Unitary representations of topological groups
... the G-stable subspaces. Certainly algebraic irreducibility is a stronger condition in general than topological irreducibility. In finite-dimensional spaces, since every subspace is closed, the distinction is meaningless, but in general in infinite-dimensional spaces there can be proper G-stable subs ...
... the G-stable subspaces. Certainly algebraic irreducibility is a stronger condition in general than topological irreducibility. In finite-dimensional spaces, since every subspace is closed, the distinction is meaningless, but in general in infinite-dimensional spaces there can be proper G-stable subs ...
Linear Algebra and Introduction to MATLAB
... – linear equations are the most elementary equations that can arise – we can (mostly) calculate explicit solutions – when studying non-linear models which cannot be solved explicitly, linear systems can serve as an approximation (calculus, Taylor polynomial) – some of the most frequently studied eco ...
... – linear equations are the most elementary equations that can arise – we can (mostly) calculate explicit solutions – when studying non-linear models which cannot be solved explicitly, linear systems can serve as an approximation (calculus, Taylor polynomial) – some of the most frequently studied eco ...
Linear Algebra I
... λ is an eigenvalue of f if Eλ (f ) 6= {0}, i.e., if there is 0 6= v ∈ V such that f (v) = λv. Such a vector v is called an eigenvector of f for the eigenvalue λ. The eigenvalues are exactly the roots (in F ) of the characteristic polynomial of f , Pf (x) = det(x idV −f ) , which is a monic polynomia ...
... λ is an eigenvalue of f if Eλ (f ) 6= {0}, i.e., if there is 0 6= v ∈ V such that f (v) = λv. Such a vector v is called an eigenvector of f for the eigenvalue λ. The eigenvalues are exactly the roots (in F ) of the characteristic polynomial of f , Pf (x) = det(x idV −f ) , which is a monic polynomia ...
Stochastic Matrices in a Finite Field Introduction Literature review
... In addition to verifying how some properties that held for stochastic matrices in the field of real numbers held in the finite field as well, we have proved analytically why in fact they did or did not hold. Our main result being the property that 2 2 stochastic matrices will have solutions to the ...
... In addition to verifying how some properties that held for stochastic matrices in the field of real numbers held in the finite field as well, we have proved analytically why in fact they did or did not hold. Our main result being the property that 2 2 stochastic matrices will have solutions to the ...
AN EFFECTIVE MATRIX GEOMETRIC MEAN SATISFYING THE
... We can interpret this iteration as a geometrical construction in the following way. To find e.g. Ā(r+1) , the algorithm is: (1) Draw the geodesic joining B̄ (r) and C̄ (r) , and take its midpoint M (r) . (2) Draw the geodesic joining Ā(r) and M (r) , and take the point lying at 2/3 of its length: t ...
... We can interpret this iteration as a geometrical construction in the following way. To find e.g. Ā(r+1) , the algorithm is: (1) Draw the geodesic joining B̄ (r) and C̄ (r) , and take its midpoint M (r) . (2) Draw the geodesic joining Ā(r) and M (r) , and take the point lying at 2/3 of its length: t ...
Linear Algebra II
... λ is an eigenvalue of f if Eλ (f ) 6= {0}, i.e., if there is 0 6= v ∈ V such that f (v) = λv. Such a vector v is called an eigenvector of f for the eigenvalue λ. The eigenvalues are exactly the roots (in F ) of the characteristic polynomial of f , Pf (x) = det(x idV −f ) , which is a monic polynomia ...
... λ is an eigenvalue of f if Eλ (f ) 6= {0}, i.e., if there is 0 6= v ∈ V such that f (v) = λv. Such a vector v is called an eigenvector of f for the eigenvalue λ. The eigenvalues are exactly the roots (in F ) of the characteristic polynomial of f , Pf (x) = det(x idV −f ) , which is a monic polynomia ...
Algebra-2: Groups and Rings
... As in the usual arithmetic, multiplication takes precedence over addition. The identity element in a ring behaves slightly differently from the identity element in the group. As a start, 1a = a does not formally imply a1 = a. Nevertheless, the identity is unique. Lemma 2.1 Let R be a ring. Then R ha ...
... As in the usual arithmetic, multiplication takes precedence over addition. The identity element in a ring behaves slightly differently from the identity element in the group. As a start, 1a = a does not formally imply a1 = a. Nevertheless, the identity is unique. Lemma 2.1 Let R be a ring. Then R ha ...
Row and Column Spaces of Matrices over Residuated Lattices 1
... (ii) hL, ⊗, 1i is a commutative monoid, i.e. ⊗ is a binary operation that is commutative, associative, and a ⊗ 1 = a for each a ∈ L; (iii) ⊗ and → satisfy adjointness, i.e. a ⊗ b ≤ c iff a ≤ b → c. Throughout the paper, L denotes an arbitrary (complete) residuated lattice. Common examples of complet ...
... (ii) hL, ⊗, 1i is a commutative monoid, i.e. ⊗ is a binary operation that is commutative, associative, and a ⊗ 1 = a for each a ∈ L; (iii) ⊗ and → satisfy adjointness, i.e. a ⊗ b ≤ c iff a ≤ b → c. Throughout the paper, L denotes an arbitrary (complete) residuated lattice. Common examples of complet ...
