ON THE NORM OF THE FOURIER-JACOBI PROJECTION

We extend the results of Pollard [7] Pollard, H. 1949. The mean convergence of orthogonal series. III. Duke Math. J., 16: 189–191. [Crossref], [Web of Science ®] , [Google Scholar] and give asymptotic estimates for the norm of the Fourier-Jacobi projection operator in the appropriate weighted L_p space.     

UNIFORM FILTERS AND SEMI-COMPATIBILITY

In [5] García Román, M., Márquez Hernández, M. and Verschoren, A. 1997. Structure Sheaves and Noncommutative Topologies. J. of Algebra, 194: 224–244. [Crossref], [Web of Science ®] , [Google Scholar] we used functors which are compositions of localization functors to construct sheaves over an arbitrary ring R. These functors share some properties with localization, and questions like when is the composition of localizations a localization functor? arise naturally. In this note we answer this question and some related ones using the key concept of semi-compatibility.

     

GRADINGS OF SIMPLE JORDAN ALGEBRAS AND THEIR RELATION TO THE GRADINGS OF SIMPLE ASSOCIATIVE ALGEBRAS

In this paper we describe all group gradings of the simple Jordan algebra of a non-degenerate symmetric form on a vector space over a field of characteristic different from 2. If we use the notion of the Clifford algebra, then we are able to recover some of the gradings on matrix algebras obtained in an entirely different way in [BSZ] Bahturin, Y., Seghal, S. and Zaicev, M. in press. Group Gradings of Associative Algebras. J. Algebra, [Web of Science ®] , [Google Scholar].

     

Homogenized Down-Up Algebras

Abstract

This paper studies two homogenizations of the down-up algebras introduced in Benkart and Roby (Benkart, G., Roby, T. (1998 Benkart, G. and Roby, T. 1998. Down-up algebras. J. Algebra, 209: 305–344. [Crossref], [Web of Science ®] , [Google Scholar]). Down-up Algebras. J. Algebra 209:305–344). We show that in all cases the homogenizing variable is not a zero-divisor, and that when the parameter β is non-zero, the homogenized down-up algebra is a Noetherian domain and a maximal order, and also Artin-Schelter regular, Auslander regular, and Cohen-Macaulay. We show that all homogenized down-up algebras have global dimension 4 and Gelfand-Kirillov dimension 4, and with one exception all homogenized down-up algebras are prime rings. We also exhibit a basis for homogenized down-up algebras and provide a necessary condition for a Noetherian homogenized down-up algebra to be a Hopf algebra.     

DECOMPOSITION OF A HIGHER DEGREE FORM

ABSTRACT

We give another proof of Harrison's decomposition result,[2] Harrison, D.K. 1975. A Grothendieck Ring of Higher Degree Forms. Journal of Algebra, 35: 123–138. [Crossref], [Web of Science ®] , [Google Scholar] Prop. 2.3 for higher degree forms over a noetherian ring, exploiting an earlier introduction of the centre. We generalise to higher degree forms over a noetherian scheme: we extend the notion of centre; we prove a decomposition result; we extend Harrison's result,[2] Harrison, D.K. 1975. A Grothendieck Ring of Higher Degree Forms. Journal of Algebra, 35: 123–138. [Crossref], [Web of Science ®] , [Google Scholar] Prop. 4.3 on the behaviour of the centre under a flat base extension; and we improve his result,[2] Harrison, D.K. 1975. A Grothendieck Ring of Higher Degree Forms. Journal of Algebra, 35: 123–138. [Crossref], [Web of Science ®] , [Google Scholar] Prop. 4.2, giving conditions on the base scheme under which the centre of the tensor product of two higher degree forms is isomorphic to the tensor product of their centres.     

