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Abstract: In 1.9 of the paper [A. Hahn, A Hermitian Morita theorem for algebras with anti-structure, J. Algebra 93 (1985) 215–235], should be replaced by . This leads to minor changes in the rest of the paper where the ring should be replaced by its opposite and vice versa. [Copyright &y& Elsevier]

In this paper, we provide generating functions and counting formulas for spanning trees and spanning forests in hypergraphs in two different ways: (1) We represent spanning trees and spanning forests in hypergraphs through Berezin-Grassmann integrals on Zeon algebra and hyper-Hafnians (orders and signs are not considered); (2) We establish a Hyper-Pfaffian-Cactus Spanning Forest Theorem through Berezin-Grassmann integrals on Grassmann algebra (orders and signs are considered), which generalizes the Hyper-Pfaffian-Cactus Theorem by Abdesselam (2004) [1] and Pfaffian matrix tree theorem by Masbaum and Vaintrob (2002) [15]. • We provide generating functions and counting formulas for spanning trees and spanning forests in hypergraphs through Berezin-Grassmann integrals on Zeon algebra and hyper- Hafnians when orders and signs are not considered. • We establish a Hyper-Pfaffian-Cactus Spanning Forest Theorem through Berezin-Grassmann integrals on Grassmann algebra (orders and signs are considered), which generalizes the Hyper-Pfaffian-Cactus Theorem by Abdesselam [Advances in Applied Mathematics (2004) Vol. 33: 51-70] and Pfaffian matrix tree theorem by Masbaum and Vaintrob [Internat. Math. Res. Notices (2002) Vol. 27: 1397-1426]. [ABSTRACT FROM AUTHOR]

Text Let m be an integer with m ≥ 2 . For A ⊆ Z m and n ∈ Z m , let R 1 ( A , n ) , R 2 ( A , n ) , R 3 ( A , n ) denote the number of solutions of the equation a + a ′ = n with ordered pairs ( a , a ′ ) ∈ A × A , unordered pairs ( a , a ′ ) ∈ A × A ( a ≠ a ′ ) and unordered pairs ( a , a ′ ) ∈ A × A , respectively. In this paper, for i ∈ { 1 , 2 , 3 } , we determine all sets A , B ⊆ Z m such that R i ( A , n ) = R i ( B , n ) for all n ∈ Z m when the cardinality of the symmetric difference of A and B is small or large. These extend some previous results. We also pose some problems for further research. Video For a video summary of this paper, please visit https://youtu.be/stBa9Uy5U0I . [ABSTRACT FROM AUTHOR]

Text Let K be a complete, algebraically closed, non-Archimedean valued field, and let ϕ ∈ K ( z ) with deg ⁡ ( ϕ ) ≥ 2 . In this paper we consider the family of functions ord Res ϕ n ( x ) , which measure the resultant of ϕ n at points x in P K 1 , the Berkovich projective line, and show that they converge locally uniformly to the diagonal values of the Arakelov–Green's function g μ ϕ ( x , x ) attached to the canonical measure of ϕ . Following this, we are able to prove an equidistribution result for Rumely's crucial measures ν ϕ n , each of which is a probability measure supported at finitely many points whose weights are determined by dynamical properties of ϕ . Video For a video summary of this paper, please visit https://youtu.be/YCCZD1iwe00 . [ABSTRACT FROM AUTHOR]

This paper is devoted to the conception of mathematical objects and methods according to d'Alembert. We first recall his vision of the place of mathematics in the knowledge of nature, then the internal hierarchy of the various fields of this science, based on their degree of abstraction from sensations (§1 and 2). Then we come to the ideas of definitions, primitive ideas , simple ideas , and their generation as well as their generalization (§3 and 4). Then, having looked at what he means by quantities, numbers, quantities, as well as his conception of the objects and rules of algebra as abstract ideas by generalization (§5), we approach the question of the reality of mathematical objects with the example of the irrational (§6). The following paragraphs of the text are devoted to the difficulties encountered in various fields and the way d'Alembert tries to solve them: algebra and negative quantities (§7); principles of geometry (§8); the notion of limit as the basis of infinitesimal calculus (§9). His reflections, even if unfinished, were not without posterity (§10). [ABSTRACT FROM AUTHOR]

