Higher Order Three-Term Recurrences and Asymptotics of Multiple Orthogonal Polynomials
The asymptotic theory is developed for polynomial sequences that are generated by the three-term higher-order recurrence. Our results generalize known results for p=1, that is, for orthogonal polynomial sequences on the real line that belong to the Blumenthal–Nevai class. As is known, for p≥2, the role of the interval is replaced by a starlike set S of p+1 rays emanating from the origin on which the Q n satisfy a multiple orthogonality condition involving p measures. Here we obtain strong asymptotics for the Q n in the complex plane outside the common support of these measures as well as on the (finite) open rays of their support. In so doing, we obtain an extension of Weyl’s famous theorem dealing with compact perturbations of bounded self-adjoint operators. Furthermore, we derive generalizations of the classical Szegő functions, and we show that there is an underlying Nikishin system hierarchy for the orthogonality measures that is related to the Weyl functions. Our results also have application to Hermite–Padé approximants as well as to vector continued fractions.
This article is dedicated to an alternative method of solving of the Chinese Remainder Theorem for polynomials. To construct the solution, a system of linear equations is constructed (using the method of undetermined coefficients) and then solved. The complexity of the proposed method is also calculated.
We introduce a unital associative algebra associated with degenerate CP1. We show that is a commutative algebra and whose Poincare' series is given by the number of partitions. Thereby, we can regard as a smooth degeneration limit of the elliptic algebra introduced by Feigin and Odesskii [Int. Math. Res. Notices 11, 531 (1997)]. Then we study the commutative family of the Macdonald difference operators acting on the space of symmetric functions. A canonical basis is proposed for this family by using and the Heisenberg representation of the commutative family studied by Shiraishi [ Commun. Math. Phys. 263, 439 (2006)]. It is found that the Ding-Iohara algebra [Lett. Math. Phys. 41, 183 (1997)] provides us with an algebraic framework for the free field construction. An elliptic deformation of our construction is discussed, showing connections with the Drinfeld quasi-Hopf twisting [Leningrad Math. J. 1, 1419 (1990)] in the sence of Babelon-Bernard-Billey [Phys. Lett. B. 375, 89 (1996)], the Ruijsenaars difference operator [Commun. Math. Phys. 110, 191 (1987)], and the operator M(q,t1,t2) of Okounkov-Pandharipande [e-print arXiv:math-ph/0411210].
We study modal logics of topological spaces in the combined language with the derivational modality and the difference modality. We give axiomatizations and prove completeness for the following classes: all spaces, T1- spaces, dense-in-themselves spaces, a zero-dimensional dense-in-itself separable metric space, R^n (n>1). We also discuss the correlation between languages with different combinations of the topological, the derivational, the universal and the difference modality in terms of definability.
In order to reduce the Gibbs phenomenon exhibited by the partial Fourier sums of a periodic function, discontinuous in 0, Driscoll and Fornberg considered so-called singular Fourier-Padé approximants constructed from the Hermite-Padé approximants. Convincing numerical experiments have been obtained by these authors, but no error estimates have been proven so far. In the present paper we study the special case of Nikishin systems and their Hermite-Padé approximants, both theoretically and numerically. We obtain rates of convergence by using orthogonality properties of the functions involved along with results from logarithmic potential theory. In particular, we address the question of how to choose the degrees of the approximants, by considering diagonal and row sequences, as well as linear Hermite-Padé approximants. Our theoretical findings and numerical experiments confirm that these Hermite-Padé approximants are more efficient than the more elementary Padé approximants, particularly around the discontinuity of the goal function.
The paper gives first quantitative estimates on the modulus of continuity of the spectral measure for weak mixing suspension flows over substitution automorphisms, which yield information about the “fractal” structure of these measures. The main results are, first, a Hölder estimate for the spectral measure of almost all suspension flows with a piecewise constant roof function; second, a log-Hölder estimate for self-similar suspension flows; and, third, a Hölder asymptotic expansion of the spectral measure at zero for such flows. Our second result implies log-Hölder estimates for the spectral measures of translation flows along stable foliations of pseudo-Anosov automorphisms. A key technical tool in the proof of the second result is an “arithmetic-Diophantine” proposition, which has other applications. In Appendix A this proposition is used to derive new decay estimates for the Fourier transforms of Bernoulli convolutions.
A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.
Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.
Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.