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Abstract  We generalize a result of F. Luca and A. Sankaranarayanan by proving that the set of n for which ϕ(1) ⃛ ϕ(n) is squareful is of zero density. A similar statement holds for σ (n) instead of ϕ(n) and for some other multiplicative functions. [ABSTRACT FROM AUTHOR]

We give counterexamples for some statements of Borwein and sufficient conditions for the validity of these. [ABSTRACT FROM AUTHOR]

Codes having higher information rates are desirable, since a higher rate code implies a more efficient use of redundancy than a lower rate code. However, when choosing a code for a particular application, we must also consider the error-correcting capabilities of the code. There is a basic trade-off between code rate and minimum distance. The smaller the code rate, the larger is the minimum distance and vice-versa. This paper proposes a simple coding scheme that can construct a code with higher information rate from an existing code. First, the paper derives a low-rate $$\mathcal {C}'(n', k', d')$$ -code from an existing $$\mathcal {C}(n, k, d)$$ -code, where $$\frac{k}{n} \ge \frac{k'}{n'}$$ . An associated decoding procedure for the newly derived class of low-rate codes is also described. Finally, the proposed coding scheme combines $$\mathcal {C}$$ and a set of $$\mathcal {C}'$$ s to obtain a $$\mathcal {C}''(n'', k'', d'')$$ -code with $$\frac{k''}{n''} \ge \frac{k}{n} \ge \frac{k'}{n'}$$ . Kronecker product is used as a basic tool in the coding procedure. [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]

In this paper, we study the unit-neighbourhood of the tree bisection and reconnection operation on unrooted binary phylogenetic trees. Specifically, we provide a recursive method to calculate the size of the unit-neighbourhood for any tree in the space $${\fancyscript{T}_n}$$ of unrooted binary phylogenetic trees with n-leaves. We also give both upper and lower bounds on this size for all trees in $${\fancyscript{T}_n}$$, and characterize those trees for which the stated upper bound is sharp. [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]

In this paper we introduce FG-coupled fixed point, which is a generalization of coupled fixed point for nonlinear mappings in partially ordered complete metric spaces. We discuss existence and uniqueness theorems of FGcoupled fixed points for different contractive mappings. Our theorems generalizes the results of Gnana Bhaskar and Lakshmikantham [1]. [ABSTRACT FROM AUTHOR]

Let IK be a complete ultrametric algebraically closed field and let A be the Banach IK-algebra of bounded analytic functions in the "open" unit disk D of IK provided with the Gauss norm. Let Mult(...) be the set of continuous multiplicative semi-norms of A provided with the topology of pointwise convergence, let Multm(...) be the subset of the ε Mult(...) whose kernel is a maximal ideal and let Mult1(...) be the subset of the φ ε Mult(...) whose kernel is a maximal ideal of the form (x-a)A with a ε D. By analogy with the Archimedean context, one usually calls ultrametric Corona problem, or ultrametric Kakutani problem the question whether Mult1(...) is dense in Multm(...). In order to recall the study of this problem that was made in several successive steps, here we first recall how to characterize the various continuous multiplicative semi-norms of A, with particularly the nice construction of certain multiplicative semi-norms of A whose kernell is neither a null ideal nor a maximal ideal, due to J. Araujo. Here we prove that multbijectivity implies density. The problem of multbijectivity will be described in a further paper. [ABSTRACT FROM AUTHOR]

The main interest of the present paper is to classify the almost cosymplectic 3-manifolds that satisfy ∥gradλ∥ = const.(≠ 0) and ∇ξh = 2ahΦ. [ABSTRACT FROM AUTHOR]

In this paper, the concept of dual normal relations on sets is introduced and generalized. Intrinsic characterizations of them are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the stability of the sets of efficient points of vector-valued optimization problems when the data of the approximate problems converges to the data of the original problem in the sense of Painlevé-Kuratowski. Our results improve the corresponding results obtained by Lucchetti and Miglierina (Optimization 53(5-6):517-528, , Section 3). [ABSTRACT FROM AUTHOR]

In this paper, the existence and multiplicity of a class of double resonant semilinear elliptic equations with the Dirichlet boundary value are studied. [ABSTRACT FROM AUTHOR]

