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Some application of difference equations in Cryptography and Coding TheoryFeb 08 2018In this paper, we present some applications of a difference equation of degree k in Cryptography and Coding Theory.

Some remarks on the non-real roots of polynomialsFeb 08 2018Let $f \in { \mathbb R} ( t) [x]$ be given by $ f(t, x) = x^n + t \cdot g(x) $ and $\beta_1 < \dots < \beta_m$ the distinct real roots of the discriminant $\Delta_{(f, x)} (t)$ of $f(t, x)$ with respect to $x$. Let $\gamma$ be the number of real roots ... More

Normal elements of completed group algebras over ${\rm SL}_3(\mathbb{Z}_p) $Feb 08 2018Let $p$ be a prime integer and $\mathbb{Z}_p$ be the ring of $p$-adic integers. By a purely computational approach we prove that each nonzero normal element of a completed group algebra over the special linear group ${\rm SL}_3(\mathbb{Z}_p)$ is a unit. ... More

Cyclicity in the Brauer group of a $p$-adic curveFeb 07 2018Let $X$ be a smooth projective curve over a finite extension $K$ of $\mathbb{Q}_p$. If $X$ has good reduction we show that every element in the Brauer group $\operatorname{Br}(X)$ of $X$ whose order is not divisible by $p$ is cyclic. In case $X$ is a ... More

Orthogonal abelian Cartan subalgebra decomposition of $\mathfrak{sl}_n$ over a finite commutative ringFeb 07 2018Orthogonal decomposition of the special linear Lie algebra over the complex numbers was studied in the early 1980s and attracted further attentions in the past decade due to its application in quantum information theory. In this paper, we study this decomposition ... More

Resolutions of DG-modules and their applications to commutative DG-algebrasFeb 06 2018A DG-version of projective resolution and injective resolution of DG-modules over DG-algebra are already known. In the first half of this paper, we introduce another DG-version for DG-modules over a connective DG-algebra and show that they behave nicely ... More

Atiyah classes of Lie bialgebrasFeb 06 2018The Atiyah class was originally introduced by M.F. Atiyah. It has many developments in recent years. One important case is the Atiyah classes of Lie algebra pairs. In this paper, we study the Atiyah class of the Lie algebra pair associated with a Lie ... More

On quiver Grassmannians and orbit closures for gen-finite modulesFeb 06 2018We show that endomorphism rings of cogenerators in the module category of a finite-dimensional algebra A admit a canonical tilting module. The tilted algebra B is related to A by a recollement. We call an A-module M gen-finite if there are only finitely ... More

Reconstruction of tensor categories from their structure invariantsFeb 03 2018In this paper, we study tensor (or monoidal) categories of finite rank over an algebraically closed field $\mathbb F$. Given a tensor category $\mathcal{C}$, we have two structure invariants of $\mathcal{C}$: the Green ring (or the representation ring) ... More

The finiteness of the genus of a finite-dimensional division algebra, and some generalizationsFeb 01 2018We prove that the genus of a finite-dimensional division algebra is finite whenever the center is a finitely generated field of any characteristic. We also discuss potential applications of our method to other problems, including the finiteness of the ... More

Total perfect codes in zero-divisor graphsFeb 01 2018Let R be a commutative ring with unity not equal to zero and let G = (V, E) be a simple, undirected graph. A total perfect code denoted by C(G), in G is a subset C(G) of V (G) such that cardinality of the set {N (v) \cap C(G)} is 1, for all v \in V (G), ... More

Solvable Leibniz algebras whose nilradical is a quasi-filiform Leibniz algebra of maximum lengthJan 26 2018We describe solvable Leibniz algebras whose nilradical is a quasi-filiform Leibniz algebra of maximum length.

Solvable Leibniz algebras with quasi-filiform Lie algebras of maximum length nilradicalsJan 26 2018In this paper solvable Leibniz algebras whose nilradical is quasi-filiform Lie algebra of maximum length, are classified. The rigidity of such Leibniz algebras with two-dimensional complemented space to nilradical is proved.

