P C R Q Q C R
E1 a) 0 ( ), 0 ( ), ( ). Further,
P t x P Q there exists constants 0 such that ( ) ( ) ( 0 ) ,
2008. februar´ 10. –14:16
134 AD´ AM´ BESENYEI
p r p2( 1 1 1)
˜ p Q j Q j j j ( 0 ) ,and ( 0 ) ( 0 ) 1 +1 ,
˜ ˜ ˜ P t x P Q t x Q ( ) ( ) ( 0 ) 0 for a.a. ( ) and every ( 0 )
n +1
R R r p where 0 1 1 1.
˜ ˜ ˜ R R L Q R R S S C R
b) Functions ( ), ( ). Further, there exists
R t x R S jS j
constants 0 such that ( ) ( ) ( 0) , ( 0) ,
p r 1(p2 1 2)
˜ p ˜ ˜ ˜ j j R t x R S
and S ( 0) 0 2 +1 , ( ) ( ) ( 0) 0for
t x Q R R r p a.a. ( ) and every ( 0) where 0 2 2 1. Assuming E1/a, E1/b one can easily verify that functions (28)–(30) fulfil
conditions A1–A4, B1–B4 (see, e.g., [4]). Further, if we make a minor
r 2( p2 1 2)
˜ p j j modification in condition E1/b, namely let S ( 0) 0 2 +1
n +1
R for every 0 , then the above functions satisfy conditions A1–A4,
B1–B3, B4 . n
Now consider the following functions for i =0
p 2
a t x P t x P j j i
(32) ( 0 0 )= ( ) ( ) i ( 0 0 )
p 2
b t x R t x R j j i (33) i ( 0 0 )= ( ) ( ) ( 0 0 )
where
P R L Q P R C R
E2. Functions ( ), ( ) and there exists a positive
P t x P R t x R t x
constant such that ( ) ( ) , ( ) ( ) for a.a. ( )
Q R P R L
and every . Further, there exist functions (Ω)
P R t x P x t x R x
such that lim ( )= ( ) and lim ( )= ( ).
t t
Suppose p and E then the above
Theorem 2 4 (32) (33)
B AB with p p p functions satisfy conditions A A B B B 1 = 2 =
Proof It is easy to see that all the conditions instead of AB are satisfied. For the proof of AB see [5].
If we consider functions
r
p 2 2
a t x j j P t x P j j
i i
i ( 0 0 )= ( 0 ) + ( ) ( ) ( 0 0 )
r
p 2 2
b t x j j R t x R j j
i i
i ( 0 0 )= ( 0 ) + ( ) ( ) ( 0 0 )
r
where 1 4 and E2 hold then it is easy to see that these functions satisfy
p p f rg conditions A1–A5, B1–B3, B4 , B5, AB with p1 = 2 = max 2 .
2008. februar´ 10. –14:16 STABILIZATION OF SOLUTIONS TO A NONLINEAR SYSTEM 135
References
Sobolev spaces [1] R A Adams , Academic Press, New York–San Francisco–
London, 1975. V Mustonen [2] J Berkovits and Topological degree for perturbations of lin- ear maximal monotone mappings and applications to a class of parabolic
problems, Rend Mat Ser VII Roma 12 (1992), 597–621.
Annales [3] A Besenyei On systems of parabolic functional differential equations,
Univ Sci Budapest 47 (2004), 143–160. [4] A Besenyei Existence of weak solutions of a nonlinear system modelling
fluid flow in porous media, Electron J Di Eqns 2006(153) (2006), 1–19.
http ejde math txstate edu Volu mes abstr htm l
Besenyei [5] A Examples for uniformly monotone operators arising in weak
forms of elliptic problems, submitted; preprint: http www cs elte hu badam publications uniform p df
[6] F E Browder Pseudo-monotone operators and nonlinear elliptic boundary
Natl Acad Sci USA value problems on unbounded domains, Proc 74 (1977), 2659–2661.