The present paper is an introduction to a combinatorial theory arising as a natural generalization of classical and virtual knot theory. There is a way to encode links by a class of "realizable" graphs. When passing to generic graphs with the same equivalence relations we get "graph-links". On one hand graph-links generalize the notion of virtual link, on the other hand they do not detect link mutations. We define the Jones polynomial for graph-links and prove its invariance. We also prove some a generalization of the Kauffman–Murasugi–Thistlethwaite theorem on "minimal diagrams" for graph-links. [ABSTRACT FROM AUTHOR]

The paper offers a matrix-based logic (relevant matrix quantum physics) for propositions which seems suitable as an underlying logic for empirical sciences and especially for quantum physics. This logic is motivated by two criteria which serve to clean derivations of classical logic from superfluous redundancies and uninformative complexities. It distinguishes those valid derivations (inferences) of classical logic which contain superfluous redundancies and complexities and are in this sense "irrelevant" from those which are "relevant" or "nonredundant" in the sense of allowing only the most informative consequences in the derivations. The latter derivations are strictly valid in RMQ, whereas the former are only materially valid. RMQ is a decidable matrix calculus which possesses a semantics and has the finite model property. It is shown in the paper how RMQ by its strictly valid derivations can avoid the difficulties with commensurability, distributivity, and Bell's inequalities when it is applied to quantum physics. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the following implicit iteration scheme recently introduced by Xu and Ori [ Numer. Funct. Anal. Optim., 22, (2001) 767–773]: and to prove several strongly and weakly convergent theorems of the iteration for a finite family of pseudocontractive mappings under condition α n ∈ (0, b] ⊂ (0, 1). [ABSTRACT FROM AUTHOR]

Pure Pattern Type Systems (P2TS) combine the frameworks and capabilities of rewriting and ?-calculus within a unified setting. Their type systems, which are adapted from Barendregt's ?-cube, are especially interesting from a logical point of view. Until now, strong normalisation, which is an essential property for logical soundness, has only been conjectured: in this paper, we give a positive answer for the simply-typed system and the dependently-typed system.The proof is based on a translation of terms and types from P2TSinto the ?-calculus. First, we deal with untyped terms, ensuring that reductions are faithfully mimicked in the ?-calculus. For this, we rely on an original encoding of the pattern matching capability of P2TSinto the System F?.Then we show how to translate types: the expressive power of System F? is needed in order to fully reproduce the original typing judgments of P2TS. We prove that the encoding is correct with respect to reductions and typing, and we conclude with the strong normalisation of simply-typed P2TSterms. The strong normalisation with dependent types is in turn obtained by an intermediate translation into simply-typed terms. [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]

Our aim in this paper is to introduce the notion of (ϵ, ϵ Vq)-fuzzy filters in MTL-algebras, which is a generalization of fuzzy filters of MTL-algebras and related properties are investigated. In particular, we extend the concept of a fuzzy subgroup with thresholds to the concept of a fuzzy filter with thresholds in MTL-algebras. [ABSTRACT FROM AUTHOR]

In this paper we present an algorithm, which helps us to define a similarity relation of fuzzy implications. This similarity relation allows us to partition by clustering fuzzy implications in equivalence classes. [ABSTRACT FROM AUTHOR]

The Cai δ-equality of fuzzy sets corresponds to the Lukasiewicz t-norm. In this paper we study the notion of (*, δ)-equality, a concept which generalizes the δ-equality to the case of the fuzzy set theory based on an arbitrary continuous t-norm *. We investigate the robustness of some fuzzy implication operators in terms of (*, δ)-equality. [ABSTRACT FROM AUTHOR]

For further studying the theories and applications of rough sets (RS), this paper proposes a new theory on RS, which mainly includes topological space, topological properties, homeomorphism, and its properties on RS by some new definitions and theorems given. The relationship between partition and countable open covering is discussed, and some applications based on the topological rough space and its topological properties are introduced. Moreover, some perspectives for future research are given. Throughout this paper, the advancements of the new theory on RS and topological algebra not only represent an important theoretical value but also exhibit significant applications of RS and topology. [ABSTRACT FROM AUTHOR]

The paper studies the existence of (n + 1)-cliques in k-connected graphs. It is proved that, in a k-connected graph G, such a clique exists, provided that the following conditions are fulfilled: (1) the vertices of every n-clique of G belong to a k-cutset; (2) the removal of certain pairs each of which consists of a vertex and an edge decreases the connectivity of the graph G by 2. Bibliography: 4 titles. [ABSTRACT FROM AUTHOR]

