Correspondences of ribbon categories

Much of algebra and representation theory can be formulated in the general framework of tensor categories. The aim of this paper is to further develop this theory for braided tensor categories. Several results are established that do not have a substantial counterpart for symmetric tensor categories. In particular, we exhibit various equivalences involving categories of modules over algebras in ribbon categories. Finally, we establish a correspondence of ribbon categories that can be applied to, and is in fact motivated by, the coset construction in conformal quantum field theory.     

The condition on the stability of stationary lines in a curvature flow in the whole plane

The long time behavior of a curve in the whole plane moving by a curvature flow is studied. Studying the Cauchy problem, we deal with moving curves represented by entire graphs on the x-axis. Here the initial curves are given by bounded functions on the x-axis. It is proved that the solution converges uniformly to the solution of the Cauchy problem of the heat equation with the same initial value. The difference is of order O(t−1/2) as time goes to infinity. The proof is based on the decay estimates for the derivatives of the solution. By virtue of the stability results for the heat equation, our result gives the sufficient and necessary condition on the stability of constant solutions that represent stationary lines of the curvature flow in the whole plane.     

Entropy and ergodic probability for differentiable dynamical systems and their bundle extensions

We answer a problem of Liao [S.T. Liao, Standard systems of differential equations and obstruction sets—from linearity to perturbations, in: System Researches, Proceedings Dedicated to the 85th Anniversary of Qian Xue-Sen, Zhejiang Education Press, Hangzhou, China, 1996, pp. 279-290 (in Chinese)]: A C¹ vector field or a C¹ diffeomorphism on an n-dimensional manifold has equal entropy with that of its bundle extensions. We also prove that each ergodic probability with simple Lyapunov spectrum has at most n²n! covering probabilities on each bundle extension.     

Density estimation and adaptive control of markov processes: Average and discounted criteria

We consider a class of discrete-time Markov control processes with Borel state and action spaces, and ^d i.i.d. disturbances with unknown distribution . Under mild semi-continuity and compactness conditions, and assuming that is absolutely continuous with respect to Lebesgue measure, we establish the existence of adaptive control policies which are (1) optimal for the average-reward criterion, and (2) asymptotically optimal in the discounted case. Our results are obtained by taking advantage of some well-known facts in the theory of density estimation. This approach allows us to avoid restrictive conditions on the state space and/or on the system's transition law imposed in recent works, and on the other hand, it clearly shows the way to other applications of nonparametric (density) estimation to adaptive control.Research partially supported by The Third World Academy of Sciences under Research Grant No. MP 898-152.     

