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.     

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].     

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.     

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]].     

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.     

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.     

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].     

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.     

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.     

Galerkin Finite Element Approximation of General Linear Second Order Hyperbolic Equations

In this article, we derive error estimates for the Galerkin approximation of a general linear second order hyperbolic equation. The results can be applied to a variety of cases, for example, vibrating systems of linked elastic bodies. The results generalize the work of Baker [1 G. A. Baker ( 1976 ). Error estimates for finite element methods for second order hyperbolic equations . SIAM J. Numer. Anal. 13 : 564 – 576 .[Crossref], [Web of Science ®] , [Google Scholar]] and also allow for viscous type damping. Splitting the proofs for the semi-discrete and fully discrete cases not only simplifies the proofs but less restrictive regularity assumptions are required.     

Optimal Control of Nonlinear Transport-Flow Mixed Variational Problems

Optimal control of nonlinear transport-flow mixed variational problems are studied qualitatively, and the solvability analysis of the transport and flow mixed state systems is performed on the basis of primal and dual evolution duality principles. Corresponding primal and dual mixed optimality conditions are established by the application of some fundamental perturbation conjugate duality results recently proposed [8 G. Alduncin (2013). Optimal control of evolution mixed variational inclusions. Applied Mathematics and Optimization 68:445–473.[Crossref], [Web of Science ®] , [Google Scholar]]. Further, for computational purposes, two- and three-field proximation penalty-duality algorithms in the resolution of the mixed optimality conditions are finally presented and discussed.     

A Simple Proof of Tyurin's Babylonian Tower Theorem

Using the method of Coand? and Trautmann [4 Coand? , I. , Trautmann , G. ( 2006 ). The splitting criterion of Kempf and the Babylonian tower theorem . Comm. Algebra 34 : 2485 – 2488 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]], we give a simple proof of a theorem due, in the smooth case, to Tyurin [9 Tyurin , A. N. ( 1976 ). Finite dimensional vector bundles over infinite varieties . Math. USSR Izv. 10 : 1187 – 1204 .[Crossref], [Web of Science ®] , [Google Scholar]]: if a vector bundle E on a c-codimensional locally Cohen–Macaulay closed subscheme X of ? ⁿ extends to a vector bundle F on a similar closed subscheme Y of ? ^N , for every N > n, then E is the restriction to X of a direct sum of line bundles on ? ⁿ . Using the same method, we also provide a proof of the Babylonian tower theorem for locally complete intersection subschemes of projective spaces.     

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.     

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.     

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.     

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]].     

Automorphism Groups of Schur Rings

In 1993, Muzychuk [23 Muzychuk , Mikhail E. ( 1993 ). The structure of rational Schur rings over cyclic groups . European Journal of Combinatorics 14 : 479 – 490 .[Crossref], [Web of Science ®] , [Google Scholar]] showed that the rational Schur rings over a cyclic group Z _n are in one-to-one correspondence with sublattices of the divisor lattice of n, or equivalently, with sublattices of the lattice of subgroups of Z _n . This can easily be extended to show that for any finite group G, sublattices of the lattice of characteristic subgroups of G give rise to rational Schur rings over G in a natural way. Our main result is that any finite group may be represented as the (algebraic) automorphism group of such a rational Schur ring over an abelian p-group, for any odd prime p. In contrast, over a cyclic group the automorphism group of any Schur ring is abelian. We also prove a converse to the well-known result of Muzychuk [24 Muzychuk , Mikhail E. ( 1994 ). On the structure of basic sets of Schur rings over cyclic groups . Journal of Algebra 169 : 655 – 678 .[Crossref], [Web of Science ®] , [Google Scholar]] that two Schur rings over a cyclic group are isomorphic if and only if they coincide; namely, we show that over a group which is not cyclic, there always exist distinct isomorphic Schur rings.     

A Matrix Approach to Generalizing Axelrod-Type Tournaments

Axelrod's (1980a Axelrod , R. ( 1980a ). Effective choice in the Prisoner's Dilemma . Journal of Conflict Resolution , 24 , 3 – 25 .[Crossref], [Web of Science ®] , [Google Scholar] 1980b Axelrod , R. ( 1980b ). More effective choice in the Prisoner's Dilemma . Journal of Conflict Resolution , 24 , 379 – 403 .[Crossref], [Web of Science ®] , [Google Scholar]) Prisoner's Dilemma computer tournaments have motivated numerous investigations of cooperation, strategy choice, and strategic evolution. By having players adopt various strategies for playing the Prisoner's Dilemma, and then programming a computer to pit the strategies against each other in a round-robin format, Axelrod uncovered important principles about the evolution of cooperation in certain contexts, and stimulated others to extend his basic method into other settings. This article presents a matrix approach for calculating the results of an Axelrod-type tournament. This approach allows one to investigate the impact of changes in the format of the tournament, the nature of the payoff matrix, or the particular strategic choices of the players.     

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]]).