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.     

Divisibility Theory of Arithmetical Rings with One Minimal Prime Ideal

Continuing the study of divisibility theory of arithmetical rings started in [1 Ánh, P. N., Márki, L., Vámos, P. (2012). Divisibility theory in commutative rings: Bezout monoidss. Trans. Amer. Math. Soc. 364:3967–3992.[Crossref], [Web of Science ®] , [Google Scholar]] and [2 Ánh, P. N., Siddoway, M. (2010). Divisibility theory of semi-hereditary rings. Proc. Amer. Math. Soc. 138:4231–4242.[Crossref], [Web of Science ®] , [Google Scholar]], we show that the divisibility theory of arithmetical rings with one minimal prime ideal is axiomatizable as Bezout monoids with one minimal m-prime filter. In particular, every Bezout monoid with one minimal m-prime filter is order-isomorphic to the partially ordered monoid with respect to inverse inclusion, of principal ideals in a Bezout ring with a smallest prime ideal. Although this result can be considered as a satisfactory answer to the divisibility theory of both semihereditary domains and valuation rings, the general representation theory of Bezout monoids is still open.     

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].     

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.     

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]).     

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]].     

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.     

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]].     

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]].     

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.     

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]].     

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.     

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]].     

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.     

Construction of CPs-Stratified Algebras

The results of [7 Dlab , V. , Ringel , C. M. ( 1992 ). The module theoretical approach to quasi-hereditary algebras. In: Tachikawa, H., Brenner, S. eds. Representations of Algebras and Related Topics, London Math. Society Lecture Note Series 168:200–224 . [Google Scholar]] and [2 Ágoston , I. , Dlab , V. , Lukács , E. ( 2011 ). Constructions of stratified algebras . Comm. Algebra 39 : 2545 – 2553 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] gave a recursive construction for all quasi-hereditary and standardly stratified algebras starting with local algebras and suitable bimodules. Using the notion of stratifying pairs of subcategories, introduced in [3 Ágoston , I. , Lukács , E. Stratifying pairs of subcategories for CPS-stratified algebras . To appear in Journal of Algebra and Its Applications , p. 11 . [Google Scholar]], we generalize these earlier results to construct recursively all CPS-stratified algebras.     

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.

     

The Thin Film Equation with Non-Zero Contact Angle: A Singular Perturbation Approach

In this paper we consider the thin film equation with prescribed non-zero contact angle condition for a large class of mobility coefficients, in dimension 1. We prove the global in time existence of weak solutions by using a diffuse approximation of the free boundary condition. This approach, which can be physically motivated by the introduction of singular disjoining/conjoining pressure forces had been suggested in particular by Bertsch et al. in [11 Bertsch , M. , Giacomelli , L. , Karali , G. ( 2005 ). Thin-film equations with “partial wetting” energy: Existence of weak solutions . Phys. D 209 : 17 – 27 .[Crossref], [Web of Science ®] , [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.     

Monoid Valuations and Value Ordered Supervaluations

We complement two papers on supertropical valuation theory ([11 Izhakian , Z. , Knebusch , M. , Rowen , L. ( 2011 ). Supertropical semirings and supervaluations . J. Pure and Applied Alg. 215 ( 10 ): 2431 – 2463 . Preprint at arXiv:1003.1101 .[Crossref], [Web of Science ®] , [Google Scholar]], [12 Izhakian , Z. , Knebusch , M. , Rowen , L. ( 2013 ). Dominance and transmissions in supertropical valuation theory . Commun. Algebra 41 ( 7 ): 2736 – 2782 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]) by providing natural examples of m-valuations (= monoid valuations), and afterwards of supervaluations and transmissions between them. These supervaluations have values in totally ordered supertropical semirings, and the transmissions discussed respect the orderings. We develop the basics of the theory of such semirings and transmissions.