Space $$ \mathcal{L} $$( X) of local dynamical systems and the Filippov space Aceu( X)

In the present paper, we establish a relationship between continuous local dynamical systems and spaces of the class A _ceu(X) of the Filippov theory. We suggest a construction method for a space of the class A _ceu(X) on the basis of a locally given dynamical system and conversely, a dynamical system is constructed locally in a specific way on the basis of a given space of the class A _ceu(X). The suggested construction method provides a homeomorphism between the space of all local dynamical systems on a locally compact metric space X and the space A _ceu(X). The obtained results generalize the Filippov theory to locally dynamical systems.     

2-dimensional simple factors ofJ 0(N)

In this paper we investigate abelian varietyA _f which is derived from a newformf ∈ S ₂(Г₀(N)) an is ℚ-simple factors ofJac(X ₀ (N)). We will develop algorithms for computing the period matrix ofA _f and for determing whenA _f is principally polarized. IfA _f is 2-dimensional principally polarized, we give an algorithm for computing the associated hyperelliptic curveC withJac(C)≊A _f.     

Construction of vertex operator algebras from commutative associative algebras

Given a commutative associative algebra A with an associative form (’), we construct a vertex operator algebra V with the weight two space V₂;? A If in addition the form (’) is nondegenerate, we show that there is a simple vertex operator algebra with V₂;? A We also show that if A is semisimple, then the vertex operator algebra constructed is the tensor products of a certain number of Virasoro vertex operator algebras.     

Notes on a generalized Fresnel class

In this paper we establish a theorem for the generalized Fresnel class F _A1,A2 ensuring that various functions are in F _A1,A2. We also prove a translation theorem for the analytic Feynman integral of functions in F _A1,A2.This research was supported in part by the Basic Science Research Program, Ministry of Education.     

The algebra ofG-relations

In this paper, we study a tower {A _n ^G: n} ≥ 1 of finite-dimensional algebras; here, G represents an arbitrary finite group,d denotes a complex parameter, and the algebraA _n ^G(d) has a basis indexed by ‘G-stable equivalence relations’ on a set whereG acts freely and has 2n orbits. We show that the algebraA _n ^G(d) is semi-simple for all but a finite set of values ofd, and determine the representation theory (or, equivalently, the decomposition into simple summands) of this algebra in the ‘generic case’. Finally we determine the Bratteli diagram of the tower {A _n ^G(d): n} ≥ 1 (in the generic case).     

On Compatibility of Interval Fuzzy Preference Relations

This paper defines the concept of compatibility degree of two interval fuzzy preference relations, and gives a compatibility index of two interval fuzzy preference relations. It is proven that an interval fuzzy preference relation B and the synthetic interval fuzzy preference relation of interval fuzzy preference relations A ₁,A ₂,...,A _s are of acceptable compatibility under the condition that the interval fuzzy preference relation B and each of the interval fuzzy preference relations A _l,A ₂,...,A _s are of acceptable compatibility, and thus a theoretic basis has been developed for the application of the interval fuzzy preference relations in group decision making.     

Gröbner–Shirshov bases for brace algebras

Let A be a brace algebra. This structure implies that A is also a pre-Lie algebra. In this paper, we establish Composition-Diamond lemma for brace algebras. For each pre-Lie algebra L, we find a Gröbner–Shirshov basis for its universal brace algebra U_b(L). As applications, we determine an explicit linear basis for U_b(L) and prove that L is a pre-Lie subalgebra of U_b(L).     

A note on skew differential operators on commutative rings

Let A be a commutative integral domain that is a finitely generated algebra over a field k of characteristic 0 and let ø be a k-algebra automorphism of A of finite order m. In this note we study the ring D(A;ø of differential operators introduced by A.D. Bell. We prove that if A is a free module over the fixed sub-ring A ^ø, with a basis containing 1, then D(A;ø) is isomorphic to the matrix ring M_m(D(A ^ø). It follows from Grothendieck's Generic