Cross-product of Bessel functions: Monotonicity patterns and functional inequalities

In this paper, we study the Dini functions and the cross-product of Bessel functions. Moreover, we are interested on the monotonicity patterns for the cross-product of Bessel and modified Bessel functions. In addition, we deduce Redheffer-type inequalities, and the interlacing property of the zeros of Dini functions and the cross-product of Bessel and modified Bessel functions. Bounds for logarithmic derivatives of these functions are also derived. The key tools in our proofs are some recently developed infinite product representations for Dini functions and cross-product of Bessel functions.     

The theory of zeta functions of several complex variables,I

The paper introduces a general class of Tate-like zeta functions and proves an analytic continuation and a general formula for the values of such zeta functions at negative integral arguments. In particular these zeta functions discussed include the Shintani zeta functions and the results generalize his formula for the value of such zeta functions at negative integral arguments.     

Regularity of boundary wavelets

The conventional way of constructing boundary functions for wavelets on a finite interval is by forming linear combinations of boundary-crossing scaling functions. Desirable properties such as regularity (i.e. continuity and approximation order) are easy to derive from corresponding properties of the interior scaling functions. In this article we focus instead on boundary functions defined by recursion relations. We show that the number of boundary functions is uniquely determined, and derive conditions for determining regularity from the recursion coefficients. We show that there are regular boundary functions which are not linear combinations of shifts of the underlying scaling functions.     

On biconjugates of infimal functions

We deliver formulae for the biconjugate functions of some infimal functions that hold provided the fulfilment of weak regularity conditions of both closedness and interiority types. As special cases, we obtain biconjugates of infimal convolutions of finitely many functions, of optimal value functions of both constrained and unconstrained optimization problems as well as of marginal functions associated with multifunctions (that can be, for instance, convex processes) and some given functions. Moreover, we rediscover or extend different results on biconjugate functions from the literature.     

A unifying theory of exactness of linear penalty functions II: parametric penalty functions

In this article, we develop a general theory of exact parametric penalty functions for constrained optimization problems. The main advantage of the method of parametric penalty functions is the fact that a parametric penalty function can be both smooth and exact unlike the standard (i.e. non-parametric) exact penalty functions that are always nonsmooth. We obtain several necessary and/or sufficient conditions for the exactness of parametric penalty functions, and for the zero duality gap property to hold true for these functions. We also prove some convergence results for the method of parametric penalty functions, and derive necessary and sufficient conditions for a parametric penalty function to not have any stationary points outside the set of feasible points of the constrained optimization problem under consideration. In the second part of the paper, we apply the general theory of exact parametric penalty functions to a class of parametric penalty functions introduced by Huyer and Neumaier, and to smoothing approximations of nonsmooth exact penalty functions. The general approach adopted in this article allowed us to unify and significantly sharpen many existing results on parametric penalty functions.     

Indefinite integrals of products of special functions

A method is given for deriving indefinite integrals involving squares and other products of functions which are solutions of second-order linear differential equations. Several variations of the method are presented, which applies directly to functions which obey homogeneous differential equations. However, functions which obey inhomogeneous equations can be incorporated into the products and examples are given of integrals involving products of Bessel functions combined with Lommel, Anger and Weber functions. Many new integrals are derived for a selection of special functions, including Bessel functions, associated Legendre functions, and elliptic integrals. A number of integrals of products of Gauss hypergeometric functions are also presented, which seem to be the first integrals of this type. All results presented have been numerically checked with Mathematica.     

Some Extensions of Loewner's Theory of Monotone Operator Functions

Several extensions of Loewner's theory of monotone operator functions are given. These include a theorem on boundary interpolation for matrix-valued functions in the generalized Nevanlinna class. The theory of monotone operator functions is generalized from scalar-to matrix-valued functions of an operator argument. A notion of κ-monotonicity is introduced and characterized in terms of classical Nevanlinna functions with removable singularities on a real interval. Corresponding results for Stieltjes functions are presented.     

Computing iterative roots of polygonal functions

Based on the iterative root theory for monotone functions, an algorithm for computing polygonal iterative roots of increasing polygonal functions was given by J. Kobza. In this paper we not only give an algorithm for roots of decreasing polygonal functions but also generalize Kobza's results to the general n. Furthermore, we extend our algorithms for polygonal PM functions, a class of non-monotonic functions.     

A construction of bent functions from plateaued functions

In this presentation, a technique for constructing bent functions from plateaued functions is introduced and analyzed. This generalizes earlier techniques for constructing bent from near-bent functions. Using this construction, we obtain a big variety of inequivalent bent functions, some weakly regular and some non-weakly regular. Classes of bent functions having some additional properties that enable the construction of strongly regular graphs are formed, and explicit expressions for bent functions with maximal degree are presented.     

The algebra of set functions I: The product theorem and duality

We give a comprehensive introduction to the algebra of set functions and its generating functions. This algebraic tool allows us to formulate and prove a product theorem for the enumeration of functions of many different kinds, in particular injective functions, surjective functions, matchings and colourings of the vertices of a hypergraph. Moreover, we develop a general duality theory for counting functions.     

