Non-expansive mappings in convex linear topological spaces
... so t h a t Q0(f(x)) < Q0(X) for x € K0. Moreover, Q0 is lower-semi continuous non-negative convex on K0. That Q0 is, indeed, normal follows from the minimality of K0\ for if C c: Ar0 is closed convex and non-trivial such t h a t Q0(X) = k = constant on C, then actually C = {x e K0\Q(X) < k} and is a ...
... so t h a t Q0(f(x)) < Q0(X) for x € K0. Moreover, Q0 is lower-semi continuous non-negative convex on K0. That Q0 is, indeed, normal follows from the minimality of K0\ for if C c: Ar0 is closed convex and non-trivial such t h a t Q0(X) = k = constant on C, then actually C = {x e K0\Q(X) < k} and is a ...
Definition 1 An AS1 system is a set, say S, with an operation S ! S
... Problem 24 Show that if G = fe; a; a2 ; :::; a5 g is a cyclic AS1 system then H = fe; a2 ; a4 g is a cyclic subsystem of G. What is a generator for H? Problem 25 What are the cosets for G and H as in the last problem? Now lets set up some more things. Suppose that f : G ! S is a morphism from some ...
... Problem 24 Show that if G = fe; a; a2 ; :::; a5 g is a cyclic AS1 system then H = fe; a2 ; a4 g is a cyclic subsystem of G. What is a generator for H? Problem 25 What are the cosets for G and H as in the last problem? Now lets set up some more things. Suppose that f : G ! S is a morphism from some ...
Chapter 9 - U.I.U.C. Math
... Let M be a semisimple R-module, and let A be the endomorphism ring EndR (M ). [Note that M is an A-module; if g ∈ A we take g • x = g(x), x ∈ M .] If m ∈ M and f ∈ EndA (M ), then there exists r ∈ R such that f (m) = rm. Before proving the lemma, let’s look more carefully at EndA (M ). Suppose that ...
... Let M be a semisimple R-module, and let A be the endomorphism ring EndR (M ). [Note that M is an A-module; if g ∈ A we take g • x = g(x), x ∈ M .] If m ∈ M and f ∈ EndA (M ), then there exists r ∈ R such that f (m) = rm. Before proving the lemma, let’s look more carefully at EndA (M ). Suppose that ...
Functional Analysis: Lecture notes based on Folland
... Next we consider linear maps from X to Y. A linear map T : X → Y is said to be bounded if there exists C < ∞ such that kT xk ≤ Ckxk. Here, of course, the norm notation refers to the norm of X for objects in X and to the norm of Y for objects in Y. Note that this notion of boundedness is not the same ...
... Next we consider linear maps from X to Y. A linear map T : X → Y is said to be bounded if there exists C < ∞ such that kT xk ≤ Ckxk. Here, of course, the norm notation refers to the norm of X for objects in X and to the norm of Y for objects in Y. Note that this notion of boundedness is not the same ...
MATH10212 Linear Algebra Lecture Notes Textbook
... equation a true identity when the values s1 , . . . , sn are substituted for x1 , . . . , xn , respectively. For example, in the system above (2, 1) is a solution. The set of all possible solutions is called the solution set of the linear system. Two linear systems are equivalent if the have the sam ...
... equation a true identity when the values s1 , . . . , sn are substituted for x1 , . . . , xn , respectively. For example, in the system above (2, 1) is a solution. The set of all possible solutions is called the solution set of the linear system. Two linear systems are equivalent if the have the sam ...
Convex Sets and Convex Functions on Complete Manifolds
... constant on any open set of M. Moreover the function F: M^R defined to be F(x) = sup[Fy(x); y(0)=p] is convex and exhaustion, where the sup is taken over all rays emanating from p. Gromoll-Meyer proved that a noncompact M is diffeomorphic to Rn if Ä>0. Then Cheeger and Gromoll proved that there exis ...
... constant on any open set of M. Moreover the function F: M^R defined to be F(x) = sup[Fy(x); y(0)=p] is convex and exhaustion, where the sup is taken over all rays emanating from p. Gromoll-Meyer proved that a noncompact M is diffeomorphic to Rn if Ä>0. Then Cheeger and Gromoll proved that there exis ...
