Generalized Differentiability of Continuous Functions
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On Semicontinuous Functions and Baire Functions
ON SEMICONTINUOUS FUNCTIONS AND BAIRE FUNCTIONS BY ROBERT E. ZINK(») 1. According to a classical theorem, every semicontinuous real-valued func- tion of a real variable can be obtained as the limit of a sequence of continuous functions. The corresponding proposition need not hold for the semicontinuous functions defined on an arbitrary topological space; indeed, it is known that the theorem holds in the general case if and only if the topology is perfectly normal [6]. For some purposes, it is important to know when a topological measure space has the property that each semicontinuous real-valued function defined thereon is almost everywhere equal to a function of the first Baire class. In the present article, we are able to resolve this question when the topology is completely regular. A most interesting by-product of our consideration of this problem is the discovery that every Lebesgue measurable function is equivalent to the limit of a sequence of approximately continuous ones. This fact enables us to answer a question, posed in an earlier work [7], concerning the existence of topological measure spaces in which every bounded measurable function is equivalent to a Baire function of the first class. 2. We begin the discussion with some pertinent background material. The Blumberg upper measurable boundary, uB, of a real-valued function / defined on Euclidean q-space, Eq, is specified in the following manner: If Ey = {x: fix) > y}, then uBi£) = inf {y: exterior metric density ofEy at £ is zero}. The lower measurable boundary, lB, is similarly defined. Blumberg proved that uB is approximately upper-semicontinuous and that lB is approximately lower-semicontinuous and then observed that approximately semicontinuous functions are Lebesgue measurable. -
The Baire Order of the Functions Continuous Almost Everywhere R
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 41, Number 2, December 1973 THE BAIRE ORDER OF THE FUNCTIONS CONTINUOUS ALMOST EVERYWHERE R. d. MAULDIN Abstract. Let í> be the family of all real-valued functions defined on the unit interval / which are continuous except for a set of Lebesgue measure zero. Let <J>0be <I>and for each ordinal a, let <S>abe the family of all pointwise limits of sequences taken from {Jy<x ÍV Then <Dm is the Baire family generated by O. It is proven here that if 0<a<co1, then O^i^ . The proof is based upon the construction of a Borel measurable function h from / onto the Hubert cube Q such that if x is in Q, then h^ix) is not a subset of an F„ set of Lebesgue measure zero. If O is a family of real-valued functions defined on a set S, then the Baire family generated by O may be described as follows: Let O0=O and for each ordinal <x>0, let Oa be the family of all pointwise limits of sequences taken from \Jy<x î>r Of course, ^o^^V+i* where mx denotes the first uncountable ordinal and 4> is the Baire family generated by O; the family O^, is the smallest subfamily of Rs containing O and which is closed under pointwise limits of sequences. The order of O is the first ordinal a such that 0>X=Q>X+X. Let C denote the family of all real-valued continuous functions on the unit interval I. -
On the Structures of Generating Iterated Function Systems of Cantor Sets
ON THE STRUCTURES OF GENERATING ITERATED FUNCTION SYSTEMS OF CANTOR SETS DE-JUN FENG AND YANG WANG Abstract. A generating IFS of a Cantor set F is an IFS whose attractor is F . For a given Cantor set such as the middle-3rd Cantor set we consider the set of its generating IFSs. We examine the existence of a minimal generating IFS, i.e. every other generating IFS of F is an iterating of that IFS. We also study the structures of the semi-group of homogeneous generating IFSs of a Cantor set F in R under the open set condition (OSC). If dimH F < 1 we prove that all generating IFSs of the set must have logarithmically commensurable contraction factors. From this Logarithmic Commensurability Theorem we derive a structure theorem for the semi-group of generating IFSs of F under the OSC. We also examine the impact of geometry on the structures of the semi-groups. Several examples will be given to illustrate the difficulty of the problem we study. 1. Introduction N d In this paper, a family of contractive affine maps Φ = fφjgj=1 in R is called an iterated function system (IFS). According to Hutchinson [12], there is a unique non-empty compact d SN F = FΦ ⊂ R , which is called the attractor of Φ, such that F = j=1 φj(F ). Furthermore, FΦ is called a self-similar set if Φ consists of similitudes. It is well known that the standard middle-third Cantor set C is the attractor of the iterated function system (IFS) fφ0; φ1g where 1 1 2 (1.1) φ (x) = x; φ (x) = x + : 0 3 1 3 3 A natural question is: Is it possible to express C as the attractor of another IFS? Surprisingly, the general question whether the attractor of an IFS can be expressed as the attractor of another IFS, which seems a rather fundamental question in fractal geometry, has 1991 Mathematics Subject Classification. -
