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In this paper, we deal with the problem of computing the sum of the [Formula: see text]th powers of all the elements of the matrix ring [Formula: see text] with [Formula: see text] and [Formula: see text] a finite commutative ring. We completely solve the problem in the case [Formula: see text] and give some results that compute the value of this sum if [Formula: see text] is an arbitrary finite commutative ring for many values of [Formula: see text] and [Formula: see text]. Finally, based on computational evidence and using some technical results proved in this paper, we conjecture that the sum of the [Formula: see text]th powers of all the elements of the matrix ring [Formula: see text] is always [Formula: see text] unless [Formula: see text], [Formula: see text], [Formula: see text] and the only element [Formula: see text] such that [Formula: see text] is idempotent, in which case the sum is [Formula: see text].

The paper is a part of an ongoing program which aims to show that the existential theory in free groups (hyperbolic groups or even toral relatively hyperbolic) is NP-complete. For that we study compression of solutions with straight-line programs (SLPs) as suggested originally by Plandowski and Rytter in the context of a single word equation. We review some basic results on SLPs and give full proofs in order to keep this fundamental part of the program self-contained. Next we study systems of equations with constraints in free groups and more generally in free products of abelian groups. We show how to compress minimal solutions with extended Parikh-constraints. This type of constraints allows to express semi linear conditions as e.g. alphabetic information. The result relies on some combinatorial analysis and has not been shown elsewhere. We show similar compression results for Boolean formula of equations over a torsion-free $\delta$-hyperbolic group. The situation is much more delicate than in free groups. As byproduct we improve the estimation of the "capacity" constant used by Rips and Sela in their paper "Canonical representatives and equations in hyperbolic groups" from a double-exponential bound in $\delta$ to some single-exponential bound. The final section shows compression results for toral relatively hyperbolic group using the work of Dahmani: We show that given a system of equations over a fixed toral relatively hyperbolic group, for every solution of length $N$ there is an SLP for another solution such that the size of the SLP is bounded by some polynomial $p(s+ \log N)$ where $s$ is the size of the system., Comment: 28 pages

This paper contributes to the theory of recursively presented (see Higman [Subgroups of finitely presented groups, Proc. R. Soc. Ser. A 262 (1961) 455–475]) infinitely generated abelian groups with solvable word problem. Mal'cev [On recursive Abelian groups, Dokl. Akad. Nauk SSSR 146 (1962) 1009–1012] and independently Rabin [Computable algebra, general theory and theory of computable fields, Trans. Amer. Math. Soc. 95 (1960) 341–360] initiated the study of such groups in the early 1960's. In this paper, we develop a technique that we call iterated effective embeddings. The significance of our new technique is that it extends existing methods from the realm of iterated 0″ arguments to iterated 0‴ ones. This is a new phenomenon in computable algebra. We use this technique to confirm a 30 year-old conjecture of Ash, Knight and Oates [Recursive abelian p-groups of small length, https://dl.dropbox.com/u/4752353/Homepage/AKO.pdf ]. More specifically, Ash, Knight and Oates [Recursive abelian p-groups of small length. https://dl.dropbox.com/u/4752353/Homepage/AKO.pdf ] conjectured that there exists a computable reduced abelian p-group of Ulm type ω such that its effective invariants, defined using limitwise monotonic functions, cannot be found uniformly. We construct a computable reduced abelian p-group of Ulm type ω where its invariants are at the maximum possible level of non-uniformity. The result confirms the conjecture in a strong way, and it provides us with an explanation of why computable reduced p-groups of Ulm type ω seem hard to classify in general. We also use p-basic trees and their iterated embeddings to solve a problem posed in [W. Calvert, D. Cenzer, V. S. Harizanov and A. Morozov, Effective categoricity of abelian p-groups, Ann. Pure Appl. Logic 159(1–2) (2009) 187–197].

In this paper, we are interested in obtaining fixed point theorems by keeping the orthogonal completeness of the orthogonal metric space and replacing the [Formula: see text]-contraction condition in theorems by another slightly modified conditions. The paper contains an example illustrating our results.