[7] S Cinca Diffusion und Transport in porosen¨ Medien bei veranderlichen¨ Poro- sitat,¨ Diplomawork, University of Heidelberg, 2000.
[8] Yu A Dubinskiy Nonlinear elliptic and parabolic equations (in Russian), in:
Modern problems in mathematics, 9, Moscow, 1976.
Quelques methodesderesolution des problemes aux limites non
[9] J L Lions
eaires
lin , Dunod, Gauthier-Villars, Paris, 1969. T S Shores
[10] J D Logan M R Petersen and Numerical study of reaction-
Mathematics and mineralogy-porosity changes in porous media, Applied
Computation 127 (2002), 149–164.
[11] L Simon On parabolic functional differential equations of general divergence
form, in: Proceedings of the Conference FSDONA , Milovy, 2004, 280–
291.
Simon Math [12] L On nonlinear hyperbolic functional differential equations,
Nachr 217 (2000), 175–186.
[13] L Simon On different types of nonlinear parabolic functional differential equa-
tions, PU M A 9 (1998), 181–192.
2008. februar´ 10. –14:16
136 AD´ AM´ BESENYEI
Nonlinear functional analysis and its applications II [14] E Zeidler , Springer, 1990.
Ad´ am´ Besenyei Department of Applied Analysis Eotv¨ os¨ Lorand´ University Pazm´ any´ Peter´ set´ any´ 1/C Budapest, H-1117
Hungary badam cs elte hu
2008. februar´ 10. –14:16 ANNALES UNIV. SCI. BUDAPEST., 49 (2006), 137–138
“CONTINUITY SPACE” = QUASI-UNIFORM SPACE
By
ISIDORE FLEISCHER Received March
Abstract In Amer. Math. Monthly 95 (1988), pp. 89–97 (and in subse- quent – as well as previous – publications) Ralph Kopperman has presented a concept he calls “continuity space”. In these structures one can define notions of limit, neighborhood and (even uniform) continuity. By establishing the equivalence of this concept with that of quasi-uniform space, one makes the constructs of either theory available to the other and so can transfer results from one to the other – e.g., that every topology can be induced from a continuity structure is seen to be Csasz´ ar’s´ (1960) theorem that every topology
is quasi-uniformizable. XxX A quasi-uniformity (on set X ) is a filter on , refined by its filter of relative composites, all of whose elements contain the diagonal. A typical example is a (strictly speaking “pseudo-”, although this term
will be suppressed) quasi-metric, which is a (real) non-negative valued func- XxX tion d on , zero on the diagonal and satisfying the triangle inequality –
the filter is that of the inverse images by d of initial intervals. These are the most general quasi–uniformities with countable base, as follows from Kelley’s 6.12 Metrization Lemma. As a consequence, every quasi–uniformity is the supremum of those induced by its quasi–uniformly continuous quasi– metrics. To formulate the triangle inequality one requires an addition and an order. Accordingly, postulate a p.o. commutative semigroup (i.e. addition order
preserving) with an identity which is the smallest element 0; the set P of
Mathematics Subject Classi cation : 54E15, 54A20
2008. februar´ 10. –13:25 138 ISIDORE FLEISCHER non-zero elements, closed for addition, will be taken down-directed (hence a filterbase) and refined by the filterbase of pairwise sums. Then a “continuity space” is (after stripping away some of Kopperman’s superfluous baggage) a set equipped with an abstract quasi-metric with values in such a structure. By taking (as was done in the real-valued case) the inverse images of
the “intervals” [0 p], one obtains a filterbase for a quasi-uniformity, whose derived convergence constructs coincide with those derived from the abstract quasi-metric. To show, conversely, that every quasi-uniform space can be construed as a continuity space, represent it as a subspace of a product of quasi-metrics via the above representation as a quasi-metric supremum.
References General Topology
[1] J L Kelley , van Nostrand, 1955. American [2] R D Kopperman All topologies come from generalized metrics,
Math Monthly 95 (1988), 89–97.
