Strongly J-Clean Matrices Over Local Rings

An element of a ring is called strongly J-clean provided that it can be written as the sum of an idempotent and an element in its Jacobson radical that commute. We investigate, in this article, a single strongly J-clean 2 × 2 matrix over a noncommutative local ring. The criteria on strong J-cleanness of 2 × 2 matrices in terms of a quadratic equation are given. These extend the corresponding results in [8 Li , B. ( 2009 ). Strongly clean matrix rings over noncommutative local rings . Bull. Korean Math. Soc. 46 : 71 – 78 .[Crossref], [Web of Science ®] , [Google Scholar], Theorems 2.7 and 3.2], [9 Li , Y. ( 2007 ). Strongly clean matrix rings over local rings . J. Algebra 312 : 397 – 404 .[Crossref], [Web of Science ®] , [Google Scholar], Theorem 2.6], and [11 Yang , X. , Zhou , Y. ( 2008 ). Strongly cleanness of the 2 × 2 matrix ring over a general local ring . J. Algebra 320 : 2280 – 2290 .[Crossref], [Web of Science ®] , [Google Scholar], Theorem 7].     

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]].     

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.     

On Cohomological Periodicity After 2-Steps

The notion of cohomological periodicity after 1-step has been studied by Talelli in [7 Talelli , O. ( 1980 ) On cohomological periodicity for infinite groups . Comment. Math. Helvetici 55 : 178 – 192 .[Crossref], [Web of Science ®] , [Google Scholar], 8 Talelli , O. ( 1983 ). On groups with periodic cohomology after 1-step . Journal of Pure and Applied Algebra 30 : 85 – 93 .[Crossref], [Web of Science ®] , [Google Scholar]], and [9 Talelli , O. ( 1996 ). On cohomological periodicity isomorphisms . Bull. London Math. Soc. 28 : 600 – 602 .[Crossref], [Web of Science ®] , [Google Scholar]]. If a group G has periodic cohomology after 1-step, then G is the fundamental group of a graph of finite groups, which have periodic cohomology of the same period. Also, the fundamental group of a tree of finite groups, which have periodic cohomology of the same period, has periodic cohomology after 1-step. In this paper, we show that if a group G has only cyclic finite subgroups and is the fundamental group of a certain tree of groups, which have -steps.     

Supercharacter Theory Constructions Corresponding to Schur Ring Products

Diaconis and Isaacs have defined the supercharacter theories of a finite group to be certain approximations to the ordinary character theory of the group [7 Diaconis , P. , Isaacs , I. M. ( 2008 ). Supercharacters and superclasses for algebra groups . Trans. Amer. Math. Soc. 360 : 2359 – 2392 .[Crossref], [Web of Science ®] , [Google Scholar]]. We make explicit the connection between supercharacter theories and Schur rings, and we provide supercharacter theory constructions which correspond to Schur ring products of Leung and Man [12 Leung , K. H. , Man , S. H. ( 1996 ). On Schur rings over cyclic groups, II . J. Algebra 183 : 273 – 285 .[Crossref], [Web of Science ®] , [Google Scholar]], Hirasaka and Muzychuk [10 Hirasaka , M. , Muzychuk , M. ( 2001 ). An elementary abelian group of rank 4 is a CI-group . J. Combin. Theory Ser. A 94 : 339 – 362 .[Crossref], [Web of Science ®] , [Google Scholar]], and Tamaschke [20 Tamaschke , O. ( 1970 ). On Schur-rings which define a proper character theory on finite groups . Math. Z. 117 : 340 – 360 .[Crossref], [Web of Science ®] , [Google Scholar]].     

Lie algebras of vector fields on smooth affine varieties

We reprove the results of Jordan [18 Jordan, D. (1986). On the ideals of a Lie algebra of derivations. J. London Math. Soc. 33:33–39.[Crossref], [Web of Science ®] , [Google Scholar]] and Siebert [30 Siebert, T. (1996). Lie algebras of derivations and a?ne algebraic geometry over fields of characteristic 0. Math. Ann. 305:271–286.[Crossref], [Web of Science ®] , [Google Scholar]] and show that the Lie algebra of polynomial vector fields on an irreducible a?ne variety X is simple if and only if X is a smooth variety. Given proof is self-contained and does not depend on papers mentioned above. Besides, the structure of the module of polynomial functions on an irreducible smooth a?ne variety over the Lie algebra of vector fields is studied. Examples of Lie algebras of polynomial vector fields on an N-dimensional sphere, non-singular hyperelliptic curves and linear algebraic groups are considered.     

Equivalences for Weak Crossed Products

In this article, we give a criterion that characterizes equivalent weak crossed products. By duality, we obtain a similar result for weak crossed coproducts and, as a consequence, we find the conditions that assures the equivalence between two weak crossed biproducts. As an application, we show that the main results proved by Panaite in [12 Panaite, F. (2014). Equivalent crossed products and cross product bialgebras. Commun. Algebr. 42:1937–1952.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] (see also [11 Panaite, F. (2012). Invariance under twisting for crossed products. Proc. Am. Math. Soc. 140:755–763.[Crossref], [Web of Science ®] , [Google Scholar]]), for Brzeziński's crossed products, admits a substantial reduction in the imposed conditions.     

