Rectifications of A∞-Algebras

In this article, in the setting of connected DG-modules, we prove that, for any A _∞-algebra (M, {m _i } _i≥1), there is a chain contraction from a DG-algebra A _M onto the DG-module M such that the A _∞-algebra structure induced by perturbation theory on M is precisely the original one. In fact, the mentioned DG-algebra can be considered a rectification of the A _∞-algebra in the sense of (Boardman and Vogt, 1973 Boardman , J. M. , Vogt , R. M. ( 1973 ). Homotopy Invariant Algebraic Structures on Topological Spaces . Lecture Notes in Mathematics . Springer-Verlag , pp. 347 . [Google Scholar]). Appropiate dual results are given for A _∞-coalgebras.     

Invertibility of ideals in Prüfer extensions

Using the general approach to invertibility for ideals in ring extensions given by Knebush and Zhang in [9 Knebush, M., Zhang, D. (2002). Manis Valuations and Prüfer Extensions I. Lecture Notes in Mathematics, Vol. 1791. Springer.[Crossref] , [Google Scholar]], we investigate about connections between faithfully flatness and invertibility for ideals in rings with zero divisors.     

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*.     

Prime Ideals in Skew Polynomial Rings of Derivation Type

ABSTRACT

It is shown that (Plesken, 1983 Plesken , W. ( 1983 ). Group rings of finite groups over the p-adic integers . Springer Lecture Notes in Mathematics 1026 . [Google Scholar], Theorem 8.5), describes blocks of cyclic defect groups up to Morita equivalence. In particular such a block is determined by its planar embedded Brauer tree. Applying the radical idealizer process, the head order of such blocks is calculated explicitly.     

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.     

Pseudo almost automorphic solutions for some partial functional differential equations with infinite delay

In this work, we study the existence of pseudo almost automorphic solution for some partial functional differential equations with infinite delay. We assume that the undelayed part is not necessarily densely defined and satisfies the Hille–Yosida condition. We use the variation of constant formula developed recently in 1 Hino, Y, Murakami, S and Naito, T. 1991. “Functional Differential Equations with Infinite Delay”. In Lecture Notes in Mathematics, Vol. 1473, New York: Springer-Verlag.  [Google Scholar] to get the existence and uniqueness of pseudo almost automorphic solution when the linear equation has an exponential dichotomy. We also give an application of the abstract results to a Lotka–Volterra model with diffusion.     

Approximations of Parabolic Equations at the Vicinity of Hyperbolic Equilibrium Point

This article is devoted to the numerical analysis of the abstract semilinear parabolic problem u′(t) = Au(t) + f(u(t)), u(0) = u ⁰, in a Banach space E. We are developing a general approach to establish a discrete dichotomy in a very general setting and prove shadowing theorems that compare solutions of the continuous problem with those of discrete approximations in space and time. In [3 W.-J. Beyn and S. Piskarev ( 2008 ). Shadowing for discrete approximations of abstract parabolic equations . Discrete Contin. Dyn. Syst., B 10 : 19 – 42 .[Crossref], [Web of Science ®] , [Google Scholar]] the discretization in space was constructed under the assumption of compactness of the resolvent. It is a well-known fact (see [10 S. Larsson ( 1999 ). Numerical analysis of semilinear parabolic problems . In: The Graduate Student's Guide to Numerical Analysis ’98 ( M. Ainsworth , ed.). Lecture Notes from the 8th EPSRC Summer School in Numerical Analysis . Leicester , UK , July 5–17. 1998. Ser. Comput. Math. 26:83–117; Springer, Berlin .[Crossref] , [Google Scholar], 11 S. Larsson and J. M. Sanz-Serna (1994). The behavior of finite element solutions of semilinear parabolic problems near stationary points. SIAM J. Numer. Anal. 31:1000–1018.[Crossref], [Web of Science ®] , [Google Scholar]]) that the phase space in the neighborhood of the hyperbolic equilibrium can be split in a such way that the original initial value problem is reduced to initial value problems with exponential bounded solutions on the corresponding subspaces. We show that such a decomposition of the flow persists under rather general approximation schemes, utilizing a uniform condensing property. The main assumption of our results are naturally satisfied, in particular, for operators with compact resolvents and condensing semigroups and can be verified for finite elements as well as finite differences methods.     

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.     

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.     

