Irreducible Fuchsian system with reducible monodromy representation

We present an example of the reducible representation χ = χ₁ ? χ₂, which, on the one hand, is the monodromy representation of a Fuchsian system. On the other hand, the representation χ₂ is a counterexample to the Riemann-Hilbert problem. Using a meromorphic gauge transformation, one cannot reduce this system to the direct sum of Fuchsian systems corresponding to the subrepresentations.     

Matrix Riemann-Hilbert problems and factorization on Riemann surfaces

The Wiener-Hopf factorization of 2×2 matrix functions and its close relation to scalar Riemann-Hilbert problems on Riemann surfaces is investigated. A family of function classes denoted C(Q₁,Q₂) is defined. To each class C(Q₁,Q₂) a Riemann surface Σ is associated, so that the factorization of the elements of C(Q₁,Q₂) is reduced to solving a scalar Riemann-Hilbert problem on Σ. For the solution of this problem, a notion of Σ-factorization is introduced and a factorization theorem is presented. An example of the factorization of a function belonging to the group of exponentials of rational functions is studied. This example may be seen as typical of applications of the results of this paper to finite-dimensional integrable systems.     

A Riemann-Hilbert Problem for the Moisil-Teodorescu System

In a bounded domain with smooth boundary in ?³ we consider the stationary Maxwell equations for a function u with values in ?³ subject to a nonhomogeneous condition (u, v)_x = u₀ on the boundary, where v is a given vector field and u₀ a function on the boundary. We specify this problem within the framework of the Riemann-Hilbert boundary value problems for the Moisil-Teodorescu system. This latter is proved to satisfy the Shapiro-Lopaniskij condition if an only if the vector v is at no point tangent to the boundary. The Riemann-Hilbert problem for the Moisil-Teodorescu system fails to possess an adjoint boundary value problem with respect to the Green formula, which satisfies the Shapiro-Lopatinskij condition. We develop the construction of Green formula to get a proper concept of adjoint boundary value problem.     

Values of twisted Artin L-functions

This note gives a simple proof that certain values of Artin’s L-function, for a representation ρ with character χ _ρ , are stable under twisting by an even Dirichlet character χ, up to the dim(ρ)th power of the Gauss sum τ(χ) and an element generated over \({\mathbb{Q}}\) by the values of χ and χ _ρ . This extends a result due to J. Coates and S. Lichtenbaum.     

Representations of the general linear group over symmetry classes of polynomials

Let V be the complex vector space of homogeneous linear polynomials in the variables x₁,..., x _m . Suppose G is a subgroup of S _m , and χ is an irreducible character of G. Let H _d (G, χ) be the symmetry class of polynomials of degree d with respect to G and χ.

For any linear operator T acting on V, there is a (unique) induced operator K _χ (T) ∈ End(H _d (G, χ)) acting on symmetrized decomposable polynomials by

$${K_\chi }\left( T \right)\left( {{f_1} * {f_2} * \cdots * {f_d}} \right) = T{f_1} * T{f_2} * \cdots * T{f_d}.$$

In this paper, we show that the representation T ? K _χ (T) of the general linear group GL(V) is equivalent to the direct sum of χ(1) copies of a representation (not necessarily irreducible) T ? B _χ ^G (T).

     

A new approach to factorization of a class of almost-periodic triangular symbols and related Riemann-Hilbert problems

The factorization of almost-periodic triangular symbols, G, associated to finite-interval convolution operators is studied for two classes of operators whose Fourier symbols are almost periodic polynomials with spectrum in the group αZ+βZ+Z (α,β∈]0,1[, α+β>1, α/β∉Q). The factorization problem is solved by a method that is based on the calculation of one solution of the Riemann-Hilbert problem GΦ₊=Φ₋ in L_∞(R) and does not require solving the associated corona problems since a second linearly independent solution is obtained by means of an appropriate transformation on the space of solutions to the Riemann-Hilbert problem. Some unexpected, but interesting, results are obtained concerning the Fourier spectrum of the solutions of GΦ₊=Φ₋. In particular it is shown that a solution exists with Fourier spectrum in the additive group αZ+βZ whether this group contains Z or not. Possible application of the method to more general classes of symbols is considered in the last section of the paper.     

