81 research outputs found
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
The Fuglede conjecture for convex domains is true in all dimensions
A set is said to be spectral if the space
has an orthogonal basis of exponential functions. A conjecture
due to Fuglede (1974) stated that 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 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 , and also in
under the a priori assumption that is a convex polytope. In higher
dimensions, this direction of the conjecture remained completely open (even in
the case when 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 is a spectral set then 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 to be spectral.Comment: To appear in Acta Mathematic
ElmĂ©let Ă©s alkalmazĂĄs a matematikai analĂzisben = Theory and applications in mathematical analysis
A kutatĂĄsi tervnek megfelelĆen több terĂŒleten is sikerĂŒlt eredmĂ©nyeket elĂ©rni: Fourier analĂzissel kapcsolatos problĂ©mĂĄkban a Fugelde sejtĂ©sre adott ellenpĂ©ldĂĄk dimenziojĂĄt 3-ra csökkentettĂŒk mindkĂ©t irĂĄnyban, valamint hatĂ©kony algoritmust adtunk ciklikus csoporotok nem-periodikus parkettĂĄzĂĄsainak megkeresĂ©sĂ©re. (A parkettĂĄzĂĄsi eredmĂ©nyek hatĂĄsĂĄra kezdtem kutatni összeghalmazok szĂĄmossĂĄgi Ă©s strukturĂĄlis kĂ©rdĂ©seit, Ă©s itt is szĂŒletett nĂ©hĂĄny kezdeti eredmĂ©ny.) Ezen kĂvĂŒl tanulmĂĄnyoztuk a pozitĂv, pozitĂv definit fĂŒggvĂ©nyek kĂșpjĂĄnak extremĂĄlisait, valamint egy sĂkbefedĂ©si problĂ©mĂĄt rĂĄcsszerƱ halmaz elforgatottjaival. Komplex Hadamard mĂĄtrixok Ă©s MUB-hĂĄrmasok Ășj csalĂĄdjait adtuk meg, valamint egy ĂgĂ©retes diszkretizĂĄciĂłs mĂłdszert a MUB-6 problĂ©ma jövĆbeni megoldĂĄsĂĄra. Ez a nevezetes nyitott problema azt kĂ©rdezi, hogy legfeljebb hĂĄny kölcsönösen torzĂtatlan bĂĄzis (MUB) adhatĂł meg komplex 6 dimenziĂłban. Ezeknek a kĂ©rdĂ©seknek kvantum-informĂĄciĂłelmĂ©letbeli gyökerei vannak. MegvizsgĂĄltuk a relativisztikus Thomas-rotĂĄcio jelensĂ©gĂ©t, valamint a GPS-ben hasznĂĄlt közelĂtĆ kĂ©plet helyett pontos formulĂĄt mutattunk a Föld bĂĄzisĂș Ă©s a GPS-szatelliteken levĆ Ăłrak idĆmĂ©rĂ©sĂ©nek differenciĂĄl hĂĄnyadosĂĄra. (Ez utĂłbbinak csak elmĂ©leti jelentĆsĂ©ge van, mert a közelĂtĆ formula mĂĄr eddig is elĂ©g pontos volt gyakorlati szempontbĂłl.) A pozitĂv realizĂĄciĂłs tĂ©makörben ĂĄltalĂĄnos Ă©s hatĂ©kony algoritmust adtunk az összes felmerĂŒlĆ transzfer fĂŒggvĂ©ny pozitĂv realizĂĄlĂĄsĂĄra. | 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
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