A Note on Dimension Modules

In [2 Camillo , V. P. , Zelmanowitz , J. M. ( 1980 ). Dimension modules . Pacific J. Math. 91 : 249 – 261 .[Crossref], [Web of Science ®] , [Google Scholar]] Camillo and Zelmanowitz stated that rings all whose modules are dimension modules are semisimple Artinian. It seem however that the proof in [2 Camillo , V. P. , Zelmanowitz , J. M. ( 1980 ). Dimension modules . Pacific J. Math. 91 : 249 – 261 .[Crossref], [Web of Science ®] , [Google Scholar]] contains a gap and applies to rings with finite Goldie dimension only. In this paper we show that the result indeed holds for all rings with a basis as well as for all commutative rings with Goldie dimension attained.     

Relaxation of Optimal Control Problems Involving Time Dependent Subdifferential Operators

This article provides in the setting of infinite dimensional Hilbert space, a result concerning the existence and uniqueness of solutions for Lipschitz single-valued perturbations of evolution problems associated with time-dependent subdifferential operators. The result is used to extend to optimal control problems associated with such equations the relaxation theorems with Young measures established in Casting et al. [7 C. Castaing , P. Raynaud de Fitte , and M. Valadier , M. ( 2004 ). Young Measures on Topological Spaces with Applications in Control Theory and Probability Theory . Kluwer Academic Publishers , Dordrecht . [Google Scholar], 11 J. F. Edmond and L. Thibault ( 2005 ). Relaxation of an optimal control problem involving a perturbed sweeping process . Math. Program Ser. B 104 : 347 – 373 .[Crossref], [Web of Science ®] , [Google Scholar]] and Edmond and Thibault [11 J. F. Edmond and L. Thibault ( 2005 ). Relaxation of an optimal control problem involving a perturbed sweeping process . Math. Program Ser. B 104 : 347 – 373 .[Crossref], [Web of Science ®] , [Google Scholar]].     

Erratum to “On α-DICC modules” Commun. Algebra. 22(6):1933–1944 (1994)

We give a correct statement for [2 Karamzadeh, O. A. S., Motamedi, M. (1994). On α-DICC modules. Commun. Algebra 22(6):1933–1944.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], Proposition 1.2]. However, this new form of the proposition needs no different proof from that of [2 Karamzadeh, O. A. S., Motamedi, M. (1994). On α-DICC modules. Commun. Algebra 22(6):1933–1944.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], Proposition 1.2].     

On L p Resolvent Estimates for Elliptic Operators on Compact Manifolds

We prove uniform L^p estimates for resolvents of higher order elliptic self-adjoint differential operators on compact manifolds without boundary, generalizing a corresponding result of [3 Dos Santos Ferreira, D., Kenig, C., and Salo, M., 2014. On L^p resolvent estimates for Laplace-Beltrami operators on compact manifolds, Forum Math. 26 (2014), pp. 815–849.[Crossref], [Web of Science ®] , [Google Scholar]] in the case of Laplace-Beltrami operators on Riemannian manifolds. In doing so, we follow the methods, developed in [1 Bourgain, J., Shao, P., Sogge, C., and Yao, X., On L^p-resolvent estimates and the density of eigenvalues for compact Riemannian manifolds, Comm. Math. Phys., to appear.[Web of Science ®] , [Google Scholar]] very closely. We also show that spectral regions in our L^p resolvent estimates are optimal.     

Virtually Biautomatic Groups

We introduce virtually biautomatic groups (groups with finite index biautomatic subgroups) and generalize results of Gersten and Short [3 Gersten , S. , Short , H. ( 1991 ). Rational subgroups of biautomatic groups . Annals of Mathematics 134 : 125 – 128 .[Crossref], [Web of Science ®] , [Google Scholar]] and Mosher [5 Mosher , L. ( 1997 ). Central quotients of biautomatic groups . Comment. Math. Helv. 72 ( 1 ): 16 – 29 .[Crossref], [Web of Science ®] , [Google Scholar]] on centralizers, normalizers, and quotients to virtually biautomatic groups.     

