Symmetric hyperbolic systems in algebras of generalized functions and distributional limits

We study existence, uniqueness, and distributional aspects of generalized solutions to the Cauchy problem for first-order symmetric (or Hermitian) hyperbolic systems of partial differential equations with Colombeau generalized functions as coefficients and data. The proofs of solvability are based on refined energy estimates on lens-shaped regions with spacelike boundaries. We obtain several variants and also partial extensions of previous results in Oberguggenberger (1989), Lafon and Oberguggenberger (1991), and Hörmann (2004) [26], [23], [16] and provide aspects accompanying related recent work in Oberguggenberger (2009), Garetto and Oberguggenberger (2011) [28], [10], [9].     

An integral characterization of random permutations. A point process approach

We consider random finite permutations and prove the following version of Thoma’s theorem in [8]: Random finite permutations which are class functions satisfy a new integration by parts formula if and only if they are given by a certain Ewens-Sütö process. The main source of inspiration for the results in this note is the fundamental work of Andras Sütö [7], from which some results are reestablished here again in the present point process approach.     

Some degenerations of Kazhdan–Lusztig ideals and multiplicities of Schubert varieties

We study Hilbert–Samuel multiplicity for points of Schubert varieties in the complete flag variety, by Gröbner degenerations of the Kazhdan–Lusztig ideal. In the covexillary case, we give a manifestly positive combinatorial rule for multiplicity by establishing (with a Gröbner basis) a reduced limit whose Stanley–Reisner simplicial complex is homeomorphic to a shellable ball or sphere. We show that multiplicity counts the number of facets of this complex. We also obtain a formula for the Hilbert series of the local ring. In particular, our work gives a multiplicity rule for Grassmannian Schubert varieties, providing alternative statements and proofs to formulae of Lakshmibai and Weyman (1990) [26], Rosenthal and Zelevinsky (2001) [37], Krattenthaler (2001) [22], Kodiyalam and Raghavan (2003) [21], Kreiman and Lakshmibai (2004) [24], Ikeda and Naruse (2009) [13] and Woo and Yong (2009) [40]. We suggest extensions of our methodology to the general case.     

On almost rc-Lindelöf sets

In [4], Dlaska introduced the class of almost rc-Lindelöf sets and studied some basic properties of such sets. In this paper, we obtain further results concerning almost rc-Lindelöf sets. We also introduce new concepts to obtain several mapping properties concerning almost rc-Lindelöf sets and almost rc-Lindelöf spaces. The property of being an almost rc-Lindelöf set is invariant under functions which are slightly continuous and weakly θ-irresolute. It is also shown that the property of being an almost rc-Lindelöf space is inverse invariant under functions which are weakly almost open, ω-regular open, and whose fibers are S-sets.     

Finite approximations of physical models over p-adic fields

We show that the p-adic Schrödinger operator, as defined in [7], can be approximated in a very strong sense by finite Schröinger operators.     

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Leindler [5] obtained certain estimates of the approximation of Fourier series by Nörlund-Voronoi means in the Hölder metric. Making use of the equality of two norms in the Hölder space and a theorem of Leindler-Meir-Totik [6], we improve these estimates.     

A remark on the Lifshitz tail for Schrödinger operator with random magnetic field

In this note, we consider the Lifshitz singularity for Schrödinger operator with ergodic random magnetic field. A key estimate is an energy bound for magnetic Schrödinger operators as discussed in Nakamura [8]. Here we remove a technical assumption in [8], namely, the uniform boundedness of the magnetic field.     

A five-dimensional modular variety

In this note we in the spirit of [1], we give ta structure theorem for a graded ring of modular forms related to the orthogonal group O(2, 5). Our results generalize some results obtained by Klöcker in his Ph.D thesis.     

Moduli spaces for covers of the Riemann sphere

Moduli spaces for covers of the Riemann sphere have been constructed in a joint work with M. Fried [FV1]. They were used to realize groups as Galois groups [FV1], [Vö1], and to determine the absolute Galois group of large fields [FV2]. Here we simplify and extend the construction of these moduli spaces.     

Almost everywhere well-posedness of continuity equations with measure initial data

The aim of this Note is to present some new results concerning “almost everywhere” well-posedness and stability of continuity equations with measure initial data. The proofs of all such results can be found in Ambrosio et al. [4], together with some application to the semiclassical limit of the Schrödinger equation.     

