Spinor equivalence of quadratic forms

Let f be an integral quadratic form in three or more variables and g any form in the genus of f. There exist an effectively determinable prime p and a form g′, belonging to the proper spinor genus of g, such that g′ is a p-neighbor of f in the graph of f. Using this, an alternative decision procedure for the spinor equivalence of quadratic forms is given.     

Motivic equivalence of quadratic forms. II

Let F be a field of characteristic ≠2 and φ be a quadratic form over F. By X _φ we denote the projective variety given by the equation φ=0. For each positive even integer d≥8 (except for d=12) we construct a field F and a pair φ, ψ of anisotropic d-dimensional forms over F such that the Chow motives of X _φ and X _ψ coincide but . For a pair of anisotropic (2 ^{n -1})-dimensional quadrics X and Y, we prove that existence of a rational morphism Y→X is equivalent to existence of a rational morphism Y→X. Received: 27 September 1999 / Revised version: 27 December 1999     

The Zariski problem for function fields of quadratic forms

By `a quadratic function field' is meant the affine function field of a nonsingular quadratic form of dimension . What quadratic function fields contain a given quadratic function field ? This problem is solved here for quadratic forms of dimensions 3 and 4, and an application to the Zariski cancellation problem for quadratic function fields is given.

     

Criteria of motivic equivalence for quadratic forms and central simple algebras

We give a short, elementary, and characteristic independent proof of the criterion for motivic isomorphism of two projective quadrics discovered by A. Vishik [24]. We also give a criterion for motivic isomorphism of two Severi-Brauer varieties. Received: 1 February 1999 / Published online: 8 May 2000     

Steinhaus tiling problem and integral quadratic forms

A lattice in is said to be equivalent to an integral lattice if there exists a real number such that the dot product of any pair of vectors in is an integer. We show that if and is equivalent to an integral lattice, then there is no measurable Steinhaus set for , a set which no matter how translated and rotated contains exactly one vector in .

     

A canonical decomposition for quadratic forms with applications to monotone convergence theorems

We prove monotone convergence theorems for quadratic forms on a Hilbert space which improve existing results. The main tool is a canonical decomposition for any positive quadratic form h = h_r + h_s where h_r is characterized as the largest closable form smaller than h. There is also a systematic discussion of nondensely defined forms.     

The Mather problem for lower semicontinuous Lagrangians

In this paper we develop the Aubry-Mather theory for Lagrangians in which the potential energy can be discontinuous. Namely we assume that the Lagrangian is lower semicontinuous in the state variable, piecewise smooth with a (smooth) discontinuity surface, as well as coercive and convex in the velocity. We establish existence of Mather measures, various approximation results, partial regularity of viscosity solutions away from the singularity, invariance by the Euler–Lagrange flow away from the singular set, and further jump conditions that correspond to conservation of energy and tangential momentum across the discontinuity.     

A criterion for equivalence of quadratic forms on a special class of riemannian manifolds

We consider two quadratic forms on a class of manifolds conformally equivalent to the euclidean space. We give necessary and sufficient conditions for the equivalence of these quadratic forms in terms of a weighted Hardy inequality. The research was supported by the Chateaubriand Scholarship 2000–2001.     

Normal forms for real quadratic forms

We investigate the non-diagonal normal forms of a quadratic form on , in particular for n = 3. For this case it is shown that the set of normal forms is the closure of a 5-dimensional submanifold in the 6-dimensional Grassmannian of 2-dimensional subspaces of . Received: 27 June 2008     

Noncommutative classical and quantum mechanics for quadratic Lagrangians (Hamiltonians)

Classical and quantum mechanics based on an extended Heisenberg algebra with additional canonical commutation relations for position and momentum coordinates are considered. In this approach additional noncommutativity is removed from the algebra by a linear transformation of coordinates and transferred to the Hamiltonian (Lagrangian). This linear transformation does not change the quadratic form of the Hamiltonian (Lagrangian), and Feynman’s path integral preserves its exact expression for quadratic models. The compact general formalism presented here can be easily illustrated in any particular quadratic case. As an important result of phenomenological interest, we give the path integral for a charged particle in the noncommutative plane with a perpendicular magnetic field. We also present an effective Planck constant ħ _eff which depends on additional noncommutativity.     

Invariants for quadratic forms

Simple systems of invariants for rational and integral quadratic forms are given, and those for rational forms are proved complete in an elementary way. Some noninvariants of quadratic forms appear, but are not concerned with invariants of objects other than quadratic forms. Our treatment of noninvariants of objects other than quadratic forms is minimal, and it is here that there is most room for further investigation.     

The norm theorem for semisingular quadratic forms

Let F be a field of characteristic 2. The aim of this paper is to give a complete proof of the norm theorem for singular F-quadratic forms which are not totally singular, i.e., we give necessary and sufficient conditions for which a normed irreducible polynomial of $F[x_1,\dots ,x_n]$ becomes a norm of such a quadratic form over the rational function field $F(x_1,\dots ,x_n)$. This completes partial results proved on this question in [8]. Combining the present work with the papers [1] and [7], we obtain the norm theorem for any type of quadratic forms in characteristic 2.     

One-dimensional Lagrangians generated by a quadratic form

In this paper we apply the classical variational calculus to the Lagrangians of particular type. Namely, we establish formulas for the 1^st integrals and propose a technique for obtaining invariant 1^st integrals. We also deduce the differential homogeneity conditions for Lagrangians with respect to multiplication of a path x(t) by a function c(t) and with respect to the change of the parameter t = t(s).