Vector measure Maurey–Rosenthal-type factorizations and -sums of -spaces

Consider a vector measure of bounded variation m with values in a Banach space and an operator T:XL¹(m), where L¹(m) is the space of integrable functions with respect to m. We characterize when T can be factorized through the space L²(m) by means of a multiplication operator given by a function of L²(|m|), where |m| is the variation of m, extending in this way the Maurey–Rosenthal Theorem. We use this result to obtain information about the structure of the space L¹(m) when m is a sequential vector measure. In this case the space L¹(m) is an ℓ-sum of L¹-spaces.     

A pathwise solution for nonlinear parabolic equations with stochastic perturbations

We analyse here a semilinear stochastic partial differential equation of parabolic type where the diffusion vector fields are depending on both the unknown function and its gradient ∂ _xu with respect to the state variable, ∈ ℝⁿ. A local solution is constructed by reducing the original equation to a nonlinear parabolic one without stochastic perturbations and it is based on a finite dimensional Lie algebra generated by the given diffusion vector fields.     

Criteria for the regularity of growth of the logarithm of modulus and the argument of an entire function

For entire functions whose zero counting functions are slowly increasing, we establish criteria for the regular growth of their logarithms of moduli and arguments in the metric of L ^p [0, 2π].     

Discrete optimization on a multivariable boolean lattice

Consider the problem of finding the minimum value of a scalar objective function whose arguments are theN components of 2 ^N vector elements partially ordered as a Boolean lattice. If the function is strictly decreasing along any shortest path from the minimum point to its logical complement, then the minimum can be located precisely after sequential measurement of the objective function atN + 1 points. This result suggests a new line of research on discrete optimization problems.This paper was presented at the 7th Mathematical Programming Symposium, The Hague, The Netherlands.This research was supported in part by U.S. Office of Saline Water Grant No. 699.     

On a linear problem of tracing a given motion

In the present paper, we consider the problem on the optimal tracing of a given vector function with the use of a generalized projection of the trajectory of a linear plant. The deviation of a given motion is measured in the metric C ^m [0, T] of continuous vector functions of the corresponding dimension m. We suggest an efficient method for the construction of an approximate solution of this optimization problem with given accuracy.     

Linear problem of tracking a given motion under an integral constraint on control

We consider the problem of optimally tracking a given vector function by means of a generalized projection of the trajectory of a linear controlled object with an integral constraint on the control. The deviation from a given motion is measured in the metric of the space C ^m [0, T] of continuous vector functions of appropriate dimension m. We describe a constructive method for solving this optimization problem with a given accuracy.     

Transformations preserving the Hausdorff-Besicovitch dimension

Continuous transformations preserving the Hausdorff-Besicovitch dimension (“DP-transformations”) of every subset of R ¹ resp. [0, 1] are studied. A class of distribution functions of random variables with independent s-adic digits is analyzed. Necessary and sufficient conditions for dimension preservation under functions which are distribution functions of random variables with independent s-adic digits are found. In particular, it is proven that any strictly increasing absolutely continuous distribution function from the above class is a DP-function. Relations between the entropy of probability distributions, their Hausdorff-Besicovitch dimension and their DP-properties are discussed. Examples are given of singular distribution functions preserving the fractal dimension and of strictly increasing absolutely continuous functions which do not belong to the DP-class.      