INTRODUCTION TO LIE GROUPS Contents Introduction - D-MATH
... second condition. This immediately implies that Iso(X) is compact if X is compact. It can be shown that Iso(X) is always locally compact. The reader may provide such a proof for proper metric spaces. As an example, recall that Iso(S n ) = O(n, R) is compact. The topological group O(p, q) is locally ...
... second condition. This immediately implies that Iso(X) is compact if X is compact. It can be shown that Iso(X) is always locally compact. The reader may provide such a proof for proper metric spaces. As an example, recall that Iso(S n ) = O(n, R) is compact. The topological group O(p, q) is locally ...
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... a finite gauge function ∥ · ∥, called norm, satisfying ∥x∥ = 0 if and only if x = 0, ∥αx∥ = |α|∥x∥ for each scalar α and ∥x + y∥ ≤ ∥x∥ + ∥y∥, x, y ∈ X and which is complete with respect to the convergence defined by the norm (a space is complete if it contains limits of all Cauchy sequences). Example ...
... a finite gauge function ∥ · ∥, called norm, satisfying ∥x∥ = 0 if and only if x = 0, ∥αx∥ = |α|∥x∥ for each scalar α and ∥x + y∥ ≤ ∥x∥ + ∥y∥, x, y ∈ X and which is complete with respect to the convergence defined by the norm (a space is complete if it contains limits of all Cauchy sequences). Example ...
Convex optimization
... Let ν1 , ν2 ∈ V and α, β ∈ R we want to show that αν1 + βν2 ∈ V which is equivalent to showing αν1 + βν2 + x0 ∈ C . The left hand side of the above display is equal to α(ν1 + x0 ) + β(ν2 + x0 ) + (1 − α − β)x0 , which is an affine combination of the points ν1 + x0 , ν2 + x0 , and x0 all of which bel ...
... Let ν1 , ν2 ∈ V and α, β ∈ R we want to show that αν1 + βν2 ∈ V which is equivalent to showing αν1 + βν2 + x0 ∈ C . The left hand side of the above display is equal to α(ν1 + x0 ) + β(ν2 + x0 ) + (1 − α − β)x0 , which is an affine combination of the points ν1 + x0 , ν2 + x0 , and x0 all of which bel ...
Slide 1
... circle of radius r is 2r. Therefore, an angle representing one complete clockwise rotation measures 2 radians. You can use the fact that 2 radians is equivalent to 360° to convert between radians and degrees. ...
... circle of radius r is 2r. Therefore, an angle representing one complete clockwise rotation measures 2 radians. You can use the fact that 2 radians is equivalent to 360° to convert between radians and degrees. ...
ISOMETRIES BETWEEN OPEN SETS OF CARNOT GROUPS AND
... In this paper we give a complete description of the space of local isometries for those homogeneous spaces that also admit dilations. These spaces, called Carnot groups, are particular nilpotent groups equipped with general left-invariant geodesic distances. Our method of proof also shows that, as i ...
... In this paper we give a complete description of the space of local isometries for those homogeneous spaces that also admit dilations. These spaces, called Carnot groups, are particular nilpotent groups equipped with general left-invariant geodesic distances. Our method of proof also shows that, as i ...
Matrices and Linear Algebra
... It follows directly from our discussion above that the range of A equals S(c1 (A), . . . , cn (A)). Row operations: To solve Ax = b we use a process called Gaussian elimination, which is based on row operations. ...
... It follows directly from our discussion above that the range of A equals S(c1 (A), . . . , cn (A)). Row operations: To solve Ax = b we use a process called Gaussian elimination, which is based on row operations. ...
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... 2. Diagrammatic Linear Algebra Linear algebra is on the one hand, the study of finite-dimensional vector spaces and linear maps between them, and on the other hand, the study of a particular class of objects (in particular, the dualizable ones) in the monoidal category (VectK , ⊗, K). We present thi ...
