Zero-Dimensional Spaces and Their Inverse Limits
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P. 1 Math 490 Notes 7 Zero Dimensional Spaces for (SΩ,Τo)
p. 1 Math 490 Notes 7 Zero Dimensional Spaces For (SΩ, τo), discussed in our last set of notes, we can describe a basis B for τo as follows: B = {[λ, λ] ¯ λ is a non-limit ordinal } ∪ {[µ + 1, λ] ¯ λ is a limit ordinal and µ < λ}. ¯ ¯ The sets in B are τo-open, since they form a basis for the order topology, but they are also closed by the previous Prop N7.1 from our last set of notes. Sets which are simultaneously open and closed relative to the same topology are called clopen sets. A topology with a basis of clopen sets is defined to be zero-dimensional. As we have just discussed, (SΩ, τ0) is zero-dimensional, as are the discrete and indiscrete topologies on any set. It can also be shown that the Sorgenfrey line (R, τs) is zero-dimensional. Recall that a basis for τs is B = {[a, b) ¯ a, b ∈R and a < b}. It is easy to show that each set [a, b) is clopen relative to τs: ¯ each [a, b) itself is τs-open by definition of τs, and the complement of [a, b)is(−∞,a)∪[b, ∞), which can be written as [ ¡[a − n, a) ∪ [b, b + n)¢, and is therefore open. n∈N Closures and Interiors of Sets As you may know from studying analysis, subsets are frequently neither open nor closed. However, for any subset A in a topological space, there is a certain closed set A and a certain open set Ao associated with A in a natural way: Clτ A = A = \{B ¯ B is closed and A ⊆ B} (Closure of A) ¯ o Iτ A = A = [{U ¯ U is open and U ⊆ A}. -
Molecular Symmetry
Molecular Symmetry Symmetry helps us understand molecular structure, some chemical properties, and characteristics of physical properties (spectroscopy) – used with group theory to predict vibrational spectra for the identification of molecular shape, and as a tool for understanding electronic structure and bonding. Symmetrical : implies the species possesses a number of indistinguishable configurations. 1 Group Theory : mathematical treatment of symmetry. symmetry operation – an operation performed on an object which leaves it in a configuration that is indistinguishable from, and superimposable on, the original configuration. symmetry elements – the points, lines, or planes to which a symmetry operation is carried out. Element Operation Symbol Identity Identity E Symmetry plane Reflection in the plane σ Inversion center Inversion of a point x,y,z to -x,-y,-z i Proper axis Rotation by (360/n)° Cn 1. Rotation by (360/n)° Improper axis S 2. Reflection in plane perpendicular to rotation axis n Proper axes of rotation (C n) Rotation with respect to a line (axis of rotation). •Cn is a rotation of (360/n)°. •C2 = 180° rotation, C 3 = 120° rotation, C 4 = 90° rotation, C 5 = 72° rotation, C 6 = 60° rotation… •Each rotation brings you to an indistinguishable state from the original. However, rotation by 90° about the same axis does not give back the identical molecule. XeF 4 is square planar. Therefore H 2O does NOT possess It has four different C 2 axes. a C 4 symmetry axis. A C 4 axis out of the page is called the principle axis because it has the largest n . By convention, the principle axis is in the z-direction 2 3 Reflection through a planes of symmetry (mirror plane) If reflection of all parts of a molecule through a plane produced an indistinguishable configuration, the symmetry element is called a mirror plane or plane of symmetry . -
The Motion of Point Particles in Curved Spacetime
Living Rev. Relativity, 14, (2011), 7 LIVINGREVIEWS http://www.livingreviews.org/lrr-2011-7 (Update of lrr-2004-6) in relativity The Motion of Point Particles in Curved Spacetime Eric Poisson Department of Physics, University of Guelph, Guelph, Ontario, Canada N1G 2W1 email: [email protected] http://www.physics.uoguelph.ca/ Adam Pound Department of Physics, University of Guelph, Guelph, Ontario, Canada N1G 2W1 email: [email protected] Ian Vega Department of Physics, University of Guelph, Guelph, Ontario, Canada N1G 2W1 email: [email protected] Accepted on 23 August 2011 Published on 29 September 2011 Abstract This review is concerned with the motion of a point scalar charge, a point electric charge, and a point mass in a specified background spacetime. In each of the three cases the particle produces a field that behaves as outgoing radiation in the wave zone, and therefore removes energy from the particle. In the near zone the field acts on the particle and gives rise toa self-force that prevents the particle from moving on a geodesic of the background spacetime. The self-force contains both conservative and dissipative terms, and the latter are responsible for the radiation reaction. The work done by the self-force matches the energy radiated away by the particle. The field’s action on the particle is difficult to calculate because of its singular nature:the field diverges at the position of the particle. But it is possible to isolate the field’s singular part and show that it exerts no force on the particle { its only effect is to contribute to the particle's inertia. -
