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In the present paper, necessary and sufficient conditions are given for the equality of the power rezidue symbols $$ {\left( {\frac{\alpha }{a}} \right)_n} $$ and $$ {\left( {\frac{\alpha }{a}} \right)_n} $$ in the cyclotomic field ℚ(ζ n), 2 ∤ n, for a ∈ ℤ, ( a, n) = 1. This result is a generalization of the classical Eisenstein reciprocity law and its continuation in a Hasse’s paper. Bibliography: 3 titles. [ABSTRACT FROM AUTHOR]

Solving high-dimensional optimal control problems and corresponding Hamilton–Jacobi PDEs are important but challenging problems in control engineering. In this paper, we propose two abstract neural network architectures which are, respectively, used to compute the value function and the optimal control for certain class of high-dimensional optimal control problems. We provide the mathematical analysis for the two abstract architectures. We also show several numerical results computed using the deep neural network implementations of these abstract architectures. A preliminary implementation of our proposed neural network architecture on FPGAs shows promising speedup compared to CPUs. This work paves the way to leverage efficient dedicated hardware designed for neural networks to solve high-dimensional optimal control problems and Hamilton–Jacobi PDEs. [ABSTRACT FROM AUTHOR]

We are concerned with the following class of equations with exponential nonlinearities: which is related to the Tzitzéica equation. Here $$h_1, h_2$$ are two smooth positive functions. The purpose of the paper is to initiate the analytical study of the above equation and to give a quite complete picture both for what concerns the blow-up phenomena and the existence issue. In the first part of the paper we provide a quantization of local blow-up masses associated to a blowing-up sequence of solutions. Next we exclude the presence of blow-up points on the boundary under the Dirichlet boundary conditions. In the second part of the paper we consider the Tzitzéica equation on compact surfaces: we start by proving a sharp Moser-Trudinger inequality related to this problem. Finally, we give a general existence result. [ABSTRACT FROM AUTHOR]

This paper presents a simple way for diagonalization of matrices in an arbitrary field, with which one can calculate functions of matrices. [ABSTRACT FROM AUTHOR]

Given a triangle functor F: A→B, the authors introduce the half image hImF, which is an additive category closely related to F. If F is full or faithful, then hImF admits a natural triangulated structure. However, in general, one can not expect that hImF has a natural triangulated structure. The aim of this paper is to prove that hImF admits a natural triangulated structure if and only if F satisfies the condition (SM). If this is the case, hImF is triangle-equivalent to the Verdier quotient A/KerF. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the formula for the Kac determinant of the algebra arising from the level N representation of the Ding-Iohara-Miki algebra. It is also discovered that its singular vectors correspond to generalized Macdonald functions (the q-deformed version of the AFLT basis). [ABSTRACT FROM AUTHOR]

The problem of maximizing non-negative monotone submodular functions under a certain constraint has been intensively studied in the last decade. In this paper, we address the problem for functions defined over the integer lattice. Suppose that a non-negative monotone submodular function f:Z+n→R+ is given via an evaluation oracle. Assume further that f satisfies the diminishing return property, which is not an immediate consequence of submodularity when the domain is the integer lattice. Given this, we design polynomial-time (1-1/e-ϵ)-approximation algorithms for a cardinality constraint, a polymatroid constraint, and a knapsack constraint. For a cardinality constraint, we also provide a (1-1/e-ϵ)-approximation algorithm with slightly worse time complexity that does not rely on the diminishing return property. [ABSTRACT FROM AUTHOR]

Generalizing both mixed-integer linear optimization and convex optimization, mixed-integer convex optimization possesses broad modeling power but has seen relatively few advances in general-purpose solvers in recent years. In this paper, we intend to provide a broadly accessible introduction to our recent work in developing algorithms and software for this problem class. Our approach is based on constructing polyhedral outer approximations of the convex constraints, resulting in a global solution by solving a finite number of mixed-integer linear and continuous convex subproblems. The key advance we present is to strengthen the polyhedral approximations by constructing them in a higher-dimensional space. In order to automate this extended formulation we rely on the algebraic modeling technique of disciplined convex programming (DCP), and for generality and ease of implementation we use conic representations of the convex constraints. Although our framework requires a manual translation of existing models into DCP form, after performing this transformation on the MINLPLIB2 benchmark library we were able to solve a number of unsolved instances and on many other instances achieve superior performance compared with state-of-the-art solvers like Bonmin, SCIP, and Artelys Knitro. [ABSTRACT FROM AUTHOR]

