228 research outputs found
Upper bounds on the rate of quantum ergodicity
We study the semiclassical behaviour of eigenfunctions of quantum systems
with ergodic classical limit. By the quantum ergodicity theorem almost all of
these eigenfunctions become equidistributed in a weak sense. We give a simple
derivation of an upper bound of order \abs{\ln\hbar}^{-1} on the rate of
quantum ergodicity if the classical system is ergodic with a certain rate. In
addition we obtain a similar bound on transition amplitudes if the classical
system is weak mixing. Both results generalise previous ones by Zelditch. We
then extend the results to some classes of quantised maps on the torus and
obtain a logarithmic rate for perturbed cat-maps and a sharp algebraic rate for
parabolic maps.Comment: 18 page
Brody curves omitting hyperplanes
A Brody curve, a.k.a. normal curve, is a holomorphic map from the complex
line to the complex projective space of dimension n, such that the family of
its translations is normal. We prove that Brody curves omitting n hyperplanes
in general position have growth order at most one, normal type. This
generalizes a result of Clunie and Hayman who proved it for n=1.Comment: 8 page
Dynamics of a higher dimensional analog of the trigonometric functions
We introduce a higher dimensional quasiregular map analogous to the
trigonometric functions and we use the dynamics of this map to define, for d>1,
a partition of d-dimensional Euclidean space into curves tending to infinity
such that two curves may intersect only in their endpoints and such that the
union of the curves without their endpoints has Hausdorff dimension one.Comment: 12 page
Analytic cliffordian functions
In classical function theory, a function is holomorphic if and only if it is
complex analytic. For higher dimensional spaces it is natural to work in the
context of Clifford algebras. The structures of these algebras depend on the
parity of the dimension n of the underlying vector space. The theory of
holomorphic Cliffordian functions reflects this dependence. In the case of odd
n the space of functions is defined by an operator (the Cauchy-Riemann
equation) but not in the case of even . For all dimensions the powers of
identity (z^n, x^n) are the foundation of function theory
On the mean square of the zeta-function and the divisor problem
Let denote the error term in the Dirichlet divisor problem, and
the error term in the asymptotic formula for the mean square of
. If with , then we obtain the
asymptotic formula where is a polynomial of degree three in
with positive leading coefficient. The exponent 7/6 in the error term
is the limit of the method.Comment: 10 page
On some elliptic transmission problems
Abstract. Boundary value problems for second order linear elliptic equations with coefficients having discontinuities of the first kind on an infinite number of smooth surfaces are studied. Existence, uniqueness and regularity results are furnished for the diffraction problem in such a bounded domain, and for the corresponding transmission problem in all of R N . The transmission problem corresponding to the scattering of acoustic plane waves by an infinitely stratified scatterer, consisting of layers with physically different materials, is also studied. 0. Introduction. In this work we study boundary value problems for linear equations of elliptic type whose coefficients have discontinuities of the first kind on an infinite number of smooth surfaces that divide a bounded domain in R N into nested layers. On those surfaces, the so-called "transmission (conjugacy, matching, linking) conditions" are imposed, that express the continuity of the medium and the equilibrium of the forces acting on it. The discontinuity of the coefficients of the equations corresponds to the fact that the medium consists of several physically different materials. From the point of view of the theory of generalized solutions-which we employ in our approach-such problems can be considered as special cases of usual boundary value problems. On the contrary, the investigation of these problems by classical methods requires the theory of integral equations, and in this context they differ essentially from the usual boundary value problems where the medium has smoothly varying characteristics. Boundary value problems with discontinuous coefficients (also known as diffraction problems) have been treated by many authors, employing a variety of approaches. I
Gradients and canonical transformations
Abstract. The main aim of this paper is to give some counterexamples to global invertibility of local diffeomorphisms which are interesting in mechanics. The first is a locally strictly convex function whose gradient is non-injective. The interest in this function is related to the Legendre transform. Then I show two non-injective canonical local diffeomorphisms which are rational: the first is very simple and related to the complex cube, the second is defined on the whole R 4 and is obtained from a recent important example by Pinchuk. Finally, a canonical transformation which is also a gradient (of a convex function) is provided
Uniform stability and semi-stability of motions in dynamical systems on metric spaces
Abstract. Some stability properties of motions in pseudo-dynamical systems and semi-systems are studied. Introduction. The purpose of the present paper is to study some properties of motions in dynamical and pseudo-dynamical systems (see definitions below) on metric spaces. We consider uniform stability and semistability (see Section 2) of motions in nonempty subsets of the phase space and discuss in particular properties of mappings of the types
The algebra generated by a pair of operator weighted shifts
Abstract. We present a model for two doubly commuting operator weighted shifts. We also investigate general pairs of operator weighted shifts. The above model generalizes the model for two doubly commuting shifts. WOT-closed algebras for such pairs are described. We also deal with reflexivity for such pairs assuming invertibility of operator weights and a condition on the joint point spectrum
How to get rid of one of the weights in a two-weight Poincaré inequality?
Abstract. We prove that if a Poincaré inequality with two different weights holds on every ball, then a Poincaré inequality with the same weight on both sides holds as well
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