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].     

Forward-backward SDEs with discontinuous coefficients

In this paper, we are interested in the well-posedness of a class of fully coupled forward-backward SDE (FBSDE) in which the forward drift coefficient is allowed to be discontinuous with respect to the backward component of the solution. Such an FBSDE is motivated by a practical issue in regime-switching term structure interest rate models, and the discontinuity makes it beyond any existing framework of FBSDEs. In a Markovian setting with non-degenerate forward diffusion, we show that a decoupling function can still be constructed and that it is a Sobolev solution to the corresponding quasilinear PDE. As a consequence we can then argue that the FBSDE admits a weak solution in the sense of [1 Antonelli, F., Ma, J. (2003). Weak solutions of forward-backward SDE’s. Stochastic Analysis and Applications 21(3):493–514.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], 2 Ma, J., Zhang, J., Zheng, Z. (2008). Weak solutions for backward stochastic differential equations, A martingale approach. The Annals of Probability 36(6):2092–2125.[Crossref], [Web of Science ®] , [Google Scholar]]. In the one-dimensional case, we further prove that the weak solution of the FBSDE is actually strong, and it is pathwisely unique. Our approach does not use the well-known Yamada–Watanabe Theorem, but instead follows the idea of Krylov for SDEs with measurable coefficients.     

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.     

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.     

Irreducible W-Graphs for Type D 4 and D 5

ABSTRACT

The “W-graph” concept was introduced by Kazhdan and Lusztig in their influential article Kazhdan and Lusztig (1979 Kazhdan , D. , Lusztig , G. ( 1979 ). Representations of Coxeter groups and Hecke algebras . Invent. Math. 53 : 165 – 184 . [CROSSREF] [CSA] [Crossref], [Web of Science ®] , [Google Scholar]). If W is a Coxeter group, then a W-graph provides a method for constructing a matrix representation of the Hecke algebra ? associated with W (the degree of the representation being the number of vertices of the W-graph). The aim of this note is to explicitly construct all the irreducible representations of ? when W is of type D ₄ and D ₅.     

Note on Conic Bundles over Henselian Discrete Valued Fields with Real Closed Residue Field

ABSTRACT

Let ? be a complete set of Sylow subgroups of a finite group G, that is, ? contains exactly one and only one Sylow p-subgroup of G for each prime p. A subgroup H of a finite group G is said to be ?-permutable if H permutes with every member of ?. The purpose of this article is to study the influence of ?-permutability of all maximal subgroups of the Sylow subgroups of the generalized Fitting subgroup of some normal subgroup of a finite group G on the structure of G. Our results improve and extend the main results of Asaad (1998 Asaad , M. ( 1998 ). On maximal subgroups of Sylow subgroups of finite groups . Comm. Algebra 26 ( 11 ): 3647 – 3652 . [CSA] [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), Asaad and Heliel (2003 Asaad , M. , Heliel , A. A. ( 2003 ). On permutable subgroups of finite groups . Arch. Math. 80 : 113 – 118 . [CROSSREF] [CSA] [Crossref], [Web of Science ®] , [Google Scholar]), Asaad et al. (1991 Asaad , M. , Ramadan , M. , Shaalan , A. ( 1991 ). Influence of π-quasinormality on maximal subgroups of Sylow subgroups of Fitting subgroup of a finite group . Arch. Math. 56 : 521 – 527 . [CROSSREF] [CSA] [Crossref], [Web of Science ®] , [Google Scholar]), Li et al. (2003 Li , Y. , Wang , Y. , Wei , H. ( 2003 ). The influence of π-quasinormality of maximal subgroups of Sylow subgroups of a finite group . Arch. Math. 81 ( 3 ): 245 – 252 . [CROSSREF] [CSA] [Crossref], [Web of Science ®] , [Google Scholar]), Ramadan (1992 Ramadan , M. ( 1992 ). Influence of normality on maximal subgroups of Sylow subgroups of a finite group . Acta Math. Hungar. 59 ( 1–2 ): 107 – 110 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]), and Srinivasan (1980 Srinivasan , S. ( 1980 ). Two sufficient conditions for supersolvability of finite groups . Israel J. Math. 35 : 210 – 214 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]).     