CONVEX PARTITIONS OF POLYHEDRA
... genus k. Similarly, polyhedra may have holes (i.e., handles), and we define the genus of a polyhedron as the genus of the surface formed by its boundary [6]. It follows from the definition that a polyhedron may not have interior boundaries. Let P be a polyhedron with n vertices vl," vn, p edges el,. ...
... genus k. Similarly, polyhedra may have holes (i.e., handles), and we define the genus of a polyhedron as the genus of the surface formed by its boundary [6]. It follows from the definition that a polyhedron may not have interior boundaries. Let P be a polyhedron with n vertices vl," vn, p edges el,. ...
Shimura.pdf
... Proposition 1.2.3. There is a canonical bijection from the set of polarized abelian varieties of type D with symplectic basis to the set of homomorphisms of real algebraic groups h1 : S1 → SpR (U, E) such that the following conditions are satisfied: (1) the complexification h1,C : Gm → Sp(U ⊗ C) giv ...
... Proposition 1.2.3. There is a canonical bijection from the set of polarized abelian varieties of type D with symplectic basis to the set of homomorphisms of real algebraic groups h1 : S1 → SpR (U, E) such that the following conditions are satisfied: (1) the complexification h1,C : Gm → Sp(U ⊗ C) giv ...
An Interpretation of Rosenbrock`s Theorem Via Local
... factorization indices at infinity, which are equal to the controllability indices of (▭) and (▭), because the controllability indices are invariant under feedback. With all this in mind it is not hard to see that Rosenbrock's Theorem on pole assignment is equivalent to finding nec‐ essary and suffic ...
... factorization indices at infinity, which are equal to the controllability indices of (▭) and (▭), because the controllability indices are invariant under feedback. With all this in mind it is not hard to see that Rosenbrock's Theorem on pole assignment is equivalent to finding nec‐ essary and suffic ...
Random Matrix Theory - Indian Institute of Science
... 2. Principal component analysis - a case for studying eigenvalues We saw some situations in which random matrices arise naturally. But why study their eigenvalues. For Wigner matrices, we made the case that eigenvalues of the Hamiltonian are important in physics, and hence one must study eigenvalues ...
... 2. Principal component analysis - a case for studying eigenvalues We saw some situations in which random matrices arise naturally. But why study their eigenvalues. For Wigner matrices, we made the case that eigenvalues of the Hamiltonian are important in physics, and hence one must study eigenvalues ...
Euclidean Spaces
... Definition 7.15 Vectors e1 , . . . , em in a Euclidean space form an orthonormal system if (ei , ej ) = 0 for i = j, ...
... Definition 7.15 Vectors e1 , . . . , em in a Euclidean space form an orthonormal system if (ei , ej ) = 0 for i = j, ...
AN ASYMPTOTIC FORMULA FOR THE NUMBER OF NON
... any of the equations sj = 0 or tk = 0. Indeed, if L ⊂ Rm+n is any hyperplane not containing the null-space of q, then the restriction q|L is positive definite and one can consider the Gaussian probability measure in L with the density proportional to e−q . We prove in Lemma 3.1 below that the expect ...
... any of the equations sj = 0 or tk = 0. Indeed, if L ⊂ Rm+n is any hyperplane not containing the null-space of q, then the restriction q|L is positive definite and one can consider the Gaussian probability measure in L with the density proportional to e−q . We prove in Lemma 3.1 below that the expect ...
Chapter 2 Matrices
... trix addition (1), Let A = [aij ], B = [bij ]. Both A and B have same size m × n, so A + B, B + A are defined. From definition A + B = [aij ] + [bij ] = [aij + bij ] and B + A = [bij ] + [aij ] = [bij + aij ]. From commutative property of addition of real numbers, we have aij + bij = bij +aij . Ther ...
... trix addition (1), Let A = [aij ], B = [bij ]. Both A and B have same size m × n, so A + B, B + A are defined. From definition A + B = [aij ] + [bij ] = [aij + bij ] and B + A = [bij ] + [aij ] = [bij + aij ]. From commutative property of addition of real numbers, we have aij + bij = bij +aij . Ther ...
3 - UCI Math
... What do an equilateral triangle and an arbitrary collection {1, 2, 3} of 3 objects have in common? Not much at first glance, but both have symmetries: rotations and reflections of the triangle and permutations of the set {1, 2, 3}. If one considers compositions of these symmetries or permuations, we ...
... What do an equilateral triangle and an arbitrary collection {1, 2, 3} of 3 objects have in common? Not much at first glance, but both have symmetries: rotations and reflections of the triangle and permutations of the set {1, 2, 3}. If one considers compositions of these symmetries or permuations, we ...
General Domain Truncated Correlation and Convolution Operators
... operator-class containing popular operators such as Toeplitz (WienerHopf), Hankel and nite interval convolution operators as well as small and big Hankel operators in several variables. We completely characterize the symbols for which such operators have nite rank, and develop methods for determin ...