Baire Sets, Borel Sets and Some Typical Semi-Continuous Functions
BAIRE SETS, BOREL SETS AND SOME TYPICAL SEMI-CONTINUOUS FUNCTIONS KEIO NAGAMI Recently Hing Tong [3], M. Katetov [4] and C. H. Dowker [2] have es- tablished two sorts of insertion theorems for semi-continuous functions defined on normal and countably paracompact normal spaces. The purpose of this paper is to give the insertion theorem for some typical semi-continuous functions defined on a T-space. The relations between Baire sets and Borel sets in some topological spaces are also studied. Throughout this paper, unless in a special context we explicitly say other- wise, any function defined on a space is real-valued and a sequence of functions means a countable sequence of them. A family of functions defined on a space R is called complete if the limit of any sequence of functions in it is also con- tained in it. The minimal complete family of functions defined on R which con- tains all continuous functions is called the family of Baire functions and its ele- ment a Baire function. A subset of R is called a Baire set if its characteristic function is a Baire function. A subset of R is called respectively elementary- open or elementary-closed if it is written as {x fix) > 0} or {x I fix) ^ 0} by a suitable continuous function fix) defined on R. The minimal completely additive class of sets which contains all open sets in R is called the family of Borel sets and its element a Borel set. It is well known that the family of Borel sets contains the family of Baire sets and that the family of Baire sets coincides with the minimal completely additive class of sets which contains all the elementary open (closed) sets. -
On Borel Measures and Baire's Class 3
proceedings of the american mathematical society Volume 39, Number 2, July 1973 ON BOREL MEASURES AND BAIRE'S CLASS 3 R. DANIEL MAULDIN Abstract. Let S be a complete and separable metric space and fi a er-finite, complete Borel measure on S. Let <I>be the family of all real-valued functions, continuous /¿-a.e. Let Bx(^) be the func- tions of Baire's class a generated by O. It is shown that if ¡x is not a purely atomic measure whose set of atoms form a dispersed subset of S, then Bt(®)^Bai(<b), where a>i denotes the first uncountable ordinal. If O is a family of real-valued functions defined on a set S, then, B(<&), the Baire system of functions generated by O is the smallest subfamily of Rx which contains O and which is closed under the process of taking pointwise limits of sequences. The family 5(0) can be generated from <I> as follows: Let £„(<!>)=O and for each ordinal a, let -Ba(0) be the family of all pointwise limits of sequences taken from IJy<a ^O^O- Thus, B^ffb) is the Baire system of functions generated by <E>,where u>1 is the first uncountable ordinal. The Baire order of a family O is the first ordinal a such that BX(®)=BX+1(4>). Kuratowski has proved that if S is a metric space and O is the family of all real-valued functions on S which are continuous except for a first category set, then the order of í> is 1 and 2?i(<J>)is the family of all functions which have the Baire property in the wide sense [1, p. -
Georg Cantor English Version
GEORG CANTOR (March 3, 1845 – January 6, 1918) by HEINZ KLAUS STRICK, Germany There is hardly another mathematician whose reputation among his contemporary colleagues reflected such a wide disparity of opinion: for some, GEORG FERDINAND LUDWIG PHILIPP CANTOR was a corruptor of youth (KRONECKER), while for others, he was an exceptionally gifted mathematical researcher (DAVID HILBERT 1925: Let no one be allowed to drive us from the paradise that CANTOR created for us.) GEORG CANTOR’s father was a successful merchant and stockbroker in St. Petersburg, where he lived with his family, which included six children, in the large German colony until he was forced by ill health to move to the milder climate of Germany. In Russia, GEORG was instructed by private tutors. He then attended secondary schools in Wiesbaden and Darmstadt. After he had completed his schooling with excellent grades, particularly in mathematics, his father acceded to his son’s request to pursue mathematical studies in Zurich. GEORG CANTOR could equally well have chosen a career as a violinist, in which case he would have continued the tradition of his two grandmothers, both of whom were active as respected professional musicians in St. Petersburg. When in 1863 his father died, CANTOR transferred to Berlin, where he attended lectures by KARL WEIERSTRASS, ERNST EDUARD KUMMER, and LEOPOLD KRONECKER. On completing his doctorate in 1867 with a dissertation on a topic in number theory, CANTOR did not obtain a permanent academic position. He taught for a while at a girls’ school and at an institution for training teachers, all the while working on his habilitation thesis, which led to a teaching position at the university in Halle. -
Fractal Geometry and Applications in Forest Science
