Representations by bilinear forms (and.p a)

Introduction Classical theory of symmetric bilinear forms and quadratic forms: Bilinear forms Quadratic forms Forms over rational function fields Function fields of quadrics Bilinear and quadratic forms and algebraic extensions $u$-invariants Applications of the Milnor conjecture On the norm residue homomorphism of degree two Algebraic cycles: Homology and cohomology Chow groups Steenrod operations Category of Chow motives Quadratic forms and algebraic cycles: Cycles on powers of quadrics The Izhboldin dimension Application of Steenrod operations The variety of maximal totally isotropic subspaces Motives of quadrics Appendices Bibliography Notation Terminology.

A metric on a differentiable manifold induces a bilinear form on the tangent space at each point of the manifold. The set of tangent vectors orthogonal, with respect to this bilinear form, to the whole tangent space is a vector subspace of the tangent space: the metric is called non-degenerate or degenerate at the point according as the subspace is or is not empty. This paper is concerned with the geometry of manifolds having everywhere degenerate metrics.

This chapter presents various results about quadratic forms over a field complete with respect to a discrete valuation. The discussion is based on the assumption that K is a field of arbitrary characteristic which is complete with respect to a discrete valuation ν‎ and uses the usual convention that ν‎(0) = infinity. The chapter starts with a notation regarding the ring of integers of K and the natural map from it to the residue field, followed by a number of propositions regarding an anisotropic quadratic space. These include an anisotropic quadratic space with residual quadratic spaces, an unramified quadratic space of finite dimension, unramified finite-dimensional anisotropic quadratic forms over K, unramified anisotropic quadratic forms and a bilinear form, and a round quadratic space over K. The chapter concludes with a theorem that there exists an anisotropic quadratic form over K, unique up to isometry, and is non-singular.

A k-representation of an integer ℓ is a representation of ℓ as sum of k powers of 2, where representations differing by the order are considered as distinct. Let [Formula: see text] be the maximum number of such representations for integers ℓ whose binary representation has exactly σ non-zero digits. [Formula: see text] can be recovered from [Formula: see text] via an explicit formula, thus in some sense [Formula: see text] is the fundamental object. In this paper we prove that [Formula: see text] tends to a computable limit as k diverges. This result improves previous bounds which were obtained with purely combinatorial tools.

Given a bilinear form on , represented by a matrix , the problem of finding the largest dimension of a subspace of such that the restriction of A to this subspace is a non-degenerate skew-symmetric bilinear form is equivalent to finding the size of the largest invertible skew-symmetric matrix B such that the equation is consistent (here denotes the transpose of the matrix X). In this paper, we provide a characterization, by means of a necessary and sufficient condition, for the matrix equation to be consistent when B is a skew-symmetric matrix. This condition is valid for most matrices . To be precise, the condition depends on the canonical form for congruence (CFC) of the matrix A, which is a direct sum of blocks of three types. The condition is valid for all matrices A except those whose CFC contains blocks, of one of the types, with size smaller than 3. However, we show that the condition is necessary for all matrices A.

If f(x 1, …, x n ) is a polynomial dependent on a large number of independent Bernoulli random variables, what can be said about the maximum concentration of f on any single value? For linear polynomials, this reduces to one version of the classical Littlewood-Offord problem: Given nonzero constants a 1, …,a n , what is the maximum number of sums of the form ±a 1 ± a 2 ± … ± a n which take on any single value? Here we consider the case where f is either a bilinear form or a quadratic form. For the bilinear case, we show that the only forms having concentration significantly larger than n −1 are those which are in a certain sense very close to being degenerate. For the quadratic case, we show that no form having many nonzero coefficients has concentration significantly larger than n −1/2. In both cases the results are nearly tight.

