Bernstein functions and rates in mean ergodic theorems for operator semigroups

We present a functional calculus approach to the study of rates of decay in mean ergodic theorems for bounded strongly continuous operator semigroups. A central role is played by operators of the form g(A, where ?A is the generator of the semigroup and g is a Bernstein function. In addition, we obtain some new results on Bernstein functions which are of independent interest.     

Differential calculus for linear operators represented by finite signed measures and applications

We introduce a differential calculus for linear operators represented by a family of finite signed measures. Such a calculus is based on the notions of g-derived operators and processes and g-integrating measures, g?being a right-continuous nondecreasing function. Depending on the choice of?g, this differential calculus works for non-smooth functions and under weak integrability conditions. For linear operators represented by stochastic processes, we provide a characterization criterion of g-differentiability in terms of characteristic functions of the random variables involved. Various illustrative examples are considered. As an application, we obtain an efficient algorithm to compute the Riemann zeta function ??(z) with a geometric rate of convergence which improves exponentially as ?(z) increases.     

Determination of the limits for multivariate rational functions

The purpose of this paper is to solve the problem of determining the limits of multivariate rational functions.It is essential to decide whether or not limxˉ→0f g=0 for two non-zero polynomials f,g∈R[x1,...,xn]with f(0,...,0)=g(0,...,0)=0.For two such polynomials f and g,we establish two necessary and sufcient conditions for the rational functionf g to have its limit 0 at the origin.Based on these theoretic results,we present an algorithm for deciding whether or not lim(x1,...,xn)→(0,...,0)f g=0,where f,g∈R[x1,...,xn]are two non-zero polynomials.The design of our algorithm involves two existing algorithms:one for computing the rational univariate representations of a complete chain of polynomials,another for catching strictly critical points in a real algebraic variety.The two algorithms are based on the well-known Wu’s method.With the aid of the computer algebraic system Maple,our algorithm has been made into a general program.In the final section of this paper,several examples are given to illustrate the efectiveness of our algorithm.     

Norm inequalities for vector functions

We study vector functions of Rn into itself, which are of the form x?g(|x|)x, where g:(0,∞)→(0,∞) is a continuous function and call these radial functions. In the case when g(t)=tc for some c∈R, we find upper bounds for the distance of image points under such a radial function. Some of our results refine recent results of L. Maligranda and S.S. Dragomir. In particular, we study quasiconformal mappings of this simple type and obtain norm inequalities for such mappings.     

The intersection number of real polynomial mappings

Let X=V(f₁,…,fn−m)⊂Rn be a compact real algebraic set and g:X→R2m be a continuous function. If the diagonal in X×X is isolated in the set of self-intersection points of g, we define the intersection number of g. In the case where X is a manifold and g is an immersion it is the intersection number defined by Whitney. In the case where g is a polynomial mapping, we present an effective formula for this number.     

A note on the implicit function theorem for quasi-linear eigenvalue problems

We consider the quasi-linear eigenvalue problem −Δ_pu=λg(u) subject to Dirichlet boundary conditions on a bounded open set Ω, where g is a locally Lipschitz continuous function. Imposing no further conditions on Ω or g, we show that for λ near zero the problem has a bounded solution which is unique in the class of all small solutions. Moreover, this curve of solutions parameterized by λ depends continuously on the parameter.     

Pseudo-linear superposition principle for the Monge-Ampère equation based on generated pseudo-operations

Through this paper, we consider generated pseudo-operations of the following form: x⊕y=g⁻¹(g(x)+g(y)), x⊙y=g⁻¹(g(x)g(y)), where g is a continuous generating function. Pseudo-linear superposition principle, i.e., the superposition principle with this type of pseudo-operations in the core, for the Monge-Ampère equation is investigated.     

C-Programming: An outline

This paper presents the essentials of a method designed to solve optimization problems whose objective functions are of the form g(x)+ ψ(u(x)), where ψ is differentiable and either concave or convex. It is shown that solutions to such problems can be obtained through the solutions of the Lagrangian problem whose objective function is of the form g(x)+ λu(x).     

Construction of concentration measures for General Lorenz curves using Riemann-Stieltjes integrals

Lorenz curves were invented to model situations of inequality in real life and applied in econometrics (distribution of wealth or poverty), biometrics (distribution of species richness), and informetrics (distribution of literature over their producers). Different types of Lorenz curves are hereby found in the literature, and in each case a theory of good concentration measures is presented. The present paper unifies these approaches by presenting one general model of concentration measure that applies to all these cases. Riemann-Stieltjes integrals are hereby needed where the integrand is a convex function and the integrator a function that generalizes the inverse of the derivative of the Lorenz function, in case this function is not everywhere differentiable.Calling this general measure C we prove that, if we have two Lorenz functions f, g such that f < g, then C(f) < C(g). This general proof contains the many partial results that are proved before in the literature in the respective special cases.     

Solving deconvolution type problems by wavelet decomposition methods

The paper considers an inverse problem associated with equations of the form Kf = g, where K is a convolution-type operator. The aim is to find a solution f for given function g. We construct approximate solutions by applying a wavelet basis that is well adapted to this problem. For this basis we calculate the elementary solutions that are the approximate preimages of the wavelets. The solution for the inverse problem is then constructed as an appropriate finite linear combination of the elementary solutions. Under certain assumptions we estimate the approximation error and discuss the advantages of the proposed scheme.     

