A generalized Henstock-Stieltjes integral involving division functions

We can consider the Riemann-Stieltjes integral dg as an integral of a point function f with respect to an interval function g. We could extend it to the Henstock-Stieltjes integral. In this paper, we extend it to a generalized Stieltjes integral dg of a point function f with respect to a function g of divisions of an interval. Then we prove for this integral the standard results in the theory of integration, including the controlled convergence theorem.      

Smooth approximations without critical points

In any separable Banach space containing c ₀ which admits a C ^k-smooth bump, every continuous function can be approximated by a C ^k-smooth function whose range of derivative is of the first category. Moreover, the approximation can be constructed in such a way that its derivative avoids a prescribed countable set (in particular the approximation can have no critical points). On the other hand, in a Banach space with the RNP, the range of the derivative of every smooth bounded bump contains a set residual in some neighbourhood of zero.     

The heat flow for subharmonic orbits

LetV: M be a smooth (potential) function on a compact Riemannian manifold. This gives rise to a second order Hamiltonian system. Assuming that the corresponding action functional is a Morse function, we will prove that the heat flow for subharmonics exists globally and converges to a critical point of the energy. As a Corollary, this shows the convergence of the geodesic heat flow (to a geodesic) without any curvature assumptions onM.     

The Dirichlet problem for Baire-one functions

Let X be a compact convex set and let ext X stand for the set of all extreme points of X. We characterize those bounded function defined on ext X which can be extended to an affine Baire-one function on the whole set X.     

<Emphasis Type="Italic">W</Emphasis><Superscript>1,<Emphasis Type="Italic">p</Emphasis></Superscript>(R<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)-boundedness for maximal functions

In this paper, we get W ^1,p (R ⁿ )-boundedness for tangential maximal function and nontangential maximal function, which improves J.Kinnunen, P.Lindqvist and Tananka’s results. Supported by the key Academic Discipline of Zhejiang Province of China under Grant No.2005 and the Zhejiang Provincial Natural Science Foundation of China.     

The wave kernel for the Laplacian on the classical locally symmetric spaces of rank one,theta functions,trace formulas and the Selberg zeta function

We calculate the wave kernels for the classical rank-one symmetric spaces. The result is employed in order to provide a meromorphic extension of the theta function of an even-dimensional compact locally symmetric space of non-compact type. Moreover we give a short derivation of the Selberg trace formula. We discuss the relation between the right hand side of the functional equation of the Selberg zeta function, the Plancherel measure, Weyl's dimension formula and the wave kernel on the non-compact symmetric space and on its compact dual in an explicit manner.The first two authors were supported by the Sonderforschungsbereich 288 Differentialgeometrie und Quantenphysik founded by the Deutsche Forschungsgemeinschaft.     

A Böchner technique for harmonic mappings from a 3-manifold to a surface

Let be a harmonic mapping from a Riemannian 3-manifold to a Riemannian 2-manifold. A smooth function on M is associated to , derived from the eigenvalues of the first fundamental form, the vanishing of which is equivalent to being a harmonic morphism. The Laplacian of this function is computed and a maximum principle applied to derive criteria when a harmonic map must be a harmonic morphism.     

Differential operators associated with holomorphic mappings

Two differential operators which act on holomorphic mappings to complex projective space are studied. One operator is of second order and characterizes projective linear mappings. The other operator is of third order and may be viewed as a curvature. The two operators together play a role analogous to the Schwarzian derivative.A canonical approximation to a holomorphic mapping is defined, and a relationship between the approximation and the operators is derived. In the one variable case, this reduces to a classical result relating the Schwarzian derivative and the best Möbius approximation to a holomorphic function.     

