Examples of Commutative Right-Nilalgebras Over Small Fields

Correa et al. (2003 Correa , I. , Hentzel , I. R. , Labra , A. ( 2003 ). On nilpotence of commutative right nilalgebras of low dimension . Int. J. Math. Game Theory Algebra 13 ( 3 ): 199 – 202 . [Google Scholar]) proved that any commutative right-nilalgebra of nilindex 4 and dimension 4 is nilpotent in characteristic ≠ 2,3. They did not assume power-associativity. In this article we will further investigate these algebras without the assumption on the dimension and providing examples in those cases that are not covered in the classification concentrating mostly on algebras generated by one element.     

A Note on the Dimension Theory of ℤ n -Graded Rings

In this note we want to generalize some of the results in [1 Brewer , J. , Montgomery , P. , Rutter E. , Heinzer , W. ( 1973 ). Krull dimension of polynomial rings in “Conference on Commutative Algebra, Lawrence 1972.” . Springer Lecture Notes in Mathematics 311 : 26 – 45 .[Crossref] , [Google Scholar]] from polynomial rings in several indeterminates to arbitrary ? ⁿ -graded commutative rings. We will prove an analogue of Jaffard's Special Chain Theorem and a similar result for the height of a prime ideal 𝔭 over its graded core 𝔭*.     

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.     

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.     

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.     

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.     

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]].     

Asymptotic Growth of Associated Primes of Certain Graph Ideals

We specify a class of graphs, H _t , and characterize the irreducible decompositions of all powers of the cover ideals. This gives insight into the structure and stabilization of the corresponding associated primes; specifically, providing an answer to the question “For each integer t ≥ 0, does there exist a (hyper) graph H _t such that stabilization of associated primes occurs at n ≥ (χ(H _t ) ?1) + t?” [4 Francisco , C. A. , Hà , H. T. , Van Tuyl , A. ( 2011 ). Colorings of hypergraphs, perfect graphs, and associated primes of powers of monomial ideals . J. Algebra 331 : 224 – 242 .[Crossref], [Web of Science ®] , [Google Scholar]]. For each t, H _t has chromatic number 3 and associated primes that stabilize at n = 2 + t.     

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.     

Classical Morita equivalence and linear topologies; applications to quasi-dualities

ABSTRACT

In this note,we answer a question of Hong et al. (2003 Hong , C. Y. , Kim , N. K. , Kwak , T. K. ( 2003 ). On skew Armendariz rings . Comm. Alg. 31 ( 1 ): 103 – 122 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) by proving that if α is a monomorphism of a reduced ring R, and R is α-skew Armendariz, then R is α-rigid.     

On quotients of generalized Euclidean group rings

Let R = ?[C] be the integral group ring of a finite cyclic group C. Dennis et al. [4 Dennis, K., Magurn, B., Vaserstein, L. (1984). Generalized Euclidean group rings. J. Reine Angew. Math. 351:113–128.[Web of Science ®] , [Google Scholar]] proved that R is a generalized Euclidean ring in the sense of Cohn [3 Cohn, P. M. (1966). On the structure of the GL₂ of a ring. Inst. Hautes Études Sci. Publ. Math. 30:5–53.[Crossref] , [Google Scholar]], i.e., SL_n(R) is generated by the elementary matrices for all n. We prove that every proper quotient of R is also a generalized Euclidean ring.     

Larson–Sweedler Theorem and Some Properties of Discrete Type in (G-Cograded) Multiplier Hopf Algebras

We extend the Larson–Sweedler theorem to group-cograded multiplier Hopf algebras introduced in Abd El-hafez et al. (2004 Abd El-hafez , A. T. , Delvaux , L. , Van Daele , A. ( 2004 ). Group-cograded multiplier Hopf (?-)algebra. Math. QA/0404026 . To appear in Algebras and Representation Theory . [CSA]  [Google Scholar]), by showing that a group-cograded multiplier bialgebra with finite-dimensional unital components is a group-cograded multiplier Hopf algebra if and only if it possesses a nondegenerate left cointegral. We also generalize the theory of multiplier Hopf algebras of discrete type in Van Daele and Zhang (1999 Van Daele , A. , Zhang , Y. ( 1999 ). Multiplier Hopf algebras of discrete type . J. Algebra 214 : 400 – 417 . [CSA] [CROSSREF]  [Google Scholar]) to group-cograded multiplier Hopf algebras. Our results are applicable to Hopf group-coalgebras in the sense of Turaev (2000 Turaev , V. G. ( 2000 ). Homotopy field theory in dimension 3 and crossed group-categories . Preprint GT/0005291. [CSA]  [Google Scholar]). Finally, we study regular multiplier Hopf algebras of η -discrete type.     

Representations of Generalized Almost-Jordan Algebras

This paper deals with the variety of commutative algebras satisfying the identity β{(yx ²)x ? ((yx)x)x} + γ{yx ³ ? ((yx)x)x} = 0, where β, γ are scalars. These algebras appeared as one of the four families of degree four identities in Carini, Hentzel, and Piacentini-Cattaneo [6 Carini , L. , Hentzel , I. R. , Piacentini-Cattaneo , J. M. ( 1988 ). Degree four identities not implies by commutativity . Comm. in Algebra 16 ( 2 ): 339 – 357 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]. We give a characterization of representations and irreducible modules on these algebras. Our results require that the characteristic of the ground field is different from 2, 3.     

Conjugacy Classes of P-Cycles of Type D in Alternating Groups

We classify the conjugacy classes of p-cycles of type D in alternating groups. This finishes the open cases in [3 Andruskiewitsch , N. , Fantino , F. , Graña , M. , Vendramin , L. ( 2011 ). Finite-dimensional pointed Hopf algebras with alternating groups are trivial . Ann. Mat. Pura Appl 190 : 225 – 245 .[Web of Science ®] , [Google Scholar]]. Also we determine all the subracks of those conjugacy classes which are not of type D.     

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]].     

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.     

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.     

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.     

Second Maximum Sum of Element Orders of Finite Nilpotent Groups

In [1 Amiri , H. , Jafarian Amiri , S. M. , Isaacs , I. M. ( 2009 ). Sums of element orders in finite groups . Communications in Algebra 37 : 2978 – 2980 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]], the authors proved that the maximum sum of element orders on finite groups of the same order occurs in cyclic group. In this paper we obtain the structure of groups having maximum sum of element orders on noncyclic nilpotent group of the same order.