Antisymetric matrices and Lotka-Volterra algebrasJan 26 2018The purpose of this paper is to study the structure of Lotka-Volterra algebras, the set of their idempotent elements and their group of automorphisms. These algebras are defined through antisymmetric matrices and they emerge in connection with biological ... More

Color Lie rings and PBW deformations of skew group algebrasJan 26 2018We investigate color Lie rings over finite group algebras and their universal enveloping algebras. We exhibit these universal enveloping algebras as PBW deformations of skew group algebras: Every color Lie ring over a finite group algebra with a particular ... More

On a New Ring Invariant - Towards a Framework for Rank IdentitiesJan 24 2018For a square matrix $A$ over a field $F$, it is known that $\mathrm{rank}(A) + \mathrm{rank}(I-A) = \mathrm{rank}(I) + \mathrm{rank}(A-A^2)$. The main goal of this article is to start a program to characterize and generalize such rank identities by constructing ... More

Algebras with finitely many conjugacy classes of left ideals versus algebras of finite representation typeJan 24 2018Let A be a finite dimensional algebra over an algebraically closed field with the radical nilpotent of index 2. It is shown that A has finitely many conjugacy classes of left ideals if and only if A is of finite representation type provided that all simple ... More

Relatively free associative algebras of ranks 2 and 3 with Lie nilpotency identity and systems of generators of some T-spacesJan 23 2018We study relatively free associative algebras $F_r^{\left(n\right)}$ of ranks $r=2,3$ with the identity $\left[x_1,\dots,x_n\right]=0$ of Lie nilpotency of step $n\ge3$ over a field $K$ of characteristic $\ne2,3$. First we prove the Theorem about the ... More

Connections between rank and dimension for subspaces of bilinear formsJan 23 2018Let $K$ be a field and let $V$ be a vector space of dimension $n$ over $K$. Let $M$ be a subspace of bilinear forms defined on $V\times V$. Let $r$ be the number of different non-zero ranks that occur among the elements of $M$. Our aim is to obtain an ... More

Counting subrings of the ring $\Bbb{Z}_m \times \Bbb{Z}_n$Jan 22 2018Let $m,n\in \Bbb{N}$. We represent the additive subgroups of the ring $\Bbb{Z}_m \times \Bbb{Z}_n$, which are also (unital) subrings, and deduce explicit formulas for $N^{(s)}(m,n)$ and $N^{(us)}(m,n)$, denoting the number of subrings of the ring $\Bbb{Z}_m ... More

Rota---Baxter Operators on Quadratic AlgebrasJan 22 2018We prove that all Rota---Baxter operators on a quadratic division algebra are trivial. For nonzero weight, we state that all Rota---Baxter operators on the simple odd-dimensional Jordan algebra of bilinear form are projections on a subalgebra along another ... More

On an extremal property of Jordan algebras of Clifford typeJan 17 2018If $V$ is a finite-dimensional unital commutative (maybe nonassociative) algebra carrying an associative positive definite bilinear form then there exist a nonzero idempotent $c\ne e$ ($e$ being the algebra unit) of the shortest possible length $|c|^2$. ... More

Modular group algebras whose group of unitary units is locally nilpotentJan 16 2018We characterize those modular group algebras FG whose group of unitary units is locally nilpotent under the classical involution of FG.

Group algebra whose unit group is locally nilpotentJan 16 2018We present a complete list of groups $G$ and fields $F$ for which: (i) the group of normalized units V(FG) of the group algebra FG is locally nilpotent; (ii) the group algebra FG has a finite number of nilpotent elements and V(FG) is an Engel group.

A characterisation of $τ$-tilting finite algebrasJan 12 2018We prove that a finite dimensional algebra is $\tau$-tilting finite if and only if it does not admit large silting modules. Moreover, we show that for a $\tau$-tilting finite algebra $A$ there is a bijection between isomorphism classes of basic support ... More

$n$-Ary generalized Lie-type color algebras admitting a quasi-multiplicative basisJan 06 2018The class of generalized Lie-type color algebras contains the ones of generalized Lie-type algebras, of $n$-Lie algebras and superalgebras, commutative Leibniz $n$-ary algebras and superalgebras, among others. We focus on the class of generalized Lie-type ... More

Completely simple endomorphism rings of modulesDec 26 2017It is proved that if A_p is a countable elementary abelian p-group, then: (i) The ring End(A_p) does not admit a nondiscrete locally compact ring topology. (ii) Under (CH) the simple ring End(A_p)/I, where I is the ideal of End(A_p) consisting of all ... More

On filiform Lie algebras. Geometric and algebraic studiesDec 01 2017A finite dimensional filiform K-Lie algebra is a nilpotent Lie algebra g whose nil index is maximal, that is equal to dim g -1. We describe necessary and sufficient conditions for a filiform algebra over an algebraically closed field of characteristic ... More