S A Naimpally Quasi Uniform Topological Spaces [3] M G Murdeshwar and ,
Series A: Preprints of Research Papers No. 4,Vol.2 (1966). Foundations of General Topology [4] W J Pervin , Academic Press, 1964.
Isidore Fleischer Centre de recherches mathematiques´ Universite´ de Montreal´ Case postale 6128 succursale Centre-ville Montreal (Quebec)´ H3C 3J7
Canada fleischi crm umontreal ca
2008. februar´ 10. –13:25 ANNALES UNIV. SCI. BUDAPEST., 49 (2006), 139–145
A THEOREM ON INTEGRABILITY OF TRIGONOMETRIC SERIES
By
LASZL´ O´ LEINDLER* Received March
1. Introduction
The basic results on integrability theorems for trigonometric series deal with monotone or positive coefficients and functions. Their generalizations weakened the monotonicity conditions gradually. Now we present a further generalization. Recently, in [3], we already proved the following result. For notions and
notations, please, see Section 2.
Theorem A Let GB V S
X
g x b nx
(1 1) ( ):= n sin
n =1
and
X
f x a nx
(1 2) ( ):= n cos
n =1
i If and
0 1
X
1
n
(1 3) n
n =1
Mathematics Subject Classi cation : 42A16, 42A32 * This research was partially supported by the Hungarian National Foundation for Scientific Research under Grant # T042462.
2008. februar´ 10. –13:26
140 LASZL´ O´ LEINDLER
x g x L L
then ( ) := (0 )
and holds then x f x L
ii If 0 1 (1.3) ( )
In this note we will replace the function x by a more general function x ( ) which belongs to one of the classes defined by Yung-Ming Chen [2] (see also H. P. Mulholland [4]).
2. Notions and notations
We preserve the following notations throughout.
f g
By we mean either an or a ; the coefficients associated with
f g
are denoted by := n .
c fc g c GB V S n A null-sequence := n ( 0) belongs to the class if
2 m
X
j c j K c c c c m
m n n Δ n ( ) (Δ := ) =1 2
n +1 m
n =
K c f g
holds, where ( ) is a positive constant and := n is a given sequence
of nonnegative numbers.
x
A not identically zero nonnegative function ( )definedon(0 )
1
x x x x belongs to the set Ψ if for some 0 ( ) is nondecreasing and ( )
is nonincreasing as x increases.
L R
We shall use the following notation: if there exists a positive
L KR K constant K such that , but not necessarily the same at each occurrence.
3. New theorem
Our result reads as follows.
Let x and GB V S If
Theorem ( ) Ψ
X 1 1
(3 1) n
n n
n =1
x x L then ( ) ( )
2008. februar´ 10. –13:26 A THEOREM ON INTEGRABILITY OF TRIGONOMETRIC SERIES 141
4. Lemma
We need the following lemma, which was proved in [3] implicitly.
Lemma If f g GB V S then
:= n
X X
1
j j k n
n
k
Δ k + =1 2
n
n k = k 2
5. Proof of Theorem
Our proof combines the methods of R. P. Boas, Jr. [1] and Yung-Ming
Chen [2] with ours [3].
a x f x n First we consider the cosine series, that is, if n = and ( )= ( ).
Then we have
1 1 X
x a n nx f ( )= [sin( +2) sin ] n +1
2 sin x
n =0
(5 1)
1 X
a x a x a a nx
= sin + sin 2 + ( )sin n 2 1 n 1 +1
sin x
n =3
x
Since ( ) Ψ, thus for the functions
1
x 1
Q x n
n ( ):= =1 2
n n
we obtain, by an easy consideration, that
1 1
x Q x x x x Q x x x n (5 2) n ( ) (0 1) ( ) ( 1)
2008. februar´ 10. –13:26 142 LASZL´ O´ LEINDLER
Utilizing (5.2) we can estimate the following integral:
Z
Z 1
1 1
x x j nx jdx Q nx x j nx jdx
( ) sin = n ( ) sin = n
0 0
n
1 Z
1
Q x x j x jdx
= n ( ) sin = n
0
n
1
Z Z
1
(5 3) 1
x x j x jdx Q
= + n ( ) sin n
0 1
1 Z
1 Z
1 1
dx x dx
x + n
0 1
1
n
x Q x
Since ( )= 1( ) (1), thus, by (5.2),
Z
x dx (5 4) ( ) 1 0 also holds.