This is the first of two articles studying the structure of left adequate and, more generally, of left Ehresmann monoids. Motivated by a careful analysis of normal forms, we introduce here a concept of proper for a left adequate monoid M. In fact, our notion is that of T-proper, where T is a submonoid of M. We show that any left adequate monoid M has an X*-proper cover for some set X, that is, there is a left adequate monoid [Formula: see text] that is X*-proper, and an idempotent separating surjective morphism [Formula: see text] of the appropriate type. Given this result, we may deduce that the free left adequate monoid on any set X is X*-proper. In a subsequent paper, we show how to construct T-proper left adequate monoids from any monoid T acting via order-preserving maps on a semilattice with identity, and prove that the free left adequate monoid is of this form. An alternative description of the free left adequate monoid will appear in a paper of Kambites. We show how to obtain the labeled trees appearing in his result from our structure theorem. Our results apply to the wider class of left Ehresmann monoids, and we give them in full generality. We also indicate how to obtain some of the analogous results in the two-sided case. This paper and its sequel, and the two of Kambites on free (left) adequate semigroups, demonstrate the rich but accessible structure of (left) adequate semigroups and monoids, introduced with startling insight by Fountain some 30 years ago.

Hedrlín and Pultr proved that for any monoid M there exists a graph G with endomorphism monoid isomorphic to M. In a previous paper, we give a construction G(M) for a graph with prescribed endomorphism monoid M known as a [Formula: see text]-graph. Using this construction, we derived bounds on the minimum number of vertices and edges required to produce a graph with a given endomorphism monoid for various classes of finite monoids. In this paper, we generalize the [Formula: see text]-graph construction and derive several new bounds for monoid classes not handled by our first paper. Among these are the so called "strong semilattices of C-semigroups" where C is one of the following: Groups, Abelian Groups, Rectangular Groups, and completely simple semigroups.

This paper extends existing relationships between automata and F-coalgebras to relationships between tree automata and (F1, F2)-coalgebras. The concept of an (F1, F2)-coalgebra generalizes that of an F-coalgebra as well as that of an F-algebra. The paper sketches some of the most important properties of the category Set(F1, F2) of (F1, F2)-coalgebras.

This paper is adjacent to [5] and motivated by the following question arising in universal algebraic geometry: when do two algebras have the same geometry? This question requires considering algebras in a variety Θ and the category Θ0 of all finitely generated free algebras in Θ. The notions of geometrically equivalent algebras and weakly geometrically equivalent algebras (cf. [2]) are intimately related to the above problem (the latter notion being related to automorphisms of the category Θ0 is replaced in the present paper by a more natural one: "automorphic equivalence"). Two algebras are said to have the same algebraic geometry if they are automorphically equivalent. If an automorphic equivalence of algebras is provided by an inner automorphism of Θ0, then this equivalence coincides with the geometric equivalence. However, not all automorphisms of Θ0 are inner (see [5, Theorem 2]). In this paper we study the situation when the automorphic equivalence of algebras is provided by a non-inner automorphism. Assuming Θ to be an IBM variety of one-sorted algebras we prove that two algebras H1 and H2 are automorphically equivalent in Θ if and only if H1 is geometrically equivalent to [Formula: see text], where [Formula: see text] is the algebra with the same domain as H2 and with the new operations induced by some system of words W = {wω(x)|ω ∈ Ω, Ω is the signature of algebras in Θ. Thus, the notion of automorphic equivalence is reduced to a more simple one of geometric equivalence.

This paper considers extended finite automata over monoids, in the sense of Dassow and Mitrana. We show that the family of languages accepted by extended finite automata over a monoid K is controlled by the word problem of K in a precisely stated manner. We also point out a critical error in the proof of the main result in the paper by Dassow and Mitrana. However as one consequence of our approach, by analyzing a certain word problem, we obtain a complete proof of this result, namely that the family of languages accepted by extended finite automata over the free group of rank two is exactly the family of context-free languages. We further deduce that along with the free group of rank two, the only finitely generated groups with this property are precisely the groups that have a nonabelian free subgroup of finite index.

This paper studies certain relations among vertex algebras, vertex Lie algebras and vertex Poisson algebras. In this paper, the notions of vertex Lie algebra (conformal algebra) and vertex Poisson algebra are revisited and certain general construction theorems of vertex Poisson algebras are given. A notion of filtered vertex algebra is formulated in terms of a notion of good filtration and it is proved that the associated graded vector space of a filtered vertex algebra is naturally a vertex Poisson algebra. For any vertex algebra V, a general construction and a classification of good filtrations are given. To each ℕ-graded vertex algebra V=∐n∈ℕV(n) with [Formula: see text], a canonical (good) filtration is associated and certain results about generating subspaces of certain types of V are also obtained. Furthermore, a notion of formal deformation of a vertex (Poisson) algebra is formulated and a formal deformation of vertex Poisson algebras associated with vertex Lie algebras is constructed.