Riemann-Hilbert and Poincaré problems with discontinuous boundary conditions for some model systems of partial differential equations

We study the solvability of the Riemann-Hilbert and Poincaré problems for systems of Cauchy-Riemann and Bitsadze equations in Sobolev spaces. For a generalized system of Cauchy-Riemann equations, we pose a boundary value problem and prove its unique solvability in the Sobolev space W ₂¹ (D). By supplementing the Riemann-Hilbert boundary conditions with some new conditions, we obtain a statement of the Poincaré problem with discontinuous boundary conditions for a system of second-order Bitsadze equations; we also prove the unique solvability of this problem in Sobolev spaces.     

Oscillatory Riemann-Hilbert problems and the corona theorem

The paper is devoted to the Riemann-Hilbert problem with matrix coefficient having in Hardy spaces [H_p^±]²,1<p?∞, on half-planes . Under the assumption of existence of a non-trivial solution of corresponding homogeneous Riemann-Hilbert problem in [H_∞^±]² we study the solvability of the non-homogeneous Riemann-Hilbert problem in [H_p^±]²,1<p<∞, and get criteria for the existence of a generalized canonical factorization and bounded canonical factorization for G as well as explicit formulas for its factors in terms of solutions of two associated corona problems (in and ). A separation principle for constructing corona solutions from simpler ones is developed and corona solutions for a number of corona problems in H_∞⁺ are obtained. Making use of these results we construct explicitly canonical factorizations for triangular bounded measurable or almost periodic 2×2 matrix functions whose diagonal entries do not possess factorizations. Such matrices arise, e.g., in the theory of convolution type equations on finite intervals.     

Fixed points for fuzzy mappings

In this paper the problem of the existence and computation of fixed points for fuzzy mappings is approached. A fuzzy mapping R over a set X is defined to be a function attaching to each x in X a fuzzy subset R_χ of X. An element x of X is called fixed point of R iff its membership degree to R_χ is at least equal to the membership degree to R_χ of any y?X, i.e. R_χ(χ)? R_χ(y)(?y?X). Two existence theorems for fixed points of a fuzzy mapping are proved and an algorithm for computing approximations of such a fixed point is described. The convergence theorem of our algorithm is proved under the restrictive assumption that for any x in X, the membership function of R_χ has a ‘complementary function’. Examples of fuzzy mappings having this property are given, but the problem of proving general criteria for a function to have a complementary remain open.     

Representations of Lie superalgebras in prime characteristic II: The queer series

The modular representation theory of the queer Lie superalgebra q(n) over characteristic p>2 is developed. We obtain a criterion for the irreducibility of baby Verma modules with semisimple p-characters χ and a criterion for the semisimplicity of the corresponding reduced enveloping algebras U_χ(q(n)). A (2p)-power divisibility of dimensions of q(n)-modules with nilpotent p-characters is established. The representation theory of q(2) is treated in detail. We formulate a Morita super-equivalence conjecture for q(n) with general p-characters which is verified for n=2.     

Hook-free colorings and a problem of Hanson

Hanson posed the following problem: What is the minimum numberχ(n) of colors needed to color all subsets of ann-set such that there is no monochromatic tripleA, B, C withA ∪B=C? It is known thatχ(n)≦[(n+1)/2], while Erd?s and Shelah provedχ(n)≧[(n+1)/4]. Their proof suggests the following notion: LetC be any finite plane point-configuration. The hook-free coloring numberχ(C) is the smallest number of colors needed forC such that no monochromatic hooks arise, i.e. if (c _x ,c _y ) are the coordinates of pointc∈C, then there are no 3 distinct pointsa, b, c∈C witha _x =b _x <c _x ,b _y =c _y <a _y . In this paperχ(R _m,n ) is determined exactly for anm×n-rectangle, and asymptotically for the triangular staircase. As a corollary one obtainsχ(n)≧0.293n.     

Characteristic polynomials of a permutation representation

Let G be a finite group. To each permutation representation (G, X) of G and class function χ of G we associate the χ-characteristic polynomialg_χ(x) of (G, X) which is a polynomial in one variable with complex numbers as coefficients. The coefficients of g_χ(x) are investigated in terms of the “exterior powers” of (G, X). If χ is the principal character 1_G of G, the coefficients of g_χ(x) are non-negative integers; and if furthermore G has odd order, the kth coefficient is the number of orbits of G acting on the subsets of X with k elements. Quadratic and linear relations among the exterior powers of (G, X) have been derived and it is shown how the polynomial g_χ(x) can be computed from the cycle index of (G, X).     

