3.3.1 Isometry
... transformation is an isometry since for any two distinct points A and B, d(I(A),I(B)) = d(A,B). Hence, we need only prove closure and the inverse property for the set of isometries under composition. Let f and g be isometries of a plane. Let A and B be two distinct points. Denote g(P) = P' and (f o ...
... transformation is an isometry since for any two distinct points A and B, d(I(A),I(B)) = d(A,B). Hence, we need only prove closure and the inverse property for the set of isometries under composition. Let f and g be isometries of a plane. Let A and B be two distinct points. Denote g(P) = P' and (f o ...
Normed vector space
... In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the "length" of a vector is intuitive and can easily be extended to any real vector space Rn. The following properties of "vector length" are crucial. 1. The zero vector, 0, has zero length; every other vector has ...
... In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the "length" of a vector is intuitive and can easily be extended to any real vector space Rn. The following properties of "vector length" are crucial. 1. The zero vector, 0, has zero length; every other vector has ...
Asymptotic cones - American Institute of Mathematics
... We had many questions about the expressive power of continuous logic in this setting. What language should we work with? What can actually be said in the theory of such a metric structure? (We will abbreviate continuous logic by cclogic.) We will work in an arbitrary complete metric space (Y, d) wit ...
... We had many questions about the expressive power of continuous logic in this setting. What language should we work with? What can actually be said in the theory of such a metric structure? (We will abbreviate continuous logic by cclogic.) We will work in an arbitrary complete metric space (Y, d) wit ...
2.1
... maximum degree k. Then any polynomial of degree larger than k can not be written as a linear combination. ...
... maximum degree k. Then any polynomial of degree larger than k can not be written as a linear combination. ...
Quasi isometries of hyperbolic space are almost isometries
... An example of how quasi-isometries arise in topology is obtained by lifting a homotopy equivalence between compact Riemannian manifolds to the universal covers. This is the starting point for Mostow's proof of the rigidity theorem for hyperbolic manifolds [Mos2]. This theorem asserts the uniqueness, ...
... An example of how quasi-isometries arise in topology is obtained by lifting a homotopy equivalence between compact Riemannian manifolds to the universal covers. This is the starting point for Mostow's proof of the rigidity theorem for hyperbolic manifolds [Mos2]. This theorem asserts the uniqueness, ...
1 Introduction 2 Compact group actions
... – Find an equivariant simplicial map h : S 3 → S 3 which is homotopically trivial. – Build the infinite mapping cylinder which is contactible and imbed it in an Euclidean space of high-dimensions where Zpq acts orthogonally. – Find the contractible neighborhood. Taking the product with the real line ...
... – Find an equivariant simplicial map h : S 3 → S 3 which is homotopically trivial. – Build the infinite mapping cylinder which is contactible and imbed it in an Euclidean space of high-dimensions where Zpq acts orthogonally. – Find the contractible neighborhood. Taking the product with the real line ...
Continuity in topological spaces and topological invariance
... Definition 9 (Everywhere continuous functions). Let (X, τ ), (Y, υ) be topological spaces and f : X → Y be a map between them. Then f is everywhere continuous on X if and only if ∀U ∈ υ, f −1 (U ) ∈ τ . Theorem 4. f is everywhere continuous on X if and only if f is continuous at every point x ∈ X. ...
... Definition 9 (Everywhere continuous functions). Let (X, τ ), (Y, υ) be topological spaces and f : X → Y be a map between them. Then f is everywhere continuous on X if and only if ∀U ∈ υ, f −1 (U ) ∈ τ . Theorem 4. f is everywhere continuous on X if and only if f is continuous at every point x ∈ X. ...
16. Subspaces and Spanning Sets Subspaces
... µu1 + νu2 ∈ U for all u1 , u2 ∈ U, µ, ν ∈ R implies that U is a subspace of V . (In other words, check all the vector space requirements for U .) 2. Let P3 [x] be the vector space of degree 3 polynomials in the variable x. Check whether x − x3 ∈ span{x2 , 2x + x2 , x + x3 } ...
... µu1 + νu2 ∈ U for all u1 , u2 ∈ U, µ, ν ∈ R implies that U is a subspace of V . (In other words, check all the vector space requirements for U .) 2. Let P3 [x] be the vector space of degree 3 polynomials in the variable x. Check whether x − x3 ∈ span{x2 , 2x + x2 , x + x3 } ...