As Lie algebra, we add the center c1 (and the outer derivation d1) to the quantum torus [Formula: see text] to give the extended torus Lie algebra [Formula: see text] (and [Formula: see text] respectively). Before the present paper, only some level 1 vertex operator representations for some [Formula: see text] (and [Formula: see text]) were constructed. In this paper, we first give vertex operator representations for [Formula: see text] where I is an arbitrary index set. By embedding some [Formula: see text] into [Formula: see text], we obtain a series of higher level vertex operator representations for [Formula: see text] and [Formula: see text]. Most of these vertex operator representations yield irreducible highest weight modules over these [Formula: see text]. Also their character formulas follow directly.

In this paper we prove certain Hurwitz equivalence properties of Bn. In particular we prove that for n = 3 every two Artin factorizations of [Formula: see text] of the form Hi1 ⋯ Hi6, Fj1 ⋯ Fj6 (with ik, jk ∈ {1, 2}) where {H1, H2}, {F1, F2} are frames, are Hurwitz equivalent. The proof provided here is geometric, based on a newly defined frame type. The results will be applied to the classification of algebraic surfaces up to deformation. It is already known that there exist surfaces that are diffeomorphic to each other but are not deformations of each other (Manetti's example). We construct a new invariant based on Hurwitz equivalence of factorizations, to distinguish among diffeomorphic surfaces which are not deformation of each other. The main result of this paper will help us to compute the new invariant.

This paper is to establish a theory of regular representations for vertex operator algebras. In the paper, for a vertex operator algebra V and a V-module W, we construct, out of the dual space W*, a family of canonical weak V ⊗ V-modules [Formula: see text] with a nonzero complex number z as the parameter. We prove that for V-modules W, W1 and W2, a P(z)-intertwining map of type [Formula: see text] in the sense of Huang and Lepowsky exactly amounts to a V ⊗ V-homomorphism from W1 ⊗ W2 into [Formula: see text]. Combining this with Huang and Lepowsky's one-to-one linear correspondence between the space of intertwining operators and the space of P(z)-intertwining maps of the same type we obtain a canonical linear isomorphism fromthe space [Formula: see text] of intertwining operators of the indicated type to [Formula: see text]. Denote by RP(z)(W) the sum of all (ordinary) V ⊗ V-submodules of [Formula: see text]. Assuming that V satisfies certain suitable conditions, we obtain a canonical decomposition of RP(z)(W) into irreducible V ⊗ V-modules. In particular, we obtain a decomposition of Peter–Weyl type for RP(z)(V). Denote by ℱP(z) the functor from the category of V-modules to the category of weak V ⊗ V-modules such that ℱP(z)(W)=RP(z)(W'). We prove that for V-modules W1, W2, a P(z)-tensor product of W1 and W2 in the sense of Huang and Lepowsky exactly amounts to a universal from W1 ⊗ W2 to the functor ℱP(z). This implies that the functor ℱP(z) is essentially a right adjoint of the Huang–Lepowsky's P(z)-tensor product functor. It is also proved that RP(z)(W) for [Formula: see text] are canonically isomorphic V ⊗ V-modules.

A commutative residuated lattice, is an ordered algebraic structure [Formula: see text], where (L, ·, e) is a commutative monoid, (L, ∧, ∨) is a lattice, and the operation → satisfies the equivalences [Formula: see text] for a, b, c ∊ L. The class of all commutative residuated lattices, denoted by [Formula: see text], is a finitely based variety of algebras. Historically speaking, our study draws primary inspiration from the work of M. Ward and R. P. Dilworth appearing in a series of important papers [9, 10, 19–22]. In the ensuing decades special examples of commutative, residuated lattices have received considerable attention, but we believe that this is the first time that a comprehensive theory on the structure of residuated lattices has been presented from the viewpoint of universal algebra. In particular, we show that [Formula: see text] is an "ideal variety" in the sense that its congruences correspond to order-convex subalgebras. As a consequence of the general theory, we present an equational basis for the subvariety [Formula: see text] generated by all commutative, residuated chains. We conclude the paper by proving that the congruence lattice of each member of [Formula: see text] is an algebraic, distributive lattice whose meet-prime elements form a root-system (dual tree). This result, together with the main results in [12, 18], will be used in a future publication to analyze the structure of finite members of [Formula: see text]. A comprehensive study of, not necessarily commutative, residuated lattices is presented in [4].

This paper generalizes the geometric part of the Esnault–Viehweg paper on Dyson's Lemma for a product of projective lines. Using the method of weak positivity from algebraic geometry, we are able to study products of smooth projective varieties of arbitrary dimension and to prove a geometric analogue of Dyson's Lemma for this case. Our main result is in fact a quantitative version of Faltings' product theorem.