Approximations of pseudo-Boolean functions; applications to game theory

This paper studies the approximation of pseudo-Boolean functions by linear functions and more generally by functions of (at most) a specified degree. Here a pseudo-Boolean function means a real valued function defined on {0,1} ⁿ , and its degree is that of the unique multilinear polynomial that expresses it; linear functions are those of degree at most one. The approximation consists in choosing among all linear functions the one which is closest to a given function, where distance is measured by the Euclidean metric onR ²ⁿ . A characterization of the best linear approximation is obtained in terms of the average value of the function and its first derivatives. This leads to an explicit formula for computing the approximation from the polynomial expression of the given function. These results are later generalized to handle approximations of higher degrees, and further results are obtained regarding the interaction of approximations of different degrees. For the linear case, a certain constrained version of the approximation problem is also studied. Special attention is given to some important properties of pseudo-Boolean functions and the extent to which they are preserved in the approximation. A separate section points out the relevance of linear approximations to game theory and shows that the well known Banzhaf power index and Shapley value are obtained as best linear approximations of the game (each in a suitably defined sense).Supported by the Air Force Office of Scientific Research (under grant number AFOSR 89-0512 and AFOSR 90-0008 to Rutgers University), as well as the National Science Foundation (under grant number DMS 89-06870).     

A new Leray formula for smooth functions on bounded domains in Cn

By means of a new technique of integral representations in C ⁿ given by the authors, we establish a new abstract formula with a vector function W for smooth functions on bounded domains in C ⁿ , which is different from the well-known Leray formula. This new formula eliminates the term that contains the parameter A from the classical Leray formula, and especially on some domains the uniform estimates for the are very simple. From the new Leray formula, we can obtain correspondingly many new formulas for smooth functions on many domains in C ⁿ , which are different from the classical ones, when we properly select the vector function W.     

Some consequences of Pisier’s approach to interpolation

A recent result of Pisier onK-functionals is exploited in the context of function spaces defined from singular integrals. The main consequences appear forH ^p spaces on the complex ball (1≤p≤∞) and spaces of differentiable functions in several variables. It turns out that the complex variable arguments used in [P1] may in certain cases be substituted for real variable methods, which of course have a wider range of applicability. We discuss interpolation of vector valued Hardy spaces on the ball and prove thatL ¹(S)/H ¹(B) satisfies Grothendieck’s theorem, similarly as in the disc case.     

Characterization of functions as radiation patterns in linear elasticity

In this paper a necessary and sufficient condition for a pair of vector functions to be radiation patterns is presented. More precisely, it is proved that two vector functions, the first in the radial direction and the second in the tangential one, are radiation patterns if and only if there are two entire harmonic vector functions whose radial and tangential projections, respectively, are identical with the previous functions on the unit sphere and whose L²-norm over a sphere of radius R is a function of exponential type in the variable R.     

New Expressions for Repeated Upper Tail Integrals of the Normal Distribution

Expressions are given for repeated upper tail integrals of the univariate normal density (and so also for the general Hermite function) for both positive and negative arguments. The expressions involve moments of the form E(x + i N) ⁿ and E1 / (x ² + N ²) ⁿ , where N is a unit normal random variable. Laplace transforms are provided for the Hermite functions and the moments. The practical use of these expressions is illustrated.     

On the second conjugate of several convex functions in general normed vector spaces

When dealing with convex functions defined on a normed vector space X the biconjugate is usually considered with respect to the dual system (X, X ^*), that is, as a function defined on the initial space X. However, it is of interest to consider also the biconjugate as a function defined on the bidual X ^**. It is the aim of this note to calculate the biconjugate of the functions obtained by several operations which preserve convexity. In particular we recover the result of Fitzpatrick and Simons on the biconjugate of the maximum of two convex functions with a much simpler proof.      

An augmented Lagrangian approach with a variable transformation in nonlinear programming

Tangent cone and (regular) normal cone of a closed set under an invertible variable transformation around a given point are investigated, which lead to the concepts of θ⁻¹-tangent cone of a set and θ⁻¹-subderivative of a function. When the notion of θ⁻¹-subderivative is applied to perturbation functions, a class of augmented Lagrangians involving an invertible mapping of perturbation variables are obtained, in which dualizing parameterization and augmenting functions are not necessarily convex in perturbation variables. A necessary and sufficient condition for the exact penalty representation under the proposed augmented Lagrangian scheme is obtained. For an augmenting function with an Euclidean norm, a sufficient condition (resp., a sufficient and necessary condition) for an arbitrary vector (resp., 0) to support an exact penalty representation is given in terms of θ⁻¹-subderivatives. An example of the variable transformation applied to constrained optimization problems is given, which yields several exact penalization results in the literature.     

