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Basic narrowing is a restricted form of narrowing which constrains narrowing steps to a set of unblocked (or basic) positions. In this work, we study the modularity of termination of basic narrowing in hierarchical combinations of TRSs, which provides new algorithmic criteria to prove termination of basic narrowing. Basic narrowing has a number of important applications including equational unification in canonical theories. Another application is analyzing termination of narrowing by checking the termination of basic narrowing, as done in pioneering work by Hullot. As a particularly interesting application, we consider solving equations modulo a theory that is given by a TRS, and then distill a number of modularity results for the decidability of equational unification via the modularity of basic narrowing (completeness and) termination. [ABSTRACT FROM PUBLISHER]

We found a minor error in the proof of paper “Universal Alignment Probability Revisited” by S.Y. Lin and Y.C. Ho (J. Optim. Theory Appl. 113(2):399–407, ). In this note, we give a counterexample and explain the reason. We also show that the conclusion of that paper is still correct despite this minor error. A new proof of the conclusion is given. [ABSTRACT FROM AUTHOR]

We consider quasihomogeneous strings with piecewise constant density and derive the quasihomogeneity condition. For a small number of string components, we present explicit formulas for reconstructing a string from its part. For an arbitrary number of string components, we construct a theory of matrices of special form (that we call distinguished) and present an algorithm for string reconstruction from its part. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to provide a rigorous mathematical proof of the existence of travelling wave solutions to the Gross-Pitaevskii equation in dimensions two and three. Our arguments, based on minimization under constraints, yield a full branch of solutions, and extend earlier results (see [3,4,8]) where only a part of the branch was built. In dimension three, we also show that there are no travelling wave solutions of small energy. [ABSTRACT FROM AUTHOR]

We give a combinatorial proof of Guo's multi-generalization of Munarini's identity, answering a question of Guo. [ABSTRACT FROM AUTHOR]

Abstract: In [E.R. van Dam, W.H. Haemers, Which graphs are determined by their spectrum? Linear Algebra Appl. 373 (2003), 241–272] we gave a survey of answers to the question of which graphs are determined by the spectrum of some matrix associated to the graph. In particular, the usual adjacency matrix and the Laplacian matrix were addressed. Furthermore, we formulated some research questions on the topic. In the meantime, some of these questions have been (partially) answered. In the present paper we give a survey of these and other developments. [Copyright &y& Elsevier]

Abstract: In this paper we consider the 2D dissipative quasi-geostrophic equations and study the regularity criterion of the solutions. By means of a commutator estimate based on frequency localization and Bony''s paraproduct decomposition, we obtain a regularity criterion which is an obvious extension of the result in [B. Dong, Z. Chen, A remark on regularity criterion for the dissipative quasi-geostrophic equations, J. Math. Anal. Appl. 329 (2007) 1212–1217] where their restrictive condition on p is . [Copyright &y& Elsevier]

Abstract: If and are linear equations, then the disjunctive Rado number of the set is the least integer n, provided that it exists, such that for every 2-coloring of the set there exists a monochromatic solution to either or . If such an integer n does not exist, then the disjunctive Rado number is infinite. In this paper, it is shown that for all integers and , the disjunctive Rado number for the equations and is if is odd and the disjunctive Rado number for these equations is infinite otherwise. It is also shown that for all integers and , the disjunctive Rado number for the equations and is if there exist natural numbers , and t such that and and is an odd integer and c is the largest such integer, and the disjunctive Rado number for these equations is infinite otherwise. [Copyright &y& Elsevier]

Abstract: This paper describes : a system for automatically translating programs written in , a simple imperative language, into a set of first-order equations. This set of equations represents a program and has a precise mathematical meaning; moreover, the standard techniques for mechanizing equational reasoning can be used for verifying properties of programs. Part of the system itself is formulated abstractly as a set of first-order rewrite rules. Then, the rewrite rules are proven to be terminating and confluent. This means that our system produces, for a given program, a unique set of equations. In our work, simple imperative programs are equational theories of a logical system within which proofs can be derived. [Copyright &y& Elsevier]

The article focuses on the extremal combinatorial problem regarding uniform hypergraphs' colorings with large girth. It states that a hypergraph refers to a pair of sets H = (V, E). It mentions random coloring's construction that is grounded on the random recoloring method of J. Beck. It suggests that the random coloring can be changed only in times of Poisson processes' jumps, corresponding to hypergraph vertices. It adds that the random coloring developed through the random recoloring process is H's proper coloring with a positive probability.

