A Mixed Boundary Value Problem for Polyanalytic Function of Order n in the Sobolev Space Wn,p (D)

We discuss the construction of a polyanalytic function Φ of order n on a simple bounded domain D. The function satisfies n prescribed generalized Riemann-Hilbert boundary conditions on the boundary ?D and n generalized jump conditions on a simple closed smooth contour γ contained in D. The boundary conditions are transformed into n classical Riemann-Hilbert problems and the n jump conditions into n Riemann problems of conjugation for some 2n holomorphic functions. These transformed problems are solved using the standard methods from the literature.     

Convex sets with semidefinite representation

We provide a sufficient condition on a class of compact basic semialgebraic sets for their convex hull co(K) to have a semidefinite representation (SDr). This SDr is explicitly expressed in terms of the polynomials g _j that define K. Examples are provided. We also provide an approximate SDr; that is, for every fixed , there is a convex set such that (where B is the unit ball of ), and has an explicit SDr in terms of the g _j ’s. For convex and compact basic semi-algebraic sets K defined by concave polynomials, we provide a simpler explicit SDr when the nonnegative Lagrangian L _f associated with K and any linear is a sum of squares. We also provide an approximate SDr specific to the convex case.      

An Application of Calculus on Measure Chains to Fourier Theory and Heisenberg's Uncertainty Principle

We present "one-dimensional" Fourier theory on commutative groups T hH , 0 h h < X , 0< H h X within the framework of the so-called calculus on measure chains (or time scales). Depending on certain values of the graininess h and length H of the group the four classical types of Fourier transform are covered: Fourier integral ( T 0 X = R ), Fourier series ( T 1 X = Z ), Fourier analysis of periodic functions ( T 0,2 ~ = S 1 (0) unit circle) and discrete Fourier transform ( T 1 N = Z N ). We will present Fourier theory on these groups in a unified manner. This also allows to closely track the roles of the graininess h and length H of the group--especially for h M 0 and H M X . In the final part of the paper, we investigate the solution of a fundamental equation on T hH , which can be considered as a generalization of the Gauss function. It finally leads to a version of the Heisenberg uncertainty principle, which extends the classical one, valid for T 0 X = R , to the case T hH , where either h >0 or H < X .     

Light leaves and Lusztig's conjecture

We define a map F with domain a certain subset of the set of light leaves (certain morphisms between Soergel bimodules introduced by the author in an earlier paper) and range the set of prime numbers. Using results of Soergel we prove the following property of F  : if the image p=F(l)$p=F(l)$ of some light leaf l under F is bigger than the Coxeter number of the corresponding Weyl group, then there is a counterexample to Lusztig's conjecture in characteristic p. We also introduce the “double leaves basis” which is an improvement of the light leaves basis that has already found interesting applications. In particular it forms a cellular basis of Soergel bimodules that allows us to produce an algorithm to find “the bad primes” for Lusztig's conjecture.     

On singularities of solution of Maxwell's equations in axisymmetric domains with conical points

Boundary value problems (BVP) in three‐dimensional axisymmetric domains can be treated more efficiently by partial Fourier analysis. Partial Fourier analysis is applied to time‐harmonic Maxwell's equations in three‐dimensional axisymmetric domains with conical points on the rotation axis thereby reducing the three dimensional BVP to an infinite sequence of 2D BVPs on the plane meridian domain Ω_a?? of . The regularity of the solutions u _n (n∈?₀:={0, 1, 2,…}) of the two dimensional BVPs is investigated and it is proved that the asymptotic behaviour of the solutions u _n near an angular point on the rotation axis can be characterized by singularity functions related to the solutions of some associated Legendre equations. By means of numerical experiments, it is shown that the solutions u _n for n∈?₀\{1} belong to the Sobolev space H² irrespective of the size of the solid angle at the conical point. However, the regularity of the coefficient u ₁ depends on the size of the solid angle at the conical point. The singular solutions of the three dimensional BVP are obtained by Fourier synthesis. Copyright © 2006 John Wiley & Sons, Ltd.     

On the Value Distribution of an Algebroid Function of Infinite Order

In this paper, we first give a fundamental inequality of algebroid functions, then using it we prove that for any x -valued algebroid functions of infinite order in the plane, there exists a sequence of filling disks and Borel direction of dealing with its multiple values, such that the number of Borel exceptional values is equal to 2ν at most.     

