Title On certain cohomology groups attached to
... we have attached to a bounded symmetric domain X and a discontinuous group Γ operating on X two kinds of cohomology groups. The first one is associated with a certain representation of Γ and the second one is associated with a so-called canonical automorphic factor. The first purpose of the present ...
... we have attached to a bounded symmetric domain X and a discontinuous group Γ operating on X two kinds of cohomology groups. The first one is associated with a certain representation of Γ and the second one is associated with a so-called canonical automorphic factor. The first purpose of the present ...
Part 1 - UBC Math
... The following questions require the big Invertible Matrix Theorem and some thinking. They will not be in the quiz, but some of them could be in the next midterm and the final. You are strongly suggested to think about them. I ...
... The following questions require the big Invertible Matrix Theorem and some thinking. They will not be in the quiz, but some of them could be in the next midterm and the final. You are strongly suggested to think about them. I ...
Matrix Operations
... AB is the sum of the 1st and 3rd rows of B, because we have 1, 0, 1, 0 in the first row in A; the second row of AB is 1st row of B minus 4th row of B because we have 1, 0, 0, −1 in the second row of A. Actually, this rule can always be applied, but is particularly effective when A is “easier” than B ...
... AB is the sum of the 1st and 3rd rows of B, because we have 1, 0, 1, 0 in the first row in A; the second row of AB is 1st row of B minus 4th row of B because we have 1, 0, 0, −1 in the second row of A. Actually, this rule can always be applied, but is particularly effective when A is “easier” than B ...
Explicit tensors - Computational Complexity
... Again, we can extend this to a linear mapping V1 ⊗ · · · ⊗ Vn → U1 ⊗ · · · ⊗ Un . We denote this mapping by h1 ⊗ · · · ⊗ hn . Fact 2. R(t) ≥ R(h1 ⊗· · ·⊗hn (t)) for any tensor t. If all hi are isomorphisms, then equality holds. Let t ∈ V1 ⊗ · · · ⊗ Vn and s ∈ U1 ⊗ · · · ⊗ Un . We can embed both tens ...
... Again, we can extend this to a linear mapping V1 ⊗ · · · ⊗ Vn → U1 ⊗ · · · ⊗ Un . We denote this mapping by h1 ⊗ · · · ⊗ hn . Fact 2. R(t) ≥ R(h1 ⊗· · ·⊗hn (t)) for any tensor t. If all hi are isomorphisms, then equality holds. Let t ∈ V1 ⊗ · · · ⊗ Vn and s ∈ U1 ⊗ · · · ⊗ Un . We can embed both tens ...
introduction to banach algebras and the gelfand
... without a unit, we consider A1 = A × F which is a vector space as a product of vector spaces and its multiplication is defined by (a, λ) (b, µ) = (ab + µa + λb, λµ) This multiplication follows supposing (a, λ) and (b, µ) to be like a + λ1, b + µ1 and multiplying as usual. A1 is obviously an algebra ...
... without a unit, we consider A1 = A × F which is a vector space as a product of vector spaces and its multiplication is defined by (a, λ) (b, µ) = (ab + µa + λb, λµ) This multiplication follows supposing (a, λ) and (b, µ) to be like a + λ1, b + µ1 and multiplying as usual. A1 is obviously an algebra ...
Finite-Dimensional Vector Spaces
... Subtracting the last equation from 2.9, we get 0 = (a1 − b1 )v1 + · · · + (an − bn )vn . This implies that each aj − bj = 0 (because (v1 , . . . , vn ) is linearly independent) and hence a1 = b1 , . . . , an = bn . We have the desired uniqueness, completing the proof in one direction. For the other ...
... Subtracting the last equation from 2.9, we get 0 = (a1 − b1 )v1 + · · · + (an − bn )vn . This implies that each aj − bj = 0 (because (v1 , . . . , vn ) is linearly independent) and hence a1 = b1 , . . . , an = bn . We have the desired uniqueness, completing the proof in one direction. For the other ...
