On the Kobayashi-Royden metric for ellipsoids

We construct a convex and strongly pseudoconvex Kobayashi positive Finsler metric on a vector bundle E under the assumption that the symmetric power of the dual $S^kE^*$ has a Griffiths negative $L^2$ -metric for some k. The proof relies on the negativity of direct image bundles and the Minkowski inequality for norms. As a corollary, we show that given a strongly pseudoconvex Kobayashi positive Finsler metric, one can upgrade to a convex Finsler metric with the same property. We also give an extremal characterization of Kobayashi curvature for Finsler metrics.

We obtain solution representation formulae for some linear initial boundary value problems posed on the half space that involve mixed spatial derivative terms via the unified transform method (UTM), also known as the Fokas method. We first implement the method on the second-order parabolic PDEs; in this case one can alternatively eliminate the mixed derivatives by a linear change of variables. Then, we employ the method to biharmonic problems, where it is not possible to eliminate the cross term via a linear change of variables. A basic ingredient of the UTM is the use of certain invariant maps. It is shown here that these maps are well defined provided that certain analyticity issues are appropriately addressed.

Center problem for Λ–Ω differential systems

We defined the quadratic Hermitian forms on a space of even dimension over a finite field; they are constructed from classical Hermitian forms. We give the exact value of exponential sums associated to the quadratic Hermitian forms. On the other hand, we define with the space of quadratic Hermitian forms a new family of subcodes of the Reed-Muller code of order two, and we give the value of the parameters of these codes and of the associated trace codes.

The understanding of chaos and strange attractors is one of the most exciting areas of mathematics today. It is the question of how the asymptotic behaviour of deterministic systems can exhibit unpredictability and apparent chaos, due to sensitive dependence upon initial conditions, and yet at the same time preserve a coherent global structure. The field represents a remarkable confluence of several different strands of thought. 1. Firstly came the influence of differential topology, giving global geometric insight and emphasis on qualitative properties. By qualitative properties I mean invariants under differentiable changes of coordinates, as opposed to quantitative properties which are invariant only under linear changes of coordinates. To give an example of this influence, I recall a year-long symposium at Warwick in 1979/80, which involved sustained interaction between pure mathematicians and experimentalists, and one of the most striking consequences of that interaction was a transformation in the way that experimentalists now present their data. It is generally in a much more translucent form: instead of merely plotting a frequency spectrum and calling the incomprehensible part ‘noise’, they began to draw computer pictures of underlying three-dimensional strange attractors.

Characterization of vector valued, gaussian, stationary, markov processes

We study the parameters of codes spanned by quadratic and Hermitian forms in projective space. The resulting codes have high minimum distance and few distinct weights. An example is a new ternary [91.9,54] code.

We introduce the notion of hyperbolic equivalence for quadric bundles and quadratic forms on vector bundles and show that hyperbolic equivalent quadric bundles share many important properties: they have the same Brauer data; moreover, if they have the same dimension over the base, they are birational over the base and have equal classes in the Grothendieck ring of varieties. Furthermore, when the base is a projective space we show that two quadratic forms are hyperbolic equivalent if and only if their cokernel sheaves are isomorphic up to twist, their fibers over a fixed point of the base are Witt equivalent, and, in some cases, certain quadratic forms on intermediate cohomology groups of the underlying vector bundles are Witt equivalent. For this we show that any quadratic form over $\mathbb{P}^n$ is hyperbolic equivalent to a quadratic form whose underlying vector bundle has many cohomology vanishings; this class of bundles, called VLC bundles in the paper, is interesting by itself.

The concept of osculating Riemannian spaces of Finsler spaces was introduced by A. Nazim [6], later O. Varga [10] studied it in details and used in the determination of Cartan's connection in Finsler spaces. The method of osculation was extended to the connections of Finsler spaces and those of the line element manifold by A. Mo6r [4], [5] and to areal spaces by L. TamAssy [9]. Roughly speaking, the osculation means that to a rather complicated structure (Finsler structure, Finsler connection), a relatively simple structure (a Riemannian metric, an affine or linear connection) is associated, which approximates the former in some sense. Based on this link there is a hope to obtain results on the more complicated structure. Our aim in this paper is to perform and investigate osculations for Finsler-type connections in a global setting. In Section 1 the necessary preliminaries on connections are given. The Finslertype connections most often used in this paper are Finsler pair connections (see [7], [1]), which are analogous to Matsumoto's Finsler connections [3]. Using the Caftan vector field C, in Section 2 the notion of C-osculating horizontal map is given, and the role of the C1 condition is investigated. In Section 3 the osculation of a connection of the Finsler vector bundle V~ along a section of r is given and its properties are studied. This osculation means the construction of a connection in the base vector bundle 4, whose parallel transport is closely related to the original connection in V4. In the last section the h-osculation, related to the connection is considered, and the link with the classical results [4] is given.