On subadditive functions and ψ-additive mappings

In [4], assuming among others subadditivity and submultiplicavity of a function ψ: [0, ∞)→[0, ∞), the authors proved a Hyers-Ulam type stability theorem for “ψ-additive” mappings of a normed space into a normed space. In this note we show that the assumed conditions of the function ψ imply that ψ=0 and, consequently, every “ψ-additive” mapping must be additive     

The strong <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-<Emphasis Type="Italic">greedy</Emphasis> property of the Walsh system

For any 0 < ? < 1 one can find a measurable set E ? [0, 1] with the measure |E| > 1 ? ? such that for each function f(x) ε L ¹ (0, 1) a function g(x) ε L ¹ (0, 1) exists such that it coincides with f (x) on E, its Fourier—Walsh series converges to it in the metric of L ¹ (0, 1), and all nonzero terms of the sequence of Fourier coefficients of the new function obtained by the Walsh system have the modulo decreasing order; consequently, the greedy algorithm for this function converges to it in the L ¹ (0, 1)-norm.     

A compound of the generalized negative binomial distribution with the generalized beta distribution

This paper presents a compound of the generalized negative binomial distribution with the generalized beta distribution. In the introductory part of the paper, we provide a chronological overview of recent developments in the compounding of distributions, including the Polish results. Then, in addition to presenting the probability function of the compound generalized negative binomial-generalized beta distribution, we present special cases as well as factorial and crude moments of some compound distributions.     

On the apostol-bernoulli polynomials

In the present paper, we obtain two new formulas of the Apostol-Bernoulli polynomials (see On the Lerch Zeta function. Pacific J. Math., 1 (1951), 161–167.), using the Gaussian hypergeometric functions and Hurwitz Zeta functions respectively, and give certain special cases and applications.     

Analysis of Fractional Covering of Some Supply Management Problems

We study a supply management problem with a linear cost function and its mixed integer program. A parametric family of such problems, possessing exponential L _k -coverings, is constructed. Besides, the NP-hard expansion of the family, which has the same property, is formulated.     

Expansion of analytic functions in q-orthogonal polynomials

A classical result on the expansion of an analytic function in a series of Jacobi polynomials is extended to a class of q-orthogonal polynomials containing the fundamental Askey–Wilson polynomials and their special cases. The function to be expanded has to be analytic inside an ellipse in the complex plane with foci at ±1. Some examples of explicit expansions are discussed.      

Equivalent Norms in Spaces of Functions of Fractional Smoothness on Arbitrary Domains

In this paper, we study the spaces B _pq ^s (G) and L _pq ^s (G) of functions f with positive exponent of smoothness s > 0 given on a domain . The norms on these spaces are defined via integral norms of the difference of the function f of order m > s treated as a function of the point of the domain and of the difference increment. For an arbitrary domain , we characterize these spaces in terms of the local approximations of the function by polynomials of degree m – 1.     

Tensor products of symmetric functions over ℤ<Subscript>2</Subscript>

We calculate the homology and the cycles in tensor products of algebras of symmetric function over ℤ₂     

Moreau–Yosida regularization of Lagrangian-dual functions for a class of convex optimization problems

In this paper, we consider the Lagrangian dual problem of a class of convex optimization problems, which originates from multi-stage stochastic convex nonlinear programs. We study the Moreau–Yosida regularization of the Lagrangian-dual function and prove that the regularized function η is piecewise C ², in addition to the known smoothness property. This property is then used to investigate the semismoothness of the gradient mapping of the regularized function. Finally, we show that the Clarke generalized Jacobian of the gradient mapping is BD-regular under some conditions.      

Extending analyticK-subanalytic functions

Letg:U→ℝ (U open in ℝⁿ) be an analytic and K-subanalytic (i. e. definable in ℝ _an ^K , whereK, the field of exponents, is any subfield ofℝ) function. Then the set of points, denoted Σ, whereg does not admit an analytic extension is K-subanalytic andg can be extended analytically to a neighbourhood of Ū\∑. Partially supported by the European RTN Network RAAG (contract no. HPRN-CT-00271)     

On the Hilbert function of curvilinear zero-dimensional subschemes of projective spaces

Here we show the existence of strong restrictions for the Hilbert function of zerodimensional curvilinear subschemes of P ⁿ with one point as support and with high regularity index.     

ε-Value Function and Dynamic Programming

In this note, we develop a dynamic programming approach for an ε-optimal control problem of Bolza. We prove that each Lipschitz continuous function satisfying the Hamilton-Jacobi inequality (less than zero and greater than −ε) is an ε-value function.