Length spectra and<Emphasis Type="Italic">p</Emphasis>-spectra of compact flat manifolds

We compare and contrast various length vs Laplace spectra of compact flat Riemannian manifolds. As a major consequence we produce the first examples of pairs of closed manifolds that are isospectral on p-forms for some p ≠ 0, but have different weak length spectrum. For instance, we give a pair of 4-dimensional manifolds that are isospectral on p-forms for p = 1, 3and we exhibit a length of a closed geodesic that occurs in one manifold but cannot occur in the other. We also exhibit examples of this kind having different injectivity radius and different first eigenvalue of the Laplace spectrum on functions. These results follow from a method that uses integral roots of the Krawtchouk polynomials. We prove a Poisson summation formula relating the p-eigenvalue spectrum with the lengths of closed geodesics. As a consequence we show that the Laplace spectrum on functions determines the lengths of closed geodesics and, by an example, that it does not determine the complex lengths. Furthermore we show that orientability is an audible property for closed flat manifolds. We give a variety of examples, for instance, a pair of manifolds isospectral on functions (resp. Sunada isospectral) with different multiplicities of length of closed geodesies and a pair with the same multiplicities of complex lengths of closed geodesies and not isospectral on p-forms for any p, or else isospectral on p-forms for only one value of p ≠ 0.     

Strongly (λ, θ)-continuous functions

In the paper, we introduce and investigate basic properties of a new class of functions between topological spaces called strongly (λ, θ)-continuous functions which is stronger than λ-continuous functions. This work was completed while the second author was visiting the Department of Mathematical Sciences, Unisa, Pretoria, South Africa partially supported by the National Research Foundation of South Africa.     

（g,f）-FACTORS WITH SPECIAL PROPERTIES IN BIPARTITE （mg,mf）-GRAPHS

Let G be a bipartite graph and g and f be two positive integer-valued functions defined on vertex set V(G) of G such that g(x)≤f(x). In this paper,some sufficient conditions related to the connectivity and edge-connectivity for a bipartite (mg,mf)-graph to have a (g,f)-factor with special properties are obtained and some previous results are generalized.Furthermore,the new results are proved to be the best possible.     

Enveloppes inférieures de fonctions admissibles sur l'espace projectif complexe. Cas dissymétrique

This paper generalizes the first author's preceding works concerning admissible functions on certain Fano manifolds [A. Ben Abdesselem, Lower bound of admissible functions on sphere, Bull. Sci. Math. 126 (2002) 675-680 [2]; A. Ben Abdesselem, Enveloppes inférieures de fonctions admissibles sur l'espace projectif complexe. Cas symétrique, Bull. Sci. Math. 130 (2006) 341-353 [3]]. Here, we study a larger class of functions which can be less symmetric than the ones studied before. When the sup of these functions is null, we prove that they admit a lower bound, giving precisely Tian invariant [G. Tian, On Kähler-Einstein metrics on certain Kähler manifolds with C₁(M)>0, Invent. Math. 89 (1987) 225-246 [7]] (see also [T. Aubin, Réduction du cas positif de l'équation de Monge-Ampère sur les variétés Kählériennes à la démonstration d'une inégalité, J. Funct. Anal. 57 (1984) 143-153 [1]]) on these manifolds.     

On the Kantorovich–Rubinstein theorem

The Kantorovich–Rubinstein theorem provides a formula for the Wasserstein metric W₁ on the space of regular probability Borel measures on a compact metric space. Dudley and de Acosta generalized the theorem to measures on separable metric spaces. Kellerer, using his own work on Monge–Kantorovich duality, obtained a rapid proof for Radon measures on an arbitrary metric space. The object of the present expository article is to give an account of Kellerer’s generalization of the Kantorovich–Rubinstein theorem, together with related matters. It transpires that a more elementary version of Monge–Kantorovich duality than that used by Kellerer suffices for present purposes. The fundamental relations that provide two characterizations of the Wasserstein metric are obtained directly, without the need for prior demonstration of density or duality theorems. The latter are proved, however, and used in the characterization of optimal measures and functions for the Kantorovich–Rubinstein linear programme. A formula of Dobrushin is proved.     

<Emphasis Type="Italic">K</Emphasis>-subanalytic rectilinearization and uniformization

We prove rectilinearization and uniformization theorems for K-subanalytic (∝ _an ^K -definable) sets and functions using the Lion-Rolin formula. Parallel reasoning gives standard results for the subanalytic case.     