Stratifying Systems via Relative Projective Modules

ABSTRACT

In this paper we point out that the “Process of standardization”, given in Dlab and Ringel (1992 Dlab , V. , Ringel , C. M. ( 1992 ). The module theoretical approach to quasi-hereditary algebras . Repr. Theory and Related Topics, London Math. Soc. LNS 168 : 200 – 224 . [Google Scholar]), and also the “Comparison method” given in Platzeck and Reiten (2001 Platzeck , M. I. , Reiten , I. ( 2001 ). Modules of finite projective dimension for standardly stratified algebras . Comm. in Algebra 29 : 973 – 986 . [CROSSREF] [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) can be generalized. To do so, we introduce the concept of relative projective stratifying system and prove a result from which the Theorem 2 in Dlab and Ringel (1992 Dlab , V. , Ringel , C. M. ( 1992 ). The module theoretical approach to quasi-hereditary algebras . Repr. Theory and Related Topics, London Math. Soc. LNS 168 : 200 – 224 . [Google Scholar]) and Proposition 2.1 in Ringel (1991 Ringel , C. M. ( 1991 ). The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences . Math. Z. 208 : 209 – 223 .[Crossref], [Web of Science ®] , [Google Scholar]) follows.

     

REPRESENTATION OF THE SPECTRAL SHIFT FUNCTION AND SPECTRAL ASYMPTOTICS FOR TRAPPING PERTURBATIONS

We obtain in the semi-classical setup of “black-box” long-range perturbations a representation for the derivative of spectral shift function ξ(λ) related to two self-adjoint operators L _j (h), j = 1,2. We show that the derivative ξ′(λ) is estimated by the norms of the cut-off resolvents of the operators L _j (h). Finally, we establish a Weyl type formula for the spectral shift function ξ(λ) generalizing the results of Robert [19] Robert, D. 1994. Relative time-delay for perturbations of elliptic operators and semiclassical asymptotics. J. Funct. Anal., 126: 36–82. [Crossref], [Web of Science ®] , [Google Scholar] and Christiansen [5] Christiansen, T. 1998. Spectral asymptotics for general compactly supported perturbations of the Laplacian on Rⁿ. Comm. P.D.E., 23: 933–947. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar].     

ON FREE PRODUCTS OF ABSOLUTE GALOIS GROUPS. II

In [1] Ershov, Yu. L. 1997. On Free Products of Absolute Galois Groups. Dokl. Math., 56(3): 915–917.  [Google Scholar], the author gave a positive solution to the problem in the survey of Jarden [2] Jarden, M. 1996. “Infinite Galois Theory”. In Handbook of Algebra I Amsterdam: Elsevier Sci.. [Crossref] , [Google Scholar] on the closedness of the class of profinite groups that are isomorphic to absolute Galois groups of fields with respect to finite free products. In [3] Mel'nikov, O. V. 1999. On Free Products of Absolute Galois Groups. Sib. Mat. J., 40(1): 95–99. [Crossref], [Web of Science ®] , [Google Scholar], O. V. Mel'nikov solved this problem for separable profinite groups ([3] Mel'nikov, O. V. 1999. On Free Products of Absolute Galois Groups. Sib. Mat. J., 40(1): 95–99. [Crossref], [Web of Science ®] , [Google Scholar] was done earlier than [1] Ershov, Yu. L. 1997. On Free Products of Absolute Galois Groups. Dokl. Math., 56(3): 915–917.  [Google Scholar]). In the same case, a more exact result on the absolute Galois groups of fields of fixed characteristic was obtained there. The proof proposed in 4-5 Koenigsmann, J. in press. Relatively Projective Groups as Absolute Galois Groups. Haran, D., Jarden, M. and Koenigsmann, J. in press. Free Products of Absolute Galois Groups.   is simpler than that in [1] Ershov, Yu. L. 1997. On Free Products of Absolute Galois Groups. Dokl. Math., 56(3): 915–917.  [Google Scholar] and, in addition, provides the results of Mel'nikov.

On February, 2000, the author (knowing nothing about 4-5 Koenigsmann, J. in press. Relatively Projective Groups as Absolute Galois Groups. Haran, D., Jarden, M. and Koenigsmann, J. in press. Free Products of Absolute Galois Groups.  ) found one more proof of these results. In the author opinion, this proof is the simplest and the construction used in the proof, as well as its properties (cf. Propositio n 1) can have other applications.     