Azumaya skew group algebras and an application to quantum Kleinian singularitiesNov 23 2017Nov 27 2017We provide easily-verified necessary and sufficient conditions for a skew group ring, or more generally, a crossed product ring, to be an Azumaya algebra. We use our results to show that (suitable localisations of) skew group rings associated to the quantum ... More

Strongly graded groupoids and strongly graded Steinberg algebrasNov 14 2017We study strongly group-graded groupoids $G$, which are topological groupoids equipped with a continuous functor $\kappa: G \to \Gamma$ to a discrete group $\Gamma$, such that $\kappa^{-1}(\gamma)\kappa^{-1}(\delta) = \kappa^{-1}(\gamma \delta)$, for ... More

On Levi-Malcev theorem for Leibniz algebrasOct 30 2017The present paper is devoted to provide conditions for the Levi--Malcev theorem to hold or not to hold (i.e. for two Levi subalgebras to be or not conjugate by an inner automorphism) in the context of finite-dimensional Leibniz algebras over a field of ... More

A Counterexample to the First Zassenhaus ConjectureOct 24 2017Nov 20 2017Hans J. Zassenhaus conjectured that for any unit $u$ of finite order in the integral group ring of a finite group $G$ there exists a unit $a$ in the rational group algebra of $G$ such that $a^{-1}\cdot u \cdot a=\pm g$ for some $g\in G$. We disprove this ... More

Maximal ideals in module categories and applicationsOct 19 2017We study the existence of maximal ideals in preadditive categories defining an order $\preceq$ between objects, in such a way that if there do not exist maximal objects with respect to $\preceq$, then there is no maximal ideal in the category. In our ... More

Acyclic cluster algebras, reflection groups, and curves on a punctured discSep 29 2017Dec 05 2017We establish a bijective correspondence between certain non-self-intersecting curves in an $n$-punctured disc and positive ${\mathbf c}$-vectors of acyclic cluster algebras whose quivers have multiple arrows between every pair of vertices. As a corollary, ... More

Locally nilpotent Lie algebras of derivations of integral domainsSep 26 2017Let $\mathbb K$ be a field of characteristic zero and $A$ an integral domain over $\mathbb K.$ The Lie algebra $\Der_{\mathbb K} A$ of all $\mathbb K$-derivations of $A$ carries very important information about the algebra $A.$ This Lie algebra is embedded ... More

On the variety of 1-dimensional representations of finite $W$-algebras in low rankAug 29 2017Let $\mathfrak g$ be a simple Lie algebra over $\mathbb C$ and let $e \in \mathfrak g$ be nilpotent. We consider the finite $W$-algebra $U(\mathfrak g,e)$ associated to $e$ and the problem of determining the variety $\mathcal E(\mathfrak g,e)$ of 1-dimensional ... More

Software for doing computations in graded Lie algebrasAug 19 2017We introduce the Macaulay2 package GradedLieAlgebras for doing computations in graded Lie algebras presented by generators and relations.

Derivations of Group AlgebrasAug 16 2017In the paper, a method of describing the outer derivations of the group algebra of a finitely presentable group is given. The description of derivations is given in terms of characters of the groupoid of the adjoint action of the group.

Nilpotence order growth of recursion operators in characteristic pJul 31 2017We prove that the killing rate of certain degree-lowering "recursion operators" on a polynomial algebra over a finite field grows slower than linearly in the degree of the polynomial attacked. We also explain the motivating application: obtaining a lower ... More

Derivatives of triangular, Toeplitz, circulant matrices and matrices of other forms over semiringsJul 15 2017In this article we construct examples of derivations in matrix semirings. We study hereditary and inner derivations, derivatives of diagonal, triangular, Toeplitz, circulant matrices and of matrices of other forms and prove theorems for derivatives of ... More

Hochschild Cohomology and the Modular GroupJul 13 2017It has been shown in previous work that the modular group acts projectively on the center of a factorizable ribbon Hopf algebra. The center is the zeroth Hochschild cohomology group. In this article, we extend this projective action of the modular group ... More

Quasi-ordered RingsJun 14 2017Jul 27 2017A quasi-order is a binary, reflexive and transitive relation. In the Journal of Pure and Applied Algebra 45 (1987), S.M. Fakhruddin introduced the notion of (totally) quasi-ordered fields and showed that each such field is either an ordered field or else ... More

Globalization of partial cohomology of groupsJun 08 2017We study the relations between partial and global group cohomology. We show, in particular, that given a unital partial action of a group $G$ on a ring $\mathcal A$, such that $\mathcal A$ is a direct product of indecomposable rings, then any partial ... More

Projections of k-simplex onto the subsimplices of arbitrary type are derivationsMay 31 2017The aim of this paper is to prove that there is a projection of an arbitrary k-simplex onto (m - l)-subsimplex, where 1 < l < m < k - 1, which is a derivation.