Suppose [Formula: see text] is a finite field extension of [Formula: see text] containing a primitive [Formula: see text]th root of unity. Let [Formula: see text] be a maximal [Formula: see text]-extension of [Formula: see text] with the Galois group of period [Formula: see text] and nilpotent class [Formula: see text]. In this paper, we develop formalism which allows us to study the structure of [Formula: see text] via methods of Lie theory. In particular, we introduce an explicit construction of a Lie [Formula: see text]-algebra [Formula: see text] and an identification [Formula: see text], where [Formula: see text] is a [Formula: see text]-group obtained from the elements of [Formula: see text] via the Campbell–Hausdorff composition law. In the next paper, we apply this formalism to describe the ramification filtration [Formula: see text] and an explicit form of the Demushkin relation for [Formula: see text].

The question of a non-[Formula: see text]-type [Formula: see text] sequence wherein the fourth term shares the property [Formula: see text] with the first term has not been investigated so far. The present paper seeks to fill up the gap in this unexplored area. Let [Formula: see text] denote the set of all natural numbers and [Formula: see text] the sequence of Fibonacci numbers. Choose two integers [Formula: see text] and [Formula: see text] with [Formula: see text] such that their product increased by [Formula: see text] is a square [Formula: see text]. Certain properties of the sequence [Formula: see text] defined by the relation [Formula: see text] are established in this paper and polynomial expressions for Diophantine quadruples from the [Formula: see text] sequence [Formula: see text] are derived. The concept of a near-Diophantine quintuple is introduced and it is proved that there exist an infinite number of near-Diophantine quintuples.

The "similarity degree" of a unital operator algebra A was defined and studied in two recent papers of ours, where in particular we showed that it coincides with the "length" of an operator algebra. This paper brings several complements: we give direct proofs (with slight improvements) of several known facts on the length which were only known via the degree, and we show that the length of a type II1 factor with property Γ is at most 5, improving on a previous bound (≤ 44) due to E. Christensen.

WOS: 000389311800021, In [Finite presentability of Bruck-Reilly extensions of groups, J. Algebra 242 (2001) 2030], Araujo and RuAtic studied finite generation and finite presentability of Bruck Reilly extension of a group. In this paper, we aim to generalize some results given in that paper to generalized Bruck Reilly s -extension of a group. In this way, we determine necessary and sufficent conditions for generalized Bruck Reilly s-extension of a group, G R. (G; beta, gamma; u), to be finitely generated and finitely presented. Let G be a group, (3,-y: be morphisms and u is an element of H-1 (H-1(*) and H-1 are the ?-G- and ?-t-classes, respectively, contains the identity element kr of T). We prove that GE R* (G; 3, 7; u) is finitely generated if and only if there exists a finite subset Xo C G such that G is generated by ((boolean OR(i >= 0) X-0,beta(i))U(Uj >= 0X0 gamma(j)). We also prove that C superset of R.* (C; beta, gamma; u) is finitely presented if and only if G is presented by (X; R), where X is a finite set and R = (U i >= 0 R-0 beta(i)) boolean OR (U-j >= 0 R0 gamma(j)) = {w(1)beta(i) = upsilon(1)beta(i) : i >= 0, w(1) = v(1) is an element of H-1* for some finite set of relations R0 subset of X* x X*.

In this paper we finished the classification of SL(2, C)×Gm almost homogeneous projective 3-folds which we started in [8, 9]. For some technical reasons we leave the proofs of Propositions 1.2 and 1.3 to [8] and that of Propositon 2.7 to [9]. But we include all other proofs, especially those of Propositions 2.1 and 2.2 in this paper and try to give an intuitive picture of the process. The proofs of Lemmas 2.1 and 2.2 as well as those of Sec. 3 (Theorems 3.1–3.4) are the main results of this paper. Most of this work was done in the fall of 1991. I thank those who had tried their best to understand my proofs, for their helpful comments and criticisms.

h(a) was studied in [2] from a combinatorial perspective. We found that it could be pictured as a generalization of the generating function for the Zeta Polynomial Z(P, n) of a finite poset (P,≥) [3; §3.11]. In this paper we continue our study of {h(a) | a ∈ S}. But this time we approach it from a more algebraic perspective. For example (i) What is the recurrence formula for {h1(a), h2(a), . . . } and how are these recurrance formulae (as a ∈ S varies) related? (See Proposition 2.4.) (ii) Is there a canonical collection {R0, R1, . . . , Rn−1} of rational functions that forms a basis of SpanQ{h(a) | a ∈ S}? (See Proposition 2.2.) (iii) What role does Q[S] (the monoid algebra) play in the discussion? (See Section 2.) We consider the example S = Mn(Fq) in detail. In a future paper we shall use the results of this paper to obtain explicit calculations for a large class of finite monoids.