Sequences Determining Mixed Multiplicities and Reductions of Ideals

Abstract

Using the notion of (FC)-sequences in Viêt (2000 Viêt, D. Q. 2000. Mixed multiplicities of arbitrary ideals in local rings. Comm. Algebra, 28(8): 3803–3821. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), this paper presents some results concerning reductions and the vanishing and non-vanishing of mixed multiplicities of finite collection of arbitrary ideals in local rings.     

PROJECTIVITY AND FLATNESS OF A MODULE OVER THE SUBRING OF INVARIANTS

Let k be a field and H a Hopf k-algebra with bijective antipode, R an H-module algebra over k and A = R#H the associated smash product. The fixed subring of R under H is denoted by S. Let P be an R#H-module. Thus P is an S-module. The aim of this paper is to study the projectivity of P as a module over S. We get a generalization of some results of J.J. Garcia and Angel Del Rio [4] Garcia, J. J. and Del Rio, A. 1995. On Flatness and Projectivity of a Ring as a Module Over a Fixed Subring. Mathem. Scandin., 76: 179–192.  [Google Scholar] of Ida Doraiswamy [8] Doraiswamy, I. 1982. Projectivity of Modules Over Rings with Suitable Group Action. Comm. Algebra, 10(8): 787–795. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar] and of ours [[7] Guédénon, T. 1997. Algèbre Homologique Dans la Catégorie Mod_(R#U(g)). J. Algebra, 197(2): 584–614.  [Google Scholar], section 5].     

GORENSTEIN TILED ORDERS

Let Λ = {O, E(Λ)} be a reduced tiled Gorenstein order with Jacobson radical R and J a two-sided ideal of Λ such that Λ ? R ² ? J ? Rⁿ (n ≥ 2). The quotient ring Λ/J is quasi-Frobenius (QF) if and only if there exists p ∈ R ² such that J = pΛ = Λp. We prove that an adjacency matrix of a quiver of a cyclic Gorenstein tiled order is a multiple of a double stochastic matrix. A requirement for a Gorenstein tiled order to be a cyclic order cannot be omitted. It is proved that a Cayley table of a finite group G is an exponent matrix of a reduced Gorenstein tiled order if and only if G = G_k = (2) × ? × (2).

Commutative Gorenstein rings appeared at first in the paper [3] Gorenstein, D. 1952. An Arithmetic Theory of Adjoint Plane Curves. Trans. AMS., 72: 414–436. [Crossref], [Web of Science ®] , [Google Scholar]. Torsion-free modules over commutative Gorenstein domains were investigated in [1] Bass, H. 1963. On the Ubiquity of Gorenstein Rings. Math. Z., 82(1): 8–27. [Crossref] , [Google Scholar]. Noncommutative Gorenstein orders were considered in [2] Drozd, Yu. A., Kirichenko, V. V. and Roiter, A. V. 1967. On Hereditary and Bass Orders. Izv. Akad. Nauk SSSR Ser. Mat., 31: 1415–1436. Math. USSR – Izvestija, 1967, 1, 1357–1375 [Google Scholar] and [10] Roggenkamp, K. W. 1970. Lattices Over Orders II Berlin, Heidelberg, New York: Springer-Verlag. [Crossref] , [Google Scholar]. Relations between Gorenstein orders and quasi-Frobenius rings were studied in [5] Kirichenko, V. V. 1978. On Quasi-Frobenius Rings and Gorenstein Orders. Trudy Math. Steklov Inst., 148: 168–174. (in Russian) [Google Scholar]. Arbitrary tiled orders were considered in [4] Jategaonkar, V. A. 1974. Global Dimension of Tiled Orders Over a Discrete Valuation Ring. Trans. AMS., 196: 313–330. [Crossref], [Web of Science ®] , [Google Scholar], 11-14 Simson, D. 1992. Linear Representations of Partially Ordered Sets and Vector Space Categories, Algebra, Logic and Appl. Vol. 4, Gordon and Breach Science Publishers. Zavadskij, A. G. 1973. The Structure of Orders with Completely Decomposable Representations. Mat. Zametki, 13: 325–335. (in Russian) Zavadskij, A. G. and Kirichenko, V. V. 1976. Torsion-free Modules over Prime Rinqs. Zap. Nauch. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 57: 100–116. J. Soviet. Math. 1979, 11, 598–612 Zavadskij, A. G. and Kirichenko, V. V. 1977. Semimaximal Rings of Finite Type. Mat. Sbornik, 103(No. 3): 323–345. Math. USSR Sbornik, 1977, 32 (3), 273–291 .     