Lecture 2
... 3. The only Hausdorff topology on a finite set is the discrete topology. In fact, since X is finite, any subset S of X is finite and so S is a finite union of singletons. But since X is also Hausdorff, the previous proposition implies that any singleton is closed. Hence, any subset S of X is closed an ...
... 3. The only Hausdorff topology on a finite set is the discrete topology. In fact, since X is finite, any subset S of X is finite and so S is a finite union of singletons. But since X is also Hausdorff, the previous proposition implies that any singleton is closed. Hence, any subset S of X is closed an ...
Valuations and discrete valuation rings, PID`s
... u ∈ R∗, say that a and b are associate. This defines an equivalence relation on R. Note: A non-zero prime element r is irreducible. This is because r = ab and Rr prime implies a ∈ Rr or b ∈ Rr. If a = ur for some u ∈ R, then a = ur = uab, so a(1 − ub) = 0. But a 6= 0 since r 6= 0, so 1 − ub = 0 and ...
... u ∈ R∗, say that a and b are associate. This defines an equivalence relation on R. Note: A non-zero prime element r is irreducible. This is because r = ab and Rr prime implies a ∈ Rr or b ∈ Rr. If a = ur for some u ∈ R, then a = ur = uab, so a(1 − ub) = 0. But a 6= 0 since r 6= 0, so 1 − ub = 0 and ...
Chapter 2: Vector spaces
... • Theorem 5. If W is a subspace of V, every linearly independent subset of W is finite and is a part of a basis of W. • W a subspace of V. dim W dim V. • A set of linearly independent vectors can be extended to a basis. • A nxn-matrix. Rows (respectively columns) of A are independent iff A is inv ...
... • Theorem 5. If W is a subspace of V, every linearly independent subset of W is finite and is a part of a basis of W. • W a subspace of V. dim W dim V. • A set of linearly independent vectors can be extended to a basis. • A nxn-matrix. Rows (respectively columns) of A are independent iff A is inv ...
Angles and a Classification of Normed Spaces
... At the end we consider products. For two spaces (A, k · kA ), (B, k · kB ) ∈ pdBW we take its Cartesian product A × B, and we get a set of balanced weights k · kp on A × B, for p > 0. ...
... At the end we consider products. For two spaces (A, k · kA ), (B, k · kB ) ∈ pdBW we take its Cartesian product A × B, and we get a set of balanced weights k · kp on A × B, for p > 0. ...
Point-Set Topology
... the interior is the largest open set contained in the subset. Similarly, one may talk about a closure operation as the smallest closed set containing the subset. Finally, another method to create a topology is to define neighborhoods of points p ∈ X. 1.1. Examples. • Our first example is the discret ...
... the interior is the largest open set contained in the subset. Similarly, one may talk about a closure operation as the smallest closed set containing the subset. Finally, another method to create a topology is to define neighborhoods of points p ∈ X. 1.1. Examples. • Our first example is the discret ...
Math 396. Gluing topologies, the Hausdorff condition, and examples
... example: Grassmannians. We begin with the algebraic development over a general field, and then we specialize to the case F = R where we can bring in some topology. (Everything we do over R works the same way over C, once one sets up the right topological theory over C.) Let F be a field and V a vect ...
... example: Grassmannians. We begin with the algebraic development over a general field, and then we specialize to the case F = R where we can bring in some topology. (Everything we do over R works the same way over C, once one sets up the right topological theory over C.) Let F be a field and V a vect ...
http://www.ms.uky.edu/~droyster/courses/spring04/classnotes/Chapter%2009.pdf
... The fractional linear transformation, T , is usually represented by a 2 × 2 matrix ...
... The fractional linear transformation, T , is usually represented by a 2 × 2 matrix ...
Introduction to Functions, Sequences, Metric and Topological
... given a metrizable topological space (X; ); one can always …nd many metrics d on X such that d = : Two metrics generating the same topology are equivalent. The Euclidean, l1 , and sup metrics on
... given a metrizable topological space (X; ); one can always …nd many metrics d on X such that d = : Two metrics generating the same topology are equivalent. The Euclidean, l1 , and sup metrics on
Math 487 Exam 2 - Practice Problems 1. Short Answer/Essay
... (g) Lines have infinite length. This statement is true in the Euclidean Plane, the Taxicab Plane, the Max-Distance Plane, the Missing Strip Plane, and the Poincare Half-Plane model. This statement is false in the Riemann Sphere, the Modified Riemann Sphere, and in Discrete Geometries. (h) the sum of ...