... 2. Diagrammatic Linear Algebra Linear algebra is on the one hand, the study of finite-dimensional vector spaces and linear maps between them, and on the other hand, the study of a particular class of objects (in particular, the dualizable ones) in the monoidal category (VectK , ⊗, K). We present thi ...
Introduction to representation theory of finite groups
... then it is easy to see, that V = U ⊕ W as a representation if and only if V = U ⊕ W as a vector space, i.e. if and only if U + W = V and U ∩ W = {0}. Definition 1.11. A representation is indecomposable if it is not a direct sum of proper subrepresentations. An irreducible representation is certainly ...
... then it is easy to see, that V = U ⊕ W as a representation if and only if V = U ⊕ W as a vector space, i.e. if and only if U + W = V and U ∩ W = {0}. Definition 1.11. A representation is indecomposable if it is not a direct sum of proper subrepresentations. An irreducible representation is certainly ...
Group theory notes
... numbers(complex) using ordinary multiplication and the other with matrices using matrix multiplication. There maybe a correspondence between the elements of two groups. The correspondence can be one-to-one , two-to-one or, many-toone. If the correspondence satisfies the same group multiplication tab ...
... numbers(complex) using ordinary multiplication and the other with matrices using matrix multiplication. There maybe a correspondence between the elements of two groups. The correspondence can be one-to-one , two-to-one or, many-toone. If the correspondence satisfies the same group multiplication tab ...
Distributions of eigenvalues of large Euclidean matrices generated
... In applications, the matrix Mn is related to Genomics [30], Phylogeny [17,23], the geometric random graphs [29] and Statistics [7,13,16]. A relevant study by Koltchinskii and Giné [21] is to use the matrix ( gn (xi , x j ))n×n to approximate the spectra of integral operators. For an n × n symmetric ...
... In applications, the matrix Mn is related to Genomics [30], Phylogeny [17,23], the geometric random graphs [29] and Statistics [7,13,16]. A relevant study by Koltchinskii and Giné [21] is to use the matrix ( gn (xi , x j ))n×n to approximate the spectra of integral operators. For an n × n symmetric ...
Exeter Math Club Competition January 25, 2014
... 3. Given two points on the plane, how many distinct regular hexagons include both of these points as vertices? 4. Jordan has six different files on her computer and needs to email them to Chad. The sizes of these files are 768, 1024, 2304, 2560, 4096, and 7680 kilobytes. Unfortunately, the email ser ...
... 3. Given two points on the plane, how many distinct regular hexagons include both of these points as vertices? 4. Jordan has six different files on her computer and needs to email them to Chad. The sizes of these files are 768, 1024, 2304, 2560, 4096, and 7680 kilobytes. Unfortunately, the email ser ...
o deliteljima nule, invertibilnosti i rangu matrica nad komutativnim
... Preliminaries on semirings In this chapter we give some basic definitions concerning semirings and matrices over semirings. For more information about these (and other) notions, we refer the reader to ...
... Preliminaries on semirings In this chapter we give some basic definitions concerning semirings and matrices over semirings. For more information about these (and other) notions, we refer the reader to ...
GENERATING SETS 1. Introduction In R
... We’ve found a pair of generators for Sn consisting of a transposition and n-cycle, and then a transposition and (n − 1)-cycle. These have orders 2 and n, and then 2 and n − 1. How small can the orders of a pair of generators of Sn be? Theorem 2.7. For n ≥ 3 except for n = 5, 6, 8, Sn is generated by ...
... We’ve found a pair of generators for Sn consisting of a transposition and n-cycle, and then a transposition and (n − 1)-cycle. These have orders 2 and n, and then 2 and n − 1. How small can the orders of a pair of generators of Sn be? Theorem 2.7. For n ≥ 3 except for n = 5, 6, 8, Sn is generated by ...
Lectures on Dirac Operators and Index Theory
... with Ω = (Ωji ) : U ∩ V → SO(n) is the transition function. Since {ei }, {e0i } (over U ∩ V ) generate the same Clifford algebra, their matrix representations are equivalent. i.e. there exists S(x) such that Sγi S −1 = γi0 = γj Ωij . Thus, for Spinor field ψ(x) for γ and spinor field ψ 0 (x) for γ 0 ...