ACKNOWLEDGMENTS Egolfs V. Bakuzis, Professor Emeritus at the University of Minnesota, College of Natural Resources, collected most of the information upon which this review is based. We express our sincere appreciation for his investment of time and energy in collecting these articles and books, in organizing the diverse material collected, and in sacrificing his personal research time to have weekly meetings with one of us (N.L.) to discuss the relevance and importance of each refer- enced paper and many not included here. Besides his interdisciplinary ap- proach to the scientific literature, his extensive knowledge of forest ecosystems and his early interest in nonlinear dynamics have helped us greatly. We express appreciation to Kevin Nimerfro for generating Diagrams 1, 3, 4, 5, and the cover using the programming package Mathematica. Craig Loehle and Boris Zeide provided review comments that significantly improved the paper. Funded by cooperative agreement #23-91-21, USDA Forest Service, North Central Forest Experiment Station, St. Paul, Minnesota. Yg._. t NAVE A THREE--PART QUE_.gTION,, F_-ACHPARToF:WHICH HA# "THREEPAP,T_.<.,EACFi PART" Of:: F_.AC.HPART oF wHIct4 HA.5 __ "1t4REE MORE PARTS... t_! c_4a EL o. EP-.ACTAL G EOPAgTI_YCoh_FERENCE I G;:_.4-A.-Ti_E AT THB Reprinted courtesy of Omni magazine, June 1994. VoL 16, No. 9. CONTENTS i_ Introduction ....................................................................................................... I 2° Description of Fractals .................................................................................... -
Fractals Lindenmayer Systems
FRACTALS LINDENMAYER SYSTEMS November 22, 2013 Rolf Pfeifer Rudolf M. Füchslin RECAP HIDDEN MARKOV MODELS What Letter Is Written Here? What Letter Is Written Here? What Letter Is Written Here? The Idea Behind Hidden Markov Models First letter: Maybe „a“, maybe „q“ Second letter: Maybe „r“ or „v“ or „u“ Take the most probable combination as a guess! Hidden Markov Models Sometimes, you don‘t see the states, but only a mapping of the states. A main task is then to derive, from the visible mapped sequence of states, the actual underlying sequence of „hidden“ states. HMM: A Fundamental Question What you see are the observables. But what are the actual states behind the observables? What is the most probable sequence of states leading to a given sequence of observations? The Viterbi-Algorithm We are looking for indices M1,M2,...MT, such that P(qM1,...qMT) = Pmax,T is maximal. 1. Initialization ()ib 1 i i k1 1(i ) 0 2. Recursion (1 t T-1) t1(j ) max( t ( i ) a i j ) b j k i t1 t1(j ) i : t ( i ) a i j max. 3. Termination Pimax,TT max( ( )) qmax,T q i: T ( i ) max. 4. Backtracking MMt t11() t Efficiency of the Viterbi Algorithm • The brute force approach takes O(TNT) steps. This is even for N = 2 and T = 100 difficult to do. • The Viterbi – algorithm in contrast takes only O(TN2) which is easy to do with todays computational means. Applications of HMM • Analysis of handwriting. • Speech analysis. • Construction of models for prediction. -
Generalizations and Properties of the Ternary Cantor Set and Explorations in Similar Sets
Generalizations and Properties of the Ternary Cantor Set and Explorations in Similar Sets by Rebecca Stettin A capstone project submitted in partial fulfillment of graduating from the Academic Honors Program at Ashland University May 2017 Faculty Mentor: Dr. Darren D. Wick, Professor of Mathematics Additional Reader: Dr. Gordon Swain, Professor of Mathematics Abstract Georg Cantor was made famous by introducing the Cantor set in his works of mathemat- ics. This project focuses on different Cantor sets and their properties. The ternary Cantor set is the most well known of the Cantor sets, and can be best described by its construction. This set starts with the closed interval zero to one, and is constructed in iterations. The first iteration requires removing the middle third of this interval. The second iteration will remove the middle third of each of these two remaining intervals. These iterations continue in this fashion infinitely. Finally, the ternary Cantor set is described as the intersection of all of these intervals. This set is particularly interesting due to its unique properties being uncountable, closed, length of zero, and more. A more general Cantor set is created by tak- ing the intersection of iterations that remove any middle portion during each iteration. This project explores the ternary Cantor set, as well as variations in Cantor sets such as looking at different middle portions removed to create the sets. The project focuses on attempting to generalize the properties of these Cantor sets. i Contents Page 1 The Ternary Cantor Set 1 1 2 The n -ary Cantor Set 9 n−1 3 The n -ary Cantor Set 24 4 Conclusion 35 Bibliography 40 Biography 41 ii Chapter 1 The Ternary Cantor Set Georg Cantor, born in 1845, was best known for his discovery of the Cantor set. -
Set Theory: Cantor