Picard sequences with every orbit containing infinitely many perfect n-powers

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Finding a function which generates a sequence via iteration whose values at one or many points in its domain satisfy certain prescribed properties, i.e., finding a function such that the Picard orbit(s) of one or many points in its domain which possess some given properties, is an interesting problem. Given any positive integer n greater than one, we construct in this paper families of functions on the natural numbers such that the sequence of the iterations of each of these functions at any positive integer s contains infinitely many perfect n-powers. In terms of Picard sequences, this amounts to constructing a function whose Picard orbit at every point in its domain contains infinitely many perfect n-powers.

Video

For a video summary of this paper, please visit http://www.youtube.com/watch?v=wJqaXyB2pdo.     

Convex Holes Produce Lower Bounds for Coefficients

Let D denote the open unit disk and $ f:D \to \bar {{\bf C}}$ be meromorphic and injective in D . Especially, we consider such f which have an expansion $$ f(z) = z + \sum \limits_{n=2}^{\infty }a_n(\;f\,)z^n $$ in a neighbourhood of the origin and map D onto a domain whose complement with respect to $\bar {{\bf C}}$ is convex. Let the set of these functions be denoted by Co . We fix | f m 1 ( X )| for f ] Co and determine the inner and outer radius of the ring domain which is the domain of variability of a 2 ( f ) for such f . Further, it is shown that f ] Co implies that $$ \phi (z) = z+2 {f'(z) \over f''(z)}$$ is holomorphic in D and maps D into itself. This implication in turn implies the inequalities | a n ( f )| S 1 for f ] Co and n = 2,3,4. In addition, we show that | a n ( f )| S 1/2 for f ] Co and all n S 2 .     

A Proof of Sharkovsky's Theorem

This paper presents a variation on the standard directed graph proof. This variation makes use of the natural representation of the symmetric group given by one-dimensional maps.     

On the Efficiency of a Global Non-differentiable Optimization Algorithm Based on the Method of Optimal Set Partitioning

The examined algorithm for global optimization of the multiextremal non-differentiable function is based on the following idea: the problem of determination of the global minimum point of the function f(x) on the set (f(x) has a finite number of local minima in this domain) is reduced to the problem of finding all local minima and their attraction spheres with a consequent choice of the global minimum point among them. This reduction is made by application of the optimal set partitioning method. The proposed algorithm is evaluated on a set of well-known one-dimensional, two-dimensional and three-dimensional test functions. Recommendations for choosing the algorithm parameters are given.     

On an alternative cauchy equation in two unknown functions. Some classes of solutions

Summary In this paper we consider the alternative Cauchy functional equationg(xy) g(x)g(y) impliesf(xy) = f(x)f(y) wheref, g are functions from a topological group (X, ·) into a group (S,·). First we prove that, ifS is a Hausdorff topological group andX satisfies some weak additional hypotheses, then (f, g) is a continuous solution if and only if eitherf org is a homomorphism. Then we describe a more general class of solutions forX =R ⁿ .Partially supported by M.U.R.S.T. Research funds (40%)Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth.     

On the Schwartz space of the basic affine space

Let G be the group of points of a split reductive algebraic group G over a local field k and let X = G / U where U is the group of k-points of a maximal unipotent subgroup of G. In this paper we construct a certain canonical G-invariant space (called the Schwartz space of X) of functions on X, which is an extension of the space of smooth compactly supported functions on X. We show that the space of all elements of , which are invariant under the Iwahori subgroup I of G, coincides with the space generated by the elements of the so called periodic Lusztig basis, introduced recently by G. Lusztig (cf. [10] and [11]). We also give an interpretation of this space in terms of a certain equivariant K-group (this was also done by G. Lusztig — cf. [12]). Finally we present a global analogue of , which allows us to give a somewhat non-traditional treatment of the theory of the principal Eisenstein series.     

A Note on the Set of Periods of Transversal Homological Sphere Self-maps

Let M be a n -dimensional manifold with the same homology than the n -dimensional sphere. A C 1 map f : M M M is called transversal if for all m ] N the graph of f m intersects transversally the diagonal of M 2 M at each point ( x , x ) such that x is a fixed point of f m . We study the minimal set of periods of f by using the Lefschetz numbers for periodic points. In the particular case that n is even, we also study the set of periods for the transversal holomorphic self-maps of M .     