... operator-class containing popular operators such as Toeplitz (WienerHopf), Hankel and nite interval convolution operators as well as small and big Hankel operators in several variables. We completely characterize the symbols for which such operators have nite rank, and develop methods for determin ...
lecture13_densela_1_.. - People @ EECS at UC Berkeley
... New lower bound for all “direct” linear algebra Let M = “fast” memory size per processor = cache size (sequential case) or n2/p (parallel case) #words_moved by at least one processor = (#flops / M1/2 ) #messages_sent by at least one processor = (#flops / M3/2 ) • Holds for • BLAS, LU, QR, eig, S ...
... New lower bound for all “direct” linear algebra Let M = “fast” memory size per processor = cache size (sequential case) or n2/p (parallel case) #words_moved by at least one processor = (#flops / M1/2 ) #messages_sent by at least one processor = (#flops / M3/2 ) • Holds for • BLAS, LU, QR, eig, S ...
Compactness of L-weakly and M-weakly compact operators on
... is termed discrete or atomic if the band generated by the discrete elements is the whole space. It turns out that the class of Banach lattices E that we need are those such that E a is discrete. Although a sublattice of a discrete vector lattice need not be discrete, an ideal must be, so that if E i ...
... is termed discrete or atomic if the band generated by the discrete elements is the whole space. It turns out that the class of Banach lattices E that we need are those such that E a is discrete. Although a sublattice of a discrete vector lattice need not be discrete, an ideal must be, so that if E i ...
Supersymmetry for Mathematicians: An Introduction (Courant
... 1.1. Introductory Remarks on Supersymmetry The subject of supersymmetry (SUSY) is a part of the theory of elementary particles and their interactions and the still unfinished quest of obtaining a unified view of all the elementary forces in a manner compatible with quantum theory and general relativ ...
... 1.1. Introductory Remarks on Supersymmetry The subject of supersymmetry (SUSY) is a part of the theory of elementary particles and their interactions and the still unfinished quest of obtaining a unified view of all the elementary forces in a manner compatible with quantum theory and general relativ ...
lecture (3) - MIT OpenCourseWare
... Indeed, a homomorphism of representations V ⊃ V ⊕ is obviously the same thing as an invariant sesquilinear form on V (i.e. a form additive on both arguments which is linear on the first one and antilinear on the second one), and an isomorphism is the same thing as a nondegenerate invariant sesquiline ...
... Indeed, a homomorphism of representations V ⊃ V ⊕ is obviously the same thing as an invariant sesquilinear form on V (i.e. a form additive on both arguments which is linear on the first one and antilinear on the second one), and an isomorphism is the same thing as a nondegenerate invariant sesquiline ...
Matrix Arithmetic
... The Invertible Matrix Theorem(IMT) Part I Let A be an n × n matrix, I the n × n identity matrix, and θ the zero vector in Rn . Then the following are equivalent. 1. A is invertible. 2. The reduced row echelon form of A is I. 3. For any b ∈ Rn the matrix equation Ax = b has exactly one solution. 4. ...
... The Invertible Matrix Theorem(IMT) Part I Let A be an n × n matrix, I the n × n identity matrix, and θ the zero vector in Rn . Then the following are equivalent. 1. A is invertible. 2. The reduced row echelon form of A is I. 3. For any b ∈ Rn the matrix equation Ax = b has exactly one solution. 4. ...
Random Involutions and the Distinct Prime Divisor Function
... It is often useful to count objects X, weighted by | Aut1(X)| , where Aut(X) denotes the group of automorphisms of X (isomorphisms from X to itself). ...
... It is often useful to count objects X, weighted by | Aut1(X)| , where Aut(X) denotes the group of automorphisms of X (isomorphisms from X to itself). ...
A proof of the multiplicative property of the Berezinian ∗
... x(yz) = (xy)z for all x,y and z in A. A unit is an even element 1 in A such that 1x = x1 = x for each x in A. It is common to reserve the name superalgebra only for an associative superalgebra A with unity. Remark 3.0.12. Note that a superalgebra A is not necessary commutative as an usual algebra. A ...
... x(yz) = (xy)z for all x,y and z in A. A unit is an even element 1 in A such that 1x = x1 = x for each x in A. It is common to reserve the name superalgebra only for an associative superalgebra A with unity. Remark 3.0.12. Note that a superalgebra A is not necessary commutative as an usual algebra. A ...
Hard Lefschetz theorem and Hodge-Riemann
... 2. We denote by A(Φ) the (also evenly graded) algebra of piecewise polynomial functions on |Φ|, i.e., f ∈ A(Φ) if fσ = f |σ is a polynomial for each σ ∈ Φ. A piecewise linear function ` ∈ A2 (Φ) is called strictly convex if for any two cones σ and τ of dimension n we have `σ (v) < `τ (v) for any v ∈ ...
... 2. We denote by A(Φ) the (also evenly graded) algebra of piecewise polynomial functions on |Φ|, i.e., f ∈ A(Φ) if fσ = f |σ is a polynomial for each σ ∈ Φ. A piecewise linear function ` ∈ A2 (Φ) is called strictly convex if for any two cones σ and τ of dimension n we have `σ (v) < `τ (v) for any v ∈ ...