Modular finite $W$-algebrasMay 17 2017Nov 03 2017Let $k$ be an algebraically closed field of characteristic $p > 0$ and let $G$ be a connected reductive algebraic group over $k$. Under some standard hypothesis on $G$, we give a direct approach to the finite $W$-algebra $U(\mathfrak g,e)$ associated ... More

Algebraic computation of genetic patterns related to three-dimensional evolution algebrasMay 11 2017The mitosis process of an eukaryotic cell can be represented by the structure constants of an evolution algebra. Any isotopism of the latter corresponds to a mutation of genotypes of the former. This paper uses Computational Algebraic Geometry to determine ... More

Periodic modules and acyclic complexesApr 21 2017We study the behaviour of modules $M$ that fit into a short exact sequence $0\to M\to C\to M\to 0$, where $C$ belongs to a class of modules $\mathcal C$, the so-called $\mathcal C$-periodic modules. We find a rather general framework to improve and generalize ... More

On monoids in the category of sets and relationsMar 10 2017The category $\mathbf{Rel}$ is the category of sets (objects) and relations (morphisms). Equipped with the direct product of sets, $\mathbf{Rel}$ is a monoidal category. Moreover, $\mathbf{Rel}$ is a locally posetal 2-category, since every homset $\mathbf{Rel}(A,B)$ ... More

Eulerian idempotent, pre-Lie logarithm and combinatorics of treesFeb 28 2017The aim of this paper is to bring together the three objects in the title. Recall that, given a Lie algebra $\mathfrak{g}$, the Eulerian idempotent is a canonical projection from the enveloping algebra $U(\mathfrak{g})$ to $\mathfrak{g}$. The Baker-Campbell-Hausdorff ... More

Some remarks on protolocalizations and protoadditive reflectionsFeb 28 2017Oct 29 2017We investigate additional properties of protolocalizations, introduced and studied by F. Borceux, M. M. Clementino, M. Gran, and L. Sousa, and of protoadditive reflections, introduced and studied by T. Everaert and M. Gran. Among other things we show ... More

On right $S$-Noetherian rings and $S$-Noetherian modulesJan 25 2017Aug 11 2017In this paper we study right $S$-Noetherian rings and modules, extending of notions introduced by Anderson and Dumitrescu in commutative algebra to noncommutative rings. Two characterizations of right $S$-Noetherian rings are given in terms of completely ... More

Symmetry of the Definition of Degeneration in Triangulated CategoriesDec 17 2016Dec 22 2016Module structures of an algebra on a fixed finite dimensional vector space form an algebraic variety. Isomorphism classes correspond to orbits of the action of an algebraic group on this variety and a module is a degeneration of another if it belongs ... More

Arithmetic of commutative semigroups with a focus on semigroups of ideals and modulesDec 09 2016Mar 04 2017Let $H$ be a commutative semigroup with unit element such that every non-unit can be written as a finite product of irreducible elements (atoms). For every $k \in \mathbb N$, let $\mathscr U_k (H)$ denote the set of all $\ell \in \mathbb N$ with the property ... More

The trace of the canonical moduleDec 08 2016The trace of the canonical module (the canonical trace) determines the non-Gorenstein locus of a local Cohen-Macaulay ring. We call a local Cohen-Macaulay ring nearly Gorenstein, if its canonical trace contains the maximal ideal. Similar definitions can ... More

I-primary submodulesDec 07 2016In this paper, we give a generalization for weakly primary submodules called $I$-primary submodule and we study some properties of it. We give some characterizations of $I$-primary submodules. Also we establish the situation of $I$-primary submodules ... More

Isometries of virtual quadratic spacesDec 07 2016In this article, we introduce a new object, a virtual quadratic space, and its group of isometries. They are presented as natural generalizations of quadratic spaces and orthogonal groups. Then we show that certain distinctions we must make over finite ... More

Comparison morphisms between projective resolutions of monomial algebrasDec 07 2016We construct comparison morphisms between two well-known projective resolutions of a monomial algebra $A$: the bar resolution and Bardzell's resolution; the first one is used to define the cup product and the Lie bracket on the Hochschild cohomology $HH^*(A)$ ... More

Finite presentation is a Morita invariant propertyDec 07 2016We prove that the property of an algebra to be finitely presented is Morita invariant.

