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.     

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.     

Homotopies for the Generalized Bar Complex Associated to Certain 3-Rowed Weyl Modules

In Buchsbaum and Rota (1994 Buchsbaum , D. , Rota , G.-C. ( 1994 ). A new construction in homological algebra . Proc. Nat. Acad. Sci. 91 : 4115 – 4119 . [CSA] [CROSSREF] [Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), the authors presented a generalized bar complex associated to certain 3-rowed Weyl modules and proved that this complex is in fact a resolution via an induction on the number of overlaps between the second and third rows and a fundamental exact sequence (Akin and Buchsbaum, 1985 Akin , K. , Buchsbaum , D. ( 1985 ). Characteristic-free representation theory of the general linear group . Adv. Math 58 ( 2 ): 149 – 200 . [CSA]  [Google Scholar]). In this article we study the structure of this resolution by constructing a splitting contracting homotopy for the complexes corresponding to certain shapes.     

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]].     

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.     

Corrigendum to: “Clifford Theory of a Finite Group that Contains a Defect 0 p-Block of a Normal Subgroup”

At some point, after publication, the author realized that the proof of [3 Harris, M. E. (2013). Clifford theory of a finite group that contains a defect 0 p-block of a normal subgroup. Comm. in Alg. 41:3509–3540.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], Theorem 5.2] is incorrect. This proof incorrectly adapts the proof of [1 Broué, M. (1990). Isométries parfaites, types de blocs, cégories dérivees. Aérisque 181–182:61–92. [Google Scholar], Theorem 4.8] since [3 Harris, M. E. (2013). Clifford theory of a finite group that contains a defect 0 p-block of a normal subgroup. Comm. in Alg. 41:3509–3540.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], (5.5)] is incorrect. Using the same proof outline, we correct the proof of [3 Harris, M. E. (2013). Clifford theory of a finite group that contains a defect 0 p-block of a normal subgroup. Comm. in Alg. 41:3509–3540.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], Theorem 5.2].     

Fuzzy Optimality Conditions for Fractional Multiobjective Bilevel Problems Under Fractional Constraints

In this article, we are concerned with fractional multiobjective optimization problems. In order to derive optimality conditions, we consider a new single level problem [12 J.J. Ye ( 2006 ). Constraint qualification and KKT conditions for bilevel programming problems . Mathematics of Operations Research 31 : 811 – 824 .[Crossref], [Web of Science ®] , [Google Scholar]], which is locally equivalent to the bilevel fractional multiobjective problem (P) at the optimal solution. Our approach consists of using another approach initiated by Mordukhovich [7 B.S. Mordukhovich ( 1976 ). Maximum principle in problems of time optimal control with nonsmooth constraints . J. Appl. Math. Mech. 40 : 960 – 969 .[Crossref], [Web of Science ®] , [Google Scholar], 8 B.S. Mordukhovich ( 1980 ). Metric approximations and necessary optimality conditions for general classes of nonsmooth extremal problems . Soviet. Math. Dokl. 22 : 526 – 530 . [Google Scholar]], which does not involve any convex approximations and convex separation arguments, called the extremal principle [5 B.S. Mordukhovich ( 2006 ). Variational Analysis and Generalized Differentiation, I: Basic Theory . Grundlehren Series (Fundamental Principles of Mathematical Sciences) , Vol. 330 , Springer , Berlin . [Google Scholar], 6 B.S. Mordukhovich ( 2006 ). Variational Analysis and Generalized Differentiation, II: Applications , Grundlehren Series (Fundamental Principles of Mathematical Sciences) , Vol. 331 , Springer , Berlin . [Google Scholar], 9 B.S. Mordukhovich ( 1994 ). Generalized differential calculus for nonsmooth and set-valued mappings . J. Math. Anal. Appl. 183 : 250 – 288 .[Crossref], [Web of Science ®] , [Google Scholar]], for the study of necessary optimality conditions in fractional vector optimization.     

Expanded Degenerations and Pairs

We provide a universal approach to the moduli of Jun Li's expanded pairs and expanded degenerations [20 Li , J. ( 2001 ). Stable morphisms to singular schemes and relative stable morphisms . J. Differential Geom. 57 : 509 – 578 .[Web of Science ®] , [Google Scholar]]. This enables us to prove algebraicity results, compare with Li's approach and with the approach of Graber and Vakil [16 Graber , T. , Vakil , R. ( 2005 ). Relative virtual localization and vanishing of tautological classes on moduli spaces of curves . Duke Math. J. 130 : 1 – 37 .[Crossref], [Web of Science ®] , [Google Scholar]], and generalize to the twisted expansions used by Abramovich and Fantechi as a basis for orbifold techniques in degeneration formulas.     

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.     

Rank Properties of the Semigroup of Singular Transformations on a Finite Set

It is known that the semigroup Sing _n of all singular self-maps of X _n  = {1,2,…, n} has rank n(n ? 1)/2. The idempotent rank, defined as the smallest number of idempotents generating Sing _n , has the same value as the rank. (See Gomes and Howie, 1987 Gomes , G. M. S. , Howie , J. M. ( 1987 ). On the rank of certain finite semigroups of transformations . Math. Proc. Cambridge Phil. Soc. 101 : 395 – 303 .[Crossref], [Web of Science ®] , [Google Scholar].) Idempotents generating Sing _n can be seen as special cases (with m = r = 2) of (m, r)-path-cycles, as defined in Ay\i k et al. (2005 Ay?k , G. , Ay?k , H. , Howie , J. M. ( 2005 ). On factorisations and generators in transformation semigroups . Semigroup Forum 70 : 225 – 237 .[Crossref], [Web of Science ®] , [Google Scholar]). The object of this article is to show that, for fixed m and r, the (m, r)-rank of Sing _n , defined as the smallest number of (m, r)-path-cycles generating Sing _n , is once again n(n ? 1)/2.     

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]].     

Auslander–Reiten Triangles in Homotopy Categories

Let A be an artin algebra. We show that the bounded homotopy category of finitely generated right A-modules has Auslander–Reiten triangles. Two applications are given: (1) we provide an alternative proof of a theorem of Happel in [14 Happel, D. (1988). Triangulated Categories in the Representation Theory of Finite-dimensional Algebras. London Mathematical Society Lecture Note Series, vol. 119. Cambridge: Cambridge University Press.[Crossref] , [Google Scholar]]; (2) we prove that over a Gorenstein algebra, the bounded homotopy category of finitely generated Gorenstein projective (resp. injective) modules, admits Auslander–Reiten triangles, which improve a main result in [12 Nan, G. (2012). Auslander-Reiten triangles on Gorenstein derived categories. Comm. Algebra 40:3912–3919.[Taylor &; Francis Online], [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.     

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].     

A note on: Quasi-linear quotients and shellability of pure simplicial complexes

The purpose of this note is to point out a careless error in the algebraic criterion of shellability of a pure simplicial complex Δ given in [1 Anwar, I., Raza, Z. (2015). Quasi-linear quotients and shellability of pure simplicial complexes. Commun. Algebra 43:4698–4704.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]].     

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.