Non-real Zeros of Derivatives of Real Entire Functions and the Pólya-Wiman Conjectures

A function f is in the class $ V_2p $ iff $ f(z) = e^{-az^{2p+2}}g(z) $ where a S 0 and g is a constant multiple of a real entire function of genus h 2 p + 1 with only real zeros. The class $ U_2p $ is defined as follows: $ U_0 = V_0 $ , $ U_{2p} = V_{2p}-V_{2p-2} $ . Functions in the class $ U_{2p}^{*} $ are represented as $ g(z) = c(z)f(z) $ where $ f\in U_{2p} $ and c is a real polynomial with no real zeros. Every real entire function g , of finite order with at most finitely many non-real zeros satisfies $ g\in U_{2p}^{*} $ for a unique p . We show the exact number of non-real zeros of f" , for $ f\in U_{2p} $ , in terms of the number of non-real zeros of f' and a geometrical condition on the components of Im Q ( z ) > 0, where $ \displaystyle Q(z) = z-({f(z)}/{f'(z)}) $ . Further, for a subclass of $ f\in U_{2p} $ , we show necessary and sufficient conditions for f" to have exactly 2 p non-real zeros. For a subclass of $ U_{2p}^{*} $ we show that if f' has only real zeros, then f" has exactly 2 p non-real zeros. For $ f\in U_{2p}^{*} $ we show that 2 p is a lower bound for the number of non-real zeros of $ f^{(k)} $ for k S 2.     

Some Theorems on Meromorphic Univalent Functions

In this paper, we will show some deviation theorems and theorems of rotational angles in classes of Σ(p) and Σ(p,q) of meromorphic univalent functions.     

A Mixed Boundary Value Problem for Generalized Polyanalytic Functions of Order n in the Sobolev Space W n,p ( D )

Given is the following boundary value problem for a generalized polyanalytic differential equation of order n in a domain D : $${{\partial ^n w} \over {\partial \bar z^n }} = F\left( {z, w, {{\partial ^{m + k} w} \over {\partial z^m \partial \bar z^k }}} \right)\quad {\rm on}\enskip D \eqno (1)$$ $$Bw = g\quad {\rm on}\enskip \gamma \cup \partial D \eqno (2)$$ $$n \ge m, k \in {\shadN}_0 \quad m + k \le n, \quad (0, 0) \ne (m, k) \ne (0, n), \quad n \in {\shadN}$$ where B is an operator acting on the boundary ‘ D of D and possibly on some closed curve n in D . First the existence of a general solution is established under certain assumptions on the right-hand side of (1). Next, a general method for solving boundary value problems or other related problems (2) is laid down. The latter is possible whenever the corresponding problem for a polyanalytic function admits a unique solution which, in addition, satisfies certain a priori estimates. The method will be illustrated for the case of a mixed boundary value problem.     

Memorandum on multiplicative bijections and order

Let denote the semigroup of continuous functions from the topological space X to , equipped with the pointwise multiplication. The paper studies semigroup homomorphisms , with emphasis on isomorphisms. The crucial observation is that, in this setting, homomorphisms preserve order, so isomorphisms preserve order in both directions and they are automatically lattice isomorphisms. Applications to uniformly continuous and Lipschitz functions on metric spaces are given. Sample result: if Y and X are complete metric spaces of finite diameter without isolated points, every multiplicative bijection has the form Tf=f○τ, where τ:X→Y is a Lipschitz homeomorphism. F. Cabello Sánchez and J. Cabello Sánchez are supported in part by DGICYT projects MTM2004-02635 and MTM2007-6994-C02-02. J. Cabello Sánchez is supported in part by a grant of the UEx (Programa Propio–Acción 2).     

Cahn-Hilliard stochastic equation: Strict positivity of the density

This paper deals with the Cahn-Hilliard stochastic equation driven by a space-time white noise with a non-linear diffusion coefficient. Using new lower estimate of the kernel, we prove the "local" existence of the density without non-degeneracy condition in a case of Hölder continuous trajectories, and we show that the density of any vector is lower bounded by a strictly positive continuous function under a non-degeneracy condition.     

High degrees in random recursive trees

For , let T_n be a random recursive tree (RRT) on the vertex set . Let be the degree of vertex v in T_n, that is, the number of children of v in T_n. Devroye and Lu showed that the maximum degree Δ_n of T_n satisfies almost surely; Goh and Schmutz showed distributional convergence of along suitable subsequences. In this work we show how a version of Kingman's coalescent can be used to access much finer properties of the degree distribution in T_n. For any , let . Also, let be a Poisson point process on with rate function . We show that, up to lattice effects, the vectors converge weakly in distribution to . We also prove asymptotic normality of when slowly, and obtain precise asymptotics for when and is not too large. Our results recover and extends the previous distributional convergence results on maximal and near‐maximal degrees in RRT.