Geometry Course Outline
GEOMETRY COURSE OUTLINE Content Area Formative Assessment # of Lessons Days G0 INTRO AND CONSTRUCTION 12 G-CO Congruence 12, 13 G1 BASIC DEFINITIONS AND RIGID MOTION Representing and 20 G-CO Congruence 1, 2, 3, 4, 5, 6, 7, 8 Combining Transformations Analyzing Congruency Proofs G2 GEOMETRIC RELATIONSHIPS AND PROPERTIES Evaluating Statements 15 G-CO Congruence 9, 10, 11 About Length and Area G-C Circles 3 Inscribing and Circumscribing Right Triangles G3 SIMILARITY Geometry Problems: 20 G-SRT Similarity, Right Triangles, and Trigonometry 1, 2, 3, Circles and Triangles 4, 5 Proofs of the Pythagorean Theorem M1 GEOMETRIC MODELING 1 Solving Geometry 7 G-MG Modeling with Geometry 1, 2, 3 Problems: Floodlights G4 COORDINATE GEOMETRY Finding Equations of 15 G-GPE Expressing Geometric Properties with Equations 4, 5, Parallel and 6, 7 Perpendicular Lines G5 CIRCLES AND CONICS Equations of Circles 1 15 G-C Circles 1, 2, 5 Equations of Circles 2 G-GPE Expressing Geometric Properties with Equations 1, 2 Sectors of Circles G6 GEOMETRIC MEASUREMENTS AND DIMENSIONS Evaluating Statements 15 G-GMD 1, 3, 4 About Enlargements (2D & 3D) 2D Representations of 3D Objects G7 TRIONOMETRIC RATIOS Calculating Volumes of 15 G-SRT Similarity, Right Triangles, and Trigonometry 6, 7, 8 Compound Objects M2 GEOMETRIC MODELING 2 Modeling: Rolling Cups 10 G-MG Modeling with Geometry 1, 2, 3 TOTAL: 144 HIGH SCHOOL OVERVIEW Algebra 1 Geometry Algebra 2 A0 Introduction G0 Introduction and A0 Introduction Construction A1 Modeling With Functions G1 Basic Definitions and Rigid -
Name: Introduction to Geometry: Points, Lines and Planes Euclid
Name: Introduction to Geometry: Points, Lines and Planes Euclid: “What’s the point of Geometry?- Euclid” http://www.youtube.com/watch?v=_KUGLOiZyK8&safe=active When do historians believe that Euclid completed his work? What key topic did Euclid build on to create all of Geometry? What is an axiom? Why do you think that Euclid’s ideas have lasted for thousands of years? *Point – is a ________ in space. A point has ___ dimension. A point is name by a single capital letter. (ex) *Line – consists of an __________ number of points which extend in ___________ directions. A line is ___-dimensional. A line may be named by using a single lower case letter OR by any two points on the line. (ex) *Plane – consists of an __________ number of points which form a flat surface that extends in all directions. A plane is _____-dimensional. A plane may be named using a single capital letter OR by using three non- collinear points. (ex) *Line segment – consists of ____ points on a line, called the endpoints and all of the points between them. (ex) *Ray – consists of ___ point on a line (called an endpoint or the initial point) and all of the points on one side of the point. (ex) *Opposite rays – share the same initial point and extend in opposite directions on the same line. *Collinear points – (ex) *Coplanar points – (ex) *Two or more geometric figures intersect if they have one or more points in common. The intersection of the figures is the set of elements that the figures have in common. -
Dimension and Dimensional Reduction in Quantum Gravity
May 2017 Dimension and Dimensional Reduction in Quantum Gravity S. Carlip∗ Department of Physics University of California Davis, CA 95616 USA Abstract , A number of very different approaches to quantum gravity contain a common thread, a hint that spacetime at very short distances becomes effectively two dimensional. I review this evidence, starting with a discussion of the physical meaning of “dimension” and concluding with some speculative ideas of what dimensional reduction might mean for physics. arXiv:1705.05417v2 [gr-qc] 29 May 2017 ∗email: [email protected] 1 Why Dimensional Reduction? What is the dimension of spacetime? For most of physics, the answer is straightforward and uncontroversial: we know from everyday experience that we live in a universe with three dimensions of space and one of time. For a condensed matter physicist, say, or an astronomer, this is simply a given. There are a few exceptions—surface states in condensed matter that act two-dimensional, string theory in ten dimensions—but for the most part dimension is simply a fixed, and known, external parameter. Over the past few years, though, hints have emerged from quantum gravity suggesting that the dimension of spacetime is dynamical and scale-dependent, and shrinks to d 2 at very small ∼ distances or high energies. The purpose of this review is to summarize this evidence and to discuss some possible implications for physics. 1.1 Dimensional reduction and quantum gravity As early as 1916, Einstein pointed out that it would probably be necessary to combine the newly formulated general theory of relativity with the emerging ideas of quantum mechanics [1]. -
