Exposed sets in potential theory

Let U be a relatively compact open subset of a harmonic space, and H(U) be the function space of all continuous functions on which are harmonic on U. We give a complete characterization of the H(U)-exposed subsets of . This extends the results of [J. Lukeš, T. Mocek, M. Smr?ka, J. Spurný, Choquet like sets in function spaces, Bull. Sci. Math. 127 (2003) 397-437].     

Spatial asymptotic behavior of homeomorphic global flows for non-Lipschitz SDEs

Let x→?s,t(x) be a -valued stochastic homeomorphic flow produced by non-Lipschitz stochastic differential equation , where W=(W¹,W²,…) is an infinite sequence of independent standard Brownian motions. We first give some estimates of modulus of continuity of {?s,t(⋅)}, then prove that the flow ?s,t(x), when x nears infinity, grows slower than for some constant c>0 and integrable random variable Z via lemma of Garsia-Rodemich-Rumsey Lemma (abbreviated as GRR Lemma) improved by Arnold and Imkeller [L. Arnold, P. Imkeller, Stratonovich calculus with spatial parameters and anticipative problems in multiplicative ergodic theory, Stochastic Process. Appl. 62 (1996) 19-54] and moment estimates for one- and two-point motions.     

On the Pythagoras numbers of real analytic surfaces

We show that (i) every positive semidefinite meromorphic function germ on a surface is a sum of 4 squares of meromorphic function germs, and that (ii) every positive semidefinite global meromorphic function on a normal surface is a sum of 5 squares of global meromorphic functions.     

Hypersurfaces Levi-plates immergées dans les surfaces complexes de courbure positive

We show that there is no immersed compact Levi-flat hypersurface of class C¹ in the complex projective plane, if the foliation by holomorphic curves carries a harmonic current which is absolutely continuous with respect to the Lebesgue measure, with a density bounded from above and below. This is a corollary of a rigidity result for immersed compact Levi-flat hypersurfaces in complex surfaces of non-negative curvature.     

Worst-case analysis of the WSPT and MWSPT rules for single machine scheduling with one planned setup period

In this article, we consider the single machine scheduling problem with one planned setup period, with the aim of minimizing the weighted sum of the completion times. We study the WSPT and MWSPT heuristics and we show that the worst-case performance ratio is 3 for the two heuristics in some cases and it is unbounded otherwise. We also show that these worst-case performance ratios are tight.     

Harnack inequality for harmonic functions relative to a nonlinear p-homogeneous Riemannian Dirichlet form

We state a Wiener criterion for the regularity of points with respect to a relaxed Dirichlet problem for a p-homogeneous Riemannian Dirichlet form.     

Bifurcations of limit cycles from quintic Hamiltonian systems with a double figure eight loop

This paper deals with Liénard equations of the form , , with P and Q polynomials of degree 5 and 4 respectively. Attention goes to perturbations of the Hamiltonian vector fields with an elliptic Hamiltonian of degree six, exhibiting a double figure eight loop. The number of limit cycles and their distributions are given by using the methods of bifurcation theory and qualitative analysis.     

Some results involving series representations of hypergeometric functions

In this paper, we prove some generalisations of several theorems given in [K.A. Driver, S.J. Johnston, An integral representation of some hypergeometric functions, Electron. Trans. Numer. Anal. 25 (2006) 115-120] and examine some special cases which correspond to a transformation given by Chaundy in [T.W. Chaundy, An extension of hypergeometric functions, I., Quart. J. Maths. Oxford Ser. 14 (1943) 55-78] and other transformations involving the Riemann zeta function and the beta function.     

Projections for harmonic Bergman spaces and applications

On the setting of general bounded smooth domains in , we construct L¹-bounded nonorthogonal projections and obtain related reproducing formulas for the harmonic Bergman spaces. In addition, we show that those projections satisfy Sobolev L^p-estimates of any order even for p=1. Among applications are Gleason's problems for the harmonic Bergman-Sobolev and (little) Bloch functions on star-shaped domains with strong reference points.     

Gleason's problem for harmonic Bergman and Bloch functions on half-spaces

On the setting of the half-spaceR ^n–1×R ₊, we investigate Gleason's problem for harmonic Bergman and Bloch functions. We prove that Gleason's problem for the harmonicL ^p -Bergman space is solvable if and only ifp>n. We also prove that Gleason's problem for the harmonic (little) Bloch space is solvable.     

Pointlike sets with respect to R and J

We present an algorithm to compute the pointlike subsets of a finite semigroup with respect to the pseudovariety of all finite R-trivial semigroups. The algorithm is inspired by Henckell’s algorithm for computing the pointlike subsets with respect to the pseudovariety of all finite aperiodic semigroups. We also give an algorithm to compute -pointlike sets, where denotes the pseudovariety of all finite J-trivial semigroups. We finally show that, in contrast with the situation for , the natural adaptation of Henckell’s algorithm to computes pointlike sets, but not all of them.     

Semi-Harmonicity,Integral Means and Euler Type Vector Fields

The Dirichlet product of functions on a semi-Riemann domain and generalized Euler vector fields, which include the radial, -Euler, and the -Neumann vector fields, are introduced. The integral means and the harmonic residues of functions on a Riemann domain are studied. The notion of semi-harmonicity of functions on a complex space is introduced. It is shown that, on a Riemann domain, the semi-harmonicity of a locally integrable function is characterized by local mean-value properties as well as by weak harmonicity. In particular, the Weyl’s Lemma is extended to a Riemann domain. Supports by Minnesota State University, Mankato and the Grant “Globale Methoden in der komplexen Geometrie” of the German research society DFG are gratefully acknowledged.     

The Möbius function of separable and decomposable permutations

We give a recursive formula for the Möbius function of an interval [σ,π] in the poset of permutations ordered by pattern containment in the case where π is a decomposable permutation, that is, consists of two blocks where the first one contains all the letters 1,2,…,k for some k. This leads to many special cases of more explicit formulas. It also gives rise to a computationally efficient formula for the Möbius function in the case where σ and π are separable permutations. A permutation is separable if it can be generated from the permutation 1 by successive sums and skew sums or, equivalently, if it avoids the patterns 2413 and 3142.We also show that the Möbius function in the poset of separable permutations admits a combinatorial interpretation in terms of normal embeddings among permutations. A consequence of this interpretation is that the Möbius function of an interval [σ,π] of separable permutations is bounded by the number of occurrences of σ as a pattern in π. Another consequence is that for any separable permutation π the Möbius function of (1,π) is either 0, 1 or −1.     

A Fatou theorem for α-harmonic functions

We study functions which are harmonic in the upper half space with respect to (−Δ)^α/2, 0<α<2. We prove a Fatou theorem when the boundary function is L^p-Hölder continuous of order β and βp>1. We give examples to show this condition is sharp.