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]]).     

Dominance and Transmissions in Supertropical Valuation Theory

This article is a sequel of [4 Izhakian , Z. , Knebusch , M. , Rowen , L. ( 2011 ). Supertropical semirings and supervaluations . J. Pure and Applied Alg. 215 ( 10 ): 2431 – 2463 .[Crossref], [Web of Science ®] , [Google Scholar]], where we defined supervaluations on a commutative semiring R and studied a dominance relation ? ≥ ψ between supervaluations ? and ψ on R, aiming at an enrichment of the algebraic tool box for use in tropical geometry.

A supervaluation ?: R → U is a multiplicative map from R to a supertropical semiring U, cf. [4 Izhakian , Z. , Knebusch , M. , Rowen , L. ( 2011 ). Supertropical semirings and supervaluations . J. Pure and Applied Alg. 215 ( 10 ): 2431 – 2463 .[Crossref], [Web of Science ®] , [Google Scholar]], [7 Izhakian , Z. , Rowen , L. ( 2011 ). Supertropical matrix algebra . Israel J. Math. 182 ( 1 ): 383 – 424 .[Crossref], [Web of Science ®] , [Google Scholar]], [8 Izhakian , Z. , Rowen , L. ( 2010 ). Supertropical polynomials and resultants . J. Alg. 324 : 1860 – 1886 . (Preprint at arXiv:0902.2155.) [Crossref], [Web of Science ®] , [Google Scholar]], [5 Izhakian , Z. , Knebusch , M. , Rowen , L. Supertropical monoids: Basics and canonical factorization . Preprint at arXiv:1108.1880 . [Google Scholar]], [9 Maclane , S. ( 1998 ). Categories for the Working Mathemtician. , 4th ed. Springer Vereag . [Google Scholar]], with further properties, which mean that ? is a sort of refinement, or covering, of an m-valuation (= monoid valuation) v: R → M. In the most important case, that R is a ring, m-valuations constitute a mild generalization of valuations in the sense of Bourbaki [1 Bourbaki , N. Algèbre Commutative VI, §3 No. 1 . [Google Scholar]], while ? ≥ ψ means that ψ: R → V is a sort of coarsening of the supervaluation ?. If ?(R) generates the semiring U, then ? ≥ ψ iff there exists a “transmission” α: U → V with ψ = α ○ ?.

Transmissions are multiplicative maps with further properties, cf. [4 Izhakian , Z. , Knebusch , M. , Rowen , L. ( 2011 ). Supertropical semirings and supervaluations . J. Pure and Applied Alg. 215 ( 10 ): 2431 – 2463 .[Crossref], [Web of Science ®] , [Google Scholar], Section 5]. Every semiring homomorphism α: U → V is a transmission, but there are others which lack additivity, and this causes a major difficulty. In the main body of the article we study surjective transmissions via equivalence relations on supertropical semirings. We put special emphasis on homomorphic equivalence relations. Even those are often much more complicated than congruences by ideals in usual commutative algebra.     

Semilocal Convergence of Steffensen-Type Algorithms for Solving Nonlinear Equations

In this article, we provide a semilocal analysis for the Steffensen-type method (STTM) for solving nonlinear equations in a Banach space setting using recurrence relations. Numerical examples to validate our main results are also provided in this study to show that STTM is faster than other methods ([7 I. K. Argyros , J. Ezquerro , J. M. Gutiérrez , M. Hernández , and S. Hilout ( 2011 ). On the semilocal convergence of efficient Chebyshev-Secant-type methods . J. Comput. Appl. Math. 235 : 3195 – 3206 .[Crossref], [Web of Science ®] , [Google Scholar], 13 J. A. Ezquerro and M. A. Hernández ( 2009 ). An optimization of Chebyshev's method . J. Complexity 25 : 343 – 361 .[Crossref], [Web of Science ®] , [Google Scholar]]) using similar convergence conditions.     

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.     

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 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.