By using tools from additive combinatorics, invariant theory and bounds on the size of the minimal generating sets of PSL2( q), we prove the following growth property. There exists ℇε > 0 such that the following holds for any finite field  q. Let G be the group SL2( q), or PSL2( q), and let A be a generating set of G. Then | A ·· A ·· A| ⩾⩾ min {| A|1 ++ ℇε, | G|}. Our work extends the work of Helfgott [[Helfgott, Ann. of Math. 167: 601--623, 2008]] who proved similar results for the family {SL2( p): p prime}. [ABSTRACT FROM AUTHOR]

We consider properties of the solution set of a word equation with one unknown. We prove that the solution set of a word equation possessing infinite number of solutions is of the form (pq)*p where pq is primitive. Next, we prove that a word equation with at most four occurrences of the unknown possesses either infinitely many solutions or at most two solutions. We show that there are equations with at most four occurrences of the unknown possessing exactly two solutions. Finally, we prove that a word equation with at most 2k occurrences of the unknown possesses either infinitely many solutions or at most 8 log k + O(1) solutions. Hence, if we consider a class εk of equations with at most 2k occurrences of the unknown, then each equation in this class possesses either infinitely many solutions or O(log k) number of solutions. Our considerations allow to construct the first alphabet independent linear time algorithm for computing the solution set of an equation in a nontrivial class of equations. [ABSTRACT FROM AUTHOR]

The article offers information on the Euclidean spaces and solvability of the spheres' chromatic numbers. It mentions that this type of equation can be seen in combinatorial geometry and cites the right formula in determining the distance between the two monochromatic points. It connotes how to ascertain the chromatic numbers as well as the forbidden distance. It highlights the applicability of the linear algebra's formula along with the combinatorics and topological methods. Furthermore, the theorems and independence numbers are also mentioned.

It is shown that (two-variable generalizations of) more than half of Slater's list of 130 Rogers–Ramanujan identities (L. J. Slater, Further identities of the Rogers–Ramanujan type, Proc. London Math Soc. (2)54 (1952) 147–167) can be easily derived using just three multiparameter Bailey pairs and their associated q-difference equations. As a bonus, new Rogers–Ramanujan type identities are found along with natural combinatorial interpretations for many of these identities. [ABSTRACT FROM AUTHOR]

The article presents a mathematical analysis which aims to demonstrate the power of the dynamical systems method (DSM) which is a tool for proving theoretical results. DSM is applied in nonlinear equations which focus on the emphasis of convergence and stability of the DSM-based algorithms in solving operator equations. In the analysis, the researchers give a proof of a Hadamard-type theorem on global hemeomorphisms and an enough condition for the nonlinear map subjectivity in a Hilbert space. In addition, part of the analysis is to consider the path in constructing a solution to the problem with an initial data.

There exist independently based semigroup varietiesand,, such thathas no cover in the interval [;]. [ABSTRACT FROM AUTHOR]

It is proved that a group G generated by a conjugacy class X of elements of order 3, so that every two non-commuting elements of X generate a subgroup isomorphic to an alternating group of degree 4 or 5, is locally finite. More precisely, either G contains a normal elementary 2-subgroup of index 3, or G is isomorphic to an alternating group of permutations on some (possibly infinite) set. [ABSTRACT FROM AUTHOR]

We survey recent algorithms for the propositional satisfiability problem. In particular, we consider algorithms having the best current worst-case upper bounds on their complexity. We also discuss some related issues: a derandomization of the algorithm of Paturi, Pudlák, Saks, and Zane, the Valiant–Vazirani lemma, and random walk algorithms with the “back button.” Bibliography: 47 titles. [ABSTRACT FROM AUTHOR]