1 Line Symmetry
Name: ______________________________________________________________ Date: _________________________ Period: _____ Chapter 4: Transformations Topic 10: Symmetry There are three types of symmetry that we will consider. #1 Line Symmetry: A line of symmetry not only cuts a figure in___________________, it creates a mirror image. In order to determine if a figure has line symmetry, a figure needs to be able to be divided into two “mirror image halves.” The line of symmetry is ____________________________from all corresponding pairs of points. Another way to think about line symmetry: If the image is reflected of the line of symmetry, it will return the same image. Examples: In the figures below, sketch all the lines of symmetry (if any): Think critically: Some have one, some have many, some have none! Remember, if the image is reflected over the line you draw, it should be identical to how it started! More Examples: Letters and numbers can also have lines of symmetry! Sketch as many lines of symmetry as you can (if any): Name: ______________________________________________________________ Date: _________________________ Period: _____ #2 Rotational Symmetry: RECALL! The total measure of degrees around a point is: _____________ A rotational symmetry of a figure is a rotation of the plane that maps the figure back to itself such that the rotation is greater than , but less than ___________________. In regular polygons (polygons in which all sides are congruent), the number of rotational symmetries equals the number of sides of the figure. If a polygon is not regular, it may have fewer rotational symmetries. We can consider increments of like coordinate plane rotations. Examples of Regular Polygons: A rotation of will always map a figure back onto itself. -
Points, Lines, and Planes a Point Is a Position in Space. a Point Has No Length Or Width Or Thickness
Points, Lines, and Planes A Point is a position in space. A point has no length or width or thickness. A point in geometry is represented by a dot. To name a point, we usually use a (capital) letter. A A (straight) line has length but no width or thickness. A line is understood to extend indefinitely to both sides. It does not have a beginning or end. A B C D A line consists of infinitely many points. The four points A, B, C, D are all on the same line. Postulate: Two points determine a line. We name a line by using any two points on the line, so the above line can be named as any of the following: ! ! ! ! ! AB BC AC AD CD Any three or more points that are on the same line are called colinear points. In the above, points A; B; C; D are all colinear. A Ray is part of a line that has a beginning point, and extends indefinitely to one direction. A B C D A ray is named by using its beginning point with another point it contains. −! −! −−! −−! In the above, ray AB is the same ray as AC or AD. But ray BD is not the same −−! ray as AD. A (line) segment is a finite part of a line between two points, called its end points. A segment has a finite length. A B C D B C In the above, segment AD is not the same as segment BC Segment Addition Postulate: In a line segment, if points A; B; C are colinear and point B is between point A and point C, then AB + BC = AC You may look at the plus sign, +, as adding the length of the segments as numbers. -
The Small Inductive Dimension Can Be Raised by the Adjunction of a Single Point
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector MATHEMATICS THE SMALL INDUCTIVE DIMENSION CAN BE RAISED BY THE ADJUNCTION OF A SINGLE POINT BY ERIC K. VAN DOUWEN (Communicated by Prof. R. TIBXMAN at the meeting of June 30, 1973) Unless otherwise indicated, all spaces are metrizable. 1. INTRODUCTION In this note we construct a metrizable space P containing a point p such that ind P\@} = 0 and yet ind P = 1, thus showing that the small inductive dimension of a metrizable space can be raised by the adjunction of a single point. (Without the metrizsbility condition such on example has already been constructed by Dowker [3, p. 258, exercise Cl). So the behavior of the small inductive dimension is in contrast with that of the dimension functions Ind and dim, which, as is well known, cannot be raised by the edjunction of a single point [3, p. 274, exercises B and C]. The construction of our example stems from the proof of the equality of ind and Ind on the class of separable metrizable spaces, as given in [lo, p. 1781. We make use of the fact that there exists a metrizable space D with ind D = 0 and Ind D = 1 [ 141. This space is completely metrizable. Hence our main example, and the two related examples, also are com- pletely metrizable. Following [7, p, 208, example 3.3.31 the example is used to show that there is no finite closed sum theorem for the small inductive dimension, even on the class of metrizable spaces. -
Metrizable Spaces Where the Inductive Dimensions Disagree
