The linear polarization constant of R^n

The present work contributes to the determination of the n-th linear polarization constant cn(H) of an n-dimensional real Hilbert space H. We provide some new lower bounds on the value of supkyk=1 | hx1, yi · · · hxn, yi |, where x1, . . . , xn are unit vectors in H. In particular, the results improve an earlier estimate of Marcus. However, the intriguing conjecture cn(H) = nn/2 remains open

On quasi-contractivity of C 0-semigroups on Banach spaces

A basic result in semigroup theory states that every C-0-semigroup is quasi-contractive with respect to some appropriately chosen equivalent norm. This paper contains a counterpart of this well-known fact. Namely, by examining the convergence of the Trotter-type formula (e(t/n) (A) p)(n) (where P denotes a bounded projection), we prove that whenever the generator A is unbounded it is possible to introduce an equivalent norm on the space with respect to which the semigroup is not quasi-contractive

COMMUTATION PROPERTIES OF THE FORM SUM OF POSITIVE, SYMMETRIC OPERATORS

A new construction for the form sum of positive, selfadjoint operators is given in this paper. The situation is a bit more general, because our aim is to add positive, symmetric operators. With the help of the used method, some commutation properties of the form sum extension are observed

Minimal positive realizations for a class of transfer functions

It is a standard result in linear-system theory that an nth-order rational transfer function of a single-input single-output system always admits a realization of order n. In some applications, however, one is restricted to realizations with nonnegative entries (i.e. a positive system), and it is known that this restriction may force the order N of realizations to be strictly larger than n. In this brief we present a class of transfer functions where positive realizations of order n do exist. With the help of our result we give improvements on some earlier results in positive-system theory

Minimal positive realizations of transfer functions with nonnegative multiple poles

This note concerns a particular case of the minimality problem in positive system theory. A standard result in linear system theory states that any nth-order rational transfer function of a discrete time-invariant linear single-input-single-output (SISO) system admits a realization of order n. In some applications, however, one is restricted to realizations with nonnegative entries (i.e., a positive system), and it is known that this restriction may force the order N of realizations to be strictly larger than n. A general solution to the minimality problem (i.e., determining the smallest possible value of N) is not known. In this note, we consider the case of transfer functions with nonnegative multiple poles, and give sufficient conditions for the existence of positive realizations of order N = n. With the help of our results we also give an improvement of an existing result in positive system theory

Algorithm for positive realization of transfer functions

The aim of this brief is to present a finite-step algorithm for the positive realization of a rational transfer function H(z). In comparision with previously described algorithms we emphasize that we do not make an a priori assumption on (but, instead, include a finite step procedure for checking) the non- negativity of the impulse response sequence of H(z). For primitive transfer functions a new method for reducing the pole order of the dominant pole is also proposed

Elmélet és alkalmazás a matematikai analízisben = Theory and applications in mathematical analysis

A kutatási tervnek megfelelően több területen is sikerült eredményeket elérni: Fourier analízissel kapcsolatos problémákban a Fugelde sejtésre adott ellenpéldák dimenzioját 3-ra csökkentettük mindkét irányban, valamint hatékony algoritmust adtunk ciklikus csoporotok nem-periodikus parkettázásainak megkeresésére. (A parkettázási eredmények hatására kezdtem kutatni összeghalmazok számossági és strukturális kérdéseit, és itt is született néhány kezdeti eredmény.) Ezen kívül tanulmányoztuk a pozitív, pozitív definit függvények kúpjának extremálisait, valamint egy síkbefedési problémát rácsszerű halmaz elforgatottjaival. Komplex Hadamard mátrixok és MUB-hármasok új családjait adtuk meg, valamint egy ígéretes diszkretizációs módszert a MUB-6 probléma jövőbeni megoldására. Ez a nevezetes nyitott problema azt kérdezi, hogy legfeljebb hány kölcsönösen torzítatlan bázis (MUB) adható meg komplex 6 dimenzióban. Ezeknek a kérdéseknek kvantum-információelméletbeli gyökerei vannak. Megvizsgáltuk a relativisztikus Thomas-rotácio jelenségét, valamint a GPS-ben használt közelítő képlet helyett pontos formulát mutattunk a Föld bázisú és a GPS-szatelliteken levő órak időmérésének differenciál hányadosára. (Ez utóbbinak csak elméleti jelentősége van, mert a közelítő formula már eddig is elég pontos volt gyakorlati szempontból.) A pozitív realizációs témakörben általános és hatékony algoritmust adtunk az összes felmerülő transzfer függvény pozitív realizálására. | According to the work-plan we have had results in various areas: In problems related to Fourier analysis we decreased the dimension of the counterexamples to Fuglede's conjecture in both direction to 3. We gave an efficient algorithm to find all non-periodic tilings of cyclic groups. (Tiling results inspired me to research structural questions of sumsets, and there have been some initial results too.) Besides this, we examined the extremal rays of the cone of positive, positive definite functions, and a covering problem of the plane with rotations of a lattice-like set. We presented new families of complex Hadamard matrices and MUB-triplets, and gave a promising discretization scheme to solve the MUB-6 problem in the future. The latter famous open problem is to find the maximal number of mutually unbiased bases (MUBs) in the complex space of dimension 6. These problems have their origin in quantum-information theory. We have given a precise mathematical description of the relativistic phenomenon of Thomas rotation, and gave an exact relation of the time-rate of Earth-based clocks and satellite clocks used in the GPS. (I remark here that this result is mainly of theoretical value, as the approximate formula being currently used in the GPS is good enough for practical purposes.) We gave an efficient general algorithm for positive realization of all transfer functions considered in the literature

Density estimates of 1-avoiding sets via higher order correlations

We improve the best known upper bound on the density of a planar measurable set A containing no two points at unit distance to 0.25442. We use a combination of Fourier analytic and linear programming methods to obtain the result. The estimate is achieved by means of obtaining new linear constraints on the autocorrelation function of A utilizing triple-order correlations in A, a concept that has not been previously studied.Comment: 10 pages, 2 figure

The Fuglede conjecture for convex domains is true in all dimensions

A set $\Omega \subset \mathbb{R}^d$ is said to be spectral if the space $L^2(\Omega)$ has an orthogonal basis of exponential functions. A conjecture due to Fuglede (1974) stated that $\Omega$ is a spectral set if and only if it can tile the space by translations. While this conjecture was disproved for general sets, it has long been known that for a convex body $\Omega \subset \mathbb{R}^d$ the "tiling implies spectral" part of the conjecture is in fact true. To the contrary, the "spectral implies tiling" direction of the conjecture for convex bodies was proved only in $\mathbb{R}^2$, and also in $\mathbb{R}^3$ under the a priori assumption that $\Omega$ is a convex polytope. In higher dimensions, this direction of the conjecture remained completely open (even in the case when $\Omega$ is a polytope) and could not be treated using the previously developed techniques. In this paper we fully settle Fuglede's conjecture for convex bodies affirmatively in all dimensions, i.e. we prove that if a convex body $\Omega \subset \mathbb{R}^d$ is a spectral set then $\Omega$ is a convex polytope which can tile the space by translations. To prove this we introduce a new technique, involving a construction from crystallographic diffraction theory, which allows us to establish a geometric "weak tiling" condition necessary for a set $\Omega \subset \mathbb{R}^d$ to be spectral.Comment: To appear in Acta Mathematic