A subgroup H of a finite group G is called a c#-normal subgroup of G if there exists a normal subgroup K of G such that G = HK and H ∩ K is a CAP-subgroup of G: In this paper, we investigate the influence of fewer c#-normal subgroups of Sylow p-subgroups on the p-supersolvability, p-nilpotency, and supersolvability of finite groups. We obtain some new sufficient and necessary conditions for a group to be p-supersolvable, p-nilpotent, and supersolvable. Our results improve and extend many known results. [ABSTRACT FROM AUTHOR]

A method to define trivariate spline quasi-interpolation operators (QIOs) is developed by blending univariate and bivariate operators whose linear functionals allow oversampling. In this paper, we construct new operators based on univariate B-splines and bivariate box splines, exact on appropriate spaces of polynomials and having small infinity norms. An upper bound of the infinity norm for a general blending trivariate spline QIO is derived from the Bernstein-Bézier coefficients of the fundamental functions associated with the operators involved in the construction. The minimization of the resulting upper bound is then proposed and the existence of a solution is proved. The quadratic and quartic cases are completely worked out and their exact solutions are explicitly calculated. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to investigate EQ-algebras with internal states and state morphism good EQ-algebras. To begin with, we introduce the notion of EQ-algebras with internal states (simplify, SEQ-algebras) and discuss the relation between SEQ-algebras and state EQ-algebras. In the following, we study state filters (simplify, S-filters) and state prefilters (simplify, S-prefilters) of SEQ-algebras and discuss subdirectly irreducible SEQ-algebras. We focus on algebraic structures of the set SPF(E,σ) of all S-prefilters on a SEQ-algebra and obtain that SPF(E,σ) forms a complete Brouwerian lattice, when E is an ℓEQ-algebra or good. Moreover, for ℓEQ-algebras, SPF(E,σ) forms a Heyting algebra if σ is faithful and preserves →. Then, we introduce the σ-co-annihilator of a non-empty set A on a SEQ-algebra. As applications, we give a characterization for minimal prime S-prefilters of state morphism good EQ-algebras and characterize the representable state morphism good EQ-algebras by minimal prime S-prefilters. [ABSTRACT FROM AUTHOR]

In this paper, we propose a sentential logic naturally associated, in the sense of Abstract Algebraic Logic, with the variety of distributive nearlattices. We show that the class of algebras canonically associated (in the sense of Abstract Algebraic Logic) with this logic is the variety of distributive nearlattices. We also present several properties of this sentential logic. [ABSTRACT FROM AUTHOR]

Quantum B-algebras as a generalization of quantales were introduced by Rump and Yang, which cover the majority of implicational algebras and provide a unified semantic for a wide class of substructural logics. Unital quantum B-algebras play an important role in the classification of implicational algebras. The main purpose of this paper is to construct unital quantum B-algebras from non-unital quantum B-algebras. [ABSTRACT FROM AUTHOR]

Oleg Viro introduced an invariant of rigid isotopy for real algebraic knots in ℝℙ3 which can be viewed as a first order Vassiliev invariant. In this paper we look at real algebraic knots of degree d with the maximal possible value of this invariant. We show that for a given d all such knots are topologically isotopic and explicitly identify their knot type. [ABSTRACT FROM AUTHOR]