This paper describes the construction of pedagogical environments in mathematics classes, centred on mathematical modelling and denominated ‘investigative scenarios’, which stimulate students to investigation, to formulation of problems and to political reflection, as well as the sharing of acquired knowledge with other persons in the community. The paper is based on the application of modelling as a teaching and learning strategy and on the pedagogical work with teenagers in an assisted freedom program. Both were accomplished in a scenario built with 10 volunteer students taking calculus in a Computer Engineering course in 2003. Among the main results we emphasise the academic maturing process for the student, how competent he gets in making models, accomplishing simulations, his perception of the relation between mathematical learning and everyday situations and political reflection about the results from working with modelling as much as about his participation in the community work. [ABSTRACT FROM AUTHOR]

This paper gives a general definition of a “kind of schema” (often called a “meta-model” in the literature, but here called a “species”) along with general definitions for the schemas of a species, and for the databases, constraints, and queries over a given schema of a species. This leads naturally to a general theory of data translation and integration over arbitrary schemas of arbitrary species, based on schema morphisms, and to a similar general theory of ontology translation and integration over arbitrary logics. Institutions provide a general notion of logic, and Grothendieck flattening provides a general tool for integrating heterogeneous schemas, species and logics, as well as theories, such as ontologies, over different logics. Many examples of our novel concepts are included, some rather detailed. An initial section introduces data integration and ontologies for readers who are not specialists, with some emphasis on challenges. A brief review of universal algebra is also given, though some familiarity with category theory is assumed in later sections. [ABSTRACT FROM PUBLISHER]

The paper studies projective stationary sets. The Projective Stationary Reflection Principle is the statement that every projective stationary set contains an increasing continuous --chain of length É1. It is shown that, if Martin's Maximum holds, then the Projective Stationary Reflection Principle holds. Also, this principle is equivalent to the Strong Reflection Principle. The paper shows that the saturation of the nonstationary ideal on É1 is equivalent to a certain kind of reflection. [ABSTRACT FROM AUTHOR]

A central result in extremal set theory is the celebrated theorem of Sperner from 1928, which gives the size of the largest family of subsets of [n] not containing a 2-chain, F1 ⊂ F2. Erdős extended this theorem to determine the largest family without a k-chain, F1 ⊂ F2 ⊂ . . . ⊂ Fk. Erdős and Katona, followed by Kleitman, asked how many chains must appear in families with sizes larger than the corresponding extremal bounds.In 1966, Kleitman resolved this question for 2-chains, showing that the number of such chains is minimized by taking sets as close to the middle level as possible. Moreover, he conjectured the extremal families were the same for k-chains, for all k. In this paper, making the first progress on this problem, we verify Kleitman's conjecture for the families whose size is at most the size of the k + 1 middle levels. We also characterize all extremal configurations. [ABSTRACT FROM AUTHOR]

In this paper, we study the class of all monadic implicational subreducts, that is, the $${\{\rightarrow, \forall,1\}}$$ -subreducts of the class of monadic MV-algebras. We prove that this class is an equational class, which we denote by $${\mathcal{ML}}$$ , and we give an equational basis for this variety. An algebra in $${\mathcal{ML}}$$ is called a monadic Łukasiewicz implication algebra. We characterize the subdirectly irreducible members of $${\mathcal{ML}}$$ and the congruences of every monadic Łukasiewicz implication algebra by monadic filters. We prove that $${\mathcal{ML}}$$ is generated by its finite members. Finally, we completely describe the lattice of subvarieties, and we give an equational basis for each proper subvariety. [ABSTRACT FROM AUTHOR]

The restriction semigroups, in both their one-sided and two-sided versions, have arisen in various fashions, meriting study for their own sake. From one historical perspective, as “weaklyE-ample” semigroups, the definition revolves around a “designated set” of commuting idempotents, better thought of as projections. This class includes the inverse semigroups in a natural fashion. In a recent paper, the author introducedP-restriction semigroups in order to broaden the notion of “projection” (thereby encompassing the regular *-semigroups). That study is continued here from the varietal perspective introduced for restriction semigroups by V. Gould. The relationship between varieties of regular *-semigroups and varieties ofP-restriction semigroups is studied. In particular, a tight relationship exists between varieties of orthodox *-semigroups and varieties of “orthodox”P-restriction semigroups, leading to concrete descriptions of the free orthodoxP-restriction semigroups and related structures. Specializing further, new, elementary paths are found for descriptions of the free restriction semigroups, in both the two-sided and one-sided cases. [ABSTRACT FROM PUBLISHER]

