The fixed point index for noncompact mappings in non locally
... on K if the following conditions are satis ed. (N1) (co M ) '( (M )) (M 2 M). (N2) (N ) (M ) = (M ) = (M [ fag) (a 2 K; M 2 M; N M ). Further let M K be nonvoid, T a topological space, H : T M ! K a continuous operator and a '-measure of noncompactness on K . H is called a ('; ...
... on K if the following conditions are satis ed. (N1) (co M ) '( (M )) (M 2 M). (N2) (N ) (M ) = (M ) = (M [ fag) (a 2 K; M 2 M; N M ). Further let M K be nonvoid, T a topological space, H : T M ! K a continuous operator and a '-measure of noncompactness on K . H is called a ('; ...
BORNOLOGICAL QUANTUM GROUPS 1. Introduction The concept
... setting is obviously limited. It is thus desirable to have a more general setup then the one provided by algebraic quantum groups. Motivated by these facts we introduce in this paper the concept of a bornological quantum group. The main idea is to replace the category of vector spaces underlying the ...
... setting is obviously limited. It is thus desirable to have a more general setup then the one provided by algebraic quantum groups. Motivated by these facts we introduce in this paper the concept of a bornological quantum group. The main idea is to replace the category of vector spaces underlying the ...
CLOSURES OF QUADRATIC MODULES In Section 1 we consider
... Consider a real vector space V . A convex set U ⊆ V is called absorbent, if for every x ∈ V there exists λ > 0 such that x ∈ λU. U is called symmetric, if λU ⊆ U for all |λ| ≤ 1. The set of all convex, absorbent and symmetric subsets of V forms a zero neighborhood base of a vector space topology on ...
... Consider a real vector space V . A convex set U ⊆ V is called absorbent, if for every x ∈ V there exists λ > 0 such that x ∈ λU. U is called symmetric, if λU ⊆ U for all |λ| ≤ 1. The set of all convex, absorbent and symmetric subsets of V forms a zero neighborhood base of a vector space topology on ...
The fixed point index for noncompact mappings in non locally
... Definition 3. Let E be a topological vector space, ∅ 6= K ⊆ E, (A, ≤) a partially ordered set, ϕ : A → A a mapping and M a system of subsets of co K such that M ∈ M implies M ∈ M, co M ∈ M, N ∈ M (N ⊆ M ) and M ∪ {a} ∈ M (a ∈ K). The mapping γ : M → A is said to be a ϕ-measure of noncompactness on K ...
... Definition 3. Let E be a topological vector space, ∅ 6= K ⊆ E, (A, ≤) a partially ordered set, ϕ : A → A a mapping and M a system of subsets of co K such that M ∈ M implies M ∈ M, co M ∈ M, N ∈ M (N ⊆ M ) and M ∪ {a} ∈ M (a ∈ K). The mapping γ : M → A is said to be a ϕ-measure of noncompactness on K ...
Part III Homomorphism and Factor Groups
... Example 13.4 (13.4). Let F = C([0, 1]) be the additive group of all continuous real valued functions and c ∈ [0, 1] be fixed point. Let ϕ : F −→ R be defined by ϕ(f ) = f (c) (to be called the evaluation map, at c). That means, ϕ(f ) = f (c) for f ∈ F . Then ϕ is a homomorphism. Example 13.5 (13.5) ...
... Example 13.4 (13.4). Let F = C([0, 1]) be the additive group of all continuous real valued functions and c ∈ [0, 1] be fixed point. Let ϕ : F −→ R be defined by ϕ(f ) = f (c) (to be called the evaluation map, at c). That means, ϕ(f ) = f (c) for f ∈ F . Then ϕ is a homomorphism. Example 13.5 (13.5) ...
Inverse Systems and Regular Representations
... where m = exp(2i/m) is a primitive mth root of unity. Choosing the corresponding dual basis for V ∗ , the ring of invariants is generated by F1 = x1 x2 and F2 = x1m + x2m . 2.5. It will be conceptually clearer for us to not to think of B as a subring of A, but rather as a polynomial ring B = k[u1 ...
... where m = exp(2i/m) is a primitive mth root of unity. Choosing the corresponding dual basis for V ∗ , the ring of invariants is generated by F1 = x1 x2 and F2 = x1m + x2m . 2.5. It will be conceptually clearer for us to not to think of B as a subring of A, but rather as a polynomial ring B = k[u1 ...