Abstract The growth of tropical geometry has generated significant interest in the tropical semiring in the past decade. However, there are other semirings in tropical algebra that provide more information, such as the symmetrized (max ⁡ , +) , Izhakian's extended and Izhakian–Rowen's supertropical semirings. In this paper we identify in which of these upper-bound semirings we can express symmetric polynomials in terms of elementary ones. We show that in the case of idempotent semirings we can do this precisely when the Frobenius property is satisfied, that in the case of supertropical semirings this is always possible, and that in non-trivial symmetrized semirings this is never possible. Our results allow us to determine the tropical algebra semirings where an analogue of the Fundamental Theorem of Symmetric Polynomials holds and to what extent. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study the representation theory of the Fuss–Catalan algebras, FC n (a , b). We prove that this algebra is cellular with a cellular basis and forms a tower of recollement, as defined by Cox, Martin, Parker, and Xi [7] , and hence, it is quasi-hereditary algebra if a , b are non-zero complex numbers. [ABSTRACT FROM AUTHOR]

In this paper we deal with a problem of Pethő related to existence of a quartic algebraic integer α for which β = 4 α 4 α 4 − 1 − α α − 1 is a quadratic algebraic number. By studying rational solutions of certain Diophantine system we prove that there are infinitely many α 's such that the corresponding β is quadratic. Moreover, we present a description of quartic numbers α such that the corresponding β is a quadratic real number. [ABSTRACT FROM AUTHOR]

This paper tackles the history of tactics, a field of investigation at the crossroads of algebra, combinatorics and recreational mathematics. Tactics was only taken up by mathematicians now and then between the 1850s and the 1900s, and its emergence was a process of mathematization of questions linked to the notions of “order” and “position”. To understand the long-term history of this field of investigation—one that became neither a theory nor a discipline—the paper analyzes the different historical configurations in which tactics took on its scientific meaning. It thus investigates how, under the banner of tactics, a continuity could be claimed by mathematicians that were, finally, working in very different scientific and historical context. [ABSTRACT FROM AUTHOR]

In this paper ideals of certain group algebra are used to prove results on k -normal elements over finite fields, including ways to determine them from normal elements by means of polynomials and circulant matrices. Alternative proofs of results appearing in [4] are also provided. [ABSTRACT FROM AUTHOR]

Solving a well-known problem of Maharam, Talagrand [18] constructed an exhaustive non uniformly exhaustive submeasure, thus also providing the first example of a Maharam algebra that is not a measure algebra. To each exhaustive submeasure one can canonically assign a certain countable ordinal, its exhaustivity rank. In this paper, we use carefully constructed Schreier families and norms derived from them to provide examples of exhaustive submeasures of arbitrary high exhaustivity rank. This gives rise to uncountably many non isomorphic separable atomless Maharam algebras. [ABSTRACT FROM AUTHOR]

In this paper Seidel matrices are studied, and their spectrum and several related algebraic properties are determined for order n ≤ 13 . Based on this Seidel matrices with exactly three distinct eigenvalues of order n ≤ 23 are classified. One consequence of the computational results is that the maximum number of equiangular lines in R 12 with common angle 1 ∕ 5 is exactly 20 . [ABSTRACT FROM AUTHOR]

Let F q denote the finite field of order q . In this paper, some new classes of permutation polynomials of the form ( x p m − x + δ ) s + x over F p 2 m are obtained by determining the number of solutions of certain equations. [ABSTRACT FROM AUTHOR]

In this paper, we propose a class of permutation polynomials over the finite field F 2 2 m for odd m . These permutations are generally quadrinomials, and some permutation trinomials can also be obtained. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the expected number of points in F q 2 of multiplicity m , for 0 ≤ m ≤ q + 1 , with respect to a randomly chosen arrangement of q + 1 lines with different slopes, is ( 1 / ( m ! e ) ) q 2 + O ( q ) , as q → ∞ . We further state that the distance between the number of such points in a randomly chosen arrangement and ( 1 / ( m ! e ) ) q 2 is lower than q ln ⁡ q with probability close to 1 for large q . [ABSTRACT FROM AUTHOR]