Bounded linear endomorphisms of rigid analytic functionsDec 06 2016Let $K$ be a field of characteristic zero complete with respect to a non-trivial, non-Archimedean valuation. We relate the sheaf $\widehat{\mathcal{D}}$ of infinite order differential operators on smooth rigid $K$-analytic spaces to the algebra $\mathcal{E}$ ... More

Unitification of Weakly p.q.-Baer *-RingsDec 06 2016In this paper, we introduce a concept of weakly principally quasi-Baer *-rings in terms of central cover. We prove that a *-rings is a principally quasi-Baer *-rings if and only if it is weakly principally quasi-Baer *-rings with unity. A partial solution ... More

Derived Recollements and Generalised AR FormulasDec 06 2016The Defect Recollement, Evaluation Recollement, Restriction Recollement, and Auslander-Gruson-Jensen-Recollement are shown to be instances of a general construction using derived functors and methods from stable module theory. The right derived functors ... More

Derived Recollements and Generalised AR FormulasDec 06 2016Dec 07 2016The Defect Recollement, Evaluation Recollement, Restriction Recollement, and Auslander-Gruson-Jensen-Recollement are shown to be instances of a general construction using derived functors and methods from stable module theory. The right derived functors ... More

Preprojective algebras of tree-type quiversDec 05 2016Let $Q$ be a tree-type quiver, $\mathbf{k}Q$ be its path algebra, and $\lambda$ be a nonzero element in a base field $\mathbf{k}$. We construct irreducible morphisms in the Auslander-Reiten quiver of the bounded derived category of $\mathbf{k}Q$ that ... More

Approximations and Mittag-Leffler conditions --- the applicationsDec 04 2016A classic result by Bass says that the class of all projective modules is covering, if and only if it is closed under direct limits. Enochs extended the if-part by showing that every class of modules $\mathcal C$, which is precovering and closed under ... More

Approximations and Mittag-Leffler conditions --- the toolsDec 04 2016Mittag-Leffler modules occur naturally in algebra, algebraic geometry, and model theory, [18], [12], [17]. If $R$ is a non-right perfect ring, then it is known that in contrast with the classes of all projective and flat modules, the class of all flat ... More

Kaplansky's zero divisor and unit conjectures on elements with supports of size $3$Dec 03 2016Kaplansky's zero divisor conjecture (unit conjecture, respectively) states that for a torsion-free group $G$ and a field $\mathbb{F}$, the group ring $\mathbb{F}[G]$ has no zero divisors (has no unit with support of size greater than $1$). In this paper, ... More

Noetherian algebras of quantum differential operatorsDec 02 2016We consider algebras of quantum differential operators, for appropriate bicharacters on a polynomial algebra in one indeterminate and for the coordinate algebra of quantum $n$-space for $n\geq 3$. In the former case a set of generators for the quantum ... More

Irreducible $W_n^+$-modules from Weyl modules and $\mathfrak{gl}_{n}$-modulesDec 01 2016For an irreducible module $P$ over the Weyl algebra $\mathcal{K}_n^+$ and an irreducible module $M$ over the general liner Lie algebra $\mathfrak{gl}_n$, using Shen's monomorphism, we make $P\otimes M$ into a module over the Witt algebra $W_n^+$. We obtain ... More

A theory of pictures for quasi-posetsDec 01 2016The theory of pictures between posets is known to encode much of the combinatorics of symmetric group representations and related topics such as Young diagrams and tableaux. Many reasons, com-binatorial (e.g. since semi-standard tableaux can be viewed ... More

Disjoint-union partial algebrasDec 01 2016Disjoint union is a partial binary operation returning the union of two sets if they are disjoint and undefined otherwise. A disjoint-union partial algebra of sets is a collection of sets closed under disjoint unions, whenever they are defined. We provide ... More

