Invariance properties of generalized monotonicity

Generalized monotone maps are studied under affine variable transformations. The results enable us to generate generalized monotone matrices of any size. Various necessary conditions and sufficient conditions for generalized monotone matrices are derived. Furthermore. admissible translations of generalized monotone linear maps are studied. Finally, the maximal domain of generalized monotonicity is characterized.     

A multiquadric quasi-interpolation with linear reproducing and preserving monotonicity

In this paper, we develop a multiquadric (MQ) quasi-interpolation which has the properties of linear reproducing and preserving monotonicity. Moreover, we give its approximation error by theoretic analysis and illustrate the effect by means of two examples. One of the examples is to approach the linear combination of two sine functions with different frequencies. Another is to approximate a function with discontinuity. From the results of the examples, we believe that the present MQ quasi-interpolation is feasible.     

A nonalgebraic patchwork

Itenberg and Shustin’s pseudoholomorphic curve patchworking is in principle more flexible than Viro’s original algebraic one. It was natural to wonder if the former method allows one to construct nonalgebraic objects. In this paper we construct the first examples of patchworked real pseudoholomorphic curves in Σ _n whose position with respect to the pencil of lines cannot be realized by any real algebraic curve of the same bidegree. Both authors are very grateful to the Max Planck Institute für Mathematik in Bonn for its financial support and excellent working conditions.     

A free analogue of Shannon's problem on monotonicity of entropy

We prove a free probability analog of a result of [S. Artstein, K. Ball, F. Barthe, A. Naor, Solution of Shannon's problem on monotonicity of entropy, J. Amer. Math. Soc. 17 (2004) 975-982]. In particular, we prove that if X₁,X₂,… are freely independent identically distributed random variables, then the function

     

A note on periodic points of order preserving subhomogeneous maps

Let be the standard closed positive cone in and let be the set of integers for which there exists a continuous, order preserving, subhomogeneous map , which has a periodic point with period . It has been shown by Akian, Gaubert, Lemmens, and Nussbaum that is contained in the set consisting of those for which there exist integers and such that , , and for some . This note shows that for all .

     

On a conjecture of Whittaker concerning uniformization of hyperelliptic curves

This article concerns an old conjecture due to E. T. Whittaker, aiming to describe the group uniformizing an arbitrary hyperelliptic Riemann surface as an index two subgroup of the monodromy group of an explicit second order linear differential equation with singularities at the values .

Whittaker and collaborators in the thirties, and R. Rankin some twenty years later, were able to prove the conjecture for several families of hyperelliptic surfaces, characterized by the fact that they admit a large group of symmetries. However, general results of the analytic theory of moduli of Riemann surfaces, developed later, imply that Whittaker's conjecture cannot be true in its full generality.

Recently, numerical computations have shown that Whittaker's prediction is incorrect for random surfaces, and in fact it has been conjectured that it only holds for the known cases of surfaces with a large group of automorphisms.

The main goal of this paper is to prove that having many automorphisms is not a necessary condition for a surface to satisfy Whittaker's conjecture.

     

On the best parametrization

The numerical solution to a system of nonlinear algebraic or transcendental equations with several parameters is examined in the framework of the parametric continuation method. Necessary and sufficient conditions are proved for choosing the best parameters, which provide the best condition number for the system of linear continuation equations. Such parameters have to be sought in the subspace tangent to the solution space of the system of nonlinear equations. This subspace is obtained if the original system of nonlinear equations is solved at the various parameter values from a given set. The parametric approximation of curves and surfaces is considered.     

Explicit maximal and minimal curves over finite fields of odd characteristics

In this work we present explicit classes of maximal and minimal Artin–Schreier type curves over finite fields having odd characteristics. Our results include the proof of Conjecture 5.9 given in [1] as a very special subcase. We use some techniques developed in [2], which were not used in [1].     

A complete monotonicity property of the gamma function

A logarithmically completely monotonic function is completely monotonic. The function is strictly completely monotonic on (0,∞). The function is strictly logarithmically completely monotonic on (0,∞).     

一些特殊函数的完全单调性

欧拉Gamma函数是一种非常重要的函数,在数学的许多分支以及物理、工程等学科中都有着十分重要的作用.而完全单调性以及对数完全单调性是Gamma函数的重要性质.主要证明了一些包含Gamma函数和Psi函数在内的特殊函数的完全单调性和对数完全单调性,并由此推出了一些重要的不等式.     

On the choice of the initial value of a quadratic spline in positive interpolation

Any quadratic spline with coinciding interpolating points and breakpoints can be uniquely determined by one initial condition. We examine how to choose the initial value in positive interpolation by using a norm of the spline. Monotone interpolation is also considered. The results are illuminated by numerical examples and by an application.     

Topology Representing Networks for the Visualization of Manifolds

In practical data mining tasks, high-dimensional data has to be analyzed. In most of the cases it is very informative to map and visualize the hidden structure of a complex data set in a low-dimensional space. In this paper a new class of mapping algorithms is defined. These algorithms combine topology representing networks and different nonlinear mapping algorithms. While the former methods aim to quantify the data and disclose the real structure of the objects, the nonlinear mapping algorithms are able to visualize the quantized data in the low-dimensional vector space. In this paper, techniques based on these methods are gathered and the results of a detailed analysis performed on them are shown. The primary aim of this analysis is to examine the preservation of distances and neighborhood relations of the objects. Preservation of neighborhood relations was analyzed both in local and global environments. To evaluate the main properties of the examined methods we show the outcome of the analysis based both on synthetic and real benchmark examples.     

On the denominator values and barycentric weights of rational interpolants

We improve upon the method of Zhu and Zhu [A method for directly finding the denominator values of rational interpolants, J. Comput. Appl. Math. 148 (2002) 341–348] for finding the denominator values of rational interpolants, reducing considerably the number of arithmetical operations required for their computation. In a second stage, we determine the points (if existent) which can be discarded from the rational interpolation problem. Furthermore, when the interpolant has a linear denominator, we obtain a formula for the barycentric weights which is simpler than the one found by Berrut and Mittelmann [Matrices for the direct determination of the barycentric weights of rational interpolation, J. Comput. Appl. Math. 78 (1997) 355–370]. Subsequently, we give a necessary and sufficient condition for the rational interpolant to have a pole.     

Boundaries of Flat Compact Surfaces in 3-Space

This paper deals with the problem `which knots or links in3-space bound flat (immersed) compact surfaces?' In aforthcoming paper by the author, it is proven that any simple closedspace curve can be deformed until it bounds a flat orientable compact(Seifert) surface. The main results of this paper are that there existknots that do not bound any flat compact surfaces. The lower bound oftotal curvature of a knot bounding an orientable nonnegatively curvedcompact surface can, for varying knot types, be arbitrarily much greaterthan the infimum of curvature needed for the knot to have its knot type.The number of 3-singular points (points of zero curvatureor if not then of zero torsion) on the boundary of a flat immersedcompact surface is greater than or equal to twice the absolute value ofthe Euler characteristic of the surface. A set of necessary and, in aweakened sense, sufficient conditions for a knot or link to be what wecall a generic boundary of a flat immersed compact surface withoutplanar regions is given.