Decomposing <Emphasis Type="Italic">k</Emphasis>-ARc-Strong Tournaments Into Strong Spanning Subdigraphs

  The so-called Kelly conjecture states that every regular tournament on 2k+1 vertices has a decomposition into k-arc-disjoint hamiltonian cycles. In this paper we formulate a generalization of that conjecture, namely we conjecture that every k-arc-strong tournament contains k arc-disjoint spanning strong subdigraphs. We prove several results which support the conjecture:If D = (V, A) is a 2-arc-strong semicomplete digraph then it contains 2 arc-disjoint spanning strong subdigraphs except for one digraph on 4 vertices.Every tournament which has a non-trivial cut (both sides containing at least 2 vertices) with precisely k arcs in one direction contains k arc-disjoint spanning strong subdigraphs. In fact this result holds even for semicomplete digraphs with one exception on 4 vertices.Every k-arc-strong tournament with minimum in- and out-degree at least 37k contains k arc-disjoint spanning subdigraphs H ₁, H ₂, . . . , H _k such that each H _i is strongly connected.The last result implies that if T is a 74k-arc-strong tournament with speci.ed not necessarily distinct vertices u ₁, u ₂, . . . , u _k , v ₁, v ₂, . . . , v _k then T contains 2k arc-disjoint branchings where is an in-branching rooted at the vertex u _i and is an out-branching rooted at the vertex v _i , i=1,2, . . . , k. This solves a conjecture of Bang-Jensen and Gutin [3].We also discuss related problems and conjectures.

Adaptive Itô-Taylor algorithm can optimally approximate the Itô integrals of singular functions

We deal with numerical approximation of stochastic Itô integrals of singular functions. We first consider the regular case of integrands belonging to the Hölder class with parameters r and ?. We show that in this case the classical Itô-Taylor algorithm has the optimal error Θ(n^−(r+?)). In the singular case, we consider a class of piecewise regular functions that have continuous derivatives, except for a finite number of unknown singular points. We show that any nonadaptive algorithm cannot efficiently handle such a problem, even in the case of a single singularity. The error of such algorithm is no less than n^{−min{1/2,r+?}}. Therefore, we must turn to adaptive algorithms. We construct the adaptive Itô-Taylor algorithm that, in the case of at most one singularity, has the optimal error O(n^−(r+?)). The best speed of convergence, known for regular functions, is thus preserved. For multiple singularities, we show that any adaptive algorithm has the error Ω(n^{−min{1/2,r+?}}), and this bound is sharp.     

Whittaker functions for generalized principal series representations of <Emphasis Type="Italic">SL</Emphasis>(3, R)

We study Whittaker functions for generalized principal series representations of the real special linear group SL(3, R) of degree 3. From the Capelli elements and Dirac-Schmid operators, we give the system of partial differential equations which is satisfied by Whittaker functions. We give six formal power series solutions of this system, which are called secondary Whittaker functions. We also give the Mellin-Barnes type integral expressions of primary Whittaker functions, i.e. the solutions having the moderate growth property.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-convergence of Lagrange interpolation on the semiaxis

In order to approximate functions defined on (0, +∞), the authors consider suitable Lagrange polynomials and show their convergence in weighted L ^p -spaces.      

Boundary Harnack Principle for <Emphasis Type="Italic">p</Emphasis>-harmonic Functions in Smooth Euclidean Domains

We establish a scale-invariant version of the boundary Harnack principle for p-harmonic functions in Euclidean C ^1,1-domains and obtain estimates for the decay rates of positive p-harmonic functions vanishing on a segment of the boundary in terms of the distance to the boundary. We use these estimates to study the behavior of conformal Martin kernel functions and positive p-superharmonic functions near the boundary of the domain. H. A. was partially supported by Grant-in-Aid for (B) (2) (No. 15340046) Japan Society for the Promotion of Science. N. S. was partially supported by NSF grant DMS-0355027. X. Z. was partially supported by the Taft foundation.     

On weakly (τ, m)-continuous functions

In this paper we introduce a new notion of weakly (τ, m)-continuous functions as functions from a topological space into a set satisfying some minimal conditions. We obtain some characterizations and several properties of such functions. This function leads to the formulation of a unified theory of weak continuity [20], almosts-continuity [33],p(θ)-continuity [10] andp-continuity [41].     