Remarks on the Global Solutions of 3-D Navier-Stokes System with One Slow Variable

By applying Wiegner's method in [16 Wiegner , M. ( 1987 ). Decay results for weak solutions to the Navier-Stokes equations on ? ⁿ . J. London Math. Soc. 35 : 303 – 313 .[Crossref], [Web of Science ®] , [Google Scholar]], we first prove the large time decay estimate for the global solutions of a 2.5 dimensional Navier-Stokes system, which is a sort of singular perturbed 2-D Navier-Stokes system in three space dimension. As an application of this decay estimate, we give a simplified proof for the global wellposedness result in [6 Chemin , J.-Y. , Gallagher , I. ( 2010 ). Large, global solutions to the Navier-Stokes equations, slowly varying in one direction . Transactions of the American Mathematical Society 362 : 2859 – 2873 .[Crossref], [Web of Science ®] , [Google Scholar]] for 3-D Navier-Stokes system with one slow variable. Let us also mention that compared with the assumptions for the initial data in [6 Chemin , J.-Y. , Gallagher , I. ( 2010 ). Large, global solutions to the Navier-Stokes equations, slowly varying in one direction . Transactions of the American Mathematical Society 362 : 2859 – 2873 .[Crossref], [Web of Science ®] , [Google Scholar]], here the assumptions in Theorem 1.3 are weaker.     

Global Existence and Uniform Boundedness of Smooth Solutions to a Cross-Diffusion System with Equal Diffusion Rates

We consider the Shigesada-Kawasaki-Teramoto cross-diffusion model for two competing species. If both species have the same random diffusion coefficients and the space dimension is less than or equal to three, we establish the global existence and uniform boundedness of smooth solutions to the model in convex domains. This extends some previous works of Kim [12 Kim, J.U. (1984). Smooth solutions to a quasilinear system of diffusion equations for a certain population model. Nonlinear Anal. 8:1121–1144.[Crossref], [Web of Science ®] , [Google Scholar]] and Shim [21 Shim, S.-A. (2002). Uniform boundedness and convergence of solutions to cross-diffusion systems. J. Diff. Eqs. 185:281–305.[Crossref], [Web of Science ®] , [Google Scholar]] in one dimensional space.     

Some Remarks on the Hyperelliptic Moduli of Genus 3

In 1967, Shioda [20 Shioda , T. ( 1967 ). On the graded ring of invariants of binary octavics . Amer. J. Math. 89 : 1022 – 1046 .[Crossref], [Web of Science ®] , [Google Scholar]] determined the ring of invariants of binary octavics and their syzygies using the symbolic method. We discover that the syzygies determined in [20 Shioda , T. ( 1967 ). On the graded ring of invariants of binary octavics . Amer. J. Math. 89 : 1022 – 1046 .[Crossref], [Web of Science ®] , [Google Scholar]] are incorrect. In this paper, we compute the correct equations among the invariants of the binary octavics and give necessary and sufficient conditions for two genus 3 hyperelliptic curves to be isomorphic over an algebraically closed field k, char k ≠ 2, 3, 5, 7. For the first time, an explicit equation of the hyperelliptic moduli for genus 3 is computed in terms of absolute invariants.     

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.     

Loewy Modules with Finite Loewy Invariants and Max Modules with Finite Radical Invariants

Over a commutative ring R, a module is artinian if and only if it is a Loewy module with finite Loewy invariants [5 Facchini , A. ( 1981 ). Loewy and artinian modules over commutative rings . Ann. Mat. Pura Appl. 128 : 359 – 374 .[Crossref], [Web of Science ®] , [Google Scholar]]. In this paper, we show that this is not necesarily true for modules over noncommutative rings R, though every artinian module is always a Loewy module with finite Loewy invariants. We prove that every Loewy module with finite Loewy invariants has a semilocal endomorphism ring, thus generalizing a result proved by Camps and Dicks for artinian modules [3 Camps , R. , Dicks , W. ( 1993 ). On semilocal rings . Israel J. Math. 81 : 203 – 211 .[Crossref], [Web of Science ®] , [Google Scholar]]. Finally, we obtain similar results for the dual class of max modules.     

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].     

A Note on Maximal Properties of Some Subsemigroups of Finite Order-Preserving Transformation Semigroups

Both maximal idempotent-generated subsemigroups and maximal idempotent-generated regular subsemigroups of O _n were studied by Yang [10 Yang , X. , Lu , C. ( 2000 ). Maximal properties of some subsemigroups in finite order-preserving transformation semigroups . Communications in Algebra 28 : 3125 – 3135 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]. The purpose of this article is to simplify the results of Yang [10 Yang , X. , Lu , C. ( 2000 ). Maximal properties of some subsemigroups in finite order-preserving transformation semigroups . Communications in Algebra 28 : 3125 – 3135 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]].     

