Scalar and matrix Riemann–Hilbert approach to the strong asymptotics of Padé approximants and complex orthogonal polynomials with varying weight

We describe methods for the derivation of strong asymptotics for the denominator polynomials and the remainder of Padé approximants for a Markov function with a complex and varying weight. Two approaches, both based on a Riemann–Hilbert problem, are presented. The first method uses a scalar Riemann–Hilbert boundary value problem on a two-sheeted Riemann surface, the second approach uses a matrix Riemann–Hilbert problem. The result for a varying weight is not with the most general conditions possible, but the loss of generality is compensated by an easier and transparent proof.     

Riemann–Hilbert Problem for Automorphic Functions and the Schottky–Klein Prime Function

The construction of analogues of the Cauchy kernel is crucial for the solution of Riemann–Hilbert problems on compact Riemann surfaces. A formula for the Cauchy kernel can be given as an infinite sum over the elements of a Schottky group, and this sum is often used for the explicit evaluation of the kernel. In this paper a new formula for a quasi-automorphic analogue of the Cauchy kernel in terms of the Schottky–Klein prime function of the associated Schottky double is derived. This formula opens the door to finding new ways to evaluate the analogue of the Cauchy kernel in cases where the infinite sum over a Schottky group is not absolutely convergent. Application of this result to the solution of the Riemann–Hilbert problem with a discontinuous coefficient for symmetric automorphic functions is discussed. Received: March 10, 2007. Accepted: April 11, 2007.     

The Riemann–Hilbert problem and the generalized Neumann kernel on multiply connected regions

This paper presents and studies Fredholm integral equations associated with the linear Riemann–Hilbert problems on multiply connected regions with smooth boundary curves. The kernel of these integral equations is the generalized Neumann kernel. The approach is similar to that for simply connected regions (see [R. Wegmann, A.H.M. Murid, M.M.S. Nasser, The Riemann–Hilbert problem and the generalized Neumann kernel, J. Comput. Appl. Math. 182 (2005) 388–415]). There are, however, several characteristic differences, which are mainly due to the fact, that the complement of a multiply connected region has a quite different topological structure. This implies that there is no longer perfect duality between the interior and exterior problems.     

A Riemann–Hilbert problem for skew-orthogonal polynomials

We find a local (d+1)×(d+1)$(d+1)\times (d+1)$ Riemann–Hilbert problem characterizing the skew-orthogonal polynomials associated to the partition function of orthogonal ensembles of random matrices with a potential function of degree d  . Our Riemann–Hilbert problem is similar to a local d×d$d\times d$ Riemann–Hilbert problem found by Kuijlaars and McLaughlin characterizing the bi-orthogonal polynomials. This gives more motivation for finding methods to compute asymptotics of high order Riemann–Hilbert problems, and brings us closer to finding full asymptotic expansions of the skew-orthogonal polynomials.     

The Supremum and Infimum of the Set of Fuzzy Numbers and Its Application

In this paper, we prove that the bounded set of fuzzy numbers must exist supremum and infimum and give the concrete representation of supremum and infimum. As an application, we obtain that the continuous fuzzy-valued function on a closed interval exists supremum and infimum and give the precise representation. We also show that the bounded fuzzy-valued function on a closed interval can define the lower and upper sums and the lower and upper integrals of Riemann and Riemann–Stieltjes by the usual way.     

Carleman-type theorems for generalized Q-holomorphic functions

In this work, we reduce the boundary condition of Riemann–Hilbert problem for generalized Q-holomorphic functions to the Vekua-type canonical form and obtain an appropriate analogue to the Carleman type representation for generalized Q-holomorphic functions.     

Wiener–Hopf factorization of a generalized Daniele–Khrapkov class of 2×2 matrix symbols

The Wiener–Hopf factorization of a class of 2×2 symbols including matrices of Daniele–Khrapkov type is studied. The partial indices and the factors are determined, both in the canonical and non‐canonical cases. A non‐linear method is used which reduces the solution of a homogeneous Riemann–Hilbert problem to a non‐linear scalar equation. Copyright © 1999 John Wiley & Sons, Ltd.     