The object of this work is to define, by analogy with algebra, the Witt group and the graded Brauer group of a topological space X.A homomorphism is defined between them analogous to the generalized Hasse invariant.Upon evaluation, the Witt group is seen to be KO(X), the graded Brauer group 1+H1(X;ZZ) + H2(X; Z2) with truncated cup product multiplication, while the homomorphism is given by Stiefel-Whitney classes: I + m^ + h^.Introduction.The Hasse invariant of a quadratic form on a vector space over a field k is an element of the Brauer group of k.It can be generalized to a homomorphism from the Witt group of classes of quadratic forms to the graded Brauer group of classes of graded ^-algebras [3], [16].Curiously enough, in this guise it is quite analogous to the homomorphism given by the first two Stiefel-Whitney classes which maps real vector bundles over a space X to the group 1 + H\X; Z2) + H2(X;Z2), where multiplication is by truncated cup product.The purpose of this paper is to display this analogy.Delzant has defined Stiefel-Whitney classes of quadratic forms [4], [13] using the discriminant and Hasse invariant.It should add insight into his definition to see that the first two topological Stiefel-Whitney classes can also be defined in an algebraic way.A vector bundle L is known to be orientable for real L-theory if its first two Stiefel-Whitney classes vanish.In this case it can be read off from the diagram of Theorem 3.1 that the Clifford bundle of L is a bundle of graded endomorphism algebras of a graded module bundle, which is then used to construct the Thorn class of L. Donovan and Karoubi in studying this orientability question have done independently much of this same work, applying it further to define L-theory with local coefficients [6], [7].The connection between algebra and topology comes from considering bundles over X whose fibers are vector spaces with quadratic forms or graded algebras and subjecting them to the same analysis the fiber alone is accorded in algebra.This is described in some detail in 1.11 is devoted to the algebraic computations and 111 to a study of the topological problems.

Given an Azumaya algebra with involution $$(A,\sigma )$$ over a commutative ring R and some auxiliary data, we construct an 8-periodic chain complex involving the Witt groups of $$(A,\sigma )$$ and other algebras with involution, and prove it is exact when R is semilocal. When R is a field, this recovers an 8-periodic exact sequence of Witt groups of Grenier-Boley and Mahmoudi, which in turn generalizes exact sequences of Parimala–Sridharan–Suresh and Lewis. We apply this result in several ways: We establish the Grothendieck–Serre conjecture on principal homogeneous bundles and the local purity conjecture for certain outer forms of $${\mathbf {GL}}_n$$ and $${\mathbf {Sp}}_{2n}$$ , provided some assumptions on R. We show that a 1-hermitian form over a quadratic étale or quaternion Azumaya algebra over a semilocal ring R is isotropic if and only if its trace (a quadratic form over R) is isotropic, generalizing a result of Jacobson. We also apply it to characterize the kernel of the restriction map $$W(R)\rightarrow W(S)$$ when R is a (non-semilocal) 2-dimensional regular domain and S is a quadratic étale R-algebra, generalizing a theorem of Pfister. In the process, we establish many fundamental results concerning Azumaya algebras with involution and hermitian forms over them.

Quadratic forms and Hermitian forms

For each positive integer n n , let g Z ( n ) g_\mathbb {Z}(n) be the smallest integer such that if an integral quadratic form in n n variables can be written as a sum of squares of integral linear forms, then it can be written as a sum of g Z ( n ) g_\mathbb {Z}(n) squares of integral linear forms. We show that every positive definite integral quadratic form is equivalent to what we call a balanced Hermite–Korkin–Zolotarev-reduced form and use it to show that the growth of g Z ( n ) g_\mathbb {Z}(n) is at most an exponential of n \sqrt {n} . Our result improves the best known upper bound on g Z ( n ) g_\mathbb {Z}(n) which is on the order of an exponential of n n . We also define an analogous number g O ∗ ( n ) g_{\mathcal O}^*(n) for writing Hermitian forms over the ring of integers O \mathcal O of an imaginary quadratic field as sums of norms of integral linear forms, and when the class number of the imaginary quadratic field is 1 1 , we show that the growth of g O ∗ ( n ) g_{\mathcal O}^*(n) is at most an exponential of n \sqrt {n} . We also improve on results of both Conway and Sloane and Kim and Oh on s s -integrable lattices.

In these lectures we give an introduction to the reduction theory of binary forms starting with quadratic forms with real coefficients, Hermitian forms, and then define the Julia quadratic for any degree n binary form. A survey of a reduction algorithm over ${\mathbb{Z}}$ is described based on recent work of Cremona and Stoll.

Hermitian forms and locally toroidal regular polytopes