In this paper, we study the polynomial Gauss sums over finite fields, and present an analogue of Davenport–Hasse's theorem for the polynomial Gauss sums, which is a generalization of the previous result obtained by Hayes. [ABSTRACT FROM AUTHOR]

Let B be a finite set of natural numbers or complex numbers. Product set corresponding to B is defined by B . B : = { a b : a , b ∈ B } . In this paper we give an upper bound for longest length of consecutive terms of a polynomial sequence present in a product set accurate up to a positive constant. We give a sharp bound on the maximum number of Fibonacci numbers present in a product set when B is a set of natural numbers and a bound which is accurate up to a positive constant when B is a set of complex numbers. [ABSTRACT FROM AUTHOR]

In this paper we focus on finding all the factorials expressible as a product of a fixed number of 2 k -nacci numbers with k ≥ 2 . We derive the 2-adic valuation of the 2 k -nacci sequence and use it to establish bounds on the solutions of the initial equation. In addition, we specify a more general family of sequences, for which we can perform a similar procedure. We also investigate a possible connection of these results with p -regular sequences. [ABSTRACT FROM AUTHOR]

Abstract: One of the particularities of information encoded as DNA strands is that a string u contains basically the same information as its Watson–Crick complement, denoted here as . Thus, any expression consisting of repetitions of u and can be considered in some sense periodic. In this paper, we give a generalization of Lyndon and Schützenberger’s classical result about equations of the form , to cases where both sides involve repetitions of words as well as their complements. Our main results show that, for such extended equations, if , then all three words involved can be expressed in terms of a common word t and its complement . Moreover, if , then is an optimal bound. These results are established based on a complete characterization of all possible overlaps between two expressions that involve only some word u and its complement , which is also obtained in this paper. [Copyright &y& Elsevier]

Abstract: This paper studies separating subsets of an invariant ring or, more generally, of any set consisting of functions. We prove that a subset of a finitely generated algebra always contains a finite separating subset. We also show that a general version of Noether’s degree bound holds for separating invariants, independently of the characteristic. While the general finiteness result is non-constructive, the Noether bound provides an easy algorithm for computing separating invariants of finite groups. The paper also contains a conceptual investigation of the difference between separating and generating subsets. [Copyright &y& Elsevier]

Abstract: Quasimodules for vertex algebras are generalizations of modules for vertex algebras. These new objects arise from a generalization of locality for fields. Quasimodules tie together module theory and twisted module theory, and both twisted and untwisted modules feature Poincaré–Birkhoff–Witt-like spanning sets. This paper generalizes these spanning set results to quasimodules for certain Möbius vertex algebras. In particular this paper presents two spanning sets, one featuring a difference-zero ordering restriction on modes and another featuring a difference-one ordering restriction. [Copyright &y& Elsevier]

Abstract: This is the final one in the series of papers where we introduce and study the -algebras associated with topological graphs. In this paper, we get a sufficient condition on topological graphs so that the associated -algebras are simple and purely infinite. Using this result, we give one method to construct all Kirchberg algebras as -algebras associated with topological graphs. [Copyright &y& Elsevier]

Abstract: Let be a class of unital C*-algebras. The class of C*-algebras which can be tracially approximated (in the Egorov-like sense first considered by Lin) by the C*-algebras in is studied (Lin considered the case that consists of finite-dimensional C*-algebras or the tensor products of such with ). In particular, the question is considered whether, for any simple separable , there is a C*-algebra B which is a simple inductive limit of certain basic homogeneous C*-algebras together with C*-algebras in , such that the Elliott invariant of A is isomorphic to the Elliott invariant of B. An interesting case of this question is answered. In the final part of the paper, the question is also considered which properties of C*-algebras are inherited by tracial approximation. (Results of this kind are obtained which are used in the proof of the main theorem of the paper, and also in the proof of the classification theorem of the second author given in [Z. Niu, A classification of tracially approximately splitting tree algebra, in preparation] and [Z. Niu, A classification of certain tracially approximately subhomogeneous C*-algebras, PhD thesis, University of Toronto, 2005]—which also uses the main result of the present paper.) [Copyright &y& Elsevier]