Consequently, (5.1), (5.3) and (5.4) imply that
Z
X 1
x jf x jdx j a j
(5 5) ( ) ( ) 1+ Δ n n
0 n =1
x f x Thus, in order to prove that ( ) ( ) is integrable, by (3.1), we have only to
verify that
X
X 1 1 1
j a j n
(5 6) Δ n
n n n n n =1 =1
First we show that
n
1 X 1 1 1
(5 7)
n k k n
k =1
2008. februar´ 10. –13:26
A THEOREM ON INTEGRABILITY OF TRIGONOMETRIC SERIES 143
o n 1 1
Since is a nondecreasing sequence, thus
k k
n n
X
X 1 1 1 1
1
k
=
k k k k k
k =1 =1
n
1 1 X
1
k
n n
k =1
1
n
o n 1 1 1
On the other hand, since is a nonincreasing sequence, we
k k
obtain that
n n X
1 1 X 1 1 1
k k k k n
n
k =1 k 2 herewith (5.7) is proved.
Using the first inequality of (5.7) and Abel’s transformation we get that
n
X X
X 1 1 1
j a j j a j n
Δ n Δ
n k k
n k n =1 =1 =1
(5 8)
X
X 1 1
j a j
= Δ n
k k
n k k =1 = Hereafter we can utilize our Lemma, later an Abel’s transformation
and (3.1) deliver the following inequalities:
X X X X 1 1 1 1
1
a j k j
n
k
Δ k +
n n n n
n
k n
n n =2 = =2 k
2
k
2 X
X 1 1
(5 9) 1
k
1+ k
n n n
k =1 =1
X 1
1
k
k k
k =1
x f x L Summing up, (3.1), (5.5), (5.6), (5.8) and (5.9) prove that ( ) ( ) .
2008. februar´ 10. –13:26
144 LASZL´ O´ LEINDLER
x g x L
Next we verify that ( ) ( ) also holds.
x g x b n
If ( )= ( )and n = , then instead of (5.1) we shall start with the
x x
following equality, using the function c ( ):=1 cos :
1 1 X
x b c n x c nx g ( )= [ (( +2) ) ( )] n +1
2 sin x
n =0
(5 10)
1 X
c x b c x b b c nx b
= ( )+ (2 )+ ( ) ( ) n 2 1 n 1 +1
sin x
n =3
x c jx x j By (5.4) the functions ( ) ( ) sin ( =12 ) are integrable, therefore it
is enough to show the existence of the following integral:
Z
X
1
x x jb b jc nx dx
( ) ( ) = n n 1 +1
0 n =3
(5 11)
Z
X
1
b b j x x c nx dx
= j ( ) ( ) n n 1 +1
n =3 0
As in (5.3), using again (5.2), we obtain that
Z
Z 1
1 1
x x c nx dx Q nx x c nx dx
( ) ( ) = n ( ) ( ) = n
0 0
n
1 Z
1
Q x x c x dx
= n ( ) ( ) n
0
1 Z
1 Z
1 1
dx x dx x
+ n
0 1
1
n
Thus, by (5.10) and (5.11), we get
Z
X 1
x jg x jdx j b j
( ) ( ) 1+ Δ n n
0 n =1
2008. februar´ 10. –13:26 A THEOREM ON INTEGRABILITY OF TRIGONOMETRIC SERIES 145
Hereafter the proof is similar to that of the cosine series, therefore we omit it. The proof of our theorem is complete.
References Quart J Math Oxford [1] R P Boas Jr Integrability of trigonometric series III,
Ser 3 (1952), 217–221. Studia Math [2] Y M Chen On two functional spaces, 24 (1964), 61–88.