In this paper, we continue to deal with singular extremal rays with a Gm x SL (2, C) action which we started in [9] (see also [6]). We shall classify all Gm x SL(2, C) actions in [10] and state our results in Sec. 5 of this paper (see also [6]). Our idea is that once we have a singular extremal ray we can obtain enough information on the whole manifold which eventually leads to a classification. In this paper we concentrate on the most difficult case (5b) in the Proposition 4. For this reason we leave the proofs of Theorem 1 and Lemma 3 to [10]. But we give the proof of Theorem 3 for intuition of the whole process. Section 4 is the core of this paper where we construct the House Models and solve the (5b) case of Proposition 4 (Theorem 6). Once again (other than the proof of the Theorem 1 in [9]) we use non-algebraic argument for Theorem 3, a method we used very often and successfully. We also find in Sec. 4 of this paper that minimal projective 3-fold house models can have an arbitrary large second Betti number. Their effective curve cones are fully understood. The last section contains a criterion for projectivity which is very useful in our situations, and I cannot find it in the literature (Theorem 8). Most of this work was finished in Fall 1991. The original work was written in one paper as in [6 Chap. 1], but we now separate it into [9, 10] and this paper to make it more readable.

The author proves for each of the operations # equalling *, °, **, □ or m, there exist pseudovarieties of finite semigroups [Formula: see text] and [Formula: see text] with decidable membership problems, such that [Formula: see text] has an undecidable membership problem. In addition, if [Formula: see text] denotes the pseudovariety of all finite aperiodic semigroups, [Formula: see text] denotes the pseudovariety of all finite groups, and [Formula: see text](E) denotes the pseudovariety of all finite aperiodic semigroups satisfying the finite number of equations E, then it is proved that there exists E such that [Formula: see text](E) has an undecidable membership problem. Note [Formula: see text] equals all semigroups of complexity ≤1. Section 6 is expanded into a joint paper with B. Steinberg, following this paper.

In this paper we compare the block product of C-varieties which we have defined in a previous paper to the two other constructions which were proposed on the one hand by D. Thérien and on the other hand by P. Jones and S. Pustejovsky. We prove that these three notions are equivalent. However, our definition is built on an explicit block product of two categories and then gives rise to a useful definition of the block product of two C-varieties which is needed for the theory of decompositions of monoids by means of wreath or block products.

Given any amalgam [Formula: see text] of inverse semigroups, we show how to construct an amalgam [Formula: see text] such that [Formula: see text] is embedded into [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text] and, for any [Formula: see text] and [Formula: see text] with [Formula: see text] in [Formula: see text], where [Formula: see text], there exists [Formula: see text] with [Formula: see text] in [Formula: see text]; that is, [Formula: see text] is a lower bounded subsemigroup of [Formula: see text] and [Formula: see text]. A recent paper by the author describes the Schützenberger automata of [Formula: see text], for an amalgam [Formula: see text] where [Formula: see text] is lower bounded in [Formula: see text] and [Formula: see text], giving conditions for [Formula: see text] to have decidable word problem. Thus we can study [Formula: see text] by considering [Formula: see text]. As an example, we generalize results by Cherubini, Jajcayová, Meakin, Piochi and Rodaro on amalgams of finite inverse semigroups.

Doty and Giaquinto constructed a certain set denoted by [Formula: see text] in this paper, and conjectured that the set [Formula: see text] forms a [Formula: see text]-basis for the integral [Formula: see text]-Schur algebra [Formula: see text], where [Formula: see text]. In this paper, we show that the set [Formula: see text] is a [Formula: see text]-basis of [Formula: see text]-Schur algebra [Formula: see text] over [Formula: see text]. The set [Formula: see text] is a truncated form of PBW basis for quantum [Formula: see text]. Moreover, we give an example to show that the set [Formula: see text] is not [Formula: see text]-basis of [Formula: see text] in general.

