On the almost Gorenstein property of determinantal rings

In this paper, we investigate the question of when the determinantal ring R over a field k is an almost Gorenstein local/graded ring in the sense of [14 Goto, S., Takahashi, R., Taniguchi, N. (2015). Almost Gorenstein rings - towards a theory of higher dimension. J. Pure Appl. Algebra 219:2666–2712.[Crossref], [Web of Science ®] , [Google Scholar]]. As a consequence of the main result, we see that if R is a non-Gorenstein almost Gorenstein local/graded ring, then the ring R has a minimal multiplicity.     

Solvable Permutation Groups and Orbits on Power Sets

We consider the class ? of finitely generated toral relatively hyperbolic groups. We show that groups from ? are commutative transitive and generalize a theorem proved by Benjamin Baumslag in [3 Baumslag, B. (1967). Residually free groups. Prceedings of the London Mathematical Society 17(3):402–418.[Crossref] , [Google Scholar]] to this class. We also discuss two definitions of (fully) residually-𝒞 groups, i.e., the classical Definition 1.1 and a modified Definition 1.4. Building upon results obtained by Ol'shanskii [18 Ol'shanskii, A. Yu. (1993). On residualing homomorphisms and G-subgroups of hyperbolic groups. International Journal of Algebra Computation 3:365–409.[Crossref] , [Google Scholar]] and Osin [22 Osin, D. V. (2010). Small cancellations over relatively hyperbolic groups and embedding theorems. Annals of mathematics 172:1–39.[Crossref], [Web of Science ®] , [Google Scholar]], we prove the equivalence of the two definitions for 𝒞 = ?. This is a generalization of the similar result obtained by Ol'shanskii for 𝒞 being the class of torsion-free hyperbolic groups. Let Γ ∈ ? be non-abelian and non-elementary. Kharlampovich and Miasnikov proved in [14 Kharlampovich, O., Myasnikov, A. (2012). Limits of relatively hyperbolic groups and Lyndon's completions. Journal of the European Math. Soc. 14:659–680.[Crossref], [Web of Science ®] , [Google Scholar]] that a finitely generated fully residually-Γ group G embeds into an iterated extension of centralizers of Γ. We deduce from their theorem that every finitely generated fully residually-Γ group embeds into a group from ?. On the other hand, we give an example of a finitely generated torsion-free fully residually-? group that does not embed into a group from ?; ? is the class of hyperbolic groups.     

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.     

Semi-Smooth Newton Methods for Optimal Control of the Dynamical Lamé System with Control Constraints

Optimal control problems governed by the dynamical Lamé system with additional constraints on the controls are analyzed. Different types of control action are considered: distributed, Neumann boundary and Dirichlet boundary control. To treat the inequality control constraints semi-smooth Newton methods are applied and their convergence is analyzed. Although semi-smooth Newton methods are widely studied in the context of PDE-constrained optimization little has been done in the context of the dynamical Lamé system. The novelty of the article is the proof that in case of distributed and Neumann boundary control the Newton method converges superlinearly. In case of Dirichlet control superlinear convergence is shown for a strongly damped Lamé system. The results are an extension of [14 A. Kröner , K. Kunisch , and B. Vexler ( 2011 ). Semi-smooth Newton methods for optimal control of the wave equation with control constraints . SIAM J. Control Optim. 49 : 830 – 858 .[Crossref], [Web of Science ®] , [Google Scholar]], where optimal control problems of the classical wave equation are considered. The control problems are discretized by finite elements and numerical examples are presented.     

Symmetric Group Decomposition Numbers for Some Three-Part Partitions

The decomposition numbers d _λμ for Specht modules S ^λ of partitions λ with three parts and whose third part is at most p ? 1 are obtained by induction and by using “node removal rules” developed in James and Williams (2000 James , G. D. , Williams , A. L. (2000). Decomposition numbers of symmetric groups by induction. J. Algebra 228:119–142. [CSA] [CROSSREF]  [Google Scholar]).     