In this paper, we consider a subsemigroup S of a real connected simple Lie group G generated by {exp tX : X ∈ Γ, t ≥ 0} for some subset Γ of L, the Lie algebra of G. It is proved that for an open class Γ = { A, ± B} and a generic pair ( A, B) in L × L, if S contains a subgroup isomorphic to SL(2, ℝ), associated to an arbitrary root, then S is the whole G. In a series of previous papers, analogous results have been obtained for the maximal root only. Recently, a similar result for complex connected simple Lie groups was proved. The proof uses special root properties that characterize some particular subalgebras of L. In control theory, this case Γ = { A, ± B} is specially important since the control system, ġ = ( A + uB) g, where u ∈ ℝ, is controllable on G if and only if S = G. [ABSTRACT FROM AUTHOR]

In this paper, we present a novel approach to the construction of new finite algebras and describe the congruence lattices of these algebras. Given a finite algebra $${\langle B_0, \ldots \rangle}$$, let $${B_1,B_2, \ldots , B_K}$$ be sets that either intersect B or intersect each other at certain points. We construct an overalgebra $${\langle A, FA \rangle}$$, by which we mean an expansion of $${\langle B_0, \ldots \rangle}$$ with universe $${A = B_0 \cup B_1 \cup \ldots \cup B_K}$$, and a certain set F of unary operations that includes mappings e satisfying $${e^2_i = e_i}$$ and e( A) = B, for $${0 \leq i \leq K}$$. We explore two such constructions and prove results about the shape of the new congruence lattices Con $${\langle A, F_A \rangle}$$ that result. Thus, descriptions of some new classes of finitely representable lattices is one contribution of this paper. Another, perhaps more significant, contribution is the announcement of a novel approach to the discovery of new classes of representable lattices, the full potential of which we have only begun to explore. [ABSTRACT FROM AUTHOR]

The present paper continues a cycle of papers of the author (same of them joint), in which the Yoneda algebras are calculated for several families of algebras of dihedral and semidihedral type (in K. Erdmam's classification). In this paper, the Yaneda algebra is described in terms of quivers with relations for algebras of semidihedral type of the family SD(2 $$ \mathcal{B} $$). Bibliography: 28 titles. [ABSTRACT FROM AUTHOR]

This paper is devoted to the discussion of some schemes of construction of radicals similar to special radicals, which generalize constructions of the basic special radicals of the algebras close to associative ones. [ABSTRACT FROM AUTHOR]

This paper takes some investigations on soft weak structures which are a generalization of weak structures and soft topological spaces with some separation axioms and compactness. Also, we give a systematic discussion on the relationship among these notions. [ABSTRACT FROM AUTHOR]

This paper extends the study of commutative rings whose ideals form an MV-algebra as carried out by Belluce and Di Nola (Math Log Q 55(5):468-486, 2009) to non-commutative rings. We study and characterize all rings whose ideals form a pseudo-MV-algebra, which shall be called here generalized Łukasiewicz rings. We obtain that up to isomorphism, these are exactly the direct sums of unitary special primary rings. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of right stabilizers in BL-algebras and define a class of BL-algebras, called RS- BL-algebra. Then we investigate the relationships between the right and left stabilizers in BL-algebras. After that we define the concept of semi RS- BL-algebras and we prove that semisimple BL-algebras and semi RS- BL-algebras are RS- BL-algebra. Moreover, every finite RS- BL-algebra is a semi RS- BL-algebra. Finally, we introduce the concept of a semi-stabilizer and a stabilizer filter in a BL-algebra and we state and prove some theorems which determine the relationships between this notion and the other types of filters of a BL-algebra. [ABSTRACT FROM AUTHOR]

The scheduling in grids is known to be a NP-hard problem. The distributed deployment of nodes, their heterogeneity and their fluctuations in terms of workload and availability make the design of an effective scheduling algorithm a very complex issue. The scientific literature has proposed several heuristics able to tackle this kind of optimization problem using techniques and strategies inspired by nature. The algorithms belonging to ant colony optimization (ACO) paradigm represent an example of these techniques: each one of these algorithms uses strategies inspired by the self-organization ability of real ants for building effective grid schedulers. In this paper, the authors propose an on line, non-clairvoyant, distributed scheduling solution for multi-broker grid based on the alienated ant algorithm (AAA), a new ACO inspired technique exploiting a 'non natural' behavior of ants and an inverse interpretation of pheromone trails. The paper introduces the proposed algorithm, explains the differences with other classical ACO approaches, and compares AAA with two different algorithms. The results of simulations show that the AAA guarantees good performance in terms of makespan, average queue waiting time and load balancing capability. [ABSTRACT FROM AUTHOR]