Superconvergence of Solution Derivatives for the Shortley–Weller Difference Approximation of Poisson's Equation. II. Singularity Problems

Abstract

This is a continued analysis on superconvergence of solution derivatives for the Shortley–Weller approximation in Li (Li, Z. C., Yamamoto, T., Fang, Q. ([2003] Li, Z. C., Yamamoto, T. and Fang, Q. 2003. Superconvergence of solution derivatives for the Shortley–Weller difference approximation of Poisson's equation, Part I. Smoothness problems. J. Comp. and Appl. Math., 152(2): 307–333.  [Google Scholar]): Superconvergence of solution derivatives for the Shortley–Weller difference approximation of Poisson's equation, Part I. Smoothness problems. J. Comp. and Appl. Math. 152(2):307–333), which is to explore superconvergence for unbounded derivatives near the boundary. By using the stretching function proposed in Yamamoto (Yamamoto, T. ([2002] Yamamoto, T. 2002. Convergence of consistant and inconsistent finite difference schemes and an acceleration technique. J. Comp. Appl. Math., 140: 849–866. [Crossref], [Web of Science ®] , [Google Scholar]): Convergence of consistant and inconsistent finite difference schemes and an acceleration technique. J. Comp. Appl. Math. 140:849–866), the second order superconvergence for the solution derivatives can be established. Moreover, numerical experiments are provided to support the error analysis made. The analytical approaches in this article are non-trivial, intriguing, and different from Li, Z. C., Yamamoto, T., Fang, Q. ([2003] Li, Z. C., Yamamoto, T. and Fang, Q. 2003. Superconvergence of solution derivatives for the Shortley–Weller difference approximation of Poisson's equation, Part I. Smoothness problems. J. Comp. and Appl. Math., 152(2): 307–333.  [Google Scholar]). This article also provides the superconvergence analysis for the bilinear finite element method and the finite difference method with nine nodes.     

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.     

Construction of Nilpotent Jordan Algebras Over any Arbitrary Field

The paper is devoted to the study of annihilator extensions of Jordan algebras and suggests new approach to classify nilpotent Jordan algebras, which is analogous to the Skjelbred–Sund method for classifying nilpotent Lie algebras [2 de Graaf, W. (2007). Classification of 6-dimensional nilpotent Lie algebras over fields of characteristic not 2. J. Algebra 309:640–653.[Crossref], [Web of Science ®] , [Google Scholar], 4 Gong, M.-P. (1998). Clasification of Nilpotent Lie Algebras of Dimension 7 [Ph.D]. Ontario, Canada: University of Waterloo. [Google Scholar], 15 Skjelbred, T., Sund, T. (1978). Sur la classification des algèbres de Lie nilpotentes. C. R. Acad. Sci. Paris Sér. A-B 286:241–242. [Google Scholar]]. Subsequently, we have classified nilpotent Jordan algebras of dimension up to four.     

Constructive Néron Desingularization of algebras with big smooth locus

An algorithmic proof of the General Néron Desingularization theorem and its uniform version is given for morphisms with big smooth locus. This generalizes the results for the one-dimensional case (cf. [10 Pfister, G., Popescu, D. (2017). Constructive General Neron Desingularization for one dimensional local rings. J. Symbolic Comput. 80:570–580.[Crossref], [Web of Science ®] , [Google Scholar]], [7 Khalid, A., Pfister, G., Popescu, D. (2018). A uniform General Neron Desingularization in dimension one. J. Algebra Appli. 16. arXiv:AC/1612.03416. [Google Scholar]]).     

Decomposition of Order Statistics of Semimartingales Using Local Times

Given a finite collection of continuous semimartingales, a semimartingale decomposition of the corresponding ranked (order-statistics) processes was derived recently in [1 Banner , A.D. , and Ghomrasni , R. 2008 . Local times of ranked continuous semimartingales . Stochastic Processes and Applications 118 : 1244 – 1253 . [Google Scholar]]. In this paper, we obtain a more general result for semimartingales (not necessarily continuous) using a simpler approach. Furthermore, we also give a generalization of Ouknine [7 Ouknine , Y. 1988 . Généralisation d'un lemme de S. Nakao et applications . Stochastics 23 : 149 – 157 .[Taylor & Francis Online] , [Google Scholar], 8 Ouknine , Y. 1990 . Temps local du produit et du sup de deux semimartingales . Séminaire de Probabilités XXIV, 1988/89 . Lecture Notes in Mathematics , Vol. 1426 , pp. 477 – 479 . [Google Scholar]] and Yan's [11 Yan , J.A. 1985 . A formula for local times of semimartingales . Northeast. Math. J. 1 : 138 – 140 . [Google Scholar]] formula for local times of ranked processes.     

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.

     

Quotients of Koszul Algebras and 2-d-Determined Algebras

Vatne [13 Vatne , J. E. ( 2012 ). Quotients of Koszul algebras with almost linear resolution. Preprint, arXiv:1103.3572 . [Google Scholar]] and Green and Marcos [9 Green , E. L. , Marcos , E. N. (2011). d-Koszul algebras, 2-d-determined algebras and 2-d-Koszul algebras. J. Pure Appl. Algebra 215(4):439–449.[Crossref], [Web of Science ®] , [Google Scholar]] have independently studied the Koszul-like homological properties of graded algebras that have defining relations in degree 2 and exactly one other degree. We contrast these two approaches, answer two questions posed by Green and Marcos, and find conditions that imply the corresponding Yoneda algebras are generated in the lowest possible degrees.     