In attempting to clear the ground for an attack on the problem of relative maxima and miniima of two quadratic or hermitian forms with aii infinite number of variables, the author found that various questions insistently arose concerning the equivalence of pairs of forms in a finite number of variables and their reduction to normal type. The methods of attack in this theory as set forth in the literature seem to lack systematic unity. Not only are different devices employed for the treatment of the various phases of the problem, but these devices vary for different cases of the same phase. In investigating the problem of maxima and minima the author uncovered fundamental principles which may serve as a new interpretation of the problem of equivalence and make possible a iiew and more unified conception of the treatment. This new foundation for the theory of bilinear forms is set forth in this memoir while the problem of relative extrema is developed in the succeeding one.t To exhibit the formal connection between the problems of relative extrema and of equivalence of pairs of forms, we note that in considering the bi linear forms i,n 1,n Atx y) a B(x, y') bx vZ, yJ _ ijt s j y. j xj jYi ii t~~~~~~~~J

13 - Matrices and related results

Generalized Hurwitz maps of the type S × V → W

As we saw in the last chapter, it is possible to obtain some non-trivial necessary conditions for the existence of orthogonal designs just by considering the equations that the coefficient matrices of an orthogonal design must satisfy. The ad-hoc procedures we give in the last chapter can, with difficulty, be pursued further. However, these procedures quickly become inadequate. To properly describe the solution to the “algebraic problem of orthogonal designs”, it is necessary to discuss the theory of quadratic and bilinear forms. Only then can the “algebraic problem” be put in proper perspective.

Introduction to Mathematics of Inversion in Remote Sensing and Indirect Measurements

The emphasis of this review is both the geometric realization of the 2-point velocity correlation tensor field Bij(x,x′,t) and isometries of the correlation space K3 equipped with a (pseudo-) Riemannian metrics ds2(t) generated by the tensor field. The special form of this tensor field for homogeneous isotropic turbulence specifies ds2(t) as the semi-reducible pseudo-Riemannian metric. This construction presents the template for the application of methods of Riemannian geometry in turbulence to observe, in particular, the deformation of length scales of turbulent motion localized within a singled out fluid volume of the flow in time. This also allows to use common concepts and technics of Lagrangian mechanics for a Lagrangian system (Mt ,ds2(t)), Mt ⊂ K3. Here the metric ds2(t), whose components are the correlation functions, evolves due to the von Kármán-Howarth equation. We review the explicit geometric realization of ds2(t) in K3 and present symmetries (or isometric motions in K3) of the metric ds2(t) which coincide with the sliding deformation of a surface arising under the geometric realization of ds2(t). We expose the fine structure of a Lie algebra associated with this symmetry transformation and construct the basis of differential invariants. Minimal generating set of differential invariants is derived. We demonstrate that the well-known Taylor microscale λg is a second-order differential invariant and show how λg can be obtained by the minimal generating set of differential invariants and the operators of invariant differentiation. Finally, we establish that there exists a nontrivial central extension of the infinite-dimensional Lie algebra constructed wherein the central charge is defined by the same bilinear skew-symmetric form c as for the Witt algebra which measures the number of internal degrees of freedom of the system. For turbulence, we give the asymptotic expansion of the transversal correlation function for the geometry generated by a quadratic form.

On the convergence of bilinear and quadratic forms in independent random variables

SUMMARY The quadratic form appearing in the likelihood function of a Gaussian stationary and invertible vector general linear process is considered in this paper. A new expression for this quadratic form is developed, which allows repeated computations to be performed very efficiently by possibly using forecast and backcast values of the series. While the quadratic form itself involves the inverse of the covariance matrix, the new expression uses the autocovariances of the 'inverse-transpose' model. The new expression outperforms alternative procedures in terms of their usefulness to compute approximate maximum likelihood estimates for the parameters in the general linear model. Although the methods developed in the paper can be applied to finite order autoregressive-moving average models, they are most useful when the process does not have a representation with finite autoregressive and moving average orders. The case of a vector autoregressive integrated moving average process in which the degree of differencing is allowed to take fractional values is considered in detail.

There are various methods in existence for the practical solution of a set of simultaneous equationsSome of these methods are appropriate to special systems, as for example to the axisymmetric “normal equations” of Least Squares. In many applications, however, as in problems of statistical correlation of many variables, it may be desired not merely to solve a given set of equations but to obtain as much knowledge as possible about the system or matrix of coefficients; perhaps to evaluate its determinant and various minors, such as principal minors, possibly also to determine the elements of the adjugate matrix, or the reciprocal matrix. The examination of the sign of successive principal minors of an axisymmetric determinant, in order to find the signature of the corresponding quadratic form, is a case in point; and there are many such applications.