Chapter 1 Theory of Matrix Functions
... 1.2. Definitions of f (A) There are many equivalent ways of defining f (A). We focus on three that are of particular interest. These definitions yield primary matrix functions; nonprimary matrix functions are discussed in Section 1.4. ...
... 1.2. Definitions of f (A) There are many equivalent ways of defining f (A). We focus on three that are of particular interest. These definitions yield primary matrix functions; nonprimary matrix functions are discussed in Section 1.4. ...
Ten ways to decompose a tensor
... Sparsity: our polynomials are always sparse (eg. for k = 3, only terms of the form xyz or x 2 y 2 z 2 or uvwxyz appear). This can be exploited. ...
... Sparsity: our polynomials are always sparse (eg. for k = 3, only terms of the form xyz or x 2 y 2 z 2 or uvwxyz appear). This can be exploited. ...
2.7.1 Euclidean Parallel Postulate
... equivalent? The exercises ask you to prove one direction on a few of the statements and to find a counterexample in the Poincaré Half-plane. Exercises 2.65. Show the Poincaré Half-plane does not satisfy the Euclidean Parallel Postulate. (a) Use dynamic geometry software to construct an example. (b) ...
... equivalent? The exercises ask you to prove one direction on a few of the statements and to find a counterexample in the Poincaré Half-plane. Exercises 2.65. Show the Poincaré Half-plane does not satisfy the Euclidean Parallel Postulate. (a) Use dynamic geometry software to construct an example. (b) ...
CHAPTER 4: PRINCIPAL BUNDLES 4.1 Lie groups A Lie group is a
... case the projection map M × V → M is simply the projection onto the first factor. A direct sum of two vector bundles E and F over the same manifold M is the bundle E ⊕ F with fiber Ex ⊕ Fx at a point x ∈ M. The tensor product bundle E ⊗ F is the vector bundle with fiber Ex ⊗ Fx at x ∈ M. Example 4.3 ...
... case the projection map M × V → M is simply the projection onto the first factor. A direct sum of two vector bundles E and F over the same manifold M is the bundle E ⊕ F with fiber Ex ⊕ Fx at a point x ∈ M. The tensor product bundle E ⊗ F is the vector bundle with fiber Ex ⊗ Fx at x ∈ M. Example 4.3 ...
Matrix Operations
... We say that matrices A1 , A2 , . . . , An of the same size are linearly independent if the only solution of the equation c1 A1 + c2 A2 + · · · + ck Ak = O is the trivial one: c1 = c2 = · · · = ck = 0, otherwise they are linearly dependent. Questions like “Is matrix B equal to a linear combination of ...
... We say that matrices A1 , A2 , . . . , An of the same size are linearly independent if the only solution of the equation c1 A1 + c2 A2 + · · · + ck Ak = O is the trivial one: c1 = c2 = · · · = ck = 0, otherwise they are linearly dependent. Questions like “Is matrix B equal to a linear combination of ...
Dense Linear Algebra
... An added Complication • What if input matrix C is transposed? a[i] += c[i][j] * b[j]; ...
... An added Complication • What if input matrix C is transposed? a[i] += c[i][j] * b[j]; ...
Sample pages 2 PDF
... where in the last equality we used orthonormality condition (2.18). The expansion (2.19) is called Fourier expansion of the vector x and the coefficients ai the Fourier coefficients of x, obtained with the basis B. Through Fourier expansion, every orthonormal basis B = {bi , i ∈ I } defines a coordi ...
... where in the last equality we used orthonormality condition (2.18). The expansion (2.19) is called Fourier expansion of the vector x and the coefficients ai the Fourier coefficients of x, obtained with the basis B. Through Fourier expansion, every orthonormal basis B = {bi , i ∈ I } defines a coordi ...
On the Kemeny constant and stationary distribution vector
... The Kemeny Constant and Stationary Distribution ...
... The Kemeny Constant and Stationary Distribution ...
Trace Inequalities and Quantum Entropy: An
... gives rise to the variational principle for Gibbs states. If one accepts the idea that equilibrium states should maximize the entropy given the expected values of macroscopic observables, such as energy, this leads to an identification of Gibbs states with thermodynamic equilibrium states. There is ...
... gives rise to the variational principle for Gibbs states. If one accepts the idea that equilibrium states should maximize the entropy given the expected values of macroscopic observables, such as energy, this leads to an identification of Gibbs states with thermodynamic equilibrium states. There is ...