Comments on Exchange Graphs in Cluster AlgebrasDec 01 2016An important problem in the theory of cluster algebras is to compute the fundamental group of the exchange graph. A non-trivial closed loop in the exchange graph, for example, generates a non-trivial identity for the classical and quantum dilogarithm ... More

D-groups and the Dixmier-Moeglin equivalenceNov 30 2016A differential-algebraic geometric analogue of the Dixmier-Moeglin equivalence is articulated, and proven to hold for $D$-groups over the constants. The model theory of differentially closed fields of characteristic zero, in particular the notion of analysability ... More

The quaternion over the ring of Colombeau's full generalized numbersNov 30 2016In this paper, we extend the results obtained in \cite{CFJ} for the quaternion over the ring Colombeau's simplified generalized numbers, denoted by $\overline{\mathbb{H}}_s$, to the quaternion over the ring of Colombeau's full generalized numbers, denoted ... More

The quaternion over the ring of Colombeau's full generalized numbersNov 30 2016Dec 06 2016In this paper, we extend the results obtained by Cortes-Ferrero-Juriaans (2009) for the quaternion over the ring Colombeau's simplified generalized numbers, denoted by $\overline{\mathbb{H}}_s$, to the quaternion over the ring of Colombeau's full generalized ... More

Horadam OctonionsNov 30 2016In this paper, first we define Horadam octonions by Horadam sequence which is a generalization of second order recurrence relations. Also, we give some fundamental properties involving the elements of that sequence. Then, we obtain their Binet-like formula, ... More

The Ziegler spectrum of the D-infinity plane singularityNov 30 2016We will describe the Cohen--Macaulay part of the Ziegler spectrum of the D-infinity plane singularity S and calculate the nilpotency index of the radical of the category of finitely generated Cohen--Macaulay S-modules.

On k-sparse numerical semigroupsNov 29 2016Given a positive integer k, we investigate the class of numerical semigroups verifying the property that every two subsequent non gaps, smaller than the conductor, are spaced by at least k. These semigroups will be called k-sparse and generalize the concept ... More

Diagonal-preserving graded isomorphisms of Steinberg algebrasNov 29 2016We study Steinberg algebras constructed from ample Hausdorff groupoids over commutative integral domains with identity. We reconstruct graded groupoids from graded Steinberg algebras and use this to characterise when there is a diagonal-preserving graded ... More

Homogeneous BandsNov 29 2016A countable band $B$ is called homogeneous if every isomorphism between finitely generated subbands extends to an automorphism of $B$. In this paper we give a complete classification of all the homogeneous bands. We prove that a homogeneous band belongs ... More

Prime spectra of ambiskew polynomial ringsNov 29 2016We determine criteria for the prime spectrum of an ambiskew polynomial algebra $R$ over an algebraically closed field $K$ to be akin to those of two of the principal examples of such an algebra, namely the universal enveloping algebra $U(sl_2)$ (in characteristic ... More

Connected quantized Weyl algebras and quantum cluster algebrasNov 29 2016For an algebraically closed field $K$, we investigate a class of noncommutative $K$-algebras called connected quantized Weyl algebras. Such an algebra has a PBW basis for a set of generators $\{x_1,\dots,x_n\}$ such that each pair satisfies a relation ... More

Nonnegative biquadratic forms with maximal number of zerosNov 29 2016We show that the maximum number of real zeros in $\mathbb{P}^2\times \mathbb{P}^2$ of a nonnegative biquadratic form that is not a sum of squares is 10. We obtain this result by constructing the first known examples of nonnegative biquadratic forms that ... More

Promiscuously Quadratic RingsNov 28 2016We register, explicitly, equivalences and dual equivalences between categories of abstract quadratic forms theories and (sub)categories of multirings and multifields.

Buchberger-Weispfenning Theory for Effective Associative RingsNov 27 2016We present here a new approach for computing Gr\"obner bases for bilateral modules over an effective ring. Our method is based on Weispfenning notion of restricted Gr\"obner bases and related multiplication.

Conrad's Partial Order on p.q.-Baer *-RingsNov 27 2016We prove that p.q.-Baer *-ring forms a pseudo lattice with Conrads partial order and also characterize p.q.-Baer *-rings which are lattices. The initial segments of a p.q.-Baer *-ring with Conrads partial order are shown to be orthomodular posets.

On just-infinite periodic locally soluble groupsNov 26 2016We construct an uncountable family of periodic locally soluble groups which are hereditary just-infinite. We also show that C*(G) is just-infinite for many groups G in this family.