Geometric properties of the functions Γ and 1/Γ

In this paper we determine the radius of convexity and the radius of starlikeness of the functions Γ and The basic tools of our work are the developments of the functions in function series.     

<Emphasis Type="Italic">K</Emphasis><Subscript>4</Subscript>-free subgraphs of random graphs revisited

In Combinatorica 17(2), 1997, Kohayakawa, ?uczak and Rödl state a conjecture which has several implications for random graphs. If the conjecture is true, then, for example, an application of a version of Szemerédi’s regularity lemma for sparse graphs yields an estimation of the maximal number of edges in an H-free subgraph of a random graph G _{n, p} . In fact, the conjecture may be seen as a probabilistic embedding lemma for partitions guaranteed by a version of Szemerédi’s regularity lemma for sparse graphs. In this paper we verify the conjecture for H = K ₄, thereby providing a conceptually simple proof for the main result in the paper cited above.     

Optimality for (<Emphasis Type="Italic">h,φ</Emphasis>)-multiobjective programming involving generalized type-I functions

Based upon Ben-Tal’s generalized algebraic operations, new classes of functions, namely (h,φ)-type-I, quasi (h,φ)-type-I, and pseudo (h,φ)-type-I, are defined for a multi-objective programming problem. Sufficient optimality conditions are obtained for a feasible solution to be a Pareto efficient solution for this problem. Some duality results are established by utilizing the above defined classes of functions, considering the concept of a Pareto efficient solution. This research is supported by National Science Foundation of China under Grant No. 69972036.     

A unified theory of<Emphasis Type="Italic">θ</Emphasis>-continuity for functions

We introduce a new notion ofθ-M-continuous functions as functions from a set satisfying some minimal conditions into a set satisfying some minimal conditions. We obtain some characterizations and several properties of such functions. The functions enable us to formulate a unified theory ofθ-continuity [15],θ-semi-continuity [4], quasi-irresoluteness [9],θ-preirresoluteness [29], and weakβ-irresoluteness [30].     

Rigidity for regular functions over Hamilton and Cayley numbers and a boundary Schwarz' Lemma

A new theory of regular functions over the skew field of Hamilton numbers (quaternions) and in the division algebra of Cayley numbers (octonions) has been recently introduced by Gentili and Struppa (Adv. Math. 216 (2007) 279–301). For these functions, among several basic results, the analogue of the classical Schwarz' Lemma has been already obtained. In this paper, following an interesting approach adopted by Burns and Krantz in the holomorphic setting, we prove some boundary versions of the Schwarz' Lemma and Cartan's Uniqueness Theorem for regular functions. We are also able to extend to the case of regular functions most of the related “rigidity” results known for holomorphic functions.     

The Vector-Valued Big <Emphasis Type="Italic">q</Emphasis>-Jacobi Transform

Big q-Jacobi functions are eigenfunctions of a second-order q-difference operator L. We study L as an unbounded self-adjoint operator on an L ²-space of functions on ℝ with a discrete measure. We describe explicitly the spectral decomposition of L using an integral transform ℱ with two different big q-Jacobi functions as a kernel, and we construct the inverse of ℱ.      

Generating functions of the (<Emphasis Type="Italic">h,q</Emphasis>) extension of twisted Euler polynomials and numbers

By using p-adic q-deformed fermionic integral on ℤ _p , we construct new generating functions of the twisted (h, q)-Euler numbers and polynomials attached to a Dirichlet character χ. By applying Mellin transformation and derivative operator to these functions, we define twisted (h, q)-extension of zeta functions and l-functions, which interpolate the twisted (h, q)-extension of Euler numbers at negative integers. Moreover, we construct the partially twisted (h, q)-zeta function. We give some relations between the partially twisted (h, q)-zeta function and twisted (h, q)-extension of Euler numbers.      

Harmonic manifolds with minimal horospheres

For a non-compact harmonic manifold M, we establish an integral formula for the derivative of a harmonic function on M. As an application we show that for the harmonic spaces having minimal horospheres, bounded harmonic functions are constant. The main result of this article states that the harmonic spaces having polynomial volume growth are flat. In other words, if the volume density function Θ of M has polynomial growth, then M is flat. This partially answers a question of Szabo namely, which density functions determine the metric of a harmonic manifold. Finally, we give some natural conditions which ensure polynomial growth of the volume function.