Abstract: This paper deals with a “naive” way of generalizing Kazhdan''s property (T) to -algebras. Our approach differs from the approach of Connes and Jones, which has already demonstrated its utility. Nevertheless, it turns out that our approach is applicable to a rather subtle question in the theory of -Hilbert modules. Namely, we prove that a separable unital -algebra A has property MI (module infinite—i.e. any countably generated self-dual Hilbert module over A is finitely generated and projective) if and only if A does not satisfy our definition of property (T). The commutative case was studied in an earlier paper. [Copyright &y& Elsevier]

Abstract: In this paper, we prove that for any given , there exists a constant such that for any prime power , there exists a primitive normal polynomial of degree n over with the first coefficients prescribed, where the first coefficient is nonzero. This result strengthens the asymptotic result of the existence of primitive polynomials with the first coefficients prescribed [S.Q. Fan, W.B. Han, p-Adic formal series and Cohen''s problem, Glasg. Math. J. 46 (2004) 47–61] in two aspects. One is that we discuss in this paper not only the primitivity but also the normality. Another is that the number of the prescribed coefficients increases from to . The estimates of character sums over Galois rings, the p-adic method introduced by the first two authors, and the computation technique used in [S.Q. Fan, W.B. Han, Primitive polynomial with three coefficients prescribed, Finite Fields Appl. 10 (2004) 506–521; D. Mills, Existence of primitive polynomials with three coefficients prescribed, J. Algebra Number Theory Appl. 4 (2004) 1–22] are the main tools to get the above result. [Copyright &y& Elsevier]

Abstract: The main result in the paper states the following: Let π be a set of odd primes. Let the finite group be the product of a π-decomposable subgroup and a π-subgroup B. Then ; equivalently the group G possesses Hall π-subgroups. In this case is a Hall π-subgroup of G. This result extends previous results of Berkovich (1966), Rowley (1977), Arad and Chillag (1981) and Kazarin (1980) where stronger hypotheses on the factors A and B of the group G were being considered. The results under consideration in the paper provide in particular criteria for the existence of non-trivial soluble normal subgroups for a factorized group G. [Copyright &y& Elsevier]

Abstract: A splitting field of a central simple algebra is said to be absolute Galois if it is Galois over some fixed subfield of the centre of the algebra. The paper proves an existence theorem for such fields over global fields with enough roots of unity. As an application, all twisted function fields and all twisted Laurent series rings over symbol algebras (or p-algebras) over global fields are crossed products. An analogous statement holds for division algebras over Henselian valued fields with global residue field. The existence of absolute Galois splitting fields in central simple algebras over global fields is equivalent to a suitable generalization of the weak Grunwald–Wang theorem, which is proved to hold if enough roots of unity are present. In general, it does not hold and counter examples have been used in noncrossed product constructions. This paper shows in particular that a certain computational difficulty involved in the construction of explicit examples of noncrossed product twisted Laurent series rings cannot be avoided by starting the construction with a symbol algebra. [Copyright &y& Elsevier]

Abstract: In the first part of the paper we give a characterization for an associative algebra to be standardly stratified in the sense of Cline, Parshall and Scott, generalizing a theorem of V. Dlab. In the second part of the paper we construct characteristic tilting modules for standardly stratified algebras and use them to estimate the finitistic dimension of such algebras. These tilting modules give rise to the Ringel duality concept for stratified algebras. We also define and investigate a generalization of the notion of properly stratified algebras to the above setup. [Copyright &y& Elsevier]

