Asymptotic Gauss–Jacobi quadrature error estimation for Schwarz–Christoffel integrals

Numerical conformal mapping packages based on the Schwarz–Christoffel formula have been in existence for a number of years. Various authors, for good reasons of practical efficiency, have chosen to use composite n-point Gauss–Jacobi rules for the estimation of the Schwarz–Christoffel path integrals. These implementations rely on an ad hoc, but experimentally well-founded, heuristic for selecting the spacing of the integration end-points relative to the position of the nearby integrand singularities. In the present paper we derive an explicitly computable estimate, asymptotic as n→∞, for the relevant Gauss–Jacobi quadrature error. A numerical example illustrates the potential accuracy of the estimate even at low values of n. It is apparent that the error estimate will allow the adaptive construction of composite rules in a manner that is more efficient than has been possible hitherto.     

A modified Schwarz–Christoffel mapping for regions with piecewise smooth boundaries

A method where polygon corners in Schwarz–Christoffel mappings are rounded, is used to construct mappings from the upper half-plane to regions bounded by arbitrary piecewise smooth curves. From a given curve, a polygon is constructed by taking tangents to the curve in a number of carefully chosen so-called tangent points. The Schwarz–Christoffel mapping for that polygon is then constructed and modified to round the corners.     

Équations du type Monge-Ampère sur les variétés Riemanniennes compactes, I

Let (V_n, g) be a C^∞ compact Riemannian manifold. For a suitable function on V_n, let us consider the change of metric: g′ = g + Hess(), and the function, as a ratio of two determinants, M() = ¦g′¦ ¦g¦⁻¹. Using the method of continuity, we first solve in C^∞ the problem: Log M() = λ + ƒ, λ > 0, ƒ ε C^∞. Then, under weak hypothesis on F, we solve the general equation: Log M() = F(P, ), F in C^∞(V_n × ¦α, β¦), using a method of iteration. Our study gives rise to an interesting a priori estimate on ¦¦, which does not occur in the complex case. This estimate should enable us to solve the equation above when λ 0, providing we can overcome difficulties related to the invertibility of the linearised operator. This open question will be treated in our next article.     

Roper–Suffridge extension operator and the lower bound for the distortion

Liczberski–Starkov gave a sharp lower bound for DΦ_n(f)(z) near the origin, where Φ_n is the Roper–Suffridge extension operator and f is a normalized convex mapping on the unit disk in C. They gave a conjecture that the sharp lower bound holds on the Euclidean unit ball Bⁿ in Cⁿ. In this paper, we will give a sharp lower bound on Bⁿ for a more general extension operator and for normalized univalent mappings f or normalized convex mappings f. We will give a lower bound for mappings f in a linear invariant family. We will also give a similar sharp lower bound on bounded convex complete Reinhardt domains in Cⁿ.     

On growth and covering theorems of quasi-convex mappings in the unit ball of a complex banach space

A class of biholomorphic mappings named “quasi-convex mapping” is introduced in the unit ball of a complex Banach space. It is proved that this class of mappings is a proper subset of the class of starlike mappings and contains the class of convex mappings properly, and it has the same growth and covering theorems as the convex mappings. Furthermore, when the Banach space is confined to ℂⁿ, the “quasi-convex mapping” is exactly the “quasi-convex mapping of type A” introduced by K. A. Roper and T. J. Suffridge.     

Uniformly Gâteaux smooth approximations on

Every Lipschitz mapping from c₀(Γ) into a Banach space Y can be uniformly approximated by Lipschitz mappings that are simultaneously uniformly Gâteaux smooth and C^∞-Fréchet smooth.     

Approximations and error bounds for computing the inverse mapping

In this paper we propose a procedure to construct approximations of the inverse of a class of C ^m differentiable mappings. First of all we determine in terms of the data a neighbourhood where the inverse mapping is well defined. Then it is proved that the theoretical inverse can be expressed in terms of the solution of a differential equation depending on parameters. Finally, using one-step matrix methods we construct approximate inverse mappings of a prescribed accuracy.     

A Distortion Theorem for Univalent Gap Series

We prove a distortion theorem for conformal mappings of the unit disk for which log f is representable as the Hadamard gap series. This theorem implies in particular that such conformal mapping is almost bounded, i.e., for every >0, there is a positive constant C such that |f,(z)|C (1–|z|) ^–.     

A Remez-Type Inequality for Non-dense Müntz Spaces with Explicit Bound

LetΛ :=(λ_k)^∞_k=0be a sequence of distinct nonnegative real numbers withλ₀ :=0 and ∑^∞_k=1 1/λ_k<∞. Let(0, 1) and(0, 1−) be fixed. An earlier work of the authors shows that [formula]is finite. In this paper an explicit upper bound forC(Λ, , ) is given. In the special caseλ_k :=k^α,α>1, our bounds are essentially sharp.     

Mixed covolume method for parabolic problems on triangular grids

In this paper, we present a mixed covolume method for parabolic equations on triangular grids. This method use the lowest order Raviart–Thomas (R–T) mixed finite element space as the trial space. We prove the optimal order of convergence for the approximate pressure and velocity in L²-norm. Furthermore, we obtain the quasi-optimal error estimates for the approximate pressure in L^∞-norm.     