Integrals of weighted maximal logarithmic kernels on bounded Vilenkin groups

The integrals of maximal Riesz and Nörlund kernels are infinite, so we have to use some weight function to “pull them back” to the finite. In this paper we give necessary and sufficient conditions for the weight function to get a finite integral on bounded Vilenkin groups. For our motivation we refer the readers to [4], [5], [6].     

The fundamental lemma for the Bessel and Novodvorsky subgroups of GSp(4)

We announce, in the case of the group GSp(4), an equality of two local integrals. One is a Kloosterman integral on the Bessel subgroups of GSp(4) and the other is a Kloosterman integral on the Novodvorsky subgroups of GSp(4). We conjecture that Jacquet's relative trace formula for GL(2) in [7], where Jacquet has given another proof of Waldspurger's result [9], generalizes to GSp(4). We believe that this approach will lead us to a proof and also a precise formulation of a conjecture of Böcherer [1]. Support for this conjecture may be found in the important paper of Böcherer and Schulze-Pillot [2]. Our result serves as the fundamental lemma for our conjectural relative trace formula for the main relevant double cosets.     

On the distribution of sample mean in finite populations. I

We obtain Bergström-type [2] asymptotic expansions for sample mean in finite population [8]. Analogues of the Cornish-Fisher transformation are obtained in the cases of limit distributions from class $\mathcal{L}$ and distributions of sums of independent identically distributed random variables (in [2], the Bergström equality is used).     

Error bounds for convex differentiable inequality systems in Banach spaces

The paper is devoted to studying the Hoffman global error bound for convex quadratic/affine inequality/equality systems in the context of Banach spaces. We prove that the global error bound holds if the Hoffman local error bound is satisfied for each subsystem at some point of the solution set of the system under consideration. This result is applied to establishing the equivalence between the Hoffman error bound and the Abadie qualification condition, as well as a general version of Wang &; Pang's result [30], on error bound of Hölderian type. The results in the present paper generalize and unify recent works by Luo &; Luo in [17], Li in [16] and Wang &; Pang in [30].     

Zur Approximation pseudoanalytischer Funktionen durch Pseudopolynome

In the theory of pseudoanalytic functions one can define (pseudoanalytic) rational functions, especially polynomials called “pseudopolynomials”. (See Bers [3], [4], Vekua [12]) Therefore it can be developed a theory of approximation and interpolation by rational functions. First results have been published by Bers [3] (Runge's theorem), Ismailov and Taglieva [8]. Let G be a domain of the complex plane bounded by a closed Jordan curve, let w(z) be pseudoanalytic in G. In this paper we deal with a relation between the behaviour of w(z) on C (Hölder-continuity) and the degree of approximation of w(z) by pseudopolynomials. The results correspond to certain theorems of Curtiss, Sewell and Walsh in the theory of analytic functions.     

An edge-coloration theorem for bipartite graphs with applications

An edge-coloration theorem for bipartite graphs, announced in [4], is proved from which some well-known theorems due to König [5] and the author [2, 3] are deduced. The theorem is further applied to prove the “dual” of a theorem due to Lovász [6].     

Gröbner–Shirshov bases for some Lie algebras

Siberian Mathematical Journal - We give Gröbner–Shirshov bases for the Drinfeld–Kohno Lie algebra L n in [1] and the Kukin Lie algebra A P in [2], where P is a semigroup. By way of...     

Collapsing estimates and the rigorous derivation of the 2d cubic nonlinear Schrödinger equation with anisotropic switchable quadratic traps

We consider the 2d and 3d many body Schrödinger equations in the presence of anisotropic switchable quadratic traps. We extend and improve the collapsing estimates in Klainerman and Machedon (2008) [25] and Kirkpatrick, Schlein and Staffilani (2011) [23]. Together with an anisotropic version of the generalized lens transform in Carles (2011) [3], we derive rigorously the cubic NLS with anisotropic switchable quadratic traps in 2d through a modified Elgart–Erdös–Schlein–Yau procedure. For the 3d case, we establish the uniqueness of the corresponding Gross–Pitaevskii hierarchy without the assumption of factorized initial data.