Derived Representation Type of Schur Superalgebras

In this note we extend the results of Bekkert and Futorny in [2 Bekkert , V. , Futorny , V. ( 2003 ). Derived categories of Schur algebras . Comm. Alg. 31 : 1799 – 1822 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] and Hemmer, Kujawa and Nakano in [10 Hemmer , D. J. , Kujawa , J. , Nakano , D. K. ( 2006 ). Representation types of Schur superalgebras . J. Group Theory 9 : 283 – 306 .[Crossref], [Web of Science ®] , [Google Scholar]] and determine the derived representation type of Schur superalgebras.     

Une Neutralisation Explicite de L'algèbre de Weyl Quantique Complétée

Soit p un nombre premier. Nous établissons l'existence de neutralisations de divers complétés de l'algèbre de Weyl quantique spécialisée en une racine de l'unité primitive d'ordre p (qui est “génériquement” une algèbre d'Azumaya) et donnons en particulier un énoncé de neutralisation explicite relevant celui construit en caractéristique p dans [3 Gros , M. , Le Stum , B. , Quiros , A. ( 2010 ). A Simpson correspondence in positive characteristic . Publ. RIMS Kyoto Univ. 46 : 1 – 35 .[Crossref], [Web of Science ®] , [Google Scholar]].

Let p be a prime number. We establish the existence of neutralizations of various completions of the quantum Weyl algebra specialized at a primitive root of unity of prime order p (which is “generically” an Azumaya algebra) and, in particular, we give a statement of explicit neutralization similar to the one built in characteristic p in [3 Gros , M. , Le Stum , B. , Quiros , A. ( 2010 ). A Simpson correspondence in positive characteristic . Publ. RIMS Kyoto Univ. 46 : 1 – 35 .[Crossref], [Web of Science ®] , [Google Scholar]].     

SOME CENTRALITY RESULTS NEW AND OLD

We prove a number of results on betweenness and closeness centrality and centralization. In particular, we prove the much used normalization expression for closeness centrality first given by Freeman (1979) Freeman, L. C. 1979. Centrality in social networks conceptual clarification. Social Networks, 1: 215–239. [Crossref], [Web of Science ®] , [Google Scholar], correcting an error in the justification given in his paper. We explore the relationship between betweenness and the cutting number and use these results to prove and correct some centrality and centralization formulae first proposed by Borgatti and Everett (1997) Borgatti, S. P. and Everett, M. G. 1997. Network analysis of 2-mode data. Social Networks, 19: 243–269. [Crossref], [Web of Science ®] , [Google Scholar].     

Semilocal Convergence of Steffensen-Type Algorithms for Solving Nonlinear Equations

In this article, we provide a semilocal analysis for the Steffensen-type method (STTM) for solving nonlinear equations in a Banach space setting using recurrence relations. Numerical examples to validate our main results are also provided in this study to show that STTM is faster than other methods ([7 I. K. Argyros , J. Ezquerro , J. M. Gutiérrez , M. Hernández , and S. Hilout ( 2011 ). On the semilocal convergence of efficient Chebyshev-Secant-type methods . J. Comput. Appl. Math. 235 : 3195 – 3206 .[Crossref], [Web of Science ®] , [Google Scholar], 13 J. A. Ezquerro and M. A. Hernández ( 2009 ). An optimization of Chebyshev's method . J. Complexity 25 : 343 – 361 .[Crossref], [Web of Science ®] , [Google Scholar]]) using similar convergence conditions.     