Cluster Categories Coming from Cyclic Posets

Cyclic posets are generalizations of cyclically ordered sets. In this article, we show that any cyclic poset gives rise to a Frobenius category over any discrete valuation ring R. The stable category of a Frobenius category is always triangulated and has a cluster structure in many cases. The continuous cluster categories of [14 Igusa , K. , Todorov , G. ( 2013 ). Continuous Frobenius categories . Proceedings of the Abel Symposium 2011 : 115 – 143 .[Crossref] , [Google Scholar]], the infinity-gon of [12 Holm , T. , Jøot rgensen , P. ( 2012 ). On a cluster category of infinite Dynkin type, and the relation to triangulations of the infinity-gon . Math. Z. 270 : 277 – 295 .[Crossref], [Web of Science ®] , [Google Scholar]], and the m-cluster category of type A _∞ (m ≥ 3) [13 Holm , T. , Jøot rgensen , P. ( 2015 ). Cluster tilting vs. weak cluster tilting in Dynkin type A infinity . Forum Math. 27 : 1117 – 1137 .[Crossref], [Web of Science ®] , [Google Scholar]] are examples of this construction as well as some new examples such as the cluster category of ?². An extension of this construction and further examples are given in [16 Igusa, K., Todorov, G. Continuous Cluster Categories of Type D. arXiv:1309.7409 . [Google Scholar]].     

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.     

Regularity Index of S + 2 Fat Points not on a Linear (S − 1)-Space

We will give a formula to compute the regularity index of s + 2 fat points not lying on a linear (s ? 1)-space in ? ⁿ , s ≤ n (Theorem 3.4). Our result generalizes a formula to compute the regularity index of fat points in general position in ? ⁿ ([3 Catalisano , M. V. , Trung , N. V. , Valla , G. ( 1993 ). A sharp bound for the regularity index of fat points in general position . Proc. Amer. Math. Soc. 118 : 717 – 724 .[Crossref], [Web of Science ®] , [Google Scholar]], Corollary 8). Our result also shows that the Segre bound is attained by s + 2 points not lying on a linear (s ? 1)-space.     

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]].     

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]].     

AF-Domains and Their Generalizations

In this article, we are concerned with the study of the dimension theory of tensor products of algebras over a field k. We introduce and investigate the notion of generalized AF-domain (GAF-domain for short) and prove that any k-algebra A such that the polynomial ring in one variable A[X] is an AF-domain is in fact a GAF-domain, in particular any AF-domain is a GAF-domain. Moreover, we compute the Krull dimension of A? _k B for any k-algebra A such that A[X] is an AF-domain and any k-algebra B generalizing the main theorem of Wadsworth in [16 Wadsworth , A. R. ( 1979 ). The Krull dimension of tensor products of commutative algebras over a field . J. London Math. Soc. 19 : 391 – 401 .[Crossref], [Web of Science ®] , [Google Scholar]].     

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.     

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.

     

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]).     

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.     

The infinitesimal and the little q-Schur superalgebras

In this paper, based on the results in [8 Du, J., Gu, H.-X. (2014). A realization of the quantum supergroup U(𝔤𝔩_m|n). J. Algebra 404:60–99.[Web of Science ®] , [Google Scholar]] we give a monomial basis for q-Schur superalgebra and then a presentation for it. The presentation is different from that in [12 El Turkey, H., Kujawa, J. (2012). Presenting Schur superalgebras. Pacific J. Math., 262(2):285–316.[Crossref], [Web of Science ®] , [Google Scholar]]. Imitating [3 Cox, A. G. (1997). On some applications of infinitesimal methods to quantum groups and related algebras. Ph.D. Thesis. University of London. [Google Scholar]] and [7 Du, J., Fu, Q., Wang, J.-P. (2005). Infinitesimal quantum 𝔤𝔩_n and little q-Schur algebras. J. Algebra 287:199–233.[Crossref], [Web of Science ®] , [Google Scholar]], we define the infinitesimal and the little q-Schur superalgebras. We give a “weight idempotent presentation” for infinitesimal q-Schur superalgebras. The BLM bases and monomial bases of little q-Schur superalgebras are obtained, and dimension formulas of infinitesimal and little q-Schur superalgebras are deduced.     

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.     

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.     

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]).