Convergence in Norm of Projection Regularized Krasnoselski-Mann Iterations for Fixed Points of Cutters

In this article, we introduce a projection regularized Krasnoselski-Mann iteration for cutters. The proposed algorithm ensures the strong convergence of the generated sequence toward the least norm element of the set of fixed points of the cutter. It is verified that the projection regularized Krasnoselski-Mann iteration converges locally faster than the regularized Krasnoselski-Mann iteration introduced by Maingé and Maruster [11 P. E. Maingé and S. Maruster ( 2011 ). Convergence in norm of modified Krasnoselski-Mann iterations for fixed points of demicontractive mappings . Appl. Math. Comput. 217 : 9864 – 9874 .[Crossref], [Web of Science ®] , [Google Scholar]]. Furthermore, we present projection regularized Krasnoselski-Mann iterations for quasi-nonexpansive and nonexpansive mappings in Hilbert spaces.     

On a Group Graded Version of BGG

A major result in Algebraic Geometry is the theorem of Bernstein–Gelfand–Gelfand that states the existence of an equivalence of triangulated categories: gr _Λ ? 𝒟^b(Coh ?ⁿ), where gr _Λ denotes the stable category of finitely generated graded modules over the n + 1 exterior algebra and 𝒟^b(Coh ?ⁿ) is the derived category of bounded complexes of coherent sheaves on projective space ?ⁿ.

Generalizations of this result were obtained in Martínez-Villa and Saorín (2004 Martínez-Villa , R. , Saorín , M. ( 2004 ). Koszul equivalences and dualities . Pacific J. Math. 214 ( 2 ): 359 – 378 . [Google Scholar]) and from a different point of view, the theorem has been extended by Yanagawa (2004 Yanagawa , K. ( 2004 ). Derived category of squarefree modules and local cohomology with monomial ideal support . J. Math. Soc. Japan 56 : 289 – 308 . [Google Scholar]) to ?ⁿ-graded modules over the polynomial algebra. This generalization has important applications in combinatorial commutative algebra.

The aim of the article is to extend the results of Martínez-Villa and Saorín (2004 Martínez-Villa , R. , Saorín , M. ( 2004 ). Koszul equivalences and dualities . Pacific J. Math. 214 ( 2 ): 359 – 378 . [Google Scholar]) to group graded algebras in order to obtain a generalization of Yanagawa's results having in mind the application to other settings (Geigle and Lenzing, 1987 Geigle , W. , Lenzing , H. ( 1987 ). A class of weghted projective curves in representation theory of finite dimensional algebras, in singularities, representations of algebras, and vector bundles . Lecture Notes in Math. 1273 : 265 – 297 .[Crossref], [Web of Science ®] , [Google Scholar]).     

Poisson deleting derivations algorithm and Poisson spectrum

Cauchon [5 Cauchon, G. (2003). Effacement des dérivations et spectres premiers des algèbres quantiques. J. Algebra 260(2):476–518.[Crossref], [Web of Science ®] , [Google Scholar]] introduced the so-called deleting derivations algorithm. This algorithm was first used in noncommutative algebra to prove catenarity in generic quantum matrices, and then to show that torus-invariant primes in these algebras are generated by quantum minors. Since then this algorithm has been used in various contexts. In particular, the matrix version makes a bridge between torus-invariant primes in generic quantum matrices, torus orbits of symplectic leaves in matrix Poisson varieties and totally non-negative cells in totally non-negative matrix varieties [12 Goodearl, K. R., Launois, S., Lenagan, T. (2011). Torus invariant prime ideals in quantum matrices, totally nonnegative cells and symplectic leaves. Math. Z. 269(1):29–45.[Crossref], [Web of Science ®] , [Google Scholar]]. This led to recent progress in the study of totally non-negative matrices such as new recognition tests [18 Launois, S., Lenagan, T. (2014). E?cient recognition of totally non-negative matrix cells. Found. Comput. Math. 14:371–387.[Crossref], [Web of Science ®] , [Google Scholar]]. The aim of this article is to develop a Poisson version of the deleting derivations algorithm to study the Poisson spectra of the members of a class 𝒫 of polynomial Poisson algebras. It has recently been shown that the Poisson Dixmier–Moeglin equivalence does not hold for all polynomial Poisson algebras [2 Bell, J., Launois, S., Sanchez, O. L., Moosa, R. Poisson algebras via model theory and differential-algebraic geometry. J. Eur. Math. Soc. (to appear). [Google Scholar]]. Our algorithm allows us to prove this equivalence for a significant class of Poisson algebras, when the base field is of characteristic zero. Finally, using our deleting derivations algorithm, we compare topologically spectra of quantum matrices with Poisson spectra of matrix Poisson varieties.