In natural duality theory, the piggybacking technique is a valuable tool for constructing dualities. As originally devised by Davey and Werner, and extended by Davey and Priestley, it can be applied to finitely generated quasivarieties of algebras having term-reducts in a quasivariety for which a well-behaved natural duality is already available. This paper presents a comprehensive study of the method in a much wider setting: piggyback duality theorems are obtained for suitable prevarieties of structures. For the first time, and within this extended framework, piggybacking is used to derive theorems giving criteria for establishing strong dualities and two-forone dualities. The general theorems specialise in particular to the familiar situation in which we piggyback on Priestley duality for distributive lattices or Hofmann-Mislove- Stralka duality for semilattices, and many well-known dualities are thereby subsumed. A selection of new dualities is also presented. [ABSTRACT FROM AUTHOR]

In this series of papers we show that there are exactly ten subfactors, other than A subfactors, of index between 4 and 5. Previously this classification was known up to index $${3+\sqrt{3}}$$. In the first paper we give an analogue of Haagerup's initial classification of subfactors of index less than $${3+\sqrt{3}}$$, showing that any subfactor of index less than 5 must appear in one of a large list of families. These families will be considered separately in the three subsequent papers in this series. [ABSTRACT FROM AUTHOR]

In this paper, we describe the BIBasis package designed for REDUCE and Macaulay2 computer algebra systems, which allows one to compute Boolean involutive bases and Gröbner bases. The implementations and user interfaces of the package for both systems are described in the respective sections of the paper. Also, we present results of comparisons of BIBasis with other packages and algorithms for constructing Boolean Gröbner bases available in the computer algebra systems. [ABSTRACT FROM AUTHOR]

This paper investigates profinite completions of residually finite algebras, drawing on ideas from the theory of natural dualities. Given a class $${\mathcal{A} = \mathbb{ISP}(\mathcal{M})}$$, where $${\mathcal{M}}$$ is a set, not necessarily finite, of finite algebras, it is shown that each $${{\bf A} \in \mathcal{A}}$$ embeds as a topologically dense subalgebra of a topological algebra $${n_{\mathcal{A}}({\bf A})}$$ (its natural extension), and that $${n_{\mathcal{A}}({\bf A})}$$ is isomorphic, topologically and algebraically, to the profinite completion of A. In addition it is shown how the natural extension may be concretely described as a certain family of relation-preserving maps; in the special case that $${\mathcal{M}}$$ is finite and $${\mathcal{A}}$$ possesses a single-sorted or multisorted natural duality, the relations to be preserved can be taken to be those belonging to a dualising set. For an algebra belonging to a finitely generated variety of lattice-based algebras, it is known that the profinite completion coincides with the canonical extension. In this situation the natural extension provides a new concrete realisation of the canonical extension, generalising the well-known representation of the canonical extension of a bounded distributive lattice as the lattice of up-sets of the underlying ordered set of its Priestley dual. The paper concludes with a survey of classes of algebras to which the main theorems do, and do not, apply. [ABSTRACT FROM AUTHOR]

class $${\mathcal {K}}$$ of algebras with a distinguished constant term 0 is called Fregean if congruences of algebras in $${\mathcal {K}}$$ are uniquely determined by their 0-cosets and Θ( 0, a) = Θ( 0, b) implies a = b for all $${a, b \in {\bf A} \in \mathcal {K}}$$ . The structure of Fregean varieties was investigated in a paper by P. Idziak, K. Słomczyńska, and A. Wroński. In particular, it was shown there that every congruence permutable Fregean variety consists of algebras that are expansions of equivalential algebras, i.e., algebras that form an algebraization of the purely equivalential fragment of the intuitionistic propositional logic. In this paper we give a full characterization of the commutator for equivalential algebras and solvable Fregean varieties. In particular, we show that in a solvable algebra from a Fregean variety, the commutator coincides with the commutator of its purely equivalential reduct. Moreover, an intrinsic characterization of the commutator in this setting is given. [ABSTRACT FROM AUTHOR]