Many commutative semisimple Banach algebras B including B = C (X), X compact, and B = L1 (G), G locally compact, have the property that every homomorphism from B into C1[0, 1] is compact. In this paper we consider this property for uniform algebras. Several examples of homomorphisms from somewhat complicated algebras of analytic functions to C1[0, 1] are shown to be compact. This, together with the fact that every homomorphism from the disc algebra and from the algebra H∞ (∆), ∆ = unit disc, to C1[0, 1] is compact, led to the conjecture that perhaps every homomorphism from a uniform algebra into C1[0, 1] is compact. The main result to which we devote the second half of this paper, is to construct a compact Hausdorff space X, a uniformly closed subalgebra [Formula: see text] of C (X), and an arc ϕ: [0, 1] → X such that the transformation T defined by Tf = f ◦ ϕ is a (bounded) homomorphism of [Formula: see text] into C1[0, 1] which is not compact.

The main purpose of this paper is to establish a Galois correspondence for a given covariant group system, its associated C*-algebra and Hopf C*-algebra. On the way to this, we first study covariance group C*-algebras and their representations, and prove a result which is simpler but yet very similar to the C*-algebra case in the main body of the paper. We then show that there is a Galois correspondence between the lattice of normal subgroups of the given covariant group system and a corresponding lattice of certain invariant *-subalgebras of the covariant group C*-algebra; in particular, there is a natural Galois correspondence for the group C*-algebra. We further study this Galois correspondence for the Hopf C*-algebras associated with covariant group systems.

This paper has twofolds. The first is to prove that there are at most two meromorphic functions sharing a small function with multiplicities truncated by 2 and other three small functions regardless of multiplicities, where all zeros with multiplicities more than a certain number are not counted. This result is an improvement of the four-value theorems of Nevanlinna, Gundersen, Fujimito, Thai–Tan and others. The second purpose of this paper is to prove that there are at most three meromorphic functions sharing four small functions ignoring multiplicity, where all zeros with multiplicities more than a certain number are omitted.

Let $G$ be a finite simple graph and $J(G)$ denote its cover ideal in a polynomial ring over a field $\mathbb{K}$. In this paper, we show that all symbolic powers of cover ideals of certain vertex decomposable graphs have linear quotients. Using these results, we give various conditions on a subset $S$ of the vertices of $G$ so that all symbolic powers of vertex cover ideals of $G \cup W(S)$, obtained from $G$ by adding a whisker to each vertex in $S$, have linear quotients. For instance, if $S$ is a vertex cover of $G$, then all symbolic powers of $J(G \cup W(S))$ have linear quotients. Moreover, we compute the Castelnuovo-Mumford regularity of symbolic powers of certain cover ideals., Comment: 15 pages, 2 figures

In this paper, we establish coupled coincidence point results in probabilistic metric spaces with a partial order. The main theorem has several corollaries and is supported with an example. The corollaries are shown to be properly contained in the main theorem. By an application of a corollary we obtained a corresponding result in metric spaces which is also supported with an example. Some results in [Nonlinear Anal. 74 (2011) 6451–6458] and [Nonlinear Anal. 71 (2009) 1833–1843] are extended in this paper. The extensions are shown to be actual with the help of an example. We use Hadžić type t-norm in this paper. The methodology of the proofs in this work is a blending of analytic and order theoretic approaches.

The purpose of this paper is to give a relation between the notion of the commutativity degree of a finite group G (denoted by d(G)) and that of isoclinism between G and an extra special p-group, where p is the smallest prime number dividing |G|. Moreover, some improvements of the results on the relative commutativity degree and relative n th nilpotency degree of a subgroup of finite groups given in [A. Erfanian, R. Rezaei and P. Lescot, On the relative commutativity degree of a subgroup of a finite group, Comm. Algebra35 (2007) 4183–4197] are also stated in this paper.

This is the third part in a series of papers developing a twisted tensor product theory for nonlocal vertex algebras and its modules. In this paper we introduce and study twisted tensor product modules over Möbius twisted tensor product nonlocal vertex algebras. Among the main results, we find the isomorphic relation between the opposite Möbius twisted tensor product nonlocal vertex algebra and twisted tensor product of opposite Möbius nonlocal vertex algebras. And we also establish the isomorphism between two twisted tensor product contragredient modules. Furthermore, we study iterated twisted tensor product modules over iterated twisted tensor product nonlocal vertex algebras and find conditions for constructing an iterated twisted tensor product module of three factors.

