The geometry of orthogonal groups over finite fields
... (see Theorem A.6). Indeed, the Witt’s extension theorem implies that the group of isometries acts transitively on the set of maximal singular subspaces. Yet another proof of this fact will be given in Appendix B. Lemma 2.2 Let K be a finite field of odd characteristic. Then for any α ∈ K there exist ...
... (see Theorem A.6). Indeed, the Witt’s extension theorem implies that the group of isometries acts transitively on the set of maximal singular subspaces. Yet another proof of this fact will be given in Appendix B. Lemma 2.2 Let K be a finite field of odd characteristic. Then for any α ∈ K there exist ...
Basic Algebra Skills
... The terms −6xy and yx are like terms. Both consist of the same variables. The order in which they appear is irrelevant. This is because multiplication is commutative. The terms 52y 2 x 3 z and −15zy 2 x 3 are like terms. Both consist of the same variables. The order in which they appear is irrelevan ...
... The terms −6xy and yx are like terms. Both consist of the same variables. The order in which they appear is irrelevant. This is because multiplication is commutative. The terms 52y 2 x 3 z and −15zy 2 x 3 are like terms. Both consist of the same variables. The order in which they appear is irrelevan ...
DOC - Brunel University London
... considered examples can be applied only to a limited range of material fatigue parameters and cannot describe the crack start delay. The non-local approach is free of the drawbacks. When the stress fields are available analytically or numerically and the strength conditions are associated with the l ...
... considered examples can be applied only to a limited range of material fatigue parameters and cannot describe the crack start delay. The non-local approach is free of the drawbacks. When the stress fields are available analytically or numerically and the strength conditions are associated with the l ...
Homological Algebra
... More generally, if I is any index set, then we can think of basis elements ei for i in I and form the free left-R-module F of formal linear combinations of the ei . So the elements of F are finite linear combinations r1 ei1 + · · · + rt eit with the rj in R. The module F has the same property with r ...
... More generally, if I is any index set, then we can think of basis elements ei for i in I and form the free left-R-module F of formal linear combinations of the ei . So the elements of F are finite linear combinations r1 ei1 + · · · + rt eit with the rj in R. The module F has the same property with r ...
Standard Monomial Theory and applications
... weight lattice X of a complex semisimple Lie algebra (or, more generallly, symmetrizable Kac-Moody algebra) g, and denote by Π the set of all piecewise linear paths π : [0, 1]Q → XQ starting in 0 and ending in an integral weight. We associate to a simple root α operators eα and fα on Π, and, using t ...
... weight lattice X of a complex semisimple Lie algebra (or, more generallly, symmetrizable Kac-Moody algebra) g, and denote by Π the set of all piecewise linear paths π : [0, 1]Q → XQ starting in 0 and ending in an integral weight. We associate to a simple root α operators eα and fα on Π, and, using t ...
Operator-valued measures, dilations, and the theory
... cannot decrease and the upper bound cannot increase). In particular, {P un } is a Parseval frame for H when {un } is an orthonormal basis for K, i.e., every orthogonal compression of an orthonormal basis (resp. Riesz basis) is a Parseval frame (resp. frame) for the projection subspace. The converse ...
... cannot decrease and the upper bound cannot increase). In particular, {P un } is a Parseval frame for H when {un } is an orthonormal basis for K, i.e., every orthogonal compression of an orthonormal basis (resp. Riesz basis) is a Parseval frame (resp. frame) for the projection subspace. The converse ...
MONOMIAL IDEALS, ALMOST COMPLETE INTERSECTIONS AND
... the form (x3 , y 3 , z 3 , f (x, y, z)), with deg f = 3, fails to have the WLP if and only if f ∈ (x3 , y 3 , z 3 , xyz). In particular, the latter ideal is the only such monomial ideal that fails to have the WLP. This paper continues the study of this question. The example of Brenner and Kaid satis ...
... the form (x3 , y 3 , z 3 , f (x, y, z)), with deg f = 3, fails to have the WLP if and only if f ∈ (x3 , y 3 , z 3 , xyz). In particular, the latter ideal is the only such monomial ideal that fails to have the WLP. This paper continues the study of this question. The example of Brenner and Kaid satis ...
Numerical solution of saddle point problems
... real coefficients, and in this paper we restrict ourselves to the real case. Complex coefficient matrices, however, do arise in some cases; see, e.g., Bobrovnikova and Vavasis (2000), Mahawar and Sarin (2003) and Strang (1986, page 117). Most of the results and algorithms reviewed in this paper admit st ...
... real coefficients, and in this paper we restrict ourselves to the real case. Complex coefficient matrices, however, do arise in some cases; see, e.g., Bobrovnikova and Vavasis (2000), Mahawar and Sarin (2003) and Strang (1986, page 117). Most of the results and algorithms reviewed in this paper admit st ...
On Boolean Ideals and Varieties with Application to
... Hilbert theorem on basis, each ideal of the ring K[x1 , . . . , xn ] is finitely generated [12] and is uniquely defined by the intersection of the powers of maximal ideals. For an ideal A ⊆ K[x1 , . . . , xn ] with variety V (A) coordinate ring K[x1 , . . . , xn ]/A is defined, whose elements are calle ...
... Hilbert theorem on basis, each ideal of the ring K[x1 , . . . , xn ] is finitely generated [12] and is uniquely defined by the intersection of the powers of maximal ideals. For an ideal A ⊆ K[x1 , . . . , xn ] with variety V (A) coordinate ring K[x1 , . . . , xn ]/A is defined, whose elements are calle ...
Centre de Recerca Matem`atica
... This is a very important notion in foliation theory. To make it clear, let us give the main examples of such structures. 2.2. Lie foliations We say that F is a Lie G-foliation, if T is a Lie group G and γij are restrictions of left translations on G. Such foliation can also be defined by a 1-form ω ...
... This is a very important notion in foliation theory. To make it clear, let us give the main examples of such structures. 2.2. Lie foliations We say that F is a Lie G-foliation, if T is a Lie group G and γij are restrictions of left translations on G. Such foliation can also be defined by a 1-form ω ...
Linear algebra
Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It includes the study of lines, planes, and subspaces, but is also concerned with properties common to all vector spaces.The set of points with coordinates that satisfy a linear equation forms a hyperplane in an n-dimensional space. The conditions under which a set of n hyperplanes intersect in a single point is an important focus of study in linear algebra. Such an investigation is initially motivated by a system of linear equations containing several unknowns. Such equations are naturally represented using the formalism of matrices and vectors.Linear algebra is central to both pure and applied mathematics. For instance, abstract algebra arises by relaxing the axioms of a vector space, leading to a number of generalizations. Functional analysis studies the infinite-dimensional version of the theory of vector spaces. Combined with calculus, linear algebra facilitates the solution of linear systems of differential equations.Techniques from linear algebra are also used in analytic geometry, engineering, physics, natural sciences, computer science, computer animation, and the social sciences (particularly in economics). Because linear algebra is such a well-developed theory, nonlinear mathematical models are sometimes approximated by linear models.