In this paper we shall introduce and study subclasses Rλ (α, s, t) and Pλ (α, s, t) of the class of λ - spirallike generalized Sakaguchi type function. Here we shall prove characterization and subordination properties for these subclasses and point out several interesting consequences of our results. [ABSTRACT FROM AUTHOR]

In this paper, we characterize the subclass of the invertible T-algebras by a second-order formula. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study the basic homogeneous -machine scheduling problem where weakly dependent unit-time jobs have to be scheduled within the time windows between their release dates and due dates so that, for any subset of machines, the set of the time units at which at least one machine is busy, is in interval. We first introduce the notions of pyramidal structure, -hole, -matching, preschedule, -schedule and schedule for this problem. Then we provide a feasibility criteria for a preschedule. The key result of the paper is then to provide a structural necessary and sufficient condition for an instance of the problem to be feasible. We conclude by giving the directions of ongoing works and by bringing open questions related to different variants of the basic non-idling -machine scheduling problem. [Copyright &y& Elsevier]

Abstract: Let be a family of subsets of an -element set. Sperner’s theorem says that if there is no inclusion among the members of then the largest family under this condition is the one containing all -element subsets. The present paper surveys certain generalizations of this theorem. The maximum size of is to be found under the condition that a certain configuration is excluded. The configuration here is always described by inclusions. More formally, let be a poset. The maximum size of a family which does not contain as a (not-necessarily induced) subposet is denoted by . The paper is based on a lecture of the author at the Jubilee Conference on Discrete Mathematics [Banasthali University, January 11–13, 2009], but it was somewhat updated in December 2010. [Copyright &y& Elsevier]

We say that an algebra $${\mathcal{A}}$$ has the retraction closure property (RCP) if the set of all retractions of $${\mathcal{A}}$$ is closed with respect to fundamental operations of $${\mathcal{A}}$$ applied pointwise. In this paper we investigate this property, both 'locally' (one algebra) and 'globally' (in some variety of algebras), especially emphasizing the case of groupoids. We compare the retraction closure property with the endomorphism closure property on both levels and prove that a necessary and sufficient condition for a variety V of algebras to have RCP is that V is a variety of entropic algebras that satisfy the diagonal identities. [ABSTRACT FROM AUTHOR]

In a natural way, we can 'lift' any operation defined on a set A to an operation on the set of all non-empty subsets of A and obtain from any algebra ( $${A, \Omega}$$) its power algebra of subsets. G. Grätzer and H. Lakser proved that for a variety $${\mathcal{V}}$$, the variety $${\mathcal{V}\Sigma}$$ generated by power algebras of algebras in $${\mathcal{V}}$$ satisfies precisely the consequences of the linear identities true in $${\mathcal{V}}$$. For certain types of algebras, the sets of their subalgebras form subalgebras of their power algebras. They are called the algebras of subalgebras. In this paper, we partially solve a long-standing problem concerning identities satisfied by the variety $${\mathcal{VS}}$$ generated by algebras of subalgebras of algebras in a given variety $${\mathcal{V}}$$. We prove that if a variety $${\mathcal{V}}$$ is idempotent and entropic and the variety $${\mathcal{V}\Sigma}$$ is locally finite, then the variety $${\mathcal{VS}}$$ is defined by the idempotent and linear identities true in $${\mathcal{V}}$$. [ABSTRACT FROM AUTHOR]

Let two nonperiodic binary morphisms ... be given. A word w is called a solution of g and h if g(w) = h(w). We say that a solution w is simple if whenever ... are prefixes of wu such that ... for some word z, then |u| = |u'| = k|w|, for some k ∈ N. In this paper we will study simple solutions and show that if a word w is a simple solution containing at least nine occurrences of the letter a and at least nine occurrences of the letter b, then either w = (ab)ia, or w = ajbk with gcd (j, k) = 1, up to the exchange of letters a and b. [ABSTRACT FROM AUTHOR]