Abstract: In this paper we study division algebras over the function fields of curves over . The first and main tool is to view these fields as function fields over nonsingular S which are projective of relative dimension 1 over the p adic ring . A previous paper showed such division algebras had index bounded by assuming the exponent was n and n was prime to p. In this paper we consider algebras of prime degree (and hence exponent) and show these algebras are cyclic. We also find a geometric criterion for a Brauer class to have index q. [Copyright &y& Elsevier]

Abstract: In this paper we use the Etingof–Kazhdan quantization of Lie bi-superalgebras to investigate some interesting questions related to Drinfeld–Jimbo type superalgebra associated to a Lie superalgebra of classical type. It has been shown that the D-J type superalgebra associated to a Lie superalgebra of type A-G, with the distinguished Cartan matrix, is isomorphic to the E-K quantization of the Lie superalgebra. The first main result in the present paper is to extend this to arbitrary Cartan matrices. This paper also contains two other main results: (1) a theorem stating that all highest weight modules of a Lie superalgebra of type A-G can be deformed to modules over the corresponding D-J type superalgebra and (2) a super version of the Drinfeld–Kohno theorem. [Copyright &y& Elsevier]

Abstract: The algebraic theory of variable-length codes was initiated by Schützenberger in the 1950s. Almost all the subsequently stated results in this theory are constructive and therefore lead to algorithms. However, there are some basic problems that are still open. For instance, we still do not know how to decide whether a finite code can be embedded in a finite maximal one. We answer this question under additional hypotheses. Precisely, let A be a finite alphabet with , let . Let , let be the number of factors in the prime factorization of n. We give an algorithm to decide whether there exists n, with , and a finite maximal code C which is also factorizing such that . We recall that a factorizing code is a finite maximal code which satisfies the factorization conjecture, proposed by Schützenberger. The above-mentioned statement is a consequence of another result proved in this paper. Namely, given a factorizing code C, it is known that the words in satisfy a property defined by using factorizations of cyclic groups. In this paper we give an algorithm to decide whether a set can be embedded in a set satisfying . Furthermore, we prove that, conversely, for each set satisfying , under additional hypotheses, there exists a factorizing code C such that and, as a consequence, is a code. In this case, C can be constructed starting with prefix/suffix codes and by using two types of operations on codes (composition and substitution). The additional required hypotheses concern the structure of the factorizations involved and are always satisfied when, for each , we have , with . In addition, we prove that there exist sets which satisfy and which are not codes. [Copyright &y& Elsevier]

Abstract: Rickard proved in his paper [J. Rickard, Equivalences of derived categories for symmetric algebras, J. Algebra 257 (2002) 460–481] that if Λ is a finite-dimensional symmetric k-algebra and if there is a set of objects in satisfying some conditions, then there is a derived equivalence taking these objects to the simple modules of another algebra Γ. In this paper we generalize Rickard''s results to finite-dimensional selfinjective k-algebras by adding an extra condition. We use the techniques of Rickard''s paper in this paper. [Copyright &y& Elsevier]

Abstract: Using new techniques of Zelmanov, C. Martinez improved on work of Jacobson, McCrimmon, and Parvathi to give a necessary and sufficient Ore-type condition for an arbitrary linear Jordan algebra (with no 2- or 3-torsion) to have an algebra of fractions. In this paper we extend to quadratic algebras the concept of algebras of outer fractions with respect to an Ore monad, and describe necessary and sufficient Ore-type conditions for the embedding in such an algebra of fractions. The details of the actual embedding will appear in a subsequent paper. [Copyright &y& Elsevier]

Abstract: This is the second in a series of four papers on fixed point ratios in non-subspace actions of finite classical groups. Our main result states that if G is a finite almost simple classical group and Ω is a faithful transitive non-subspace G-set then either for all elements of prime order, or is one of a small number of known exceptions. In this paper we record a number of preliminary results and prove the main theorem in the case where the stabiliser is contained in a maximal non-subspace subgroup which lies in one of the Aschbacher families , where . [Copyright &y& Elsevier]