A viscosity of extragradient approximation method for finding equilibrium problems, variational inequalities and fixed point problems for nonexpansive mappings

In this paper, we investigate the problem for finding the set of solutions for equilibrium problems, the set of solutions of the variational inequalities for k-Lipschitz continuous mappings and fixed point problems for nonexpansive mappings in a Hilbert space. We introduce a new viscosity extragradient approximation method which is based on the so-called viscosity approximation method and extragradient method. We show that the sequence converges strongly to a common element of the above three sets under some parameters controlling conditions. Finally, we utilize our results to study some convergence problems for finding the zeros of maximal monotone operators. Our results are generalization and extension of the results of Kumam [P. Kumam, Strong convergence theorems by an extragradient method for solving variational inequalities and equilibrium problems in a Hilbert space, Turk. J. Math. 33 (2009) 85–98], Wangkeeree [R. Wangkeeree, An extragradient approximation method for equilibrium problems and fixed point problems of a countable family of nonexpansive mappings, Fixed Point Theory and Applications, 2008, Article ID 134148, 17 pages, doi:10.1155/2008/134148], Yao et al. [Y. Yao, Y.C. Liou, R. Chen, A general iterative method for an finite family of nonexpansive mappings, Nonlinear Analysis 69 (5–6) (2008) 1644–1654], Qin et al. [X. Qin, M. Shang, Y. Su, A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, Nonlinear Analysis (69) (2008) 3897–3909], and many others.     

Violation of Symmetry in the System of Three Laterally Coupled Quantum Waveguides,and Resonance Asymptotics

The system of three two-dimensional laterally coupled quantum waveguides with the Dirichlet boundary condition is considered. Violation of geometrical symmetry is studied. The behaviour of the resonance asymptotics and the transition “eigenvalue — resonance” are investigated. Bibliography: 16 titles.__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 300, 2003, pp. 221–227.     

The Moduli Space of Boundary Compactifications of SL (2, R)

In an earlier paper, the authors introduced the notion of a boundary compactification of SL(2, R) and SL(2, C), a normal projective embedding of PSL2 arising as the Zariski closure of an orbit in (P1)n under the diagonal action of SL2. Here the moduli space of such boundary compactifications of SL(2, R) is shown to be a contractible hyperbolic orbifold, by using the Schwarz–Christoffel transformation to identify it with a quotient of the moduli space of equi-angular planar polygons.     

Quasiconformal harmonic mappings between \fancyscriptC1,m{{\fancyscript{C}}^{1,\mu}} Euclidean surfaces

The conformal deformations are contained in two classes of mappings quasiconformal and harmonic mappings. In this paper we consider the intersection of these classes. We show that, every K quasiconformal harmonic mapping between surfaces with boundary is a Lipschitz mapping. This extends some recent results of several authors where the same problem has been considered for plane domains. As an application it is given an explicit Lipschitz constant of normalized isothermal coordinates of a disk-type minimal surface in terms of boundary curve only. It seems that this kind of estimates are new for conformal mappings of the unit disk onto a Jordan domain as well.     

Completeness of Trigonometric System with Integer Indices {e; x }

Necessary and sufficient conditions are given which ensure the completeness of the trigonometric systems with integer indices; {e^inx; x }^∞_n=−∞ or {e^inx; x }^∞_n=1 in L^α(μ,  ), α1. If there exists a support Λ of the measure μ which is a wandering set, that is, Λ+2kπ, k=0, ±1, ±2, … are mutually disjoint for different k's, then the linear span of our trigonometric system {e^inx; x }^∞_n=−∞ is dense in L^α(μ,  ) α1. The converse statement is also true.     

The Faber Operator and its Boundedness

Let G be a domain bounded by a Jordan curve Γ, and let A(G) be the Banach space of functions continuous on G and holomorphic in G. The Faber operator T is a linear mapping from A( ) to A(G) mapping wⁿ onto the nth Faber polynomial F_n(z) (n=0, 1, 2, …). We show that T<∞ if Γ is piecewise Dini-smooth, and give an example of a quasicircle Γ for which T=∞.     

Continuous Approximations of Multivalued Mappings and Fixed Points

In the present paper, we prove a fixed-point theorem for completely continuous multivalued mappings defined on a bounded convex closed subset X of the Hilbert space H which satisfies the tangential condition , where T _X (x) is the cone tangent to the set X at a point x. The proof of this theorem is based on the method of single-valued approximations to multivalued mappings. In this paper, we consider a simple approach for constructing single-valued approximations to multivalued mappings. This approach allows us not only to simplify the proofs of already-known theorems, but also to obtain new statements needed to prove the main theorem in this paper.__________Translated from Matematicheskie Zametki, vol. 78, no. 2, 2005, pp. 212–222.Original Russian Text Copyright © 2005 by B. D. Gel’man.     

Estimates of Freud-Christoffel functions for some weights with the whole real line as support

Upper and lower bounds for generalized Christoffel functions, called Freud-Christoffel functions, are obtained. These have the form λ_n,p(W,j,x) = infPW_Lp(R)/|P^(j)(X)| where the infimum is taken over all polynomials P(x) of degree at most n − 1. The upper and lower bounds for λ_n,p(W,j,x) are obtained for all 0 < p ∞ and J = 0, 1, 2, 3,… for weights W(x) = exp(−Q(x)), where, among other things, Q(x) is bounded in [− A, A], and Q″ is continuous in β(−A, A) for some A > 0. For p = ∞, the lower bounds give a simple proof of local and global Markov-Bernstein inequalities. For p = 2, the results remove some restrictions on Q in Freud's work. The weights considered include W(x) = exp(− ¦x¦^α/2), α > 0, and W(x) = exp(− exp(¦x¦)), > 0.     

The Classification of Some Perfect Codes

Perfect 1-error correcting codes C in Z ₂ ⁿ , where n=2 ^m–1, are considered. Let ; denote the linear span of the words of C and let the rank of C be the dimension of the vector space . It is shown that if the rank of C is n–m+2 then C is equivalent to a code given by a construction of Phelps. These codes are, in case of rank n–m+2, described by a Hamming code H and a set of MDS-codes D _h , h H, over an alphabet with four symbols. The case of rank n–m+1 is much simpler: Any such code is a Vasil'ev code.