Recursive Structures and Ershov's Hierarchy

Ash and Nerode [2] gave natural definability conditions under which a relation is intrinsically r. e. Here we generalize this to arbitrary levels in Ershov's hierarchy of Δ sets, giving conditions under which a relation is intrinsically α-r. e. Mathematics Subject Classification: 03C57, 03D55.     

Interplay of Correlation,Randomness and Dimensionality Effects in Weakly-Coupled Half-Filled Random Hubbard Chains

Abstract

We study interplay of electronic correlation, randomness and dimensionality effects in half-filled random Hubbard chains weakly coupled via an interchain one-particle hopping. Based on the two-loop renormalization-group approach, phase diagrams are given in terms of temperature vs. strengths of the intrachain electron-electron umklapp scattering, the random scattering and the interchain one-particle hopping.     

A nonsymmetric matrix with integer eigenvalues

A nonsymmetric N?×?N matrix with elements as certain simple functions of N distinct real or complex numbers r ₁, r ₂, …, r_N is presented. The matrix is special due to its eigenvalues???the consecutive integers 0,1,2, …, N?1. Theorems are given establishing explicit expressions of the right and left eigenvectors and formulas for recursive calculation of the right eigenvectors. A special case of the matrix has appeared in sampling theory where its right eigenvectors, if properly normalized, give the inclusion probabilities of the conditional Poisson sampling design.     

Finite-stage stochastic decision processes with recursive reward structure I: optimality equations and deterministic strategies

In this paper we examine an N-stage stochastic decision model with a recursive reward structure whose state and action spaces are standard Borel ones. The central results relate to the validity of the optimality equations and to the sufficiency of deterministic strategies. The results expand statements known for classical dynamic programming problems with additive total rewards to a wide class of recursive reward functions under certain monotonicity and continuity assumptions. The existence of an expected utility representation of the total rewards is generally not presupposed     

Lattice-valued fuzzy Turing machines: Computing power, universality and efficiency

In this paper we study fuzzy Turing machines with membership degrees in distributive lattices, which we called them lattice-valued fuzzy Turing machines. First we give several formulations of lattice-valued fuzzy Turing machines, including in particular deterministic and non-deterministic lattice-valued fuzzy Turing machines (l-DTMcs and l-NTMs). We then show that l-DTMcs and l-NTMs are not equivalent as the acceptors of fuzzy languages. This contrasts sharply with classical Turing machines. Second, we show that lattice-valued fuzzy Turing machines can recognize n-r.e. sets in the sense of Bedregal and Figueira, the super-computing power of fuzzy Turing machines is established in the lattice-setting. Third, we show that the truth-valued lattice being finite is a necessary and sufficient condition for the existence of a universal lattice-valued fuzzy Turing machine. For an infinite distributive lattice with a compact metric, we also show that a universal fuzzy Turing machine exists in an approximate sense. This means, for any prescribed accuracy, there is a universal machine that can simulate any lattice-valued fuzzy Turing machine on it with the given accuracy. Finally, we introduce the notions of lattice-valued fuzzy polynomial time-bounded computation (lP) and lattice-valued non-deterministic fuzzy polynomial time-bounded computation (lNP), and investigate their connections with P and NP. We claim that lattice-valued fuzzy Turing machines are more efficient than classical Turing machines.     

Finite-Time Ruin Probabilities for Discrete, Possibly Dependent, Claim Severities

An important question in insurance is how to evaluate the probabilities of (non-) ruin of a company over any given horizon of finite length. This paper aims to present some (not all) useful methods that have been proposed so far for computing, or approximating, these probabilities in the case of discrete claim severities. The starting model is the classical compound Poisson risk model with constant premium and independent and identically distributed claim severities. Two generalized versions of the model are then examined. The former incorporates a non-constant premium function and a non-stationary claim process. The latter takes into account a possible interdependence between the successive claim severities. Special attention will be paid to a recursive computational method that enables us to tackle, in a simple and unified way, the different models under consideration. The approach, still relatively little known, relies on the use of remarkable families of polynomials which are of Appell or generalized Appell (Sheffer) types. The case with dependent claim severities will be revisited accordingly.      

Combinatorial identities related to 2 × 2 submatrices of recursive matrices

Recursive matrices are ubiquitous in combinatorics, which have been extensively studied. We focus on the study of the sums of 2 × 2 minors of certain recursive matrices, the alternating sums of their 2 × 2 minors, and the sums of their 2 × 2 permanents. We obtain some combinatorial identities related to these sums, which generalized the work of Sun and Ma (2014) [23,24]. With the help of the computer algebra package HolonomicFunctions, we further get some new identities involving Narayana polynomials.     

An integral recursive inequality and applications

An integral recursive inequality for two functions is obtained. It is used to describe the equality cases in the related inequalities. The applications involve some bi-Hermitian forms, integral transformations, and confluent hypergeometric functions.      

Intrinsical Randomness of Kolmogorov ℤd-Actions on a Lebesgue Space

We introduce a concept of an intrinsically weak and strong randomness of a ^d -action on a Lebesgue space and we show that Kolmogorov ^d -actions are intrinsically weak random, and Bernoulli ^d -actions are intrinsically strong random.