MATLAB TOOLS FOR SOLVING PERIODIC EIGENVALUE
... computed with the square matrices T22,1 , . . . , T22,p in unreduced form. The orthogonal transformation matrices and the transformed matrices are contained in the cell arrays Q and T, respectively. 3.2.2. Reduction to periodic HT form [{Q,}T] = per hess(A{,s,l}) Reduces a general matrix product (4) ...
... computed with the square matrices T22,1 , . . . , T22,p in unreduced form. The orthogonal transformation matrices and the transformed matrices are contained in the cell arrays Q and T, respectively. 3.2.2. Reduction to periodic HT form [{Q,}T] = per hess(A{,s,l}) Reduces a general matrix product (4) ...
ALTERNATING TRILINEAR FORMS AND GROUPS OF EXPONENT
... If U and V are T-spaces then we may conside their direct sum U © V in the usual sense of linear algebra. Of course, this will not have the structure of a T-space unless the values of(u1,v1, v2) and (u 1( U2, I>I) are defined for all uuu2eU and v1,v2 e V: e.g. these values are defined if U and V are ...
... If U and V are T-spaces then we may conside their direct sum U © V in the usual sense of linear algebra. Of course, this will not have the structure of a T-space unless the values of(u1,v1, v2) and (u 1( U2, I>I) are defined for all uuu2eU and v1,v2 e V: e.g. these values are defined if U and V are ...
Vector Spaces
... These are essentially the same properties enjoyed by geometric vectors and algebraic or coordinate vectors. Actually, functions have more properties: you can multiply them, differentiate them, and so on. But many properties of functions just rely on addition and scalar multiplication. Polynomials beh ...
... These are essentially the same properties enjoyed by geometric vectors and algebraic or coordinate vectors. Actually, functions have more properties: you can multiply them, differentiate them, and so on. But many properties of functions just rely on addition and scalar multiplication. Polynomials beh ...
Linear Algebra and Matrices
... If A is invertible and maps U into V, both n dimensional, then we define A−1 , which maps V into U, by the equation A−1 (y0 ) = x0 where y0 is in V and x0 is in U and satisfies Ax0 = y0 (such an x0 exists by (2) and is unique by (1)). Thus defined A−1 is a legitimate transformation from V to U. That ...
... If A is invertible and maps U into V, both n dimensional, then we define A−1 , which maps V into U, by the equation A−1 (y0 ) = x0 where y0 is in V and x0 is in U and satisfies Ax0 = y0 (such an x0 exists by (2) and is unique by (1)). Thus defined A−1 is a legitimate transformation from V to U. That ...
Group-theoretic algorithms for matrix multiplication
... In fact, the reader familiar with Strassen’s 1987 paper [10] and Coppersmith and Winograd’s paper [3] (or the presentation of this material in, for example, [1]) will recognize that our exponent bounds of 2.48 and 2.41 match bounds derived in those works. It turns out that with some effort the algor ...
... In fact, the reader familiar with Strassen’s 1987 paper [10] and Coppersmith and Winograd’s paper [3] (or the presentation of this material in, for example, [1]) will recognize that our exponent bounds of 2.48 and 2.41 match bounds derived in those works. It turns out that with some effort the algor ...
LINEAR ALGEBRA
... 3. The set of numbers between −1 and 1 is not a linear space since the sum of two elements of this set (e.g.,0.5+0.6) may not be an element of this set (Find also the other violations). 4. The set of polynomials of degree n is not a linear space since the sum of two polynomials of degree n may give ...
... 3. The set of numbers between −1 and 1 is not a linear space since the sum of two elements of this set (e.g.,0.5+0.6) may not be an element of this set (Find also the other violations). 4. The set of polynomials of degree n is not a linear space since the sum of two polynomials of degree n may give ...
EXERCISE SET 5.1
... scalar multiplication, Axioms 2, 3, 5, 7, 8, 9, 10 will hold automatically. However, for Axiom 4 to hold, we need the zero vector (0, 0) to be in V. Thus a0 + b0 = c, which forces c = 0. In this case, Axioms 1 and 6 are also satisfied. Thus, the set of all points in R2 lying on a line is a vector spa ...