Topological classification of systems of bilinear and sesquilinear formsNov 26 2016Let $\cal A$ and $\cal B$ be two systems consisting of the same vector spaces $\mathbb C^{n_1},\dots,\mathbb C^{n_t}$ and bilinear or sesquilinear forms $A_i,B_i:\mathbb C^{n_{k(i)}}\times\mathbb C^{n_{l(i)}}\to\mathbb C$, for $i=1,\dots,s$. We prove ... More

Some conjectures on generalized cluster algebras via the cluster formula and $D$-matrix patternNov 26 2016In the theory of generalized cluster algebras, we build the so-called cluster formula and $D$-matrix pattern. Then as applications, some fundamental conjectures of generalized cluster algebras are solved affirmatively.

Species with potential arising from surfaces with orbifold points of order 2, Part II: arbitrary weightsNov 24 2016Let (\Sigma,M,O) be a surface with marked points and order-2 orbifold points which is either unpunctured or once-punctured closed, and let \omega be a function from O to {1,4}. For each triangulation \tau of (\Sigma,M,O) we construct a cochain complex ... More

Refined Inertia of Matrix PatternsNov 24 2016We explore how the combinatorial arrangement of prescribed zeros in a matrix affects the possible eigenvalues that the matrix can obtain. We demonstrate that there are inertially arbitrary patterns having a digraph with no 2-cycle, unlike what happens ... More

A combinatorial divisibility question from noncommutative algebraNov 23 2016We present a general conjecture on the divisibility of a certain expression in terms of Kostka numbers and their close variants. This conjecture is closely related to a variant of the period-index problem of noncommutative algebra, with partial implications ... More

Leavitt path algebras with bounded index of nilpotence and simple modules over themNov 23 2016In this paper we completely describe graphically Leavitt path algebras with bounded index of nilpotence and show that each graded simple module $S$ over a Leavitt path algebra with bounded index of nilpotence is graded $\Sigma$-injective, that is, $S^{(\alpha)}$ ... More

Classification of extended Clifford algebrasNov 23 2016We define so-called extended Clifford algebras as tensor products of special commutative algebras and general Clifford algebras. We have found that there are tree types of extended Clifford algebras. The class of extended Clifford algebras is closed with ... More

On a Generalization for Quaternion SequencesNov 23 2016In this study, we introduce a new classes of quaternion numbers. We show that this new classes quaternion numbers include all of quaternion numbers such as Fibonacci, Lucas, Pell, Jacobsthal, Pell-Lucas, Jacobsthal-Lucas quaternions have been studied ... More

Boolean graphs are unmixed and vertex decomposableNov 22 2016Dec 06 2016For each Boolean graph $B_n$, it is proved that both $B_n$ and its complement graph $\overline{B_n}$ are vertex decomposable. It is also proved that $B_n$ is an unmixed graph, thus it is also Cohen-Macaulay.

Boolean graphs are unmixed and vertex decomposableNov 22 2016For each Boolean graph $B_n$, it is proved that both $B_n$ and its complement graph $\overline{B_n}$ are vertex decomposable. It is also proved that $B_n$ is an unmixed graph, thus it is also Cohen-Macaulay.

Boolean graphs are unmixed and vertex decomposableNov 22 2016Nov 27 2016For each Boolean graph $B_n$, it is proved that both $B_n$ and its complement graph $\overline{B_n}$ are vertex decomposable. It is also proved that $B_n$ is an unmixed graph, thus it is also Cohen-Macaulay.

Two new direct linear solvers in the QR familyNov 21 2016Two new algorithms for computing a direct solution to a system of linear equations are presented. A variation on the orthonormalization step in the usual QR method allows one to bypass subsequent elimination or matrix inversion steps. The algorithms are ... More

Meet-reducible submaximal clones determined by nontrivial equivalence relationsNov 20 2016The structure of the lattice of clones on a finite set has been proven to be very complex. To better understand the top of this lattice, it is important to provide a characterization of submaximal clones in the lattice of clones. It is known that the ... More

Koszul duality between Higgs and Coulomb categories $\mathcal{O}$Nov 20 2016We prove a Koszul duality theorem between the category of weight modules over the quantized Coulomb branch (as defined by Braverman, Finkelberg and Nakajima) attached to a group $G$ and representation $V$ and a category of $G$-equivariant D-modules on ... More