TRANSACTIONS OF THE AMERICAN MATHEMATICALSOCIETY Volume 318, Number 2, April 1990 METRIZABLE SPACES WHERE THE INDUCTIVE DIMENSIONS DISAGREE JOHN KULESZA Abstract. A method for constructing zero-dimensional metrizable spaces is given. Using generalizations of Roy's technique, these spaces can often be shown to have positive large inductive dimension. Examples of N-compact, complete metrizable spaces with ind = 0 and Ind = 1 are provided, answering questions of Mrowka and Roy. An example with weight c and positive Ind such that subspaces with smaller weight have Ind = 0 is produced in ZFC. Assuming an additional axiom, for each cardinal X a space of positive Ind with all subspaces with weight less than A strongly zero-dimensional is constructed. I. Introduction There are three classical notions of dimension for a normal topological space X. The small inductive dimension, denoted ind(^T), is defined by ind(X) = -1 if X is empty, ind(X) < n if each p G X has a neighborhood base of open sets whose boundaries have ind < n - 1, and ind(X) = n if ind(X) < n and n is as small as possible. The large inductive dimension, denoted Ind(Jf), is defined by Ind(X) = -1 if X is empty, \t\c\(X) < n if each closed C c X has a neighborhood base of open sets whose boundaries have Ind < n — 1, and Ind(X) = n if Ind(X) < n and n is as small as possible. The covering dimension, denoted by dim(X), is defined by dim(A') = -1 if X is empty, dim(A') < n if for each finite open cover of X there is a finite open refining cover such that no point is in more than n + 1 of the sets in the refining cover and, dim(A") = n if dim(X) < n and n is as small as possible. -
Quasiorthogonal Dimension
Quasiorthogonal Dimension Paul C. Kainen1 and VˇeraK˚urkov´a2 1 Department of Mathematics and Statistics, Georgetown University Washington, DC, USA 20057 [email protected] 2 Institute of Computer Science, Czech Academy of Sciences Pod Vod´arenskou vˇeˇz´ı 2, 18207 Prague, Czech Republic [email protected] Abstract. An interval approach to the concept of dimension is pre- sented. The concept of quasiorthogonal dimension is obtained by relax- ing exact orthogonality so that angular distances between unit vectors are constrained to a fixed closed symmetric interval about π=2. An ex- ponential number of such quasiorthogonal vectors exist as the Euclidean dimension increases. Lower bounds on quasiorthogonal dimension are proven using geometry of high-dimensional spaces and a separate argu- ment is given utilizing graph theory. Related notions are reviewed. 1 Introduction The intuitive concept of dimension has many mathematical formalizations. One version, based on geometry, uses \right angles" (in Greek, \ortho gonia"), known since Pythagoras. The minimal number of orthogonal vectors needed to specify an object in a Euclidean space defines its orthogonal dimension. Other formalizations of dimension are based on different aspects of space. For example, a topological definition (inductive dimension) emphasizing the recur- sive character of d-dimensional objects having d−1-dimensional boundaries, was proposed by Poincar´e,while another topological notion, covering dimension, is associated with Lebesgue. A metric-space version of dimension was developed -
Lecture 2: Analyzing Algorithms: the 2-D Maxima Problem
Lecture Notes CMSC 251 Lecture 2: Analyzing Algorithms: The 2-d Maxima Problem (Thursday, Jan 29, 1998) Read: Chapter 1 in CLR. Analyzing Algorithms: In order to design good algorithms, we must first agree the criteria for measuring algorithms. The emphasis in this course will be on the design of efficient algorithm, and hence we will measure algorithms in terms of the amount of computational resources that the algorithm requires. These resources include mostly running time and memory. Depending on the application, there may be other elements that are taken into account, such as the number disk accesses in a database program or the communication bandwidth in a networking application. In practice there are many issues that need to be considered in the design algorithms. These include issues such as the ease of debugging and maintaining the final software through its life-cycle. Also, one of the luxuries we will have in this course is to be able to assume that we are given a clean, fully- specified mathematical description of the computational problem. In practice, this is often not the case, and the algorithm must be designed subject to only partial knowledge of the final specifications. Thus, in practice it is often necessary to design algorithms that are simple, and easily modified if problem parameters and specifications are slightly modified. Fortunately, most of the algorithms that we will discuss in this class are quite simple, and are easy to modify subject to small problem variations. Model of Computation: Another goal that we will have in this course is that our analyses be as independent as possible of the variations in machine, operating system, compiler, or programming language.