This paper deals with an extremal problem concerning hypergraph colorings. Let k be an integer. The problem is to find the value m( n) equal to the minimum number of edges in an n-uniform hypergraph not admitting two-colorings of the vertex set such that every edge of the hypergraph contains k vertices of each color. In this paper, we obtain the exact values of m(5) and m(4), and the upper bounds for m(7) and m(9). [ABSTRACT FROM AUTHOR]

The present paper continues a cycle of papers, in which the Yoneda algebras were calculated for several families of algebras of dihedral and semidihedral type in the classification by K. Erdmann. Using the technique of a previous paper, a description of the Yoneda algebras for both families of local algebras occurring in this classification is given. Namely, a conjecture about the structure of the minimal free resolution of a (unique) simple module is stated, which is based on some empirical observations, and after establishing this conjecture, “cohomology information" is derived from the resolution discovered, and, as a result, this allows us to describe the Yoneda algebras of the algebras under consideration, It is noted that a similar technique was applied in computation of the Hochschild cohomology algebra for some finite-dimensional algebras. Bibliography: 23 titles. [ABSTRACT FROM AUTHOR]

In this paper, we prove the Hyers-Ulam-Rassias stability of isometric homomorphisms in proper CQ*-algebras for the following Cauchy-Jensen additive mapping: The concept of Hyers-Ulam-Rassias stability originated from the Th.M. Rassias’ stability theorem that appeared in the paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297–300. This is applied to investigate isometric isomorphisms between proper CQ*-algebras. [ABSTRACT FROM AUTHOR]

Abstract.  This paper studies diophantine properties of sets definable in an o-minimal structure over the real field. The main theorem of the author’s recent paper with A. J. Wilkie is refined, and used to deduce a strong result on the density of algebraic points of such sets. [ABSTRACT FROM AUTHOR]

Abstract  The study and research of evolutionary algorithms (EAs) is getting great attention in recent years. Although EAs have earned extensive acceptance through numerous successful applications in many fields, the problem of finding the best combination of evolutionary parameters especially for population size that need the manual settings by the user is still unresolved. In this paper, our system is focusing on differential evolution (DE) and its control parameters. To overcome the problem, two new systems were carried out for the self-adaptive population size to test two different methodologies (absolute encoding and relative encoding) in DE and compared their performances against the original DE. Fifty runs are conducted for every 20 well-known benchmark problems to test on every proposed algorithm in this paper to achieve the function optimization without explicit parameter tuning in DE. The empirical testing results showed that DE with self-adaptive population size using relative encoding performed well in terms of the average performance as well as stability compared to absolute encoding version as well as the original DE. [ABSTRACT FROM AUTHOR]

The paper continues the series of papers devoted to surveying and developing methods for solving algebraic problems for two-parameter polynomial and rational matrices of general form. It considers linearization methods, which allow one to reduce the problem of solving an equation F(λ, µ)x = 0 with a polynomial two-parameter matrix F(λ, µ) to solving an equation of the form D(λ, µ)y = 0, where D(λ, µ) = A(µ)-λB(µ) is a pencil of polynomial matrices. Consistent pencils and their application to solving spectral problems for the matrix F(λ, µ) are discussed. The notion of reducing subspace is generalized to the case of a pencil of polynomial matrices. An algorithm for transforming a general pencil of polynomial matrices to a quasitriangular pencil is suggested. For a pencil with multiple eigenvalues, algorithms for computing the Jordan chains of vectors are developed. Bibliography: 8 titles. [ABSTRACT FROM AUTHOR]