Abstract: We construct a new equivariant cohomology theory for a certain class of differential vertex algebras, which we call the chiral equivariant cohomology. A principal example of a differential vertex algebra in this class is the chiral de Rham complex of Malikov–Schechtman–Vaintrob of a manifold with a group action. The main idea in this paper is to synthesize the algebraic approach to classical equivariant cohomology due to H. Cartan, 2 with [2] Cartan''s theory was further developed by Duflo–Kumar–Vergne [M. Duflo, S. Kumar, M. Vergne, Sur la cohomologie équivariante des variétés différentiables, Astérisque 215 (1993)] and Guillemin–Sternberg [V. Guillemin, S. Sternberg, Supersymmetry and Equivariant de Rham Theory, Springer, 1999]. This paper follows closely the latter approach. the theory of differential vertex algebras, by using an appropriate notion of invariant theory. We also construct the vertex algebra analogues of the Mathai–Quillen isomorphism, the Weil and the Cartan models for equivariant cohomology, and the Chern–Weil map. We give interesting cohomology classes in the new theory that have no classical analogues. [Copyright &y& Elsevier]

Abstract: This paper is devoted to the conjecture saying that, for any connected locally finite graph and any vertex-transitive group of automorphisms of , at least one of the following assertions holds: (1) There exists an imprimitivity system of on with finite (maybe one-element) blocks such that the stabilizer of a vertex of the factor graph in the induced group of automorphisms is finite. (2) The graph is hyperbolic (i.e., for some positive integer , the graph defined by and contains the regular tree of valency 3). Our approach to the conjecture consists in fixing a finite permutation group and considering the conjecture under the assumption that the stabilizer of a vertex of in induces on the neighborhood of the vertex a group permutation isomorphic to . In the paper we elaborate a method (the modified track method) which allows us to prove the conjecture for many groups . The paper consists of two parts. The present first part of the paper involves results on which the modified track method arguments are based, and a few first applications of the method. The second part is devoted to applications of the modified track method. [Copyright &y& Elsevier]

Abstract: Let A be a symmetric k-algebra over a perfect field k. Külshammer defined for any integer n a mapping on the degree 0 Hochschild cohomology and a mapping on the degree 0 Hochschild homology of A as adjoint mappings of the respective p-power mappings with respect to the symmetrising bilinear form. In an earlier paper it is shown that is invariant under derived equivalences. In the present paper we generalise the definition of to higher Hochschild homology and show the invariance of κ and its generalisation under derived equivalences. This provides fine invariants of derived categories. [Copyright &y& Elsevier]

Abstract: In this paper we complete a distributional Fourier analysis developed by Howell in a serie of papers. We investigate convolution operators in the corresponding distribution spaces. [Copyright &y& Elsevier]

Abstract: The verbally prime algebras are well understood in characteristic 0 while over a field of positive characteristic little is known about them. In previous papers we discussed some sharp differences between these two cases for the characteristic, and we showed that the so-called Tensor Product Theorem is in part no longer valid in the second case. In this paper we study the Gelfand–Kirillov dimension of the relatively free algebras of verbally prime and related algebras. We compute the GK dimensions of several algebras and thus obtain a new proof of the fact that the algebras and are not PI equivalent in characteristic . Furthermore we show that the following algebras are not PI equivalent in positive characteristic: and ; and when , , and ; and finally, and . Here E stands for the infinite-dimensional Grassmann algebra with 1, and is the subalgebra of of the block matrices with blocks and on the main diagonal with entries from , and off-diagonal entries from ; is the natural grading on E. [Copyright &y& Elsevier]

Abstract: This paper studies Frobenius powers of parameter ideals in a commutative Noetherian local ring R of prime characteristic p. For a given ideal of R, there is a power Q of p, depending on , such that the Qth Frobenius power of the Frobenius closure of is equal to the Qth Frobenius power of . The paper addresses the question as to whether there exists a uniform which ‘works’ in this context for all parameter ideals of R simultaneously. In a recent paper, Katzman and Sharp proved that there does exists such a uniform when R is Cohen–Macaulay. The purpose of this paper is to show that such a uniform exists when R is a generalized Cohen–Macaulay local ring. A variety of concepts and techniques from commutative algebra are used, including unconditioned strong d-sequences, cohomological annihilators, modules of generalized fractions, and the Hartshorne–Speiser–Lyubeznik Theorem employed by Katzman and Sharp in the Cohen–Macaulay case. [Copyright &y& Elsevier]