Exponentially small asymptotics of solutions to the defocusing nonlinear Schrödinger equation

The matrix Riemann-Hilbert factorization approach is used to derive the leading-order, exponentially small asymptotics as t → ± ∞ such that x/t ∼ O(1) of solutions to the Cauchy problem for the defocusing nonlinear Schrödinger equation, i∂_tu + ∂_x²u − 2(|u|² − 1)u = 0, with finite density initial data u(x,0) = _x→±∞exp(i(1 ∓ 1)φ/2)(1+o(1)), φ ϵ [0, 2π).     

Initial boundary value problems for the nonlinear Schrödinger equation

A new spectral method for solving initial boundary value problems for linear and integrable nonlinear partial differential equations in two independent variables is applied to the nonlinear Schrödinger equation and to its linearized version in the domain {x≥l(t), t≥0}. We show that there exist two cases: (a) if l″(t)<0, then the solution of the linear or nonlinear equations can be obtained by solving the respective scalar or matrix Riemann-Hilbert problem, which is defined on a time-dependent contour; (b) if l″(t)>0, then the Riemann-Hilbert problem is replaced by a respective scalar or matrix $\bar \partial $ problem on a time-independent domain. In both cases, the solution is expressed in a spectrally decomposed form.     

Central invariants and Frobenius-Schur indicators for semisimple quasi-Hopf algebras

In this paper, we obtain a canonical central element ν_H for each semi-simple quasi-Hopf algebra H over any field k and prove that ν_H is invariant under gauge transformations. We show that if k is algebraically closed of characteristic zero then for any irreducible representation of H which affords the character takes only the values 0, 1 or −1, moreover if H is a Hopf algebra or a twisted quantum double of a finite group then χ(ν_H) is the corresponding Frobenius-Schur indicator. We also prove an analog of a theorem of Larson-Radford for split semi-simple quasi-Hopf algebras over any field k. Using this result, we establish the relationship between the antipode S, the values of χ(ν_H), and certain associated bilinear forms when the underlying field k is algebraically closed of characteristic zero.     

Inequalities and identities for generalized matrix functions

Let χ be a character on the symmetric group S_n, and let A = (a_ij) be an n-by-n matrix. The function d_χ(A) = Σ_σ?Snχ(σ)Πⁿ_{t = 1}a_tσ(t) is called a generalized matrix function. If χ is an irreducible character, then d_χ is called an immanent. For example, if χ is the alternating character, then d_χ is the determinant, and if χ ≡ 1, then d_χ is called the permanent (denoted per). Suppose that A is positive semidefinite Hermitian. We prove that the inequality $\text{(1/χ(}\text{id}\text{))d}{}_{\mathrm{\chi }}\text{(A) ?}\text{per}\text{A}$ holds for a variety of characters χ including the irreducible ones corresponding to the partitions (n ? 1,1) and (n ? 2,1,1) of n. The main technique used to prove these inequalities is to express the immanents as sums of products of principal subpermanents. These expressions for the immanents come from analogous expressions for Schur polynomials by means of a correspondence of D.E. Littlewood.     

The inverse scattering problem of quantum theory

The inverse phase-type scattering problem for the boundary-value problem?y″+q(x)y=k ² y (0?x<∞), (1)y′ (0)=hy (0) (2) is studied. It is shown that, for each function δ(k) satisfying the hypotheses of Levinson's theorem, there exists a problem (1)–(2) with h≠∞ and another problem (1)–(2) with h=∞ (i.e., with the boundary condition o (0)=0). The solvability condition for the Riemann-Hilbert problem is used more directly than has been done heretofore by others in deriving boundary condition (2).     

Sums with convolutions of Dirichlet characters

We bound short sums of the form ${\sum_{n\le X}(\chi_1{*}\chi_2)(n)}$ , where χ ₁*χ ₂ is the convolution of two primitive Dirichlet characters χ ₁ and χ ₂ with conductors q ₁ and q ₂, respectively.     

Stabilität Mehrfach Zusammenhängender Minimalflächen

Multiply connected minimal surfaces of genus 0 with only simple interior branch points, for which the corresponding boundary value problem $$\Delta h - K|x_z |^2 h = 0; h_{|\partial \Omega } = 0$$ (K is the Gauss curvature and x_z is the complex gradient of the surface x) is uniquely solvable and which have the property, that the condition K|x_z|²≠0 holds in the branch points, are always isolated and stable solutions of the Plateau problem, corresponding to their boundary curves. To achieve these results one has to consider the conformal type as a variable. We give a method to perform the variation of the conformal type for holomorphic functions. Using the Weierstrass representation we thus obtain a differentiable structure on the set of multiply connected minimal surfaces. We find interesting connections between the classical Riemann-Hilbert problem and Fredholm properties of a projection operator on this manifold.