... with Ω = (Ωji ) : U ∩ V → SO(n) is the transition function. Since {ei }, {e0i } (over U ∩ V ) generate the same Clifford algebra, their matrix representations are equivalent. i.e. there exists S(x) such that Sγi S −1 = γi0 = γj Ωij . Thus, for Spinor field ψ(x) for γ and spinor field ψ 0 (x) for γ 0 ...
Linear Transformations
... A linear transformatio n T : V W that is one to one and onto is called an isomorphis m. Moreover, if V and W are vector spaces such that there exists an isomorphis m from V to W , then V and W are said to be isomorphic to each other. Thm 6.9: (Isomorphic spaces and dimension) Two finite-dimensiona ...
... A linear transformatio n T : V W that is one to one and onto is called an isomorphis m. Moreover, if V and W are vector spaces such that there exists an isomorphis m from V to W , then V and W are said to be isomorphic to each other. Thm 6.9: (Isomorphic spaces and dimension) Two finite-dimensiona ...
Lie Groups and Algebraic Groups
... for V such that the matrix [B(vi , vj )] = J (call such a basis a B-symplectic basis). Proof. Let v be a nonzero element of V . Since B is nondegenerate, there exists w ∈ V with B(v, w) 6= 0. Replacing w with B(v, w)−1 w, we may assume that B(v, w) = 1. Let W = {x ∈ V : B(v, x) = 0 and B(w, x) = 0}. ...
... for V such that the matrix [B(vi , vj )] = J (call such a basis a B-symplectic basis). Proof. Let v be a nonzero element of V . Since B is nondegenerate, there exists w ∈ V with B(v, w) 6= 0. Replacing w with B(v, w)−1 w, we may assume that B(v, w) = 1. Let W = {x ∈ V : B(v, x) = 0 and B(w, x) = 0}. ...
October 28, 2014 EIGENVALUES AND EIGENVECTORS Contents 1.
... Note that a matrix with real entries can also act on Cn , since for any x ∈ Cn also M x ∈ Cn . But a matrix with complex non real entries cannot act on Rn , since for x ∈ Rn the image M x may not belong to Rn (while certainly M x ∈ Cn ). Definition 1. Let M be an n×n matrix acting on the vector spac ...
... Note that a matrix with real entries can also act on Cn , since for any x ∈ Cn also M x ∈ Cn . But a matrix with complex non real entries cannot act on Rn , since for x ∈ Rn the image M x may not belong to Rn (while certainly M x ∈ Cn ). Definition 1. Let M be an n×n matrix acting on the vector spac ...
Abstract ordered compact convex sets and the algebras of the (sub
... The following lemma has been proved by Nachbin for spaces with a closed partial order [15, Proposition 4 and Theorem 4]. His proof carries over to arbitrary closed binary relations: Lemma 2.2. Let X be a topological space with a binary relation the graph G of which is closed. (a) For any compact sub ...
... The following lemma has been proved by Nachbin for spaces with a closed partial order [15, Proposition 4 and Theorem 4]. His proof carries over to arbitrary closed binary relations: Lemma 2.2. Let X be a topological space with a binary relation the graph G of which is closed. (a) For any compact sub ...
Homework assignment, Feb. 18, 2004. Solutions
... 5. Prove that if A : X → Y and V is a subspace of X then dim AV ≤ dim V . Deduce from here that rank(AB) ≤ rank B. Proof. Let dim V = r and let v1 , v2 , . . . , vr be a basis in V . Then the system of vectors Av1 , Av2 , . . . , Avr isa generating system for AV . Indeed, every vector v ∈ V can be ...
... 5. Prove that if A : X → Y and V is a subspace of X then dim AV ≤ dim V . Deduce from here that rank(AB) ≤ rank B. Proof. Let dim V = r and let v1 , v2 , . . . , vr be a basis in V . Then the system of vectors Av1 , Av2 , . . . , Avr isa generating system for AV . Indeed, every vector v ∈ V can be ...