Quasi-Deformations of 𝔰𝔩2(𝔽) Using Twisted Derivations

In this article we apply a method devised in Hartwig, Larsson, and Silvestrov (2006 Hartwig , J. T. , Larsson , D. , Silvestrov , S. D. ( 2006 ). Deformations of Lie algebras using σ-derivations . J. Algebra 295 : 314 – 361 .[Crossref], [Web of Science ®] , [Google Scholar]) and Larsson and Silvestrov (2005a Larsson , D. , Silvestrov , S. D. (2005a). Quasi-hom-Lie algebras, Central extensions and 2-cocycle-like identities. J. Algebra 288:321–344.[Crossref], [Web of Science ®] , [Google Scholar]) to the simple 3-dimensional Lie algebra 𝔰𝔩₂(𝔽). One of the main points of this deformation method is that the deformed algebra comes endowed with a canonical twisted Jacobi identity. We show in the present article that when our deformation scheme is applied to 𝔰𝔩₂(𝔽) we can, by choosing parameters suitably, deform 𝔰𝔩₂(𝔽) into the Heisenberg Lie algebra and some other 3-dimensional Lie algebras in addition to more exotic types of algebras, this being in stark contrast to the classical deformation schemes where 𝔰𝔩₂(𝔽) is rigid.     

Corrigendum to: Goldie Extending Modules

This corrigendum is written to correct an error in Corollary 2.5(ii) and an error in the proof of the converse of Theorem 2.7 of Akalan, Birkenmeier, and Tercan [1 Akalan , E. , Birkenmeier , G. F. , Tercan , A. ( 2009 ). Goldie extending modules . Comm. Algebra 37 : 663 – 683 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]].     

Graded Left Symmetric Pseudo-algebras

In the present article, we introduce G-graded left symmetric H-pseudoalgebras, where G is a grading group, and H is a cocommutative Hopf algebra. Some results about associative H-pseudoalgebras in [23 Retakh , A. ( 2004 ). Unital associative pseudoalgebras and their representations . J. Algebra 227 : 769 – 805 .[Crossref] , [Google Scholar]] are generalized. The commutator algebras of the G-graded left symmetric H-pseudo-algebras are Lie H-pseudoalgebras, which are classified when the grading group is trivial in [3 Bakalov , B. , D'Andrea , A. , Kac , V. G. ( 2001 ). Theory of finite pseudoalgebras . Adv. in Math. 162 : 1 – 140 .[Crossref], [Web of Science ®] , [Google Scholar]]. We investigate the left symmetric structure of Lie H-pseudoalgebras W(𝔟), S(𝔟), and He defined in [3 Bakalov , B. , D'Andrea , A. , Kac , V. G. ( 2001 ). Theory of finite pseudoalgebras . Adv. in Math. 162 : 1 – 140 .[Crossref], [Web of Science ®] , [Google Scholar]].     

Simple Modules and Primitive Ideals of Non-Noetherian Generalized Down-Up Algebras

Generalized down-up algebras were first introduced in Cassidy and Shelton (2004 Cassidy , T. , Shelton , B. ( 2004 ). Basic properties of generalized down-up algebras . J. Algebra 279 : 402 – 421 .[Crossref], [Web of Science ®] , [Google Scholar]). Their simple weight modules were classified in Cassidy and Shelton (2004 Cassidy , T. , Shelton , B. ( 2004 ). Basic properties of generalized down-up algebras . J. Algebra 279 : 402 – 421 .[Crossref], [Web of Science ®] , [Google Scholar]) in the noetherian case, and in Praton (2007 Praton , I. ( 2007 ). Simple weight modules of non-noetherian generalized down-up algebras . Comm. Algebra 35 : 325 – 337 .[Taylor &; Francis Online] , [Google Scholar]) in the non-noetherian case. Here we concentrate on non-noetherian down-up algebras. We show that almost all simple modules are weight modules. We also classify the corresponding primitive ideals.     