Algebra Structures Arising from Yang–Baxter Systems

Yang–Baxter operators from algebra structures appeared for the first time in [11 D?sc?lescu , S. , Nichita , F. ( 1999 ). Yang–Baxter operators arising from (co)algebra structures . Comm. Algebra 27 : 5833 – 5845 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], 22 Nichita , F. ( 1999 ). Self-inverse Yang–Baxter operators from (co)algebra structures . J. Algebra 218 : 738 – 759 .[Crossref], [Web of Science ®] , [Google Scholar], 23 Nuss , P. ( 1997 ). Noncommutative descent and non-abelian cohomology . K-Theory 12 ( 1 ): 23 – 74 .[Crossref] , [Google Scholar]]. Later, Yang–Baxter systems from entwining structures were constructed in [8 Brzeziński , T. , Nichita , F. F. ( 2005 ). Yang–Baxter systems and Entwining Structures . Comm. Algebra 33 : 1083 – 1093 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]. In fact, Yang–Baxter systems are equivalent with braid systems. In this paper we show that braidings and entwinings of various algebraic structures—in particular, algebra factorisations—can be constructed from a braid system, whence from a Yang–Baxter system as well.     

Large time behavior of solutions of a dissipative semilnear heat equation

We study the current of the Pauli operator in a strong constant magnetic field. We prove that in the semi-classical limit the persistent current and the current from the interaction of the spin with the magnetic field cancel, in the case where the magnetic field is very strong. Furthermore we calculate the next term in the asymptotics and estimate the error. Finally, we discuss the connection between this work and the semi-classical estimate of the energy in strong magnetic fields proved by Lieb, Solovej and Yngvason [8] Lieb, E., Solovej, J. P. and Yngvason, J. 1994. Asymptotics of heavy atoms in high magnetic fields: II. Semiclassical regions. Commun. Math. Phys., 161: 77–124. [Crossref], [Web of Science ®] , [Google Scholar].

     

Prüfer v-Multiplication Domains and w-Multiplicative Canonical Ideals

Let R be an integral domain. A w-ideal I of R is called a w-multiplicative canonical ideal if (I: (I: J)) = J for each w-ideal J of R. In particular, if R is a w-multiplicative canonical ideal of R, then R is a w-divisorial domain. These are the w-analogues of the concepts of a multiplicative canonical ideal and a divisorial domain, respectively. Motivated by the articles [8 El Baghdadi S., Gabelli , S. ( 2005 ). w-Divisorial domains . J. Algebra 285 : 335 – 355 .[Crossref], [Web of Science ®] , [Google Scholar], 10 Heinzer , W. , Huckaba , J. A. , Papick , I. J. ( 1998 ). m-Canonical ideals in integral domains . Comm. Algebra 26 ( 9 ): 3021 – 3043 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]], we study the domains possessing w-multiplicative canonical ideals; in particular, we consider Prüfer v-multiplication domains.     

Centralizers of Subgroups in Direct Limits of Symmetric Groups with Strictly Diagonal Embedding

