Local-global principles for representations of quadratic forms

We prove a local-global principle for the problem of representations of quadratic forms by quadratic forms over ℤ, in codimension ≥5. The proof uses the ergodic theory of p-adic groups, together with a fairly general observation on the structure of orbits of an arithmetic group acting on integral points of a variety.     

Extensions of representations of integral quadratic forms

Let N and M be quadratic ?-lattices, and K be a sublattice of N. A representation σ:K→M is said to be extensible to N if there exists a representation ρ:N→M such that ρ | _K =σ. We prove in this paper a local–global principle for extensibility of representation, which is a generalization of the main theorems on representations by positive definite ?-lattices by Hsia, Kitaoka and Kneser (J. Reine Angew. Math. 301:132–141, 1978) and Jöchner and Kitaoka (J. Number Theory 48:88–101, 1994). Applications to almost n-universal lattices and systems of quadratic equations with linear conditions are discussed.     

Local minima of quadratic forms on convex cones

We study the local minima and the critical values of a quadratic form on the trace of a convex cone. This variational problem leads to the development of a spectral theory that combines matrix algebra and facial analysis of convex cones.      

Multiplicative factorizations of integral representations by quadratic forms

Multiplicative dependence of integral representations of integers by positive definite quadratic forms in an odd number of variables on square factors of the represented numbers is studied. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 212, 1994, pp. 15–55. Translated by A. S. Goloubeva.     

Limit theorems for convex hulls

It is shown that the process of vertices of the convex hull of a uniform sample from the interior of a convex polygon converges locally, after rescaling, to a strongly mixing Markov process, as the sample size tends to infinity. The structure of the limiting Markov process is determined explicitly, and from this a central limit theorem for the number of vertices of the convex hull is derived. Similar results are given for uniform samples from the unit disk.     

Strong approximation of quadrics and representations of quadratic forms

Let V be an indefinite quadratic space over a number field F and U be a nondegenerate subspace of V. Suppose that M is a lattice on V, and that N is a lattice on U which is represented by M locally everywhere. The main result of this paper is a necessary and sufficient condition for which there exists a representation of N by M that approximates a given family of local representations. This is applied to determine when the variety of representations of U by V has strong approximation with respect to a finite set of primes of F that contains all the archimedean primes.     

On simultaneous representations of primes by binary quadratic forms

Let p ≡ ± 1 (mod 8) be a prime which is a quadratic residue modulo 7. Then p = M² + 7N², and knowing M and N makes it possible to “predict” whether p = A² + 14B² is solvable or p = 7C² + 2D² is solvable. More generally, let q and r be distinct primes, and let an integral solution of H²p = M² + qN² be known. Under appropriate assumptions, this information can be used to restrict the possible values of K for which K²q = A² + qrB² is solvable and the possible values of K′ for which K′²p = qC² + rD² is solvable. These restrictions exclude some of the binary quadratic forms in the principal genus of discriminant ?4qr from representing p.     

Limit theorems for functionals of convex hulls

Summary In [4] a central limit theorem for the number of vertices of the convex hull of a uniform sample from the interior of a convex polygon is derived. This is done by approximating the process of vertices of the convex hull by the process of extreme points of a Poisson point process and by considering the latter process of extreme points as a Markov process (for a particular parametrization). We show that this method can also be applied to derive limit theorems for the boundary length and for the area of the convex hull. This extents results of Rényi and Sulanke (1963) and Buchta (1984), and shows that the boundary length and the area have a strikingly different probabilistic behavior.     

Metric entropy of convex hulls

LetT be a precompact subset of a Hilbert space. The metric entropy of the convex hull ofT is estimated in terms of the metric entropy ofT, when the latter is of order ε^ℒ2. The estimate is best possible. Thus, it answers a question left open in [CKP].     

Interior points of convex hulls

If a setX ⊂E ⁿ has non-emptyk-dimensional interior, or if some point isk-dimensional surrounded, then the classic theorem of E. Steinitz may be extended. For example ifX ⊂E ⁿ has int _k X ≠ 0, (0 ≦k≦n) and ifp ɛ int conX, thenp ɛ int conY for someY ⊂X with cardY≦2n−k+1.     

Approximating largest convex hulls for imprecise points

Assume that a set of imprecise points in the plane is given, where each point is specified by a region in which the point will lie. Such a region can be modelled as a circle, square, line segment, etc. We study the problem of maximising the area of the convex hull of such a set. We prove NP-hardness when the imprecise points are modelled as line segments, and give linear time approximation schemes for a variety of models, based on the core-set paradigm.     

Number of the representations of integers by certain ternary quadratic forms

Exact formulas are derived for the number of the representations of positive integers by certain positive ternary quadratic forms, belonging to multiclass genera. The Fourier coefficients of the corresponding cusp forms (on the basis of the known results of Waldspurger and Tunnell) are expressed in terms of the values of the Hasse-Weil L-functions of certain elliptic curves at the center of the critical strip.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 154, pp. 154–162, 1986.     

Evaluating inclusion functionals for random convex hulls

Summary The inclusion functional of a random convex set, evaluated at a fixed convex set K, measures the probability that the random convex set contains K. This functional is an analogue of the complement of the distribution function of an ordinary random variable. A methodology is described for evaluating the inclusion functional for the case where the random convex set is generated as the convex hull of n i.i.d. points from a distribution function F in the plane. For general K and F, the inclusion probability is difficult to compute in closed form. The case where K is a straight line segment is examined in detail and, in this situation, a simple answer is given for an interesting class of distributions F.