Quadratic Hermite–Padé Approximation to the Exponential Function: A Riemann–Hilbert Approach

We investigate the asymptotic behavior of the polynomials p, q, r of degrees n in type I Hermite–Padé approximation to the exponential function, defined by p(z)e^-z + q(z) + r(z) e^z = O(z³ⁿ⁺²) as z 0. These polynomials are characterized by a Riemann–Hilbert problem for a 3 × 3 matrix valued function. We use the Deift–Zhou steepest descent method for Riemann–Hilbert problems to obtain strong uniform asymptotics for the scaled polynomials p(3nz), q(3nz), and r(3nz) in every domain in the complex plane. An important role is played by a three-sheeted Riemann surface and certain measures and functions derived from it. Our work complements the recent results of Herbert Stahl.     

Vector functional-difference equation in electromagnetic scattering

A vector functional-difference equation of the first order witha special matrix coefficient is analysed. It is shown how itcan be converted into a Riemann–Hilbert boundary-valueproblem on a union of two segments on a hyper-elliptic surface.The genus of the surface is defined by the number of zeros andpoles of odd order of a characteristic function in a strip.An even solution of a symmetric Riemann–Hilbert problemis also constructed. This is a key step in the procedure fordiffraction problems. The proposed technique is applied forsolving in closed form a new model problem of electromagneticscattering of a plane wave obliquely incident on an anisotropicimpedance half-plane (all the four impedances are assumed tobe arbitrary).     

The Riemann–Roch theorem without denominators in motivic homotopy theory

The Riemann–Roch theorem without denominators for the Chern class maps on higher algebraic K-groups with values in motivic cohomology groups in the context of motivic homotopy theory is proved.     

Bernhard Riemann Remarks on his Life and Work

The first part contains a few facts about Riemann taken from correspondence (of Riemann and of Dedekind), and then describes Riemann’s letter to Weierstrass concerning his paper on the distribution of primes. It is followed by a brief outline of the paper itself.     

An analytical formula for the effective conductivity of 2D domains with cracks of high density

We construct the conformal mapping of the square with circular non-overlapping holes onto the square with non-overlapping slits. It is constructed as a solution of the Riemann–Hilbert problem for a multiply connected domain in a class of double periodic functions. The Riemann–Hilbert problem is reduced to a system of functional equations which is solved with an arbitrary order of approximations. On the basis of this conformal mapping, an analytical formula for the effective conductivity of randomly distributed cracks in 2D media is deduced. This formula extends the known before formulae to high density fractures in 2D media.     

Asymptotic Tian�CYau�CZelditch expansions on singular Riemann surfaces

In this paper, we give both lower and upper bound estimates of the Bergman kernel for a degeneration of Riemann surfaces with constant curvature −1. As a result, we give a geometric proof of the Riemann–Rock theorem for a singular Riemann surface.     

The Riemann–Hilbert Problem for Analytic Description of the DM Solitons

A simple exact formula is derived for the profile of the optical pulse propagating over a DM fiber with zero mean dispersion. The dissipation is neglected, and dispersion is assumed to be constant along the adjacent legs of the waveguide, thus providing the applicability of the integrable NLS models within each leg. The formula describes a class of solutions called dispersion-managed solitons (DM solitons), which are periodic along the waveguide and exponentially localized in time. The DM solitons are parameterized by a certain class of spectral data, specified from numerical simulations. Using a related Riemann–Hilbert problem, we reconstruct a profile of the DM soliton from the given spectral data. For sufficiently long legs, the leading term of DM soliton is found in explicit form by asymptotic undressing of the Riemann–Hilbert problem. The analytic results are compared with numerical simulations.     

A numeric–analytic method for time-fractional Swift–Hohenberg (S–H) equation with modified Riemann–Liouville derivative

In this paper, the fractional variational iteration method (FVIM) was applied to obtain the approximate solutions of time-fractional Swift–Hohenberg (S–H) equation with modified Riemann–Liouville derivative. A new application of fractional variational iteration method (FVIM) was extended to derive analytical solutions in the form of a series for these equations. Numerical results showed the FVIM is powerful, reliable and effective method when applied strongly nonlinear equations with modified Riemann–Liouville derivative.     