Let ξ = (p ₁, p ₂,…) be a given infinite sequence of not necessarily distinct primes. In 1976, the structure of locally finite groups S(ξ) (respectively A(ξ) ) which are obtained as a direct limit of finite symmetric (finite alternating) groups are investigated in [7 Kegel , O. H. , Wehrfritz , B. A. F. ( 1973 ). Locally Finite Groups . Amsterdam : North-Holland Publishing Company . [Google Scholar]]. The countable locally finite groups A(ξ) gives an important class in the theory of infinite simple locally finite groups. The classification of these groups using the lattice of Steinitz numbers is completed by Kroshko and Sushchansky in 1998 see [8 Kroshko , N. V. , Sushchansky , V. I. ( 1998 ). Direct Limits of symmetric and alternating groups with strictly diagonal embeddings . Arch. Math. 71 : 173 – 182 .[Crossref], [Web of Science ®] , [Google Scholar]]. Here we extend the results on the structure of centralizers of elements to centralizers of arbitrary finite subgroups and correct some of the errors in the section of centralizers of elements in [8 Kroshko , N. V. , Sushchansky , V. I. ( 1998 ). Direct Limits of symmetric and alternating groups with strictly diagonal embeddings . Arch. Math. 71 : 173 – 182 .[Crossref], [Web of Science ®] , [Google Scholar]]. We construct for each infinite cardinal κ, a new class of uncountably many simple locally finite groups of cardinality κ as a direct limit of finitary symmetric groups. We investigate the centralizers of elements and finite subgroups in this new class of simple locally finite groups, and finally, we characterize this class by the lattice isomorphism with the cardinality of the group and the Steinitz numbers.     

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.     

Approximation theorems for set-valued stochastic integrals

The article is devoted to new properties of Aumann, Lebesgue, and Itô set-valued stochastic integrals considered in papers [1 Kisielewicz, M. (2014). Properties of generalized set-valued stochastic integrals. Discuss. Math. (DICO) 34:131–147. [Google Scholar],2 Kisielewicz, M., Michta, M. (2017). Integrably bounded set-valued stochastic integrals. J. Math. Anal. Appl. 449:1893–1910.[Crossref], [Web of Science ®] , [Google Scholar]]. In particular, it contains some approximation theorems for Aumann and Itô set-valued stochastic integrals. Hence, in particular, it follows that Aumann and Lebesgue set-valued stochastic integrals cover a.s., both for measurable and IF-nonanticipative integrably bounded set-valued stochastic processes.     

Fractional Gamma and Gamma-Subordinated Processes

We introduce and study fractional generalizations of the well-known Gamma process, in the following sense: the corresponding densities are proved to satisfy the same differential equation as the usual Gamma process, but with the shift operator replaced by its fractional version of order ν > 0. In the case ν > 1, the solution corresponds to the density of a Gamma process time-changed by an independent stable subordinator of index 1/ν. For ν less than one an analogous result holds, with the subordinator replaced by the inverse. In this case the fractional Gamma process is proved to be a non-stationary version of the standard one, with power law behavior of the expected value. Hence it can be considered a useful tool in modelling stochastic deterioration in the non-linear cases, a situation which often occurs in real data (see i.e., [42 Van Noortwijk, J.M., 2009. A survey of the application of Gamma processes in maintenance. Reliability Engineering System Safety 94: 2–21.[Crossref], [Web of Science ®] , [Google Scholar]] and the references therein).

As a consequence of the previous results, the fractional generalizations of some Gamma subordinated processes (i.e. the Variance Gamma, the Geometric Stable and the Negative Binomial) are introduced and the corresponding fractional differential equations are obtained. These processes are particularly relevant for a wide range of financial and technological applications.     

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.     

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.     

A note on some structure theorems of modules in category 0

ABSTRACT

Using the local subgroup strategy of An and O'Brien (1997 An , J. , O'Brien , E. A. ( 1997 ). A local strategy to decide the Alperin and Dade conjectures . J. Alg. 189 : 34 – 57 . [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]), An and O'Brien (1999 An , J. , O'Brien , E. A. ( 1999 ). The Alperin and Dade conjectures for the Fischer simple group Fi₂₃ . Internat. J. Alg. Comput. 9 : 621 – 670 . [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]), we classify the radical subgroups and chains of the Fischer simple group Fi ₂₂ and verify the Alperin weight conjecture and the Uno reductive conjecture for this group; the latter is a refinement of the Dade reductive and Isaacs–Navarro conjectures.

     

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.