A Braided Version of Some Results of Skryabin

We extend the main result of Skryabin in [9 Skryabin , S. ( 2007 ). Projectivity and freeness over comodule algebras . Trans. Amer. Math. Soc. 359 : 2597 – 2623 .[Crossref], [Web of Science ®] , [Google Scholar]] to Yetter-Drinfeld Hopf algebras over an arbitrary Hopf algebra H with bijective antipode.     

Uniqueness Criteria for the Adjoint Equation in State-Constrained Elliptic Optimal Control

The article considers linear elliptic equations with regular Borel measures as inhomogeneity. Such equations frequently appear in state-constrained optimal control problems. By a counter example of Serrin [18 J. Serrin ( 1964 ). Pathological solutions of elliptic differential equations . Ann. Scuola Norm. Sup. Pisa Cl. Sci. 18 : 385 – 388 . [Google Scholar]], it is known that, in the presence of non-smooth data, a standard weak formulation does not ensure uniqueness for such equations. Therefore several notions of solution have been developed that guarantee uniqueness. In this note, we compare different definitions of solutions, namely the ones of Stampacchia [19 G. Stampacchia ( 1965 ). Le probléme de Dirichlet pour les équations elliptiques du second ordre à coéffcients discontinus . Ann. Inst. Fourier 15 : 189 – 258 .[Crossref] , [Google Scholar]] and Boccardo-Galouët [4 L. Boccardo and T. Gallouët ( 1989 ). Nonlinear elliptic and parabolic equations involving measure data . J. Func. Anal. 87 : 149 – 169 .[Crossref], [Web of Science ®] , [Google Scholar]] and the two notions of solutions of [2 J.-J. Alibert and J.-P. Raymond ( 1997 ). Boundary control of semilinear elliptic equations with discontinuous leading coefficients and unbounded controls . Numer. Func. Anal. Optim. 18 : 235 – 250 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], 7 E. Casas (1993). Boundary control of semilinear elliptic equations with pointwise state constraints. SIAM J. Control Optim. 31:993–1006.[Crossref], [Web of Science ®] , [Google Scholar]], and show that they are equivalent. As side results, we reformulate the solution in the sense of [19 G. Stampacchia ( 1965 ). Le probléme de Dirichlet pour les équations elliptiques du second ordre à coéffcients discontinus . Ann. Inst. Fourier 15 : 189 – 258 .[Crossref] , [Google Scholar]], and prove the existence of solutions in the sense of [2 J.-J. Alibert and J.-P. Raymond ( 1997 ). Boundary control of semilinear elliptic equations with discontinuous leading coefficients and unbounded controls . Numer. Func. Anal. Optim. 18 : 235 – 250 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], 4 L. Boccardo and T. Gallouët ( 1989 ). Nonlinear elliptic and parabolic equations involving measure data . J. Func. Anal. 87 : 149 – 169 .[Crossref], [Web of Science ®] , [Google Scholar], 7 E. Casas (1993). Boundary control of semilinear elliptic equations with pointwise state constraints. SIAM J. Control Optim. 31:993–1006.[Crossref], [Web of Science ®] , [Google Scholar]] in case of mixed boundary conditions.     

Regularity of Analytic Algebra of Prime Characteristic

Let A be an analytic algebra over a field k of characteristic p > 0. In this article, for an analytic k-algebra we introduce the concept of analytic pⁿ-basis which generalizes the pⁿ-basis defined in [1 Furuya , M. , Niitsuma , H. ( 2002 ). Regularity criterion of Noetherian local rings of prime characteristic . J. Algebra 247 : 219 – 230 .[Crossref], [Web of Science ®] , [Google Scholar]], and the concept of an pⁿ-admissible field for an algebraic function field, we give regularity criteria and absolute regularity criteria for an analytic algebra A/k in terms of the higher differential algebra and analytic pⁿ-basis. The results are partial extension of our previous results for affine algebras to the case of analytic algebras (cf. [1 Furuya , M. , Niitsuma , H. ( 2002 ). Regularity criterion of Noetherian local rings of prime characteristic . J. Algebra 247 : 219 – 230 .[Crossref], [Web of Science ®] , [Google Scholar], 3 Furuya , M. , Niitsuma , H. (2006). Regular local rings essentially of finite type over fields of prime characteristic. J. Algebra 306:703–711.[Crossref], [Web of Science ®] , [Google Scholar]]), and these are partial generalization of results of Orbanz in the analytic case (cf. [9 Orbanz , U. ( 1973 ). Höhere Derivationen und Regularität . J. Reine Angew. Mathematik 262/263 : 194 – 204 .[Crossref] , [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.     

Automorphisms of Incidence Algebras

We study automorphisms of the incidence algebra of a finite quasiordered set M. In particular, we describe explicitly the group of outer automorphisms and give a criterion for any automorphism of this algebra to be a product of an inner one and an automorphism of M, which corrects some results of Spiegel (2001 Spiegel , E. ( 2001 ). On the automorphisms of incidence algebras . J. Algebra 239 : 615 – 623 .[Crossref], [Web of Science ®] , [Google Scholar]).