Differential Algebras with Banach-Algebra Coefficients II: The Operator Cross-Ratio Tau-Function and the Schwarzian Derivative

Several features of an analytic (infinite-dimensional) Grassmannian of (commensurable) subspaces of a Hilbert space were developed in the context of integrable PDEs (KP hierarchy). We extended some of those features when polarized separable Hilbert spaces are generalized to a class of polarized Hilbert modules and then consider the classical Baker and τ-functions as operator-valued. Following from Part I we produce a pre-determinant structure for a class of τ-functions defined in the setting of the similarity class of projections of a certain Banach *-algebra. This structure is explicitly derived from the transition map of a corresponding principal bundle. The determinant of this map leads to an operator τ-function. We extend to this setting the operator cross-ratio which had previously been used to produce the scalar-valued τ-function, as well as the associated notion of a Schwarzian derivative along curves inside the space of similarity classes of a given projection. We link directly this cross-ratio with Fay’s trisecant identity for the τ-function. By restriction to the image of the Krichever map, we use the Schwarzian to introduce the notion of an operator-valued projective structure on a compact Riemann surface: this allows a deformation inside the Grassmannian (as it varies its complex structure). Lastly, we use our identification of the Jacobian of the Riemann surface in terms of extensions of the Burchnall–Chaundy C*-algebra (Part I) provides a link to the study of the KP hierarchy.     

On the convergence of circle packings to the Riemann map

Rodin and Sullivan (1987) proved Thurston’s conjecture that a scheme based on the Circle Packing Theorem converges to the Riemann mapping, thereby providing a refreshing geometric view of Riemann’s Mapping Theorem. We now present a new proof of the Rodin–Sullivan theorem. This proof is based on the argument principle, and has the following virtues. 1. It applies to more general packings. The Rodin–Sullivan paper deals with packings based on the hexagonal combinatorics. Later, quantitative estimates were found, which also worked for bounded valence packings. Here, the bounded valence assumption is unnecessary and irrelevant. 2. Our method is rather elementary, and accessible to non-experts. In particular, quasiconformal maps are not needed. Consequently, this gives an independent proof of Riemann’s Conformal Mapping Theorem. (The Rodin–Sullivan proof uses results that rely on Riemann’s Mapping Theorem.) 3. Our approach gives the convergence of the first and second derivatives, without significant additional difficulties. While previous work has established the convergence of the first two derivatives for bounded valence packings, now the bounded valence assumption is unnecessary. Oblatum 15-V-1995 & 13-XI-1995     

An operator van der Corput estimate arising from oscillatory Riemann–Hilbert problems

We study an operator analogue of the classical problem of finding the rate of decay of an oscillatory integral on the real line. This particular problem arose in the analysis of oscillatory Riemann–Hilbert problems associated with partial differential equations in the Ablowitz–Kaup–Newell–Segur hierarchy, but is interesting in its own right as a question in harmonic analysis and oscillatory integrals. As was the case in earlier work of the first author [9], the approach is general and purely real-variable. The resulting estimates we achieve are strongly uniform as a function of the phase and can simultaneously accommodate phases with low regularity (as low as C^1,α$C^{1,\alpha }$), local singularities, and essentially arbitrary sets of stationary points that degenerate to finite or infinite order.     

Asymptotics of Orthogonal Polynomials: Some Old, Some New, Some Identities

We briefly review some asymptotics of orthonormal polynomials. Then we derive the Bernstein–Szeg, the Riemann–Hilbert (or Fokas–Its–Kitaev), and Rakhmanov projection identities for orthogonal polynomials and attempt a comparison of their applications in asymptotics.     

On extensions of a theorem of Baxter

We combine the Riemann–Hilbert approach with the techniques of Banach algebras to obtain an extension of Baxter's Theorem for polynomials orthogonal on the unit circle. This is accomplished by using the link between the negative Fourier coefficients of the scattering function and the coefficients in the recurrence formula satisfied by these polynomials.