Orthogonal polynomials and analytic functions associated to positive definite matrices

For a positive definite infinite matrix A, we study the relationship between its associated sequence of orthonormal polynomials and the asymptotic behaviour of the smallest eigenvalue of its truncation An of size n×n. For the particular case of A being a Hankel or a Hankel block matrix, our results lead to a characterization of positive measures with finite index of determinacy and of completely indeterminate matrix moment problems, respectively.     

Properties of positive harmonic functions on the half-space with a nonlinear boundary condition

In this paper we study existence and properties of solutions of the problem Δw=0 on the half-space with nonlinear boundary condition ∂w/∂η+w=|w|^p−2w where 2<p<2(N−1)/(N−2) and N?3. We obtain a ground state solution w=w(x₁,…,xN−1,t) which is radial and has exponential decay in the first N−1 variables. Moreover, w has sharp polynomial decay in the variable t.     

Interior and exterior functions of positive Boolean functions

The interior and exterior functions of a Boolean function f were introduced in Makino and Ibaraki (Discrete Appl. Math. 69 (1996) 209–231), as stability (or robustness) measures of the f. In this paper, we investigate the complexity of two problems -INTERIOR and -EXTERIOR, introduced therein. We first answer the question about the complexity of -INTERIOR left open in Makino and Ibaraki (Discrete Appl. Math. 69 (1996) 209–231); it has no polynomial total time algorithm even if is bounded by a constant, unless P=NP. However, for positive h-term DNF functions with h bounded by a constant, problems -INTERIOR and -EXTERIOR can be solved in (input) polynomial time and polynomial delay, respectively. Furthermore, for positive k-DNF functions, -INTERIOR for two cases in which k=1, and and k are both bounded by a constant, can be solved in polynomial delay and in polynomial time, respectively.     

Spectral representations and global maps of cellular automata dynamics

We present a spectral representation of any computation performed by a Cellular Automaton (CA) of arbitrary topology and dimensionality via an appropriate coding scheme in Fourier space that can be implemented in an analog machine ideally circumventing part of the overall waste heat production. We explore further consequences of this encoding and we provide a simple example based on the “Game-of-Life” where we find global maps for small lattices indicating an interesting underlying recursive structure.     

Metric properties of functions defined by partial automata

We characterize natural categories in which morphisms are defined by partial automata of the following three types: asynchronous automata, window automata, and automata synchronous over finite alphabets. We distinguish subcategories whose morphisms are defined by finite automata.     

Norm estimates of interpolation matrices and their inverses associated with strictly positive definite functions

In this paper, we estimate the norms of the interpolation matrices and their inverses that arise from scattered data interpolation on spheres with strictly positive definite functions.

     

The dynamics of expansive invertible onesided cellular automata

Using textile systems, we prove the conjecture of Boyle and Maass that the dynamical system defined by an expansive invertible onesided cellular automaton is topologically conjugate to a topological Markov shift. We also study expansive leftmost-permutive onesided cellular automata and bipermutive endomorphisms of mixing topological Markov shifts.

     

Limit averages of continuous functions under the action of cellular automata

We consider averages of continuous functions under the action of cellular automata, which were proposed by Boyle, Lind and Rudolph in 1988. It is proved that, at non-shift-periodic points, the averages converge point-wisely to the integration with respect to the uniform Bernoulli measure.     

A remark on the transformation formula of theta functions associated to positive definite quadratic forms

Let L be a positive Z-lattice with level N = cd, (c, d) = 1. Then the Fourier expansion at cusp $\text{1}\text{d}$ of the theta function associated to L is a theta function associated to L₁, where a lattice L₁ is defined by Z_pL₁ = Z_pL for $\text{p?c}$, Z_pL₁ = the dual of Z_pL for p | c.     

Extension of positive definite functions

Let Ω⊂Rⁿ$\Omega \subset {\mathbb{R}}^n$ be an open, connected subset of Rⁿ${\mathbb{R}}^n$, and let F:Ω−Ω→C$F:\Omega -\Omega \to \mathbb{C}$, where Ω−Ω={x−y:x,y∈Ω}$\Omega -\Omega =\{x-y:x,y\in \Omega \}$, be a continuous positive definite function. We give necessary and sufficient conditions for F   to have an extension to a continuous positive definite function defined on the entire Euclidean space Rⁿ${\mathbb{R}}^n$. The conditions are formulated in terms of existence of a unitary representations of Rⁿ${\mathbb{R}}^n$ whose generators extend a certain system of unbounded Hermitian operators defined on a Hilbert space associated to F. Different positive definite extensions correspond to different unitary representations.     

Unbounded functions and positive linear operators

The approximation of unbounded functions by positive linear operators under multiplier enlargement is investigated. It is shown that a very wide class of positive linear operators can be used to approximate functions with arbitrary growth on the real line. Estimates are given in terms of the usual quantities which appear in the Shisha-Mond theorem. Examples are provided.     

On totally positive sequences and functions

Let a>0 be a fixed number. A function f:R→R is said to be a-shift-generating (a-SG) if for every x∈R, is a totally positive sequence and it does not coincide with a sequence of the form , where A?0 and λ>0. In this paper, we describe all a-SG functions and obtain a new characterization of totally positive functions in the terms of a-SG functions. In addition, using characteristic properties of a-SG functions, we generalize the famous Jacobian identity in theory of elliptic functions.     

On right congruences associated with the input semigroupS of automata

LetA=(M, S, δ) be an automaton without outputs whereM is a nonemptyset andS is a nonempty semigroup. Then the right congruences μM and μ _m associated withS have been expressed in many different ways (μ _M is called the Myhill-Nerode congruence onS). Also, their algebraic properties have been investigated. We have introduced the right congruences μ _S and μ_α onM and we have obtained necessary and sufficient conditions thatS/μ andM/w have nontrivialS-homomorphisms where μ andw are any right congruences onS andM respectively. The faithfulness ofS has been introduced.