Lattice point theory in polyhedra

The exact order of the remainder term is determined in the formula for the number of lattice points in the region α₁∥u₁ + b₁∥ + α₂∥u₂ + b₂∥ + … + α_r∥u_r + b_r∥ ≤ x in dependence on the arithmetical properties of the coefficients α₁, α₂,…, α_r.     

A sum formula related to ellipsoids with applications to lattice point theory

In this article we consider finite and infinitep-dimensional sums over functionsf, where the argument off is represented by a positive definite quadratic form. We develop a sum formula like theEuler-Maclaurin orPoisson sum formula. Applications to exponential sums and lattice point problems are given.     

Lattice points in rational ellipsoids

We combine exponential sums, character sums and Fourier coefficients of automorphic forms to improve the best known upper bound for the lattice error term associated to rational ellipsoids.     

One-loop effective action for an arbitrary theory

Using the diagram technique, we obtain formulas for the divergent part of the one-loop effective action for an arbitrary minimal operator in four-dimensional curved space and for an arbitrary nonminimal operator in flat space.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 109, No. 2, pp. 215–231, November, 1996.     

On maximization of quadratic form over intersection of ellipsoids with common center

We demonstrate that if A ₁,...,A _m are symmetric positive semidefinite n×n matrices with positive definite sum and A is an arbitrary symmetric n×n matrix, then the relative accuracy, in terms of the optimal value, of the semidefinite relaxation of the optimization program is not worse than . It is shown that this bound is sharp in order, as far as the dependence on m is concerned, and that a~feasible solution x to (P) with can be found efficiently. This somehow improves one of the results of Nesterov [4] where bound similar to (*) is established for the case when all A_i are of rank 1. Received August 13, 1998 / Revised version received May 25, 1999? Published online September 15, 1999     

A reduced Hsieh-Clough-Tocher element with splitting based on an arbitrary interior point

We present formulas for a reduced Hsieh-Clough-Tocher (rHCT) element with splitting based on an arbitrary interior point. These formulas use local barycentric coordinates in each of the subtriangles and are not significantly more complicated than formulas for an rHCT element with splitting based on the centroid.     

Involutions fixing an arbitrary product of spheres and a point

Summary In this paper we identify up to cobordism all involutions whose fixed point set is the disjoint union of an arbitrary product of spheres and a point. This article was processed by the author using the Springer-Verlag T_EX macro package 1991     

Realizations for Schur upper triangular operators centered at an arbitrary point

Reproducing kernel spaces introduced by L. de Branges and J. Rovnyak provide isometric, coisometric and unitary realizations for Schur functions, i.e. for matrix-valued functions analytic and contractive in the open unit disk. In our previous paper [12] we showed that similar realizations exist in the nonstationary setting, i.e. when one considers upper triangular contractions (which appear in time-variant system theory as transfer functions of dissipative systems) rather than Schur functions and diagonal operators rather than complex numbers. We considered in [12] realizations centered at the origin. In the present paper we study realizations of a more general kind, centered at an arbitrary diagonal operator. Analogous realizations (centered at a point of the open unit disk) for Schur functions were introduced and studied in [3] and [4].     

Lattice inequalities for convex bodies and arbitrary lattices

Two isoperimetric inequalities with lattice constraints for arbitrary lattices are proved, where the last successive minimum of the lattice is used. The results generalize previous results by Hadwiger et al. for the special lattice ^d to general lattices.     

Lattice theory and metric geometry

K. Menger and G. Birkhoff recognized 70 years ago that lattice theory provides a framework for the development of incidence geometry (affine and projective geometry). We show in this article that lattice theory also provides a framework for the development of metric geometry (including the euclidean and classical non-euclidean geometries which were first discovered by A. Cayley and F. Klein). To this end we introduce and study the concept of a Cayley–Klein lattice. A detailed investigation of the groups of automorphisms and an algebraic characterization of Cayley–Klein lattices are included. The authors would like to thank an unknown referee for his helpful suggestions.     

Lattice point generating functions and symmetric cones

We show that a recent identity of Beck–Gessel–Lee–Savage on the generating function of symmetrically constrained compositions of integers generalizes naturally to a family of convex polyhedral cones that are invariant under the action of a finite reflection group. We obtain general expressions for the multivariate generating functions of such cones, and work out their general form more specifically for all symmetry groups of type A (previously known) and types B and D (new). We obtain several applications of these expressions in type B, including identities involving permutation statistics and lecture hall partitions.     

Coincidence and common fixed point results on metric spaces endowed with an arbitrary binary relation and applications

In this paper, we establish coincidence and common fixed point theorems for contractive mappings on a metric space endowed with an amorphous binary relation. The presented theorems extend the results of Samet and Turinici in [Commun. Math. Anal. 12 (2012), 82– 97] and generalize many existing results on metric and ordered metric spaces. We apply also our main results to derive coincidence and common fixed point theorems for cyclic contractive mappings.     

Lattice point problems and values of quadratic forms

For d-dimensional ellipsoids E with d5 we show that the number of lattice points in rE is approximated by the volume of rE, as r tends to infinity, up to an error of order for general ellipsoids and up to an error of order o(r^d-2) for irrational ones. The estimate refines earlier bounds of the same order for dimensions d9. As an application a conjecture of Davenport and Lewis about the shrinking of gaps between large consecutive values of Q[m],m^d of positive definite irrational quadratic forms Q of dimension d5 is proved. Finally, we provide explicit bounds for errors in terms of certain Minkowski minima of convex bodies related to these quadratic forms. Mathematics Subject Classification (1991) 11P21     

Stochastic distributed systems with point observations and boundary control: an abstract theory

There already exists a fairly complete theory for the problems of estimation and stochastic optimal control for linear distributed parameter systems, with Gaussian or non Gaussian noise disturbance. In [8] and [12] generalizations of the familiar finite dimensional results of the Kalman-Bucy filter and the separation principle are obtained using an abstract input-output Hilbert space representation for the system. However, in [8] and [12] all the input operators are assumed to be bounded and so it does not cover the important practical cases of control and noise on submanifolds of the spatial domain or point observations. Here we introduce unbounded system operators in the abstract input-output Hilbert space representation and thus extend all the results of [8] and [12] to allow for point observations and noise and control on submanifolds including the boundary. The theory is illustrated by several examples.