Jensen measures and boundary values of plurisubharmonic functions

We study different classes of Jensen measures for plurisubharmonic functions, in particular the relation between Jensen measures for continuous functions and Jensen measures for upper bounded functions. We prove an approximation theorem for plurisubharmonic functions inB-regular domain. This theorem implies that the two classes of Jensen measures coincide inB-regular domains. Conversely we show that if Jensen measures for continuous functions are the same as Jensen measures for upper bounded functions and the domain is hyperconvex, the domain satisfies the same approximation theorem as above. The paper also contains a characterisation in terms of Jensen measures of those continuous functions that are boundary values of a continuous plurisubharmonic function.     

逐段单调函数的高度与拓扑共轭

迭代运算下,函数值可以交叉于不同的子区间,使得逐段单调函数的高度异常复杂.本文考虑一个非单调点的连续函数类.首先给出高度的充分必要条件,以此获得此类函数的一种划分.其次针对函数类的一个非空子集,给出判定拓扑共轭的充分必要条件和构造拓扑共轭的新方法.进一步地,我们阐明这样的事实:两个逐段单调函数拓扑共轭是其高度相等的充分不必要条件,最后举例说明.     

Exact distributions of MLEs of regression coefficients in GMANOVA-MANOVA model

This paper studies the exact distributions of the MLEs of the regression coefficient matrices in a GMANOVA-MANOVA model with normal error. The unique conditions for linear functions of the MLEs of regression coefficient matrices are presented, and the exact density functions or characteristic functions for these linear functions are derived.     

A function class of strictly positive definite and logarithmically completely monotonic functions related to the modified Bessel functions

In this paper, we give some conditions for a class of functions related to Bessel functions to be positive definite or strictly positive definite. We present some properties and relationships involving logarithmically completely monotonic functions and strictly positive definite functions. In particular, we are interested with the modified Bessel functions of the second kind. As applications, we prove the logarithmically monotonicity for a class of functions involving the modified Bessel functions of second kind and we established new inequalities for this function.     

Reflections on and of minor-closed classes of multisorted operations

The minor relation of functions is generalized to multisorted functions. Pippenger’s Galois theory for minor-closed sets of functions is extended to multisorted functions and multisorted relation pairs. Reflections of minor-closed sets are again minor-closed, and the effect of reflections on the invariant relation pairs of minor-closed sets of multisorted functions is described.     

Minimizing the Difference of Dual Functions of Two Coradiant Functions

In this article, we solve the problem of minimizing the difference of dual functions of two coradiant functions. We do this by applying a type of duality, that is used in microeconomic theory. Indeed, the dual function of a co-radiant function is decreasing and inverse coradiant. So, we first give various characterizations for the maximal elements of the support sets of this class of functions. Next, by using these results, we obtain the necessary and sufficient conditions for the global minimizers of the difference of two decreasing and inverse coradiant functions. Finally, as an application, we present the necessary and sufficient conditions for the global minimizers of the difference of dual functions of two co-radiant functions.     

Properties of boolean functions with a tree decomposition

Boolean functions that have a multiple disjoint decomposition scheme in the form of a tree are considered. Properties of such functions are given for the case that the functions are increasing, unate, and/or have no vacuous variables. The functions with a binary decomposition scheme are of special interest. The modulus of sensitivity is defined, and evaluated for some classes of functions. The modulus of sensitivity is interesting from the point of view of semantic information processing. It is found that the sensitivity for the class of functions with a given disjoint binary decomposition scheme is much smaller than for the unrestricted class of boolean functions. This indicates that these functions are potentially useful in pattern recognition of discrete data.The authors gratefully acknowledge the financial support of the National Research Council of Canada through a postdoctoral fellowship and an operating grant respectively.     

Unification of Filled Function and Tunnelling Function in Global Optimization

In this paper,two auxiliary functions for global optimization are proposed.These two auxiliaryfunctions possess all characters of tunnelling functions and filled functions under certain general assumptions.Thus,they can be considered as the unification of filled function and tunnelling function.Moreover,the processof tunneling or filling for global optimization can be unified as the minimization of such auxiliary functions.Result of numerical experiments shows that such two auxiliary functions are effective.     

Game representations of classes of piecewise definable functions

We present a general way of defining various reduction games on ω which “represent” corresponding topologically defined classes of functions. In particular, we will show how to construct games for piecewise defined functions, for functions which are pointwise limit of certain sequences of functions and for Γ ‐measurable functions. These games turn out to be useful as a combinatorial tool for the study of general reducibilities for subsets of the Baire space [10] (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Functions of a quaternion variable which are gradients of real-valued functions

We consider the collection of functions of one quaternion variable which can be expressed asG(Y) whereY is a real-valued quaternion function andG is a differential operator which corresponds to the gradient of real variable theory. Integral theorems for such functions are given, together with necessary and sufficient conditions for a function to be a gradient function, in terms of its Frechet derivative. The extended complex analytic functions, the Fueter functions, and the momentum-energy density functions are seen to be gradient functions which correspond to biharmonic, harmonic, and wave functions respectively.