NIVEAU HERMITIEN D'UNE ALGÈBRE DE QUATERNIONS À DIVISION MUNIE D'UNE INVOLUTION DE DEUXIÈME ESPÈCE

ABSTRACT

In this note we show that the hermitian level of a quaternion division algebra with involution of second kind, is always a power of 2, when it is finite. This result holds for a field with trivial or non-trivial involution, and quaternion division algebras with involution of first kind [6] Pfister, A. 1965. Darstellung von -1 als Summe Von Quadraten in Einem Körper. J. London Math. Soc., 40: 159–165. [Crossref], [Web of Science ®] , [Google Scholar], [5] Lewis, D.W. 1988. Sums of Hermitian Squares. Journal of Algebra, 115(2): 446–480.  [Google Scholar], [9] Serhir, A. 1997. Niveau Hermitien de Certaines Algèbres de Quaternions. Communications in Algebra, 25(8): 2531–2538. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar].     

ON RESULTANT CRITERIA AND FORMULAS FOR THE INVERSION OF A POLYNOMIAL MAP

Concerning the inversion of a polynomial map F: K ² ? K ² over an arbitrary field K, it is natural to consider the following questions: (1) Can we find a necessary and sufficient criterion in terms of resultants for F to be invertible with polynomial ((2) resp. rational) inverse such that, this criterion gives an explicit formula to compute the inverse of F in this case? MacKay and Wang [5] McKay, J. and Wang, S. S. 1986. An Inversion Formula for Two Polynomials in Two Variables. J. of Pure and Appl. Algebra., 40: 245–257. [Crossref], [Web of Science ®] , [Google Scholar] gave a partial answer to question (1), by giving an explicit expression of the inverse of F, when F is invertible without constant terms. On the other hand, Adjamagbo and van den Essen [3] Adjamagbo, K. and van den Essen, A. 1990. A Resultant Criterion and Formula for the Inversion of a Polynomial Map in Two Variables. J. of Pure and Appl. Algebra., 64: 1–6. North-Holland [Google Scholar] have fully answered question (2) and have furnished a necessary and sufficient criterion which relies on the existence of some constants λ₁, λ₂ in K ^*. We improve this result by giving an explicit relation between λ₁, λ₂ and constants of the Theorem of MacKay and Wang [5] McKay, J. and Wang, S. S. 1986. An Inversion Formula for Two Polynomials in Two Variables. J. of Pure and Appl. Algebra., 40: 245–257. [Crossref], [Web of Science ®] , [Google Scholar].

Concerning question (2), Adjamagbo and Boury [2] Adjamagbo, K. and Boury, P. 1992. A Resultant Criterion and Formula for the Inversion of a Rational Map in Two Variables. J. of Pure and Appl. Algebra., 79: 1–13. North-Holland [Google Scholar] give a criterion for rational maps which relies on the existence of two polynomials λ₁, λ₂. We also improve this result, by expliciting the relations between these λ₁, λ₂ and the coefficients of F. This improvement enables us, first to give an explicit proof of the corresponding Theorem of Abhyankhar [1] Abhyankar, S. S. 1990. Algebraic Geometry for Scientists and Engineers. Math. Surveys and Monographs., 5: 267–273.  [Google Scholar], and secondly, to give a counter example where these λ₁, λ₂ are not in K ^*, contrary to claim of Yu [6] Yu, J.-T. 1993. Computing Minimal Polynomials and the Inverse via GCP. Comm. Algebra, 21(No.7): 2279–2294.  [Google Scholar].     

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.     

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]).     

