Hypergeometric Solutions of Soliton Equations

We consider multivariable hypergeometric functions related to Schur functions and show that these hypergeometric functions are tau functions of the KP hierarchy and are simultaneously the ratios of Toda lattice tau functions evaluated at certain values of higher Toda lattice times. The variables of the hypergeometric functions are related to the higher times of those hierarchies via a Miwa change of variables. The discrete Toda lattice variable shifts the parameters of the hypergeometric functions. We construct the determinant representation and the integral representation of a special type for the KP tau functions. We write a system of linear differential and difference equations on these tau functions, which play the role of string equations.     

Structure of rings of functions with Riemann Stieltjes convolution products

In this note three sets of complex valued functions with pointwise addition and a Riemann Stieltjes convolution product are considered. The functions considered are discrete analytic functions, sequences, and continuous functions of bounded variation defined on the nonnegative real numbers. Each forms a commutative algebra with identity. The discrete analytic functions form a principal ideal ring with five maximal ideals, nine prime ideals, and is essentially a direct sum of four discrete valuation rings. The ring of sequences is isomorphic to an ideal of the ring of discrete analytic functions; it has two maximal and three prime ideals. Both contain divisors of zero. The units, associates, irreducible elements and primes in these two rings are described. The results are used to study the continuous functions; partial results are obtained concerning units and divisors of zero. The product satisfies a convolution theorem.     

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.     

Construction of functions with uniform sublevel sets

In this paper, we study extended real-valued functions with uniform sublevel sets. The sublevel sets are defined by a linear shift of a set in a specified direction. We prove that the class of these functions coincides with the class of Gerstewitz functionals. In this way, we obtain a formula for the construction of such functions. The sublevel sets of Gerstewitz functionals are characterized and illustrated by examples. The results contain statements for translative functions, which are just the functions with uniform sublevel sets considered. The investigated functions are defined on an arbitrary real vector space without assuming any topology or convexity.     

Properties and construction of NCP functions

The nonlinear complementarity or NCP functions were introduced by Mangasarian and these functions are proved to be useful in constrained optimization and elsewhere. Interestingly enough there are only two general methods to derive such functions, while the known or used NCP functions are either individual constructions or modifications of the few individual NCP functions such as the Fischer-Burmeister function. In the paper we analyze the elementary properties of NCP functions and the various techniques used to obtain such functions from old ones. We also prove some new nonexistence results on the possible forms of NCP functions. Then we develop and analyze several new methods for the construction of nonlinear complementarity functions that are based on various geometric arguments or monotone transformations. The appendix of the paper contains the list and source of the known NCP functions.     

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.     

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 a Class of Generalized Sampling Functions

In this note, we discuss a class of so-called generalized sampling functions. These functions are defined to be the inverse Fourier transform of a family of piecewise constant functions that are either square integrable or Lebegue integrable on the real number line. They are in fact the generalization of the classic sinc function. Two approaches of constructing the generalized sampling functions are reviewed. Their properties such as cardinality, orthogonality, and decaying properties are discussed. The interactions of those functions and Hilbert transformer are also discussed.     

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.     

Approximation by translates of refinable functions

Summary. The functions are refinable if they are combinations of the rescaled and translated functions . This is very common in scientific computing on a regular mesh. The space of approximating functions with meshwidth is a subspace of with meshwidth . These refinable spaces have refinable basis functions. The accuracy of the computations depends on , the order of approximation, which is determined by the degree of polynomials that lie in . Most refinable functions (such as scaling functions in the theory of wavelets) have no simple formulas. The functions are known only through the coefficients in the refinement equation – scalars in the traditional case, matrices for multiwavelets. The scalar "sum rules" that determine are well known. We find the conditions on the matrices that yield approximation of order from . These are equivalent to the Strang–Fix conditions on the Fourier transforms , but for refinable functions they can be explicitly verified from the . Received August 31, 1994 / Revised version received May 2, 1995     

Characterizations and Applications of Prequasi-Invex Functions

In this paper, two new types of generalized convex functions are introduced. They are called strictly prequasi-invex functions and semistrictly prequasi-invex functions. Note that prequasi-invexity does not imply semistrict prequasi-invexity. The characterization of prequasi-invex functions is established under the condition of lower semicontinuity, upper semicontinuity, and semistrict prequasi-invexity, respectively. Furthermore, the characterization of semistrictly prequasi-invex functions is also obtained under the condition of prequasi-invexity and lower semicontinuity, respectively. A similar result is also obtained for strictly prequasi-invex functions. It is worth noting that these characterizations reveal various interesting relationships among prequasi-invex, semistrictly prequasi-invex, and strictly prequasi-invex functions. Finally, prequasi-invex, semistrictly prequasi-invex, and strictly prequasi-invex functions are used in the study of optimization problems.     

The ranks of Maiorana-McFarland bent functions

In this paper, the ranks of a special family of Maiorana-McFarland bent functions are discussed. The upper and lower bounds of the ranks are given and those bent functions whose ranks achieve these bounds are determined. As a consequence, the inequivalence of some bent functions are derived. Furthermore, the ranks of the functions of this family are calculated when t 6.     

Construction and Dimension Analysis for a Class of Fractal Functions

In this paper, we construct a class of nowhere differentiable continuous functions by means of the Cantor series expression of real numbers. The constructed functions include some known nondifferentiable functions, such as Bush type functions. These functions are fractal functions since their graphs are in general fractal sets. Under certain conditions, we investigate the fractal dimensions of the graphs of these functions, compute the precise values of Box and Packing dimensions, and evaluate the Hausdorff dimension. Meanwhile, the Holder continuity of such functions is also discussed.     

多值函数在复变函数中的应用

通过讨论初等多值函数的单值解析分枝问题,重点研究了初等多值函数在复变函数中的具体应用.     

On the analytic model of a class of semi-hyponormal operators

The analytic model of a class of semi-hyponormal operators is derived using three kernal functions. In addition, explicit forms of the kernal functions are given and the Pincus principal functions of the operators are calculated.     

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.     

Binary terms in polynomial representations of Boolean functions

Polynomial representations of Boolean functions by binary terms are considered. The construction of terms involves variables and residual functions. Special cases of such representations are the decomposition of a function with respect to variables, Zhegalkin polynomials, and representations of functions as sums of conjunctions of residual functions.     

Quasidifferentiability of nonsmooth quasiconvex functions

The paper deals with nonsmooth quasiconvex functions and develops a quasidifferential analysis for this class of functions. Therefore, in terms of sub and superdifferentials, first order approximations of the functions are derived, optimality conditions are stated and directions of descent (either simple feasible or of steepest descent) are determined. Moreover, a relation among positively homogeneous convex and quasiconvex functions is established     

Objective comparisons of the optimal portfolios corresponding to different utility functions

This paper considers the effects of some frequently used utility functions in portfolio selection by comparing the optimal investment outcomes corresponding to these utility functions. Assets are assumed to form a complete market of the Black–Scholes type. Under consideration are four frequently used utility functions: the power, logarithm, exponential and quadratic utility functions. To make objective comparisons, the optimal terminal wealths are derived by integration representation. The optimal strategies which yield optimal values are obtained by the integration representation of a Brownian martingale. The explicit strategy for the quadratic utility function is new. The strategies for other utility functions such as the power and the logarithm utility functions obtained this way coincide with known results obtained from Merton’s dynamic programming approach.     

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.