This paper discusses the twisted conjugacy classes in loop groups. We restrict to classical groups and give some explicit classifications. [ABSTRACT FROM AUTHOR]

In this paper, we define a soft intersection group (soft int-group) on a soft set. This new concept functions as a bridge among soft set theory, set theory and group theory and shows the effect of soft sets on a group structure in the sense of intersection and inclusion of sets. We then derive the basic properties of soft int-groups and give its applications to group theory. [ABSTRACT FROM AUTHOR]

In this paper, from the viewpoint of quandle construction we prove that for any link type L and any diagram of it, there exists another diagram of L such that these two diagrams are Ω1-dependent, Ω2-dependent and Ω3-dependent. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to show that every GTS can be realized as a μ-closed subspace of a generalized hyperconnected space. Also, we give more characterizations of generalized hyperconnected spaces. [ABSTRACT FROM AUTHOR]

In this paper, we compare the virtual fundamental classes of the stacks of (g, β, μ)-stable ramified maps ,Ug, μ(X, β) and of (g, β, μ)-log stable ramified maps Ulog g,μ(X, β). For that we will see how they are identified via virtual normalization and then apply Costello's push-forward formula. [ABSTRACT FROM AUTHOR]

For the extended mixed linear complementarity problem (EML CP), we first present the characterization of the solution set for the EMLCP. Based on this, its global error bound is also established under milder conditions. The results obtained in this paper can be taken as an extension for the classical linear complementarity problems. [ABSTRACT FROM AUTHOR]

Skew lattices are a noncommutative generalization of lattices. In the paper we study the varieties of symmetric, strongly symmetric and cancellative skew lattices, and characterize them in terms of certain laws regarding the coset structure of a skew lattice. As a consequence, combinatorial results connecting powers of ${\mathcal{D}}$-classes and indices are derived. [ABSTRACT FROM AUTHOR]

Property testing is a rapid growing field in theoretical computer science. It considers the following task: given a function f over a domain D, a property ℘ and a parameter 0< ε<1, by examining function values of f over o(| D|) elements in D, determine whether f satisfies ℘ or differs from any one which satisfies ℘ in at least ε| D| elements. An algorithm that fulfills this task is called a property tester. We focus on tree-likeness of quartet topologies, which is a combinatorial property originating from evolutionary tree construction. The input function is f, which assigns one of the three possible topologies for every quartet over an n-taxon set S. We say that f satisfies tree-likeness if there exists an evolutionary tree T whose induced quartet topologies coincide with f. In this paper, we prove the existence of a set of quartet topologies of error number at least $c{n\choose 4}$ for some constant c>0, and present the first property tester for tree-likeness of quartet topologies. Our property tester makes at most O( n/ ε) queries, and is of one-sided error and non-adaptive. [ABSTRACT FROM AUTHOR]

A set Q⊆ V is a hub set of a graph G=( V, E) if, for every pair of vertices u, v∈ V∖ Q, there exists a path from u to v such that all intermediate vertices are in Q. The hub number of G is the minimum size of a hub set in G. This paper derives the hub numbers of Sierpiński-like graphs including: Sierpiński graphs, extended Sierpiński graphs, and Sierpiński gasket graphs. Meanwhile, the corresponding minimum hub sets are also obtained. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to introduce and study generalized hyperconnected spaces. Various characterizations of generalized hyperconnected spaces and preservation theorems are discussed. [ABSTRACT FROM AUTHOR]

In this paper we give the definition of [λ ᵘ]-continuity of real-valued functions defined on an open interval, which is an example of path continuity. We give some properties of [λ, ᵘ]-continuous functions. The aim of the paper is to find the maximal additive class and the maximal multiplicative class for the family of [λ, ᵘ]-continuous functions. [ABSTRACT FROM AUTHOR]

It is proved in the paper that the multiplication operation on an arbitrary E-algebra can be extended to an E-algebra A-morphism. As a corollary, it is proved that every May algebra defined by an E-algebra is a Cartan object in the category of May algebras. [ABSTRACT FROM AUTHOR]

In this paper, we define and study a special type of trisections in a module category, namely the compact trisections which characterize quasi-directed components. We apply this notion to the study of laura algebras and we use it to define a class of algebras with predictable Auslander-Reiten components. [ABSTRACT FROM AUTHOR]