On the solvability of the simultaneous Pell equations x2 − ay2 = 1 and y2 − bz2 = v12

Let [Formula: see text] be fixed positive integers such that [Formula: see text] is not a perfect square and [Formula: see text] is squarefree, and let [Formula: see text] denote the number of distinct prime divisors of [Formula: see text]. Let [Formula: see text] denote the least solution of Pell equation [Formula: see text]. Further, for any positive integer [Formula: see text], let [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text]. In this paper, using the basic properties of Pell equations and some known results on binary quartic Diophantine equations, a necessary and sufficient condition for the system of equations [Formula: see text] and [Formula: see text] to have positive integer solutions [Formula: see text] is obtained. By this result, we prove that if [Formula: see text] has a positive integer solution [Formula: see text] for [Formula: see text] or [Formula: see text] according to [Formula: see text] or not, then [Formula: see text] and [Formula: see text], where [Formula: see text] is a positive integer, [Formula: see text] or [Formula: see text] and [Formula: see text] or [Formula: see text] according to [Formula: see text] or not, [Formula: see text] is the integer part of [Formula: see text], except for [Formula: see text]

DEVELOPMENT OF METHODOLOGY FOR CADASTRAL ACTIVITY

The purpose of the research is the formalization of cadastral activity from systemic viewpoint. The article discusses the task of ensuring good quality of USRER data and provided services, which is solved by conducting high-quality cadastral works. The article deals defines the notion of quality, its factors and formulates cadastral results, which should contribute to "efficient" and least solution for goal achievement, which is legally provided, least resource-consuming and implemented in a given situation. As a result, the article offers a new methodology for cadastral activity, aimed at solving existing problems.

ON INTERPRETATION OF TYPED AND UNTYPED FUNCTIONAL PROGRAMS

In this paper the interpretation algorithms for typed and untyped functional programs are considered. Typed functional programs use variables of any order and constants of order $ \leq 1 $, where constants of order 1 are strongly computable, monotonic functions with indeterminate values of arguments. The basic semantics of the typed functional program is a function with indeterminate values of arguments, which is the main component of its least solution. The interpretation algorithms of typed functional programs are based on substitutions, $ \beta $-reduction and canonical $ \delta $-reduction. The basic semantics of the untyped functional program is the untyped $ \lambda $-term, which is defined by means of the fixed point combinator. The interpretation algorithms of untyped functional programs are based on substitutions and $ \beta $-reduction. Interpretation algorithms are examined for completeness and comparability. It is investigated how the “behavior” of the interpretation algorithm changes after translation of typed functional program into untyped functional program.

Parametric Strategy Iteration

Program behavior may depend on parameters, which are either configuredbefore compilation time, or provided at runtime, e.g., by sensors or other input devices.Parametric program analysis explores how different parameter settings may affect theprogram behavior.In order to infer invariants depending on parameters, we introduce parametric strategy iteration.This algorithm determines the precise least solution of systems of integer equations dependingon surplus parameters. Conceptually, our algorithm performs ordinary strategy iterationon the given integer system for all possible parameter settings in parallel.This is made possible by means of region trees to represent the occurring piecewise affine functions.We indicate that each required operation on these trees is polynomial-time if only constantly manyparameters are involved.Parametric strategy iteration for systems of integer equationsallows to construct parametric integer interval analysisas well as parametric analysis of differences of integer variables.It thus provides a general technique to realize precise parametricprogram analysis if numerical properties of integer variables are of concern.

A linear algebraic approach to datalog evaluation

We propose a fundamentally new approach to Datalog evaluation. Given a linear Datalog program DB written usingNconstants and binary predicates, we first translate if-and-only-if completions of clauses in DB into a setEq(DB) of matrix equations with a non-linear operation, where relations inMDB, the least Herbrand model of DB, are encoded as adjacency matrices. We then translateEq(DB) into another, but purely linear matrix equationsẼq(DB). It is proved that the least solution ofẼq(DB) in the sense of matrix ordering is converted to the least solution ofEq(DB) and the latter givesMDBas a set of adjacency matrices. Hence, computing the least solution ofẼq(DB) is equivalent to computingMDBspecified by DB. For a class of tail recursive programs and for some other types of programs, our approach achievesO(N3) time complexity irrespective of the number of variables in a clause since only matrix operations costingO(N3) or less are used. We conducted two experiments that compute the least Herbrand models of linear Datalog programs. The first experiment computes transitive closure of artificial data and real network data taken from the Koblenz Network Collection. The second one compared the proposed approach with the state-of-the-art symbolic systems including two Prolog systems and two ASP systems, in terms of computation time for a transitive closure program and the same generation program. In the experiment, it is observed that our linear algebraic approach runs 101~ 104times faster than the symbolic systems when data is not sparse. Our approach is inspired by the emergence of big knowledge graphs and expected to contribute to the realization of rich and scalable logical inference for knowledge graphs.

On Dependence of Interpretation Algorithms of Typed Functional Programs on Canonical Notion of δ-Reduction

In this paper the interpretation algorithms of typed functional programs are considered. The interpretation algorithm is based on substitutions, β-reduction and canonical δ-reduction. The basic semantics of typed functional program is a function with indeterminate values of arguments, which is the main component of its least solution. If the value of the basic semantics for some values of arguments is indeterminate, then the interpretation algorithm either stops with the value ┴, or works endlessly. It is shown that seven known interpretation algorithms are ┴-depend on canonical notion of δ-reduction. Here are these algorithms: FS (of full substitution), PES (of parallel external substitution), LES (of left external substitution), PIS (of parallel inner substitution), LIS (of left inner substitution), ACT (active algorithm), PAS (passive algorithm).

On the Odd Prime Solutions of the Diophantine Equationxy+yx=zz

Using the elementary method and some properties of the least solution of Pell’s equation, we prove that the equationxy+yx=zzhas no positive integer solutions (x,y,z) withxandybeing odd primes.