[3] L Leindler A newer class of numerical sequences and its applications to sine
and cosine series, Analysis Math 33(1) (2007), 37–43.
[4] H P Mulholland The generalization of certain inequality theorems involv-
ing powers, Proc London Math Soc 33 (1932), 481–516.
Laszl´ o´ Leindler Department of Analysis Bolyai Institute University of Szeged
Szeged, Hungary leindler math u szeged hu
2008. februar´ 10. –13:26 2008. februar´ 10. –15:13
INDEX
uller U Baron S Liflyand E Stadtm : The Fourier integrals of func-
tions of bounded variation 43
A Besenyei : Stabilization of solutions to a nonlinear system modelling fluid
flow in porous media 115
M Ekici E Jafari S Noiri T Caldas : On the class of contra -
continuous functions 75
R
Caldas M Jafari S Noiri T : A weaker form of pre- 0 103
aszar A
Cs : Intersections of open sets in generalized topological spaces 53
aszar A
Cs : -open sets in generalized topological spaces 59
E Caldas M Jafari S Noiri T Ekici : On the class of contra -
continuous functions 75
E Jafari S E F
Ekici :On -sets and -sets and decompositions of -
continuity, quasi-continuity and A-continuity 65
Fleischer I : “Continuity space” = quasi-uniform space 137 Fried E : Corrections to my paper “Discriminator or dual discriminator?” 95
Gyarmati K : A note to the paper “On a fast version of a pseudorandom
generator” 87
S Caldas M Ekici E Noiri T Jafari : On the class of contra -
continuous functions 75
S Ekici E E F
Jafari :On -sets and -sets and decompositions of -
continuity, quasi-continuity and A-continuity 65
R
Jafari S Noiri T Caldas M : A weaker form of pre- 0 103
acs K
Kov : On triples of consecutive Legendre symbols 109
Leindler L : A theorem on integrability of trigonometric series 139
uller U Liflyand E Baron S Stadtm : The Fourier integrals of func-
tions of bounded variation 43
2008. februar´ 10. –14:52
Mocanu M : Semi-separated sets in generalized closure spaces 25
Nagy Gy : Tessellation-like rod-joint frameworks 3
T Caldas M Ekici E Jafari S Noiri : On the class of contra -
continuous functions 75
R
Noiri T Jafari S Caldas M : A weaker form of pre- 0 103
er D Recski A Salamon G Szeszl : Improving size-bounds for subcases
of square-shaped switchbox routing 15
er D Salamon G Recski A Szeszl : Improving size-bounds for subcases
of square-shaped switchbox routing 15
uller U Baron S Liflyand E Stadtm : The Fourier integrals of func-
tions of bounded variation 43
er D Recski A Salamon G Szeszl : Improving size-bounds for subcases
of square-shaped switchbox routing 15
ISSN 0524-9007 Address: MATHEMATICAL INSTITUTE, EOTV¨ OS¨ LORAND´ UNIVERSITY ELTE, TTK, KARI KONYVT¨ AR,´ MATEMATIKAI TUDOMANY´ AGI´ SZAKGYUJTEM˝ ENY´ PAZM´ ANY´ PETER´ SET´ ANY´ 1/c, 1117 BUDAPEST, HUNGARY Muszaki˝ szerkeszto:˝ Fried Katalin PhD Akiadas´ ert´ felelos:˝ az Eotv¨ os¨ Lorand´ Tudomanyegyetem´ rektora Akezirat´ a nyomdaba´ erkezett:´ 2008. februar.´ Megjelent: 2008. marcius´ Terjedelem: 16,25 A/4 ´ıv. Peld´ anysz´ am:´ 500 Kesz´ ult¨ az EMTEX szedoprogram˝ felhasznal´ as´ aval´ az MSZ 5601–59 es´ 5602–55 szabvanyok´ szerint Az elektronikus tipografal´ as´ Bori Tamas´ munkaja´ Nyomda: Haxel Bt. Felelos˝ vezeto:˝ Korandi´ Jozsef´
2008. februar´ 10. –14:52