The umbral calculus and orthogonal polynomials

We determine all orthogonal polynomials having Boas-Buck generating functions g(t)(xf(t)), where% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqqHOo% qwcaGGOaGaamiDaiaacMcacqGH9aqpruqqYLwySbacfaGaa8hiamaa% BeaaleaacaaIWaaabeaakiaadAeacaqGGaWaaSbaaSqaaiaabgdaae% qaaOGaaeikaiaadggacaGGSaGaa8hiaiaadshacaqGPaGaaeilaiaa% bccacaqGGaGaaeiiaiaadggacqGHGjsUcaaIWaGaaiilaiaa-bcacq% GHsislcaaIXaGaaiilaiaa-bcacqGHsislcaaIYaGaaiilaiablAci% ljaacUdaaeaacqqHOoqwcaGGOaGaamiDaiaacMcacqGH9aqpcaWFGa% WaaSraaSqaaiaaicdaaeqaaOGaamOraiaabccadaWgaaWcbaGaaeOm% aaqabaGccaGGOaWaaSqaaSqaaiaaigdaaeaacaaIZaaaaOGaaiilai% aa-bcadaWcbaWcbaGaaGOmaaqaaiaaiodaaaGccaGGSaGaa8hiaiaa% dshacaGGPaGaa8hiamaaBeaaleaacaaIWaaabeaakiaadAeacaqGGa% WaaSbaaSqaaiaabkdaaeqaaOGaaeikamaaleaaleaacaaIYaaabaGa% aG4maaaakiaacYcacaWFGaWaaSqaaSqaaiaaisdaaeaacaaIZaaaaO% Gaaiilaiaa-bcacaWG0bGaaiykaiaacYcacaWFGaWaaSraaSqaaiaa% icdaaeqaaOGaamOraiaabccadaWgaaWcbaGaaeOmaaqabaGccaGGOa% WaaSqaaSqaaiaaisdaaeaacaaIZaaaaOGaaiilaiaa-bcadaWcbaWc% baGaaGynaaqaaiaaiodaaaGccaGGSaGaa8hiaiaadshacaGGPaGaai% 4oaaqaaiabfI6azjaacIcacaWG0bGaaiykaiabg2da9iaa-bcadaWg% baWcbaGaaGimaaqabaGccaWGgbGaaeiiamaaBaaaleaacaqGZaaabe% aakiaacIcadaWcbaWcbaGaaGymaaqaaiaaisdaaaGccaGGSaGaa8hi% amaaleaaleaacaaIYaaabaGaaGinaaaakiaacYcacaWFGaWaaSqaaS% qaaiaaiodaaeaacaaI0aaaaOGaaiilaiaa-bcacaWG0bGaaiykaiaa% -bcadaWgbaWcbaGaaGimaaqabaGccaWGgbGaaeiiamaaBaaaleaaca% qGZaaabeaakiaabIcadaWcbaWcbaGaaGOmaaqaaiaaisdaaaGccaGG% SaGaa8hiamaaleaaleaacaaIZaaabaGaaGinaaaakiaacYcacaWFGa% WaaSqaaSqaaiaaiwdaaeaacaaI0aaaaOGaaiilaiaa-bcacaWG0bGa% aiykaiaacYcaaeaadaWgbaWcbaGaaGimaaqabaGccaWGgbGaaeiiam% aaBaaaleaacaqGZaaabeaakiaacIcadaWcbaWcbaGaaG4maaqaaiaa% isdaaaGccaGGSaGaa8hiamaaleaaleaacaaI1aaabaGaaGinaaaaki% aacYcacaWFGaWaaSqaaSqaaiaaiAdaaeaacaaI0aaaaOGaaiilaiaa% -bcacaWG0bGaaiykaiaacYcacaGGUaGaa8hiamaaBeaaleaacaaIWa% aabeaakiaadAeacaqGGaWaaSbaaSqaaiaabodaaeqaaOGaaeikamaa% leaaleaacaaI1aaabaGaaGinaaaakiaacYcacaWFGaWaaSqaaSqaai% aaiAdaaeaacaaI0aaaaOGaaiilaiaa-bcadaWcbaWcbaGaaG4naaqa% aiaaisdaaaGccaGGSaGaa8hiaiaadshacaGGPaGaaiOlaaaaaa!C1F3!\[\begin{gathered}\Psi (t) = {}_0F{\text{ }}_{\text{1}} {\text{(}}a, t{\text{), }}a \ne 0, - 1, - 2, \ldots ; \hfill \\\Psi (t) = {}_0F{\text{ }}_{\text{2}} (\tfrac{1}{3}, \tfrac{2}{3}, t) {}_0F{\text{ }}_{\text{2}} {\text{(}}\tfrac{2}{3}, \tfrac{4}{3}, t), {}_0F{\text{ }}_{\text{2}} (\tfrac{4}{3}, \tfrac{5}{3}, t); \hfill \\\Psi (t) = {}_0F{\text{ }}_{\text{3}} (\tfrac{1}{4}, \tfrac{2}{4}, \tfrac{3}{4}, t) {}_0F{\text{ }}_{\text{3}} {\text{(}}\tfrac{2}{4}, \tfrac{3}{4}, \tfrac{5}{4}, t), \hfill \\{}_0F{\text{ }}_{\text{3}} (\tfrac{3}{4}, \tfrac{5}{4}, \tfrac{6}{4}, t),. {}_0F{\text{ }}_{\text{3}} {\text{(}}\tfrac{5}{4}, \tfrac{6}{4}, \tfrac{7}{4}, t). \hfill \\\end{gathered}\]We also determine all Sheffer polynomials which are orthogonal on the unit circle. The formula for the product of polynomials of the Boas-Buck type is obtained.     

A new Pythagorean functional equation

The purpose of this paper is to solve the following Pythagorean functional equation:(e ^p(x,y) ) ² ) = q(x,y) ² + r(x, y) ², where each ofp(x,y), q(x, y) andr(x, y) is a real-valued unknown harmonic function of the real variablesx, y on the wholexy-planeR ².The result is as follows.     

Finding numerical solution of linear partial differential equations of first order with real coefficients

Applying Bittner's operational calculus we present a method to give approximate solutions of linear partial differential equations of first order

Variations on the theme of repeated distances

We give an asymptotically sharp estimate for the error term of the maximum number of unit distances determined byn points in ^{d, d}4. We also give asymptotically tight upper bounds on the total number of occurrences of the favourite distances fromn points in ^{d, d}4. Related results are proved for distances determined byn disjoint compact convex sets in ².At the time this paper was written, both authors were visiting the Technion — Israel Institute of Technology.     

Dependence of “Reststrahlen” bands in far-infrared reflectivity on configuration of GaAs/AlAs multiple quantum well heterostructures

Far infrared reflectivity measurements are performed on a series of GaAs/AlAs multiple quantum well (MQW) heterostructures with systematically varied thicknesses of the constituent layers. In addition to the artificial anisotropy we observe two distinct bulk-like Reststrahlen regions. The widths of the GaAs-like and the AlAs-like Reststrahlen bands strongly depend on the relative thicknesses of the constituent layers of the MQW heterostructures, in excellent agreement with the predictions of the effective-medium theory.Prof. Aldo Cingolani passed away just before the publication of this article. We would like to dedicate this paper to his memory     

Sphere orders

Ann-sphere order is a finite partially ordered set representable by containment ofn-spheres in Euclidean (n+1)-space. We present a sequence {P _i } of ordered sets such that eachP _i is ann-sphere order only forni; one consequence is that we are able to determine the dimension of a Euclidean space-time manifold from the finite suborders of its causality order.Research supported by ONR grant N00014 85-K-0769.