Common fixed point of mappings satisfying implicit contractive
... Huang and Zhang [14] reintroduced such spaces under the name of cone metric spaces and reintroduced definition of convergent and Cauchy sequences in the terms of interior points of the underlying cone. They also proved some fixed point results in framework of cone metric spaces. Subsequently, severa ...
... Huang and Zhang [14] reintroduced such spaces under the name of cone metric spaces and reintroduced definition of convergent and Cauchy sequences in the terms of interior points of the underlying cone. They also proved some fixed point results in framework of cone metric spaces. Subsequently, severa ...
Proximity Searching and the Quest for the Holy Grail
... A. Gionis, P. Indyk, R. Motwani. Similarity search in high dimensions via ...
... A. Gionis, P. Indyk, R. Motwani. Similarity search in high dimensions via ...
Trigonometric functions and Fourier series (Part 1)
... coset space of its period group. That’s because by its very definition, the period group is the group such that the function is constant on every coset. Since R is an Abelian group, the coset space is actually a group. Thus, the study of periodic continuous functions on R is equivalent to the study ...
... coset space of its period group. That’s because by its very definition, the period group is the group such that the function is constant on every coset. Since R is an Abelian group, the coset space is actually a group. Thus, the study of periodic continuous functions on R is equivalent to the study ...
Section 13.1 Vectors in the Plane
... 13.1 Vectors in the Plane Imagine a raft drifting down a river, carried by the current. The speed and direction of the raft at a point may be represented by an arrow (Figure 13.1). The length of the arrow represents the speed of the raft at that point; longer arrows correspond to greater speeds. The ...
... 13.1 Vectors in the Plane Imagine a raft drifting down a river, carried by the current. The speed and direction of the raft at a point may be represented by an arrow (Figure 13.1). The length of the arrow represents the speed of the raft at that point; longer arrows correspond to greater speeds. The ...
topology : notes and problems
... Let (X, Ω) be a topological space and let Y be a subset of X. Then ΩY := {U ∩ Y : U ∈ Ω} is a topology on Y, called the subspace topology. Remark 6.1 : If U ∈ Ω and U ⊆ Y then U is open in the subspace topology. Remark 6.2 : If V ∈ ΩY and Y ∈ Ω then V ∈ Ω. Example 6.3 : Consider the real numbers R w ...
... Let (X, Ω) be a topological space and let Y be a subset of X. Then ΩY := {U ∩ Y : U ∈ Ω} is a topology on Y, called the subspace topology. Remark 6.1 : If U ∈ Ω and U ⊆ Y then U is open in the subspace topology. Remark 6.2 : If V ∈ ΩY and Y ∈ Ω then V ∈ Ω. Example 6.3 : Consider the real numbers R w ...
Weakly Perfect Generalized Ordered Spaces
... Remark. In the light of Example 2.1, it is unlikely that there is an interesting characterization of paracompactness in weakly perfect GO-spaces that goes beyond repeating the familiar characterization of paracompactness in arbitrary ordered spaces, namely that the space in question does not contain ...
... Remark. In the light of Example 2.1, it is unlikely that there is an interesting characterization of paracompactness in weakly perfect GO-spaces that goes beyond repeating the familiar characterization of paracompactness in arbitrary ordered spaces, namely that the space in question does not contain ...
8-1 - Cloudfront.net
... It is given that AC DC and that AB DB. By the Reflexive Property of Congruence, BC BC. Therefore ∆ABC ∆DBC by SSS. GEOMETRY ...
... It is given that AC DC and that AB DB. By the Reflexive Property of Congruence, BC BC. Therefore ∆ABC ∆DBC by SSS. GEOMETRY ...
x - Cinvestav
... Fundamental Theorem of Arithmetic • Proof: existence [sketch] Suppose there exist positive integers that are not product of primes. Let n be the smallest such integer. Then n cannot be 1 or a prime, so n must be composite. Therefore n = ab with 1 < a, b < n. Since n is the smallest positive integer ...