This paper is one of the series of survey papers dedicated to the development of methods for solving problems of algebra for two-parameter polynomial matrices of general form. The paper considers the AB-algorithm and the ∇V-2 factorization algorithm, together with their applications. Bibliography: 4 titles. [ABSTRACT FROM AUTHOR]

Starting from an abstract setting for the Lüders-von Neumann quantum measurement process and its interpretation as a probability conditionalization rule in a non-Boolean event structure, the author derived a certain generalization of operator algebras in a preceding paper. This is an order-unit space with some specific properties. It becomes a Jordan operator algebra under a certain set of additional conditions, but does not own a multiplication operation in the most general case. A major objective of the present paper is the search for such examples of the structure mentioned above that do not stem from Jordan operator algebras; first natural candidates are matrix algebras over the octonions and other nonassociative rings. Therefore, the case when a nonassociative commutative multiplication exists is studied without assuming that it satisfies the Jordan condition. The characteristics of the resulting algebra are analyzed. This includes the uniqueness of the spectral resolution as well as a criterion for its existence, subalgebras that are Jordan algebras, associative subalgebras, and more different levels of compatibility than occurring in standard quantum mechanics. However, the paper cannot provide the desired example, but contribute to the search by the identification of some typical differences between the potential examples and the Jordan operator algebras and by negative results concerning some first natural candidates. The possibility that no such example exists cannot be ruled out. However, this would result in an unexpected new characterization of Jordan operator algebras, which would have a significant impact on quantum axiomatics since some customary axioms (e.g., power-associativity or the sum postulate for observables) might turn out to be redundant then. [ABSTRACT FROM AUTHOR]

This review paper is devoted to some questions related to investigations of bases in PI-algebras. The central point is generalization and refinement of the Shirshov height theorem, of the Amitsur–Shestakov hypothesis, and of the independence theorem. The paper is mainly inspired by the fact that these topics shed some light on the analogy between structure theory and constructive combinatorial reasoning related to the “microlevel,” to relations in algebras and straightforward calculations. Together with the representation theory of monomial algebras, height and independence theorems are closely connected with combinatorics of words and of normal forms, as well as with properties of primary algebras and with combinatorics of matrix units. Another aim of this paper is an attempt to create a kind of symbolic calculus of operators defined on records of transformations. [ABSTRACT FROM AUTHOR]

This paper deals with a parametric family of convex semi-infinite optimization problems for which linear perturbations of the objective function and continuous perturbations of the right-hand side of the constraint system are allowed. In this context, Cánovas et al. (SIAM J. Optim. 18:717–732, []) introduced a sufficient condition (called ENC in the present paper) for the strong Lipschitz stability of the optimal set mapping. Now, we show that ENC also entails high stability for the minimal subsets of indices involved in the KKT conditions, yielding a nice behavior not only for the optimal set mapping, but also for its inverse. Roughly speaking, points near optimal solutions are optimal for proximal parameters. In particular, this fact leads us to a remarkable simplification of a certain expression for the (metric) regularity modulus given in Cánovas et al. (J. Glob. Optim. 41:1–13, []) (and based on Ioffe (Usp. Mat. Nauk 55(3):103–162, []; Control Cybern. 32:543–554, [])), which provides a key step in further research oriented to find more computable expressions of this regularity modulus. [ABSTRACT FROM AUTHOR]

The theory of nonnegative matrices is an example of a theory motivated in its origins and development by purely mathematical concerns that later proved to have a remarkably broad spectrum of applications to such diverse fields as probability theory, numerical analysis, economics, dynamical programming, and demography. At the heart of the theory is what is usually known as the Perron–Frobenius Theorem. It was inspired by a theorem of Oskar Perron on positive matrices, usually called Perron’s Theorem. This paper is primarily concerned with the origins of Perron’s Theorem in his masterful work on ordinary and generalized continued fractions (1907) and its role in inspiring the remarkable work of Frobenius on nonnegative matrices (1912) that produced, inter alia, the Perron–Frobenius Theorem. The paper is not at all intended exclusively for readers with expertise in the theory of nonnegative matrices. Anyone with a basic grounding in linear algebra should be able to read this article and come away with a good understanding of the Perron–Frobenius Theorem as well as its historical origins. The final section of the paper considers the first major application of the Perron–Frobenius Theorem, namely, to the theory of Markov chains. When he introduced the eponymous chains in 1908, Markov adumbrated several key notions and results of the Perron–Frobenius theory albeit within the much simpler context of stochastic matrices; but it was by means of Frobenius’ 1912 paper that the linear algebraic foundations of Markov’s theory for nonpositive stochastic matrices were first established by R. Von Mises and V.I. Romanovsky. [ABSTRACT FROM AUTHOR]