Extrapolating the mean-values of multiplicative functions

It is shown that certain commonly occurring conditions may be factored out of sums of multiplicative arithmetic functions.A function is arithmetic if it is defined on the positive integers. Those complex-valued arithmetic functions g which satisfy the relation g(ab) = g(a)g(b) for all coprime pairs of positive integers a, b are here called multiplicative. In this paper g will be a multiplicative function which satisfies |g(n)| ≤ 1 for all positive integers n.     

C-programming problems: a class of non-linear optimization problems

A method is introduced for the solution of minimization problems possessing objective functions of the form g(x)+?(u(x)), where ? is differentiable and concave. It is shown that a solution to such problems can be obtained through the solution of the Lagrangian problem whose objective function has the form g(x)+λu(x).     

Countably additive gambling with a general expected reward

Summary Optimal strategies and the optimal return function are characterized for a Borel gambling problem in which the utility of a strategy is the expectation under the strategy of a general, measurable function g defined on the space of all infinite histories. These results are based on a previous paper with Lester Dubins where g was assumed to be shift-invariant.Research supported by National Science Foundation Grant MCS77-28424     

Bounds on the degree of APN polynomials: the case of x ?1?+?g(x)

In this paper we consider APN functions ${f:\mathcal{F}_{2^m}\to \mathcal{F}_{2^m}}$ of the form f(x) = x ^?1 + g(x) where g is any non ${\mathcal{F}_{2}}$ -affine polynomial. We prove a lower bound on the degree of the polynomial g. This bound in particular implies that such a function f is APN on at most a finite number of fields ${\mathcal{F}_{2^m}}$ . Furthermore we prove that when the degree of g is less than 7 such functions are APN only if m ?? 3 where these functions are equivalent to x ³.     

An Empirical Study of MAX-2-SAT Phase Transitions

The decision version of the maximum satisfiability problem (MAX-SAT) is stated as follows: Given a set S of propositional clauses and an integer g, decide if there exists a truth assignment that falsifies at most g clauses in S, where g is called the allowance for false clauses. We conduct an extensive experiment on over a million of random instances of 2-SAT and identify statistically the relationship between g, n (number of variables) and m (number of clauses). In our experiment, we apply an efficient decision procedure based on the branch-and-bound method. The statistical data of the experiment confirm not only the “scaling window” of MAX-2-SAT discovered by Chayes, Kim and Borgs, but also the recent results of Coppersmith et al. While there is no easy-hard-easy pattern for the complexity of 2-SAT at the phase transition, we show that there is such a pattern for the decision problem of MAX-2-SAT associated with the phase transition. We also identify that the hardest problems are among those with high allowance for false clauses but low number of clauses.     

Derivative and factorization of holomorphic functions

For a holomorphic function f of bounded type on a complex Banach space E, we show that its derivative df:E→E^∗ takes bounded sets into certain families of sets if and only if f may be factored in the form f=g○S, where S is in some associated operator ideal, and g is a holomorphic function of bounded type. We also prove that the multilinear and polynomial mappings factor in an analogous way if and only if they are “K-bounded.”     

Commuting Toeplitz operators on the Segal-Bargmann space

Consider two Toeplitz operators Tg, Tf on the Segal-Bargmann space over the complex plane. Let us assume that g is a radial function and both operators commute. Under certain growth condition at infinity of f and g we show that f must be radial, as well. We give a counterexample of this fact in case of bounded Toeplitz operators but a fast growing radial symbol g. In this case the vanishing commutator [Tg,Tf]=0 does not imply the radial dependence of f. Finally, we consider Toeplitz operators on the Segal-Bargmann space over Cn and n>1, where the commuting property of Toeplitz operators can be realized more easily.     

Sums over numbers with restricted prime factors

For k a non-negative integer, let P_k(n) denote the kth largest prime factor of n where P₀(n) = +∞ and if the number of prime factors of n is less than k, then P_k(n) = 1. We shall study the asymptotic behavior of the sum Ψ_k(x, y; g) = Σ_{1 ≤ n ≤ x, Pk(n) ≤ y}g(n), where g(n) is an arithmetic function satisfying certain general conditions regarding its behavior on primes. The special case where g(n) = μ(n), the Möbius function, is discussed as an application.     

Equality of two-variable functional means generated by different measures

We consider two-variable functional means of the form $$M_{f,g;\mu}(x,y) := \left(\frac{f}{g}\right)^{-1}\left(\frac{\int\nolimits_{[0,1]} f(tx+(1-t)y)\,d\mu(t)}{\int\nolimits_{[0,1]}g(tx+(1-t)y)\,d\mu(t)}\right),$$ where f, g are continuous functions on a real interval such that g is positive, f/g is strictly monotonic and??? is a measure over the Borel sets of [0,1]. The main results concern the functional equation M _f,g;?? ?=?M _f,g;?? for the unknown functions f, g, where??? and ?? are given measures. Depending on the symmetry properties of the measures, various necessary conditions and sufficient conditions are established.     

Density condensation of Boolean formulas

The following problem is considered: given a Boolean formula f, generate another formula g such that: (i) If f is unsatisfiable then g is also unsatisfiable. (ii) If f is satisfiable then g is also satisfiable and furthermore g is “easier” than f. For the measure of this easiness, we use the density of a formula f which is defined as (the number of satisfying assignments)/2ⁿ, where n is the number of Boolean variables of f. In this paper, we mainly consider the case that the input formula f is given as a 3-CNF formula and the output formula g may be any formula using Boolean AND, OR and negation. Two different approaches to this problem are presented: one is to obtain g by reducing the number of variables and the other by increasing the number of variables, both of which are based on existing SAT algorithms. Our performance evaluation shows that, a little surprisingly, better SAT algorithms do not always give us better density-condensation algorithms.