     

F-Monoids

A semigroup S is called F-monoid if S has an identity and if there exists a group congruence ρ on S such that each ρ-class of S contains a greatest element with respect to the natural partial order of S (see Mitsch, 1986 Mitsch , H. ( 1986 ). A natural partial order for semigroups . Proc. Amer. Math. Soc. 97 : 384 – 388 .[Crossref], [Web of Science ®] , [Google Scholar]). Generalizing results given in Giraldes et al. (2004 Giraldes , E. , Marques-Smith , P. , Mitsch , H. ( 2004 ). F-regular semigroups . J. Algebra 274 : 491 – 510 .[Crossref], [Web of Science ®] , [Google Scholar]) and specializing some of Giraldes et al. (Submitted Giraldes , E. , Marques-Smith , P. , Mitsch , H. F-semigroups. Submitted . [Google Scholar]) five characterizations of such monoids S are provided. Three unary operations “?”, “○”, and “ ? ” on S defined by means of the greatest elements in the different ρ-classes of S are studied. Using their properties a charaterization of F-monoids S by their regular part S° = {a°|a ? S} and the associates of elements in S° is given. Under the hypothesis that S ^? = {a ^?|a ? S} is a subsemigroup it is shown that S is regular, whence of a known structure (see Giraldes et al., 2004 Giraldes , E. , Marques-Smith , P. , Mitsch , H. ( 2004 ). F-regular semigroups . J. Algebra 274 : 491 – 510 .[Crossref], [Web of Science ®] , [Google Scholar]).     

On the Convergence of Broyden-Like Methods Using Recurrent Functions

In this article, we provide a finer semilocal convergence analysis of a certain class of Broyden-like methods for solving equations containing a nondifferentiable term on the p-dimensional Euclidean space (p ≥ 1, a natural number) [2 I.K. Argyros (2004). On the convergence of Broyden's method. Comm. Appl. Nonlinear Anal. 11:77–86. [Google Scholar], 4 I.K. Argyros ( 2007 ). Computational Theory of Iterative Methods, Series: Studies in Computational Mathematics, 15 ( C.K. Chui and L. Wuytack , eds.). Elsevier Publ. Co. , New York , USA . [Google Scholar], 5 I.K. Argyros ( 2007 ). On the convergence of Broyden–like methods . Acta Math. Sin. (Engl. Ser.) 23 : 2087 – 2096 .[Crossref] , [Google Scholar], 8 X. Chen ( 1990 ). On the convergence of Broyden–like methods for nonlinear equations with nondifferentiable terms . Ann. Inst. Statist. Math. 42 : 387 – 401 . [Google Scholar], 9 X. Chen and T. Yamamoto ( 1989 ). Convergence domains of certain iterative methods for solving nonlinear equations . Numer. Funct. Anal. Optim. 10 : 37 – 48 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]. We achieve the goal by using our new idea of recurrent functions. Special cases and applications are also provided.     

Non–existence of theta–shaped self–similarly shrinking networks moving by curvature

We prove that there are no networks homeomorphic to the Greek “Theta” letter (a double cell) embedded in the plane with two triple junctions with angles of 120 degrees, such that under the motion by curvature they are self–similarly shrinking.

This fact completes the classification of the self–similarly shrinking networks in the plane with at most two triple junctions, see [5 Chen, X., Guo, J.-S. (2007). Self-similar solutions of a 2-D multiple-phase curvature flow. Phys. D. 229(1):22–34.[Crossref], [Web of Science ®] , [Google Scholar], 10 Hättenschweiler, J. (2007). Mean curvature flow of networks with triple junctions in the plane. Master’s thesis. ETH Zürich. [Google Scholar], 25 Schnürer, O. C., Azouani, A., Georgi, M., Hell, J., Nihar, J., Koeller, A., Marxen, T., Ritthaler, S., Sáez, M., Schulze, F., Smith, B. (2011). Evolution of convex lens–shaped networks under the curve shortening flow. Trans. Am. Math. Soc. 363(5):2265–2294.[Crossref], [Web of Science ®] , [Google Scholar], 2 Baldi, P., Haus, E., Mantegazza, C. (2016). Networks self-similarly moving by curvature with two triple junctions. Networks self-similarly moving by curvature with two triple junctions. 28(2017):323–338. [Google Scholar]].     

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]).     