... Fundamental Theorem of Arithmetic • Proof: existence [sketch] Suppose there exist positive integers that are not product of primes. Let n be the smallest such integer. Then n cannot be 1 or a prime, so n must be composite. Therefore n = ab with 1 < a, b < n. Since n is the smallest positive integer ...
Introduction to Point-Set Topology
... Let Fn2 be the set of all “words” of length n, where every “letter” must be either a “0” or a “1”. For instance, (0, 1, 1, 1, 0, 1) and (1, 1, 0, 0, 1, 1) are elements of F62 . Then Fn2 is a metric space under the metric d(x, y ) = the number of entries in which the words x and y differ. Proof: Let ...
... Let Fn2 be the set of all “words” of length n, where every “letter” must be either a “0” or a “1”. For instance, (0, 1, 1, 1, 0, 1) and (1, 1, 0, 0, 1, 1) are elements of F62 . Then Fn2 is a metric space under the metric d(x, y ) = the number of entries in which the words x and y differ. Proof: Let ...
1 Weakly Perfect Generalized Ordered Spaces by Harold R Bennett
... of paracompactness in weakly perfect GO-spaces that goes beyond repeating the familiar characterization of paracompactness in arbitrary ordered spaces, namely that the space in question does not contain a closed subspace that is a topological copy of a stationary subset of an uncountable regular car ...
... of paracompactness in weakly perfect GO-spaces that goes beyond repeating the familiar characterization of paracompactness in arbitrary ordered spaces, namely that the space in question does not contain a closed subspace that is a topological copy of a stationary subset of an uncountable regular car ...
derived smooth manifolds
... Theorem 2.6). We did not include that as an axiom here, however, because it does not seem to us to be an inherently necessary aspect of a good intersection theory. The category of smooth manifolds does not have the general cup product formula because it does not satisfy Condition (2). Indeed, suppos ...
... Theorem 2.6). We did not include that as an axiom here, however, because it does not seem to us to be an inherently necessary aspect of a good intersection theory. The category of smooth manifolds does not have the general cup product formula because it does not satisfy Condition (2). Indeed, suppos ...
Non-archimedean analytic spaces
... the set AnA = M(A[T1 , . . . , Tn ]) of all real semivaluations on A[T ] that are bounded on A. Topology on X = AnA is defined by functions |f | : X → R+ , f ∈ A[T ]. X is provided with the structure sheaf OFX of analytic functions: f ∈ OX (U) is a map f : U → x∈U H(x) which locally at each x ∈ U is ...
... the set AnA = M(A[T1 , . . . , Tn ]) of all real semivaluations on A[T ] that are bounded on A. Topology on X = AnA is defined by functions |f | : X → R+ , f ∈ A[T ]. X is provided with the structure sheaf OFX of analytic functions: f ∈ OX (U) is a map f : U → x∈U H(x) which locally at each x ∈ U is ...
Mathematics 205A Topology — I Course Notes Revised, Fall 2005
... is the inclusion mapping sending an element of the unit interval into itself. One particular context in which it is necessary to make such distinctions is the construction of the fundamental group as in Chapter 9 of Munkres. We should also note that the need to specify codomains is important in cert ...
... is the inclusion mapping sending an element of the unit interval into itself. One particular context in which it is necessary to make such distinctions is the construction of the fundamental group as in Chapter 9 of Munkres. We should also note that the need to specify codomains is important in cert ...
Vector Algebra
... It' there is no such set of scalars that the linear combination of vectors with scalar multiplications yield a zero vector then the vectors a, b and c are linearly independent. In such a case the linear combination r will be a zero vector only if all the scalars x, y, z are zero. Xl. Position Vector ...
... It' there is no such set of scalars that the linear combination of vectors with scalar multiplications yield a zero vector then the vectors a, b and c are linearly independent. In such a case the linear combination r will be a zero vector only if all the scalars x, y, z are zero. Xl. Position Vector ...