... (g) Lines have infinite length. This statement is true in the Euclidean Plane, the Taxicab Plane, the Max-Distance Plane, the Missing Strip Plane, and the Poincare Half-Plane model. This statement is false in the Riemann Sphere, the Modified Riemann Sphere, and in Discrete Geometries. (h) the sum of ...
X → Y must be constant. .... Let T
... the relation between the two topologies (are they comparable, and if yes, which is finer and which coarser)? Any two metrics on the same space can be majorized by a third metric. But it is not true that they can be minorized. [Hint: The sum majorizes. For a counter example take the reals with d1 the ...
... the relation between the two topologies (are they comparable, and if yes, which is finer and which coarser)? Any two metrics on the same space can be majorized by a third metric. But it is not true that they can be minorized. [Hint: The sum majorizes. For a counter example take the reals with d1 the ...
Taxicab Angles and Trigonometry - Department of Physics | Oregon
... a commonly used method for estimating the distance to a nearby object. The method of stellar parallax was used extensively to find the distances to nearby stars in the 19th and early 20th centuries. We now wish to explore the method and results of parallax in taxicab geometry and examine how these d ...
... a commonly used method for estimating the distance to a nearby object. The method of stellar parallax was used extensively to find the distances to nearby stars in the 19th and early 20th centuries. We now wish to explore the method and results of parallax in taxicab geometry and examine how these d ...
The Space of Metric Spaces
... A metric space (X, d) is a Length Space if d(x, y) = inf {L(γ) | γ(a) = x, γ(b) = y} . In particular, X is path connected. A curve γ in a length space is called Length minimizing or a Geodesic of L(γ) = d(γ(a), γ(b)). A Lengths space X is called Strictly intrinsic or Geodesic if for all pairs of poi ...
... A metric space (X, d) is a Length Space if d(x, y) = inf {L(γ) | γ(a) = x, γ(b) = y} . In particular, X is path connected. A curve γ in a length space is called Length minimizing or a Geodesic of L(γ) = d(γ(a), γ(b)). A Lengths space X is called Strictly intrinsic or Geodesic if for all pairs of poi ...
Functional Analysis
... and PY : X × Y → Y be the projections on the respective components. d) Prove that the product topology on X × Y is the F-initial topology with F = {PX , PY }. Hint: You have to generalize the above definition of the F-initial topology to different target spaces of the functions in F in the obvious w ...
... and PY : X × Y → Y be the projections on the respective components. d) Prove that the product topology on X × Y is the F-initial topology with F = {PX , PY }. Hint: You have to generalize the above definition of the F-initial topology to different target spaces of the functions in F in the obvious w ...
Chapter 4 (version 3)
... meanings in mathematics. It may mean any change in an equation or expression to simplify an operation such as computing a derivative or an integral. Another meaning expresses a functional relationship because the notion of a function is often introduced in terms of a mapping f:A → B between sets A a ...
... meanings in mathematics. It may mean any change in an equation or expression to simplify an operation such as computing a derivative or an integral. Another meaning expresses a functional relationship because the notion of a function is often introduced in terms of a mapping f:A → B between sets A a ...
Euclidean space
In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces. It is named after the Ancient Greek mathematician Euclid of Alexandria. The term ""Euclidean"" distinguishes these spaces from other types of spaces considered in modern geometry. Euclidean spaces also generalize to higher dimensions.Classical Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates, while the other properties of these spaces were deduced as theorems. Geometric constructions are also used to define rational numbers. When algebra and mathematical analysis became developed enough, this relation reversed and now it is more common to define Euclidean space using Cartesian coordinates and the ideas of analytic geometry. It means that points of the space are specified with collections of real numbers, and geometric shapes are defined as equations and inequalities. This approach brings the tools of algebra and calculus to bear on questions of geometry and has the advantage that it generalizes easily to Euclidean spaces of more than three dimensions.From the modern viewpoint, there is essentially only one Euclidean space of each dimension. With Cartesian coordinates it is modelled by the real coordinate space (Rn) of the same dimension. In one dimension, this is the real line; in two dimensions, it is the Cartesian plane; and in higher dimensions it is a coordinate space with three or more real number coordinates. Mathematicians denote the n-dimensional Euclidean space by En if they wish to emphasize its Euclidean nature, but Rn is used as well since the latter is assumed to have the standard Euclidean structure, and these two structures are not always distinguished. Euclidean spaces have finite dimension.