... scalar multiplication, Axioms 2, 3, 5, 7, 8, 9, 10 will hold automatically. However, for Axiom 4 to hold, we need the zero vector (0, 0) to be in V. Thus a0 + b0 = c, which forces c = 0. In this case, Axioms 1 and 6 are also satisfied. Thus, the set of all points in R2 lying on a line is a vector spa ...
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 ...
Complex vector spaces, duals, and duels: Fun
... so that J applied twice multiplies everything by −1. This is just like multiplication by i, as we should expect. Thus, we can perhaps say (in our incorrigibly unrigorous way) that while R2 ∼ = C is true, it doesn’t capture all the structure of C, and in fact that (R2 , J) ∼ = (C, i) is better, whate ...
... so that J applied twice multiplies everything by −1. This is just like multiplication by i, as we should expect. Thus, we can perhaps say (in our incorrigibly unrigorous way) that while R2 ∼ = C is true, it doesn’t capture all the structure of C, and in fact that (R2 , J) ∼ = (C, i) is better, whate ...
Exterior algebra
In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogs. The exterior product of two vectors u and v, denoted by u ∧ v, is called a bivector and lives in a space called the exterior square, a vector space that is distinct from the original space of vectors. The magnitude of u ∧ v can be interpreted as the area of the parallelogram with sides u and v, which in three dimensions can also be computed using the cross product of the two vectors. Like the cross product, the exterior product is anticommutative, meaning that u ∧ v = −(v ∧ u) for all vectors u and v. One way to visualize a bivector is as a family of parallelograms all lying in the same plane, having the same area, and with the same orientation of their boundaries—a choice of clockwise or counterclockwise.When regarded in this manner, the exterior product of two vectors is called a 2-blade. More generally, the exterior product of any number k of vectors can be defined and is sometimes called a k-blade. It lives in a space known as the kth exterior power. The magnitude of the resulting k-blade is the volume of the k-dimensional parallelotope whose edges are the given vectors, just as the magnitude of the scalar triple product of vectors in three dimensions gives the volume of the parallelepiped generated by those vectors.The exterior algebra, or Grassmann algebra after Hermann Grassmann, is the algebraic system whose product is the exterior product. The exterior algebra provides an algebraic setting in which to answer geometric questions. For instance, blades have a concrete geometric interpretation, and objects in the exterior algebra can be manipulated according to a set of unambiguous rules. The exterior algebra contains objects that are not just k-blades, but sums of k-blades; such a sum is called a k-vector. The k-blades, because they are simple products of vectors, are called the simple elements of the algebra. The rank of any k-vector is defined to be the smallest number of simple elements of which it is a sum. The exterior product extends to the full exterior algebra, so that it makes sense to multiply any two elements of the algebra. Equipped with this product, the exterior algebra is an associative algebra, which means that α ∧ (β ∧ γ) = (α ∧ β) ∧ γ for any elements α, β, γ. The k-vectors have degree k, meaning that they are sums of products of k vectors. When elements of different degrees are multiplied, the degrees add like multiplication of polynomials. This means that the exterior algebra is a graded algebra.The definition of the exterior algebra makes sense for spaces not just of geometric vectors, but of other vector-like objects such as vector fields or functions. In full generality, the exterior algebra can be defined for modules over a commutative ring, and for other structures of interest in abstract algebra. It is one of these more general constructions where the exterior algebra finds one of its most important applications, where it appears as the algebra of differential forms that is fundamental in areas that use differential geometry. Differential forms are mathematical objects that represent infinitesimal areas of infinitesimal parallelograms (and higher-dimensional bodies), and so can be integrated over surfaces and higher dimensional manifolds in a way that generalizes the line integrals from calculus. The exterior algebra also has many algebraic properties that make it a convenient tool in algebra itself. The association of the exterior algebra to a vector space is a type of functor on vector spaces, which means that it is compatible in a certain way with linear transformations of vector spaces. The exterior algebra is one example of a bialgebra, meaning that its dual space also possesses a product, and this dual product is compatible with the exterior product. This dual algebra is precisely the algebra of alternating multilinear forms, and the pairing between the exterior algebra and its dual is given by the interior product.