COMPLETION FUNCTORS FOR CAUCHY SPACES
... denotes the filter generated on X by- (considered as a filter base on X). If (X, C) is a complete Cauchy space (i.e. convergence space), then it will be necessary to distinguish between a convergence subspace (a subspace in the usual convergence space sense) and a Cauchy subspace (with the meaning d ...
... denotes the filter generated on X by- (considered as a filter base on X). If (X, C) is a complete Cauchy space (i.e. convergence space), then it will be necessary to distinguish between a convergence subspace (a subspace in the usual convergence space sense) and a Cauchy subspace (with the meaning d ...
Some applications of the ultrafilter topology on spaces of valuation
... finite subsets of K. When no confusion can arise, we will simply denote by Bx the basic open set B{x} of Z. This topology is now called the Zariski topology on Z and the set Z, equipped with this topology, denoted also by Z zar , is usually called the (abstract) Zariski-Riemann surface of K over A. ...
... finite subsets of K. When no confusion can arise, we will simply denote by Bx the basic open set B{x} of Z. This topology is now called the Zariski topology on Z and the set Z, equipped with this topology, denoted also by Z zar , is usually called the (abstract) Zariski-Riemann surface of K over A. ...
Lectures on Lie groups and geometry
... Theorem 1 Given a finite dimensional real Lie algebra g there is a Lie group G = Gg with Lie algebra g and the universal property that for any Lie group H with Lie algebra h and Lie algebra homomorphism ρ : g → h there is a unique group homomorphism G → H with derivative ρ. We will discuss the proof ...
... Theorem 1 Given a finite dimensional real Lie algebra g there is a Lie group G = Gg with Lie algebra g and the universal property that for any Lie group H with Lie algebra h and Lie algebra homomorphism ρ : g → h there is a unique group homomorphism G → H with derivative ρ. We will discuss the proof ...
LECTURE NOTES IN TOPOLOGICAL GROUPS 1. Lecture 1
... (10) For every Banach space (V, ||·||) the group Isolin (V ) of all linear onto isometries V → V endowed with the pointwise topology inherited from V V . For example, if V := Rn is the Euclidean space then Isolin (V ) = On (R) the orthogonal group. Note that, in contrast to the case of Rn , for infi ...
... (10) For every Banach space (V, ||·||) the group Isolin (V ) of all linear onto isometries V → V endowed with the pointwise topology inherited from V V . For example, if V := Rn is the Euclidean space then Isolin (V ) = On (R) the orthogonal group. Note that, in contrast to the case of Rn , for infi ...
Smooth Manifolds
... The empty set satisfies the definition of a topological n-manifold for every n. For the most part, we will ignore this special case (sometimes without remembering to say so). But because it is useful in certain contexts to allow the empty manifold, we choose not to exclude it from the definition. Th ...
... The empty set satisfies the definition of a topological n-manifold for every n. For the most part, we will ignore this special case (sometimes without remembering to say so). But because it is useful in certain contexts to allow the empty manifold, we choose not to exclude it from the definition. Th ...
Projective Geometry on Manifolds - UMD MATH
... a group G of transformations of X. For example Euclidean geometry is the geometry of n-dimensional Euclidean space Rn invariant under its group of rigid motions. This is the group of transformations which tranforms an object ξ into an object congruent to ξ. In Euclidean geometry can speak of points, ...
... a group G of transformations of X. For example Euclidean geometry is the geometry of n-dimensional Euclidean space Rn invariant under its group of rigid motions. This is the group of transformations which tranforms an object ξ into an object congruent to ξ. In Euclidean geometry can speak of points, ...
Lecture 1: Introduction to bordism Overview Bordism is a notion
... (1.1) Convention. All manifolds in this course are smooth, or smooth manifolds with boundary or corners, so we omit the modifier ‘smooth’ from now on. In bordism theory the manifolds are almost always compact, though we retain that modifier to be clear. ...
... (1.1) Convention. All manifolds in this course are smooth, or smooth manifolds with boundary or corners, so we omit the modifier ‘smooth’ from now on. In bordism theory the manifolds are almost always compact, though we retain that modifier to be clear. ...
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.