Abstract: In [I. Gelfand, V. Retakh, S. Serconek, R.L. Wilson, On a class of algebras associated to directed graphs, Selecta Math. (N.S.) 11 (2005), math.QA/0506507] I. Gelfand and the authors of this paper introduced a new class of algebras associated to directed graphs. In this paper we show that these algebras are Koszul for a large class of layered graphs. [Copyright &y& Elsevier]

Abstract: Consider the enveloping algebra of a complex semisimple Lie algebra . The heart of a prime ideal I of is the center of the total ring of fractions of . This is an extension field of the field of fractions of the center of . Let d be the degree of this field extension. An old problem of J. Dixmier asked whether . A recent paper of the second author [R. Rentschler, A negative answer to the problem of Dixmier on hearts of prime quotients of enveloping algebras, preprint, 2004] gave a negative answer by an example in . The present paper provides many more examples, involving the so-called sheets of primitive ideals introduced and studied by A. Joseph and the first author in [W. Borho, A. Joseph, Sheets and topology of primitive spectra for semisimple Lie algebras, J. Algebra 244 (2001) 76–167]. A sheet corresponds to a prime ideal I which has a heart of degree d. The main result of this paper is that d equals the covering degree of the sheet as introduced in [W. Borho, A. Joseph, Sheets and topology of primitive spectra for semisimple Lie algebras, J. Algebra 244 (2001) 76–167, 8.7]. [Copyright &y& Elsevier]

Abstract: The main aim of this paper is the adaptation of the results of the papers [Y.A. Bahturin, S.K. Sehgal, M.V. Zaicev, Group gradings on associative algebras, J. Algebra 241 (2) (2001) 677–698; Y.A. Bahturin, S.K. Sehgal, M.V. Zaicev, Finite-dimensional graded simple algebras, preprint, and M.V. Zaĭtsev, S.K. Segal, Finite gradings of simple Artinian rings, Vestnik Moskov. Univ. Ser. I Mat. Mekh., vol. 3, 2001, pp. 21–24, 77 (in Russian); translation in Moscow Univ. Math. Bull. 56 (3) (2001) 21–24] about the structure of graded simple rings from the gradings by groups to the gradings by cancellative semigroups. [Copyright &y& Elsevier]

Abstract: Any finite-dimensional Hopf algebra has a left and a right integral. Conversely, Larsen and Sweedler showed that, if a finite-dimensional algebra with identity and a comultiplication with counit has a faithful left integral, it has to be a Hopf algebra. In this paper, we generalize this result to possibly infinite-dimensional algebras, with or without identity. We have to leave the setting of Hopf algebras and work with multiplier Hopf algebras. Moreover, whereas in the finite-dimensional case, there is a complete symmetry between the bialgebra and its dual, this is no longer the case in infinite dimensions. Therefore we consider a direct version (with integrals) and a dual version (with cointegrals) of the Larson–Sweedler theorem. We also add some results about the antipode. Furthermore, in the process of this paper, we obtain a new approach to multiplier Hopf algebras with integrals. [Copyright &y& Elsevier]

Abstract: This paper distinguishes two categories of questions that the Price equation can help us answer. The two different types of questions require two different disciplines that are related, but nonetheless move in opposite directions. These disciplines are probability theory on the one hand and statistical inference on the other. In the literature on the Price equation this distinction is not made. As a result of this, questions that require a probability model are regularly approached with statistical tools. In this paper, we examine the possibilities of the Price equation for answering questions of either type. By spending extra attention on mathematical formalities, we avoid the two disciplines to get mixed up. After that, we look at some examples, both from kin selection and from group selection, that show how the inappropriate use of statistical terminology can put us on the wrong track. Statements that are ‘derived’ with the help of the Price equation are, therefore, in many cases not the answers they seem to be. Going through the derivations in reverse can, however, be helpful as a guide how to build proper (probabilistic) models that do give answers. [Copyright &y& Elsevier]