Certain Cyclically Presented Groups with the Same Polynomial

We consider three infinite families of cyclic presentations of groups, depending on a finite set of integers and having the same polynomial. Then we prove that the corresponding groups with the same parameters are isomorphic, and that the groups are almost all infinite. Finally, we completely compute the maximal Abelian quotients of such groups, and show that their HNN extensions are high-dimensional knot groups. Our results contain as particular cases the main theorems obtained in two nice articles: Johnson et al. (1999 Johnson , D. L. , Kim , A. C. , O'Brien , E. A. ( 1999 ). Certain cyclically presented groups are isomorphic . Comm. Algebra 27 ( 7 ): 3531 – 3536 . [CSA] [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) and Havas et al. (2001 Havas , G. , Holt , D. F. , Newman , M. F. ( 2001 ). Certain cyclically presented groups are infinite . Comm. Algebra 29 ( 11 ): 5175 – 5178 . [CSA] [CROSSREF] [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]).     

The Generalized C.M.Z.-Theorem and a Drinfel'd Double Construction for WT-Coalgebras and Graded Quantum Groupoids

Let π be a group. In this article, we introduce the notions of a weak Doi–Hopf π-module and a weak π-twisted smash product. We show that the Yetter–Drinfel'd π-modules over a weak crossed Hopf π-coalgebra (WT-coalgebra) are special cases as these new weak Doi–Hopf π-modules, generalizing the main result by Caenepeel et al. (1997 Caenepeel , S. , Militaru , G. , Zhu , S. ( 1997 ). Crossed modules and Doi–Hopf modules . Israel J. Math. 100 : 221 – 248 .[Crossref] , [Google Scholar]) and that the Drinfel'd double for WT-coalgebras (Van Daele and Wang, 2008 Van Daele , A. , Wang , S. H. ( 2008 ). New braided crossed categories and Drinfel'd quantum double for weak Hopf group-coalgebras . Comm. Algebra 36 : 2341 – 2386 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) appears as, a type of such a weak π -twisted smash product, respectively. Finally, starting from a weak Hopf algebra endowed with an action of a group π by weak Hopf automorphisms, we construct a quasitriangular weak Hopf π -coalgebra by a twisted double method, generalizing the main result in Virelizier (2005 Virelizier , A. ( 2005 ). Graded quantum groups and quasitriangular Hopf group-coalgebras . Comm. Algebra 33 ( 9 ): 3029 – 3050 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]). This method allows us to obtain nontrivial examples of quasitriangular weak Hopf π-coalgebras.     

The Complement Graph for Gaussian Integers Modulo n

This article is a continuation for the work done in [1 Abu Osba , E. , Al-Addasi , S. , Abu Jaradeh , N. ( 2008 ). Zero divisor graph for the ring of Gaussian integers modulo . n. Comm. Algebra 36 ( 10 ): 3865 – 3877 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], 2 Abu Osba , E. , Al-Addasi , S. , Al-Khamaiseh , B. ( 2011 ). Some properties of the zero divisor graph for the ring of gaussian integers modulo . n. Glasgow Journal of Mathematics 53 : 391 – 399 .[Crossref], [Web of Science ®] , [Google Scholar]] on the zero divisor graph for the ring of Gaussian integers modulo n. It investigates when the complement graph of the zero divisor graph for the Gaussian integers modulo n connected, planar, regular, or Eulerian. The girth and diameter were also studied.     

Classification of Pointed Fusion Categories of dimension 8 up to weak Morita equivalence