The famous finitistic dimension conjecture says that every finite-dimensional 𝕂-algebra over a field 𝕂 should have finite finitistic dimension. This conjecture is equivalent to the following statement: If B is a subalgebra of a finite-dimensional 𝕂-algebra A such that the radical of B is a left ideal in A, and if A has finite finitistic dimension, then B has finite finitistic dimension. In the paper, we shall work with a more general setting of Artin algebras. Let B be a subalgebra of an Artin algebra A such that the radical of B is a left ideal in A. (1) If the category of all finitely generated (A, B)-projective A-modules is closed under taking A-syzygies, then fin.dim (B) ≤ fin.dim (A) + fin.dim (BA) + 3, where fin.dim (A) denotes the finitistic dimension of A, and where fin.dim (BA) stands for the supremum of the projective dimensions of those direct summands of BA that have finite projective dimension. (2) If the extension B ⊆ A is n-hereditary for a non-negative integer n, then gl.dim (A) ≤ gl.dim (B) + n. Moreover, we show that the finitistic dimension of the trivially twisted extension of two algebras of finite finitistic dimension is again finite. Also, a new formulation of the finitistic dimension conjecture in terms of relative homological dimension is given. Our approach in this paper is completely different from the one in our earlier papers.

In this paper the Lebesgue covering lemma is extended from the setting of metric spaces to the setting of admissible spaces. An admissible space is a topological space endowed with an admissible family of open coverings, and need not be metric. The paper contains applications to uniform continuity and dimension theory.

L systems generalise context-free grammars by incorporating parallel rewriting, and generate languages such as EDT0L and ET0L that are strictly contained in the class of indexed languages. In this paper we show that many of the languages naturally appearing in group theory, and that were known to be indexed or context-sensitive, are in fact ET0L and in many cases EDT0L. For instance, the language of primitives in the free group on two generators, the Bridson-Gilman normal forms for the fundamental groups of 3-manifolds or orbifolds, and the co-word problem of Grigorchuk's group can be generated by L systems. To complement the result on primitives in free groups, we show that the language of primitives, and primitive sets, in free groups of rank higher than two is context-sensitive. We also show the existence of EDT0L and ET0L languages of intermediate growth., Revised following referees suggestions. 21 pages, 2 figures

Dendrimers are highly branched monodisperse, macromolecules and are considered in nanotechnology with a variety of suitable applications. In this paper, the matching polynomial and some results of the matchings for three classes of nanostar dendrimers are obtained. Furthermore, we express the recursive formulas of the Hosoya index for these structures of dendrimers by their matching polynomials.

In this paper, we study the spectral property of the self-affine measure [Formula: see text] generated by an expanding real matrix [Formula: see text] and the four-element digit set [Formula: see text]. We show that [Formula: see text] is a spectral measure, i.e. there exists a discrete set [Formula: see text] such that the collection of exponential functions [Formula: see text] forms an orthonormal basis for [Formula: see text], if and only if [Formula: see text] for some [Formula: see text]. A similar characterization for Bernoulli convolution is provided by Dai [X.-R. Dai, When does a Bernoulli convolution admit a spectrum? Adv. Math. 231(3) (2012) 1681–1693], over which [Formula: see text]. Furthermore, we provide an equivalent characterization for the maximal bi-zero set of [Formula: see text] by extending the concept of tree-mapping in [X.-R. Dai, X.-G. He and C. K. Lai, Spectral property of Cantor measures with consecutive digits, Adv. Math. 242 (2013) 187–208]. We also extend these results to the more general self-affine measures.

In this paper, we propose Jensen–Shannon divergence (JSD) based on horizontal visibility graph (HVG) to measure the time series irreversibility for both stationary and non-stationary series efficiently. Numerical simulations are first conducted to show the validity of the proposed method and then empirical applications to the financial time series and traffic time series are investigated. It can be found that JSD shows better robustness than Kullback–Leibler divergence (KLD) on quantifying time series irreversibility and correctly distinguishes the different type of simulated series. For the empirical analysis, JSD based on HVG is able to detect the significant time irreversibility of stock indices and reveal the relationship between different stock indices. JSD results show the time irreversibility of speed time series for different detectors and present better accuracy and robustness than KLD. The hierarchical clustering based on their behavior of time irreversibility obtained by JSD classifies the detectors into four groups.

We study some properties of the generalized local cohomology modules [Formula: see text] of [Formula: see text] with respect to a pair of ideals [Formula: see text] in Serre subcategories. Some results concerning to [Formula: see text]-cofinite modules are also given in this paper.