Abstract: Double Hurwitz numbers count branched covers of with fixed branch points, with simple branching required over all but two points 0 and , and the branching over 0 and specified by partitions of the degree (with m and n parts, respectively). Single Hurwitz numbers (or more usually, Hurwitz numbers) have a rich structure, explored by many authors in fields as diverse as algebraic geometry, symplectic geometry, combinatorics, representation theory, and mathematical physics. The remarkable ELSV formula relates single Hurwitz numbers to intersection theory on the moduli space of curves. This connection has led to many consequences, including Okounkov and Pandharipande''s proof of Witten''s conjecture. In this paper, we determine the structure of double Hurwitz numbers using techniques from geometry, algebra, and representation theory. Our motivation is geometric: we give evidence that double Hurwitz numbers are top intersections on a moduli space of curves with a line bundle (a universal Picard variety). In particular, we prove a piecewise-polynomiality result analogous to that implied by the ELSV formula. In the case (complete branching over one point) and n is arbitrary, we conjecture an ELSV-type formula, and show it to be true in genus 0 and 1. The corresponding Witten-type correlation function has a richer structure than that for single Hurwitz numbers, and we show that it satisfies many geometric properties, such as the string and dilaton equations, and an Itzykson–Zuber-style genus expansion ansatz. We give a symmetric function description of the double Hurwitz generating series, which leads to explicit formulae for double Hurwitz numbers with given m and n, as a function of genus. In the case where m is fixed but not necessarily 1, we prove a topological recursion on the corresponding generating series, which leads to closed-form expressions for double Hurwitz numbers and an analogue of the Goulden–Jackson polynomiality conjecture (an early conjectural variant of the ELSV formula). In a later paper (Faber''s intersection number conjecture and genus 0 double Hurwitz numbers, 2005, in preparation), the formulae in genus 0 will be shown to be equivalent to the formulae for “top intersections” on the moduli space of smooth curves . For example, three formulae we give there will imply Faber''s intersection number conjecture (in: Moduli of Curves and Abelian Varieties, Aspects of Mathematics, vol. E33, Vieweg, Braunschweig, 1999, pp. 109–129) in arbitrary genus with up to three points. [Copyright &y& Elsevier]

Abstract: In this paper we describe software for an efficient factorization of polynomials over global fields . The algorithm for function fields was recently incorporated into our system KANT. The method is based on a generic algorithm developed by the second author in an earlier paper in this journal. Besides algorithmic aspects not contained in that paper we give details about the current implementation and about some complexity issues as well as a few illustrative examples. Also, a generalization of the application of LLL reduction for factoring polynomials over arbitrary global fields is developed. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate certain ideals in the center of a symmetric algebra A over an algebraically closed field of characteristic . These ideals include the Higman ideal and the Reynolds ideal. They are closely related to the p-power map on A. We generalize some results concerning these ideals from group algebras to symmetric algebras, and we obtain some new results as well. In case , these ideals detect odd diagonal entries in the Cartan matrix of A. In a sequel to this paper, we will apply our results to group algebras. [Copyright &y& Elsevier]

Abstract: In an earlier paper [T. Košir, B.A. Sethuraman, Determinantal varieties over truncated polynomial rings, J. Pure Appl. Algebra 195 (2005) 75–95] we had begun a study of the components and dimensions of the spaces of th order jets of the classical determinantal varieties: these are the varieties obtained by considering generic () matrices over rings of the form , and for some fixed r, setting the coefficients of powers of t of all minors to zero. In this paper, we consider the case where , and provide a Groebner basis for the ideal which defines the tangent bundle to the classical determinantal variety. We use the results of these Groebner basis calculations to describe the components of the varieties where r is arbitrary. (The components of and were already described in the above cited paper.) [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to characterize QM-abelian surfaces which has a model of -type over . The “special” involutions on a corresponding indefinite quaternion algebra (which is defined in Section 2) play an essential role in this paper. [Copyright &y& Elsevier]