In this paper we give a complete classification of pointed fusion categories over ? of global dimension 8. We first classify the equivalence classes of pointed fusion categories of dimension 8, and then we proceed to determine which of these equivalence classes have equivalent categories of modules following the procedure presented in [9 Naidu, D. (2007). Categorical Morita equivalence for group-theoretical categories. Commun. Algebra 35(11):3544–3565.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], 11 Uribe, B. (2017). On the classification of pointed fusion categories up to weak Morita equivalence. Pac. J. Math. 290(2):437–466.[Crossref], [Web of Science ®] , [Google Scholar]]. The results of this paper permit to recover the classification of twisted quantum doubles of groups of order 8 up to gauge equivalence of braided quasi-Hopf algebras that was previously done in [6 Mason, C., Ng, S.-H (2001). Group cohomology and gauge equivalence of some twisted quantum doubles. Trans. Am. Math. Soc. 353(9):3465–3509.[Crossref], [Web of Science ®] , [Google Scholar]] and [5 Goff, C., Mason, G., Ng, S.-H (2007). On the gauge equivalence of twisted quantum doubles of elementary abelian and extra-special 2-groups. J. Algebra 312(2):849–875.[Crossref], [Web of Science ®] , [Google Scholar]].     

A New Infeasible Interior-Point Method Based on a Non-Coercive Kernel Function with Improved Centering Steps for Second-Order Cone Optimization

In this article, we present a new full Nesterov-Todd step infeasible interior-point method for second-order cone optimization based on a non-coercive kernel function. The main iteration consists of a so-called feasibility step and one centering step, whereas the earlier versions, in [4 S. Bouali and S. Kabbaj ( 2014 ). Full-NT step infeasible interior-point method for SOCO based on a specific kernel function . Afr. Mat. 25 : 549 – 565 .[Crossref] , [Google Scholar], 21 M. Zangiabadi , G. Gu , and C. Roos ( 2013 ). A full Nesterov-Todd step infeasible interior-point method for second-order cone optimization . J. Optim. Theory Appl. 158 : 816 – 858 .[Crossref], [Web of Science ®] , [Google Scholar]], needed two additional centering steps. We use a kernel function to induce the feasibility step. The new algorithm reduces the searching steps in each iteration and tenders an interesting analysis for complexity bound.     

Double Ore Extensions Versus Iterated Ore Extensions

Motivated by the construction of new examples of Artin–Schelter regular algebras of global dimension four, Zhang and Zhang [6 Zhang , J. J. , Zhang , J. ( 2008 ). Double Ore extensions . J. Pure Appl. Algebra 212 ( 12 ): 2668 – 2690 .[Crossref], [Web of Science ®] , [Google Scholar]] introduced an algebra extension A _P [y ₁, y ₂; σ, δ, τ] of A, which they called a double Ore extension. This construction seems to be similar to that of a two-step iterated Ore extension over A. The aim of this article is to describe those double Ore extensions which can be presented as iterated Ore extensions of the form A[y ₁; σ₁, δ₁][y ₂; σ₂, δ₂]. We also give partial answers to some questions posed in Zhang and Zhang [6 Zhang , J. J. , Zhang , J. ( 2008 ). Double Ore extensions . J. Pure Appl. Algebra 212 ( 12 ): 2668 – 2690 .[Crossref], [Web of Science ®] , [Google Scholar]].     

A new family of set-theoretic solutions of the Yang–Baxter equation

A new family of non-degenerate involutive set-theoretic solutions of the Yang–Baxter equation is constructed. Two subfamilies, consisting of irretractable square-free solutions, are new counterexamples to Gateva-Ivanova’s Strong Conjecture [7 Gateva-Ivanova, T. (2004). A combinatorial approach to the set-theoretic solutions of the Yang-Baxter equation. J. Math. Phys. 45(10):3828–3858.[Crossref], [Web of Science ®] , [Google Scholar]]. They are in addition to those obtained by Vendramin [15 Vendramin, L. (2016). Extensions of set-theoretic solutions of the Yang-Baxter equation and a conjecture of Gateva-Ivanova. J. Pure Appl. Algebra 220:2064–2076.[Crossref], [Web of Science ®] , [Google Scholar]] and [1 Bachiller, D., Cedó, F., Jespers, E., Okniński, J. (2017). A family of irretractable square-free solutions of the Yang-Baxter equation. Forum Math. (to appear). [Google Scholar]].