Ribet`s lemma, generalizations, and pseudocharacters
... of any point X is |k| + 1 (by Lemma 1.2), so X is a homogeneous tree. In a tree, we define the segment [x, x0 ] as the set {x} is x = x0 , and as the set of points in the unique injective path from x to x0 otherwise. A subset C of X is called convex if for every x, x0 ∈ C, the segment [x, x0 ] is in ...
... of any point X is |k| + 1 (by Lemma 1.2), so X is a homogeneous tree. In a tree, we define the segment [x, x0 ] as the set {x} is x = x0 , and as the set of points in the unique injective path from x to x0 otherwise. A subset C of X is called convex if for every x, x0 ∈ C, the segment [x, x0 ] is in ...
Matrices for which the squared equals the original
... expansion of the identity matrix. Simply put, if a square matrix contains only zeros, with the exception of filling any or all of the entries along the main diagonal with ones, then it will return itself upon squaring. There are other matrices that also form a solution to this problem, but in a more ...
... expansion of the identity matrix. Simply put, if a square matrix contains only zeros, with the exception of filling any or all of the entries along the main diagonal with ones, then it will return itself upon squaring. There are other matrices that also form a solution to this problem, but in a more ...
History of Numerical Linear Algebra, a Personal View
... Discretizing using centered schemes on a uniform mesh, the matrix associated with the linear system is: ...
... Discretizing using centered schemes on a uniform mesh, the matrix associated with the linear system is: ...
MATLAB TOOLS FOR SOLVING PERIODIC EIGENVALUE
... elements can contain data of a different type, size and dimension. For cell arrays, storage is allocated dynamically. In particular, for the matrices in (4), we may use a code snippet as A = cell(1,p); for k=1:p, A{k} = A(:,:,k);, end to create the cell array A and store the matrices A1 , A2 , . . . ...
... elements can contain data of a different type, size and dimension. For cell arrays, storage is allocated dynamically. In particular, for the matrices in (4), we may use a code snippet as A = cell(1,p); for k=1:p, A{k} = A(:,:,k);, end to create the cell array A and store the matrices A1 , A2 , . . . ...
3.IV. Matrix Operations - National Cheng Kung University
... Matrix dimensions: m r times r n equals m n. ...
... Matrix dimensions: m r times r n equals m n. ...
Chapter VI. Inner Product Spaces.
... 1.3. Exercise. Verify that both inner products in the last example actually satisfy the inner product axioms. In particular, explain why the L2 -inner product (f, h)2 has ∥f ∥2 > 0 when f is not the zero function (f (t) ≡ 0 for all t). We now take up the basic properties common to all inner product ...
... 1.3. Exercise. Verify that both inner products in the last example actually satisfy the inner product axioms. In particular, explain why the L2 -inner product (f, h)2 has ∥f ∥2 > 0 when f is not the zero function (f (t) ≡ 0 for all t). We now take up the basic properties common to all inner product ...
MATH 2030: EIGENVALUES AND EIGENVECTORS Eigenvalues
... Proof. We proceed by induction on n; for the base-case n = 1 the result is what has been given, for the inductive-assumption we assume the result is true for n = k, Ak x = λk x. To prove this for arbitrary n, we must show this holds for n = k + 1. As the identity Ak+1 x = A(Ak x) = A(λk x) - using t ...
... Proof. We proceed by induction on n; for the base-case n = 1 the result is what has been given, for the inductive-assumption we assume the result is true for n = k, Ak x = λk x. To prove this for arbitrary n, we must show this holds for n = k + 1. As the identity Ak+1 x = A(Ak x) = A(λk x) - using t ...
THE DIRAC OPERATOR 1. First properties 1.1. Definition. Let X be a
... We observe that if a nonzero tangent vector t ∈ Tx X is fixed the morphism Vx → Vx given by Clifford multiplication by t is an isomorphism of vector spaces (in fact, the inverse is a scalar multiple of Clifford multiplication by t again). Consequently: Corollary 2. The Dirac operator is elliptic. We ...
... We observe that if a nonzero tangent vector t ∈ Tx X is fixed the morphism Vx → Vx given by Clifford multiplication by t is an isomorphism of vector spaces (in fact, the inverse is a scalar multiple of Clifford multiplication by t again). Consequently: Corollary 2. The Dirac operator is elliptic. We ...