A new family of set-theoretic solutions of the Yang–Baxter equation

A new family of non-degenerate involutive set-theoretic solutions of the Yang–Baxter equation is constructed. Two subfamilies, consisting of irretractable square-free solutions, are new counterexamples to Gateva-Ivanova’s Strong Conjecture [7 Gateva-Ivanova, T. (2004). A combinatorial approach to the set-theoretic solutions of the Yang-Baxter equation. J. Math. Phys. 45(10):3828–3858.[Crossref], [Web of Science ®] , [Google Scholar]]. They are in addition to those obtained by Vendramin [15 Vendramin, L. (2016). Extensions of set-theoretic solutions of the Yang-Baxter equation and a conjecture of Gateva-Ivanova. J. Pure Appl. Algebra 220:2064–2076.[Crossref], [Web of Science ®] , [Google Scholar]] and [1 Bachiller, D., Cedó, F., Jespers, E., Okniński, J. (2017). A family of irretractable square-free solutions of the Yang-Baxter equation. Forum Math. (to appear). [Google Scholar]].     

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.     

Une Neutralisation Explicite de L'algèbre de Weyl Quantique Complétée

Soit p un nombre premier. Nous établissons l'existence de neutralisations de divers complétés de l'algèbre de Weyl quantique spécialisée en une racine de l'unité primitive d'ordre p (qui est “génériquement” une algèbre d'Azumaya) et donnons en particulier un énoncé de neutralisation explicite relevant celui construit en caractéristique p dans [3 Gros , M. , Le Stum , B. , Quiros , A. ( 2010 ). A Simpson correspondence in positive characteristic . Publ. RIMS Kyoto Univ. 46 : 1 – 35 .[Crossref], [Web of Science ®] , [Google Scholar]].

Let p be a prime number. We establish the existence of neutralizations of various completions of the quantum Weyl algebra specialized at a primitive root of unity of prime order p (which is “generically” an Azumaya algebra) and, in particular, we give a statement of explicit neutralization similar to the one built in characteristic p in [3 Gros , M. , Le Stum , B. , Quiros , A. ( 2010 ). A Simpson correspondence in positive characteristic . Publ. RIMS Kyoto Univ. 46 : 1 – 35 .[Crossref], [Web of Science ®] , [Google Scholar]].     

Hurwitz Generation of PSp 6(q)

We show that the symplectic groups PSp₆(q) are Hurwitz for all q = p ^m  ≥ 5, with p an odd prime. The result cannot be improved since, for q even and q = 3, it is known that PSp₆(q) is not Hurwitz. In particular, n = 6 turns out to be the smallest degree for which a family of classical simple groups of degree n, over 𝔽 _{p ^m} , contains Hurwitz groups for infinitely many values of m. This fact, for a given (possibly large) p, also follows from [9 Larsen , M. , Lubotzky , A. , Marion , C. ( 2014 ). Deformation theory and finite simple quotients of triangle groups I . J. Eur. Math. Soc. (JEMS) 16 ( 7 ): 1349 – 1375 .[Crossref], [Web of Science ®] , [Google Scholar]] and [10 Larsen , M. , Lubotzky , A. , Marion , C. ( 2014 ). Deformation theory and finite simple quotients of triangle groups II . Groups Geom. Dyn. 8 ( 3 ): 811 – 836 .[Crossref], [Web of Science ®] , [Google Scholar]].     

The Complement Graph for Gaussian Integers Modulo n

This article is a continuation for the work done in [1 Abu Osba , E. , Al-Addasi , S. , Abu Jaradeh , N. ( 2008 ). Zero divisor graph for the ring of Gaussian integers modulo . n. Comm. Algebra 36 ( 10 ): 3865 – 3877 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], 2 Abu Osba , E. , Al-Addasi , S. , Al-Khamaiseh , B. ( 2011 ). Some properties of the zero divisor graph for the ring of gaussian integers modulo . n. Glasgow Journal of Mathematics 53 : 391 – 399 .[Crossref], [Web of Science ®] , [Google Scholar]] on the zero divisor graph for the ring of Gaussian integers modulo n. It investigates when the complement graph of the zero divisor graph for the Gaussian integers modulo n connected, planar, regular, or Eulerian. The girth and diameter were also studied.     

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]].     