This paper is devoted to relations between the Kurosh problem and the Shirshov height theorem. The central point and main technical tool is the identity of algebraicity. The main result of this paper is the following. Let A be a finitely generated PI-algebra and Y be a finite subset of A. For any Noetherian associative and commutative ring {ie125-01}, let any factor of R ⊗ A such that all projections of elements from Y are algebraic over π( R) be a Noetherian R-module. Then A has bounded essential height over Y. If, furthermore, Y generates A as an algebra, then A has bounded height over Y in the Shirshov sense. This paper also contains a new proof of the Razmyslov-Kemer-Braun theorem on radical nilpotence of affine PI-algebras. This proof allows one to obtain some constructive estimates. The main goal of the paper is to develop a “virtual operator calculus.” Virtual operators (pasting, deleting, and transfer) depend not only on an element of the algebra but also on its representation. [ABSTRACT FROM AUTHOR]

This paper is devoted to the introduction of a D ∞-differential analog of the notion of an E ∞-(co)algebra and to the construction of generalized Steenrod operations in terms of multiplicative spectral sequences. In this paper, we investigate basic homotopy properties of D ∞-differential E ∞-(co)algebras and construct a spectral sequence of a D ∞-differential E ∞-(co)algebra. [ABSTRACT FROM AUTHOR]

In the present paper, we consider estimates for the orders of zeros of polynomials in functions satisfying a system of algebraic differential equations and possessing a special D-property defined in the paper. The main result obtained in the paper consists of two theorems for the two cases in which these estimates are given. These estimates are improved versions of a similar estimate proved earlier in the case of algebraically independent functions and a single point. They are derived from a more general theorem concerning the estimates of absolute values of ideals in the ring of polynomials, and the proof of this theorem occupies the main part of the present paper. The proof is based on the theory of ideals in rings of polynomials. Such estimates may be used to prove the algebraic independence of the values of functions at algebraic points. [ABSTRACT FROM AUTHOR]

In the paper, the solvability of the word problem and the conjugacy problem is proved for a wide class of finitely presented groups defined by periodic defining relations of a sufficiently large odd degree. In the proof, we use a certain simplified version of the classification of periodic words and transformations of these words, which was presented in detail in the author’s monograph devoted to the well-known Burnside problem. The result is completed by the proof of an interesting result of Sarkisyan on the existence of a group, given by defining relations of the form E = 1, for which the word problem is unsolvable. This result was first published in abstracts of papers of the 13th All-Union Algebra Symposium in Gomel in 1975. [ABSTRACT FROM AUTHOR]

Abstract.  In this paper we consider the category Q-Mod of modules over a given quantale Q. The paper is motivated by constructions and results from the category of modules over a ring. We show that the category Q-Mod is monadic, consider its relation to the category Q-Top of Q-topological spaces and generalize a method of completion of partially ordered sets. [ABSTRACT FROM AUTHOR]

This is the second of the author’s three papers on stability in the Liouville theorem on the Heisenberg group. The aim is to prove that each mapping with bounded distortion of a John domain on the Heisenberg group is close to a conformal mapping with order of closeness $$\sqrt {K - 1} $$ in the uniform norm and order of closeness K − 1 in the Sobolev norm L for all $$p < \tfrac{C}{{K - 1}}$$ . In this paper we prove a local variant of the desired result: each mapping on a ball with bounded distortion and distortion coefficient K near to 1 is close on a smaller ball to a conformal mapping with order of closeness $$\sqrt {K - 1} $$ in the uniform norm and order of closeness K − 1 in the Sobolev norm L for all $$p < \tfrac{C}{{K - 1}}$$ . We construct an example that demonstrates the asymptotic sharpness of the order of closeness of a mapping with bounded distortion to a conformal mapping in the Sobolev norm. [ABSTRACT FROM AUTHOR]