We show that, given an equation over a finitely generated free group, the set of all solutions in reduced words forms an effectively constructible EDT0L language. In particular, the set of all solutions in reduced words is an indexed language in the sense of Aho. The language characterization we give, as well as further questions about the existence or finiteness of solutions, follow from our explicit construction of a finite directed graph which encodes all the solutions. Our result incorporates the recently invented recompression technique of Je\.z, and a new way to integrate solutions of linear Diophantine equations into the process. As a byproduct of our techniques, we improve the complexity from quadratic nondeterministic space in previous works to $\mathsf{NSPACE}(n\log n)$ here., Comment: 38 pages, 3 figures. A conference version of this paper was presented at ICALP 2015, Kyoto (Japan), July 4-10, 2015, see http://arxiv.org/abs/1502.03426

In this paper it is shown that a classical continuation principle due to Granas for contractions holds under weaker contractive assumptions. This leads to a Leray–Schauder principle for such contractions in hyperbolic spaces. Some applications to nonexpansive mappings in hyperbolic geodesic spaces are also discussed.

Let [Formula: see text] be a nondegenerate projective integral curve of degree [Formula: see text] which is not linearly normal. In this paper, we continues the study begun in [E. Park, Projective curves of degree=codimension+2, Math. Z. 256 (2007) 685–697] for the minimal free resolution of [Formula: see text]. It is well-known that [Formula: see text] is an isomorphic projection of a rational normal curve [Formula: see text] from a point [Formula: see text]. Our main result is about how the graded Betti numbers of [Formula: see text] are determined by the rank of [Formula: see text] with respect to [Formula: see text], which is a measure of the relative location of [Formula: see text] from [Formula: see text].

In this paper, we define a new family of Gauss [Formula: see text]-Lucas numbers. We give the relationships between the family of the Gauss [Formula: see text]-Lucas numbers and the known Gauss Lucas numbers. We also define the generalized polynomials for these numbers. We obtain some interesting properties of the polynomials. We also give the relationships between the generalized Gauss [Formula: see text]-Lucas polynomials and the known Gauss Lucas polynomials. Furthermore, we find new generalizations of these families and the polynomials in matrix representation. Then we prove Cassini’s identities for the families and their polynomials.

Given a positive integer [Formula: see text], in this paper, we investigate some more properties of the [Formula: see text]-capability of groups. For instance, the relationship between [Formula: see text]-capability and the varietal capability is determined. Moreover, we introduce the notion of [Formula: see text]-epicenter for a group and then we obtain some criteria for the [Formula: see text]-capability of groups. Finally, as an application, we characterize all [Formula: see text]-capable extra-special [Formula: see text]-groups when [Formula: see text] is a power of [Formula: see text]

In this paper, we present a fixed point theorem for multivalued mappings satisfying new inequalities without using the Pompeiu–Hausdorff distance. The results generalize the famous Feng and Liu’s result [Feng and Liu, J. Math. Anal. Appl. 317 (2006) 103–112] and some known results in the literature. Some examples are given to illustrate our results.

Given a set of points in the complex plane, an incomplete polynomial is defined as one which has these points as zeros except one of them. Recently, the classical result known as Gauss–Lucas theorem on the location of zeros of polynomials and their derivatives was extended to the linear combinations of incomplete polynomials. In this paper, a simple proof of this result is given, and some results concerning the critical points of polynomials due to Jensen and others have extended the linear combinations of incomplete polynomials.

In this paper, we study nonbinary quantum codes from cyclic codes over the ring [Formula: see text] for odd prime [Formula: see text]. We give the constructional properties of linear and cyclic codes over the ring [Formula: see text]. As an applications of these classes of codes, we obtain quantum codes over [Formula: see text] using self-orthogonal property. Moreover, we find some new nonbinary quantum codes with better parameters than the known quantum codes over [Formula: see text].

Let R be a ring, and let S be a strictly ordered monoid. The generalized power series ring R[[S]] is a common generalization of polynomial rings, Laurent polynomial rings, power series rings, Laurent series rings, Mal'cev–Neumann series rings, monoid rings and group rings. In this paper, we examine which conditions on R and S are necessary and which are sufficient for the generalized power series ring R[[S]] to be semilocal right Bézout or semilocal right distributive. In the case where S is a strictly totally ordered monoid we characterize generalized power series rings R[[S]] that are semilocal right distributive or semilocal right Bézout (the latter under the additional assumption that S is not a group).

In this paper, we record a proof that a relative moduli stack of principal G-bundles over a moduli stack of genus g, n-pointed twisted curves is a smooth Artin stack locally of finite type with the universal family. As an application, we reprove that the stack 𝒦g, n(𝒳, β) of Abramovich, Graber and Vistoli is a Deligne–Mumford stack of finite type when 𝒳 is a quotient stack.