DIRECT SUM DECOMPOSITION OF THE PRODUCT OF PREINJECTIVE MODULES OVER RIGHT PURE SEMISIMPLE HEREDITARY RINGS

ABSTRACT

In his recent work, [1] Simson, D. 2000. An Artin Problem for Division Ring Extensions and the Pure Semisimplicity Conjecture, II. J. Algebra, 227: 670–705. [Crossref], [Web of Science ®] , [Google Scholar] and [2] Simson, D. 2001. On Small Right Pure Semisimple Rings and the Structure of their Auslander-Reiten Quiver. Communic. in Algebra, 29 in press[Web of Science ®] , [Google Scholar], on the pure semisimplicity conjecture Simson raised two problems about the structure of the direct sum decomposition of the direct product modulo the direct sum of indecomposable preinjective modules over right pure semisimple hereditary rings. The main goal of this paper is the proof of a theorem that resolves one of these problems and provides a partial answer to the other.     

Evaluating a model of forest succession using fuzzy analysis

In this paper we discuss how to select the optimal policy from a set of possible policies for a model of forest succession, which can be characterized by a set of trees and the corresponding average life-span with each possible tree transition. The transition probabilities are estimated by counting the numbers of sapling trees of each species under a canopy tree. [1] Horn, Henry S. 1975. Forest Succession. Sci. Amer., : 90–98.  [Google Scholar]. In our setting the transition matrix is defined by using the linguistic terms and as a consequence, the expected longevity of each tree is fuzzy. We use the Dempster–Shafer theory [8] Shafer, G. 1976. A Mathematical Theory of Evidence Princeton University Press.  [Google Scholar] ('76) together with techniques of Norton [7] Norton, J. 1988. Limit Theorems for Dempster's Rule of Combination. Theory and Decision, 25(3): 287–313. [Crossref], [Web of Science ®] , [Google Scholar] ('88) and Smetz [9] Smetz, P. 1990. Belief Functions versus Probability Functions. Uncertainty in Artificial Intelligence, 5: 1–8.  [Google Scholar] ('76) to approximate the transition probabilities.     

Lie Algops and Simple Lie Algebras

ABSTRACT

Pairs (A, L) with A a commutative algebra and L a Lie algebra acting on A by derivations, called Lie algops, are studied as algebraic structures over arbitrary fields of arbitrary characteristic. Lie algops possess modules and tensor products—and are considered with respect to a central simple theory.

The simplicity problem of determining the faithful unital simple Lie algops ( A, L ) is of interest since the corresponding Lie algebras AL are usually simple (Jordan, 2000 Jordan , D. A. ( 2000 ). On the simplicity of Lie algebras of derivations of commutative algebras . J. Algebra 228 : 580 – 585 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]). For locally finite Lie algops, and up to purely inseparable descent, this problem reduces by way of closures to the closed central simplicity problem of determining those which are closed central simple.

The simplicity and representation theories for locally nilpotent separably triangulable unital Lie algops are of particular interest because they relate to the problems of classifying simple Lie algebras of Witt type and their representations. Of these, the simplicity theory reduces to that of Jordan Lie algops.

The main Theorems 7.3 and 7.4 reduce the simplicity and representation theories for Jordan Lie algops to the simplicity and representation theories for simple nil and toral Lie algops.     

Self-Dual Hopf Quivers

ABSTRACT

We study self-dual coradically graded pointed Hopf algebras with a help of the dual Gabriel theorem for pointed Hopf algebras (van Oystaeyen and Zhang, 2004 van Oystaeyen , F. , Zhang , P. ( 2004 ). Quiver Hopf algebras . J. Algebra 280 ( 2 ): 577 – 589 . [CSA] [CROSSREF]  [Google Scholar]). The co-Gabriel Quivers of such Hopf algebras are said to be self-dual. An explicit classification of self-dual Hopf quivers is obtained. We also prove that finite dimensional pointed Hopf algebras with self-dual graded versions are generated by group-like and skew-primitive elements as associative algebras. This partially justifies a conjecture of Andruskiewitsch and Schneider (2000 Andruskiewitsch , N. , Schneider , H.-J . ( 2000 ). Finite quantum groups and Cartan matrices . Adv. Math. 154 : 1 – 45 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) and may help to classify finite dimensional self-dual coradically graded pointed Hopf algebras.