Continuous Cohomology of Permutation Groups on Profinite Modules

ABSTRACT

Model theorists have made use of low-dimensional continuous cohomology of infinite permutation groups on profinite modules, see Ahlbrandt and Ziegler (1991 Ahlbrandt , G. , Ziegler , M. ( 1991 ). What's so special about (?/4?)^ω? Archive for Math. Logic 31 : 115 – 132 . [CSA] [Crossref] , [Google Scholar]), Evans (1997b Evans , D. M. ( 1997b ). Computation of first cohomology groups of finite covers . J. Algebra 193 : 214 – 238 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]), Evans et al. (1997 Evans , D. M. , Ivanov , A. A. , Macpherson , H. D. ( 1997 ). Finite covers . In: Evans , D. M. , ed. Model Theory of Groups and Automorphism Groups . London Mathematical Society Lecture Notes 244 . Cambridge : Cambridge Univ Press , pp. 1 – 72 .[Crossref] , [Google Scholar]), and Hodges and Pillay (1994 Hodges , W. , Pillay , A. ( 1994 ). Cohomology of structures and some problems of Ahlbrandt and Ziegler . J. London Math. Soc. 50 ( 2 ): 1 – 16 . [CSA] [Crossref] , [Google Scholar]), for example. We expand the module category in order to widen the cohomological toolkit. For an important class of groups we use these tools to establish criteria for finiteness of cohomology.     

Graded Left Symmetric Pseudo-algebras

In the present article, we introduce G-graded left symmetric H-pseudoalgebras, where G is a grading group, and H is a cocommutative Hopf algebra. Some results about associative H-pseudoalgebras in [23 Retakh , A. ( 2004 ). Unital associative pseudoalgebras and their representations . J. Algebra 227 : 769 – 805 .[Crossref] , [Google Scholar]] are generalized. The commutator algebras of the G-graded left symmetric H-pseudo-algebras are Lie H-pseudoalgebras, which are classified when the grading group is trivial in [3 Bakalov , B. , D'Andrea , A. , Kac , V. G. ( 2001 ). Theory of finite pseudoalgebras . Adv. in Math. 162 : 1 – 140 .[Crossref], [Web of Science ®] , [Google Scholar]]. We investigate the left symmetric structure of Lie H-pseudoalgebras W(𝔟), S(𝔟), and He defined in [3 Bakalov , B. , D'Andrea , A. , Kac , V. G. ( 2001 ). Theory of finite pseudoalgebras . Adv. in Math. 162 : 1 – 140 .[Crossref], [Web of Science ®] , [Google Scholar]].     

An Application of Darbo Fixed-Point Theorem to a Class of Functional Integral Equations

In this article, using Darbo's fixed-point theorem associated with the measure of noncompactness, we prove a theorem on the existence of the solutions of some nonlinear functional integral equations in the space of continuous functions on interval [0, a]. Our existence results include several existence results obtained earlier by Maleknejad et al. [7 K. Maleknejad , K. Nouri , and R. Mollapourasl ( 2009 ). Investigation on the existence of solutions for some nonlinear functional-integral equations . Nonlinear Anal. 71 : 1575 – 1578 .[Crossref], [Web of Science ®] , [Google Scholar]] and Özdemir et al. [8 ?. Özdemir , Ü. Çakan , and B. ?lhan ( 2013 ). On the existence of the solutions for some nonlinear Volterra integral equations . Abstr. Appl. Anal. Article ID 689234.  [Google Scholar]] as special cases under some weaker conditions. We give also some examples which show that our results are applicable.     

Relaxation of Optimal Control Problems Involving Time Dependent Subdifferential Operators

This article provides in the setting of infinite dimensional Hilbert space, a result concerning the existence and uniqueness of solutions for Lipschitz single-valued perturbations of evolution problems associated with time-dependent subdifferential operators. The result is used to extend to optimal control problems associated with such equations the relaxation theorems with Young measures established in Casting et al. [7 C. Castaing , P. Raynaud de Fitte , and M. Valadier , M. ( 2004 ). Young Measures on Topological Spaces with Applications in Control Theory and Probability Theory . Kluwer Academic Publishers , Dordrecht . [Google Scholar], 11 J. F. Edmond and L. Thibault ( 2005 ). Relaxation of an optimal control problem involving a perturbed sweeping process . Math. Program Ser. B 104 : 347 – 373 .[Crossref], [Web of Science ®] , [Google Scholar]] and Edmond and Thibault [11 J. F. Edmond and L. Thibault ( 2005 ). Relaxation of an optimal control problem involving a perturbed sweeping process . Math. Program Ser. B 104 : 347 – 373 .[Crossref], [Web of Science ®] , [Google Scholar]].