Recurrence set-relations in stochastic multiobjective dynamic decision processes

Finite horizon stochastic dynamic decision processes with R^p valued additive returns are considered. The optimization criterion is a partial-order preference relation induced from a convex cone in R^p . The state space is a countable set, and the action space is a compact metric spaces. The optimal value function, which is of a set-valued mapping, is defined. Under certain assumptions on the continuity of the reward vector and the transition probability, a system of a recurrence set-relations concerning the optimal value functions is given.     

On generalized weak subdifferentials and some properties

In this article, some characterizations for gw-subdifferentiability of functions from ? ⁿ to ? ^m are stated. Some criteria for gw-subdifferentiability of generalized lower locally Lipschitz functions and positively homogeneous functions are given. Furthermore, it is proved that every Lipschitz function is gw-subdifferentiable at any point in its domain. Finally, the relationship between directional derivative and gw-subdifferential is given and a convexity criteria for Fréchet differentiable function is given by using gw-subdifferential.     

Multiwavelet Frames from Refinable Function Vectors

Starting from any two compactly supported d-refinable function vectors in (L ₂(R)) ^r with multiplicity r and dilation factor d, we show that it is always possible to construct 2rd wavelet functions with compact support such that they generate a pair of dual d-wavelet frames in L ₂(R) and they achieve the best possible orders of vanishing moments. When all the components of the two real-valued d-refinable function vectors are either symmetric or antisymmetric with their symmetry centers differing by half integers, such 2rd wavelet functions, which generate a pair of dual d-wavelet frames, can be real-valued and be either symmetric or antisymmetric with the same symmetry center. Wavelet frames from any d-refinable function vector are also considered. This paper generalizes the work in [5,12,13] on constructing dual wavelet frames from scalar refinable functions to the multiwavelet case. Examples are provided to illustrate the construction in this paper.     

Solutions in Sobolev spaces of vector refinement equations with a general dilation matrix

In this paper, we present a necessary and sufficient condition for the existence of solutions in a Sobolev space W_p^k(ℝ^s) (1≤p≤∞) to a vector refinement equation with a general dilation matrix. The criterion is constructive and can be implemented. Rate of convergence of vector cascade algorithms in a Sobolev space W_p^k(ℝ^s) will be investigated. When the dilation matrix is isotropic, a characterization will be given for the L_p (1≤p≤∞) critical smoothness exponent of a refinable function vector without the assumption of stability on the refinable function vector. As a consequence, we show that if a compactly supported function vector φ∈L_p(ℝ^s) (φ∈C(ℝ^s) when p=∞) satisfies a refinement equation with a finitely supported matrix mask, then all the components of φ must belong to a Lipschitz space Lip(ν,L_p(ℝ^s)) for some ν>0. This paper generalizes the results in R.Q. Jia, K.S. Lau and D.X. Zhou (J. Fourier Anal. Appl. 7 (2001) 143–167) in the univariate setting to the multivariate setting. Dedicated to Professor Charles A. Micchelli on the occasion of his 60th birthday Mathematics subject classifications (2000) 42C20, 41A25, 39B12. Research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC Canada) under Grant G121210654.     

The p-Rank of the Incidence Matrix of Intersecting Linear Subspaces

Let V be a vector space of dimension n+1 over a field of p ^t elements. A d-dimensional subspace and an e-dimensional subspace are considered to be incident if their intersection is not the zero subspace. The rank of these incidence matrices, modulo p, are computed for all n, d, e and t. This result generalizes the well-known formula of Hamada for the incidence matrices between points and subspaces of given dimensions in a finite projective space. A generating function for these ranks as t varies, keeping n, d and e fixed, is also given. In the special case where the dimensions are complementary, i.e., d+e=n+ 1, our formula improves previous upper bounds on the size of partial m-systems (as defined by Shult and Thas).     

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.