Notes prepared by Stanley Burris March 13, 2001 Set Theory: Cantor As we have seen, the naive use of classes, in particular the connection be- tween concept and extension, led to contradiction. Frege mistakenly thought he had repaired the damage in an appendix to Vol. II. Whitehead & Russell limited the possible collection of formulas one could use by typing. Another, more popular solution would be introduced by Zermelo. But ¯rst let us say a few words about the achievements of Cantor. Georg Cantor (1845{1918) 1872 - On generalizing a theorem from the theory of trigonometric series. 1874 - On a property of the concept of all real algebraic numbers. 1879{1884 - On in¯nite linear manifolds of points. (6 papers) 1890 - On an elementary problem in the study of manifolds. 1895/1897 - Contributions to the foundation to the study of trans¯nite sets. We include Cantor in our historical overview, not because of his direct contribution to logic and the formalization of mathematics, but rather be- cause he initiated the study of in¯nite sets and numbers which have provided such fascinating material, and di±culties, for logicians. After all, a natural foundation for mathematics would need to talk about sets of real numbers, etc., and any reasonably expressive system should be able to cope with one- to-one correspondences and well-orderings. Cantor started his career by working in algebraic and analytic number theory. Indeed his PhD thesis, his Habilitation, and ¯ve papers between 1867 and 1880 were devoted to this area. At Halle, where he was employed after ¯nishing his studies, Heine persuaded him to look at the subject of trigonometric series, leading to eight papers in analysis. -
Sets on Which Measurable Functions Are Determined by Their Range
Sets on which measurable functions are determined by their range Maxim R. Burke∗ Krzysztof Ciesielski† Department of Mathematics and Computer Science, Department of Mathematics, University of Prince Edward Island, West Virginia University, Charlottetown, P.E.I., Canada C1A4P3 Morgantown, WV 26506-6310, USA [email protected] [email protected] Version of 96/01/14 Abstract We study sets on which measurable real-valued functions on a measurable space with negligibles are determined by their range. 1 Introduction In [BD, theorem 8.5], it is shown that, under the Continuum Hypothesis (CH), in any separable Baire topological space X there is a set M such that for any two continuous real-valued functions f and g on X,iff and g are not constant on any nonvoid open set then f[M] ⊆ g[M] implies f = g. In particular, if f[M]=g[M] then f = g. Sets havingthis last property with respect to entire (analytic) functions in the complex plane were studied in [DPR] where they were called sets of range uniqueness (SRU’s).We study the properties of such sets in measurable spaces with negligibles. (See below for the definition.) We prove a generalization of the aforementioned result [BD, theorem 8.5] to such spaces (Theorem 4.3) and answer Question 1 from [BD] by showingthat CH cannot be omitted from the hypothesis of their theorem (Example 5.17). We also study the descriptive nature of SRU’s for the nowhere constant continuous functions on Baire Tychonoff topological spaces. When X = R, the result of [BD, theorem 8.5] states that, under CH, there is a set M ⊆ R such that for any two nowhere constant continuous functions f,g: R → R,iff[M] ⊆ g[M] then f = g.Itis shown in [BD, theorem 8.1] that there is (in ZFC) a set M ⊆ R such that for any continuous functions f,g: R → R,iff has countable level sets and g[M] ⊆ f[M] then g is constant on the connected components of {x ∈ R: f(x) = g(x)}. -
Generators, Recursion, and Fractals
Generators, Recursion, and Fractals 1 Generators computing a list of Fibonacci numbers defining a generator with yield putting yield in the function fib 2 Recursive Functions computing factorials recursively computing factorials iteratively 3 Recursive Images some examples recursive definition of the Cantor set recursive drawing algorithm MCS 260 Lecture 41 Introduction to Computer Science Jan Verschelde, 22 April 2016 Intro to Computer Science (MCS 260) generators and recursion L-41 22 April 2016 1 / 36 Generators, Recursion, and Fractals 1 Generators computing a list of Fibonacci numbers defining a generator with yield putting yield in the function fib 2 Recursive Functions computing factorials recursively computing factorials iteratively 3 Recursive Images some examples recursive definition of the Cantor set recursive drawing algorithm Intro to Computer Science (MCS 260) generators and recursion L-41 22 April 2016 2 / 36 the Fibonacci numbers The Fibonacci numbers are the sequence 0, 1, 1, 2, 3, 5, 8,... where the next number in the sequence is the sum of the previous two numbers in the sequence. Suppose we have a function: def fib(k): """ Computes the k-th Fibonacci number. """ and we want to use it to compute the first 10 Fibonacci numbers. Intro to Computer Science (MCS 260) generators and recursion L-41 22 April 2016 3 / 36 the function fib def fib(k): """ Computes the k-th Fibonacci number. """ ifk==0: return 0 elif k == 1: return 1 else: (prevnum, nextnum) = (0, 1) for i in range(1, k): (prevnum, nextnum) = (nextnum, \ prevnum + nextnum) return nextnum Intro to Computer Science (MCS 260) generators and recursion L-41 22 April 2016 4 / 36 themainprogram def main(): """ Prompts the user for a number n and prints the first n Fibonacci numbers.