In the paper, it is proved that the radical of a relatively free associative algebra of countable rank over an infinite field of characteristic p > 0 is a nil ideal of bounded index if the complexity of the corresponding variety is less than p. Moreover, a description of a basis for trace identities for the matrix algebra M n over an infinite field of characteristic p > 0, n < p, is obtained in the paper. [ABSTRACT FROM AUTHOR]

This paper considers the Steiner Minimal Tree (SMT) problem in the rectilinear and octilinear planes. The study is motivated by the physical design of VLSI: The rectilinear case corresponds to the currently used M-architecture, which uses either horizontal or vertical routing, while the octilinear case corresponds to a new routing technique, X-architecture, that is based on the pervasive use of diagonal directions. The experimental studies show that the X-architecture demonstrates a length reduction of more than 10–20%. In this paper, we make a theoretical study on the lengths of SMTs in these two planes. Our mathematical analysis confirms that the length reduction is significant as the previous experimental studies claimed, but the reduction for three points is not as significant as for two points. We also obtain the lower and upper bounds on the expected lengths of SMTs in these two planes for arbitrary number of points. [ABSTRACT FROM AUTHOR]

When n − k systems of an n-partite permutation-invariant state are traced out, the resulting state can be approximated by a convex combination of tensor product states. This is the quantum de Finetti theorem. In this paper, we show that an upper bound on the trace distance of this approximation is given by $${2\frac{kd^2}{n}}$$ , where d is the dimension of the individual system, thereby improving previously known bounds. Our result follows from a more general approximation theorem for representations of the unitary group. Consider a pure state that lies in the irreducible representation $${U_{\mu +\nu} \subset U_\mu \otimes U_\nu}$$ of the unitary group U( d), for highest weights μ, ν and μ + ν. Let ξμ be the state obtained by tracing out U ν. Then ξμ is close to a convex combination of the coherent states $${U_\mu(g)|{v_\mu\rangle}}$$ , where $${g\in U(d)}$$ and $${|v_\mu\rangle}$$ is the highest weight vector in U μ. For the class of symmetric Werner states, which are invariant under both the permutation and unitary groups, we give a second de Finetti-style theorem (our “half” theorem). It arises from a combinatorial formula for the distance of certain special symmetric Werner states to states of fixed spectrum, making a connection to the recently defined shifted Schur functions [1]. This formula also provides us with useful examples that allow us to conclude that finite quantum de Finetti theorems (unlike their classical counterparts) must depend on the dimension d. The last part of this paper analyses the structure of the set of symmetric Werner states and shows that the product states in this set do not form a polytope in general. [ABSTRACT FROM AUTHOR]

In this paper, we provide theoretical analysis for a cubic regularization of Newton method as applied to unconstrained minimization problem. For this scheme, we prove general local convergence results. However, the main contribution of the paper is related to global worst-case complexity bounds for different problem classes including some nonconvex cases. It is shown that the search direction can be computed by standard linear algebra technique. [ABSTRACT FROM AUTHOR]

Let R be an arbitrary finite commutative local ring. In this paper, we obtain a necessary and sufficient condition for a function over R to be a polynomial function. Before this paper, necessary and sufficient conditions for a function to be a polynomial function over some special finite commutative local rings were obtained. [ABSTRACT FROM AUTHOR]

In this paper, we first prove that the θ-deformation U θ (2) of U(2) constructed by Connes and Violette is our special case of the quantum group U q (2) constructed in our previous paper. Then we will show that the set of traces on the C*–algebra U θ , θ irrational, is determined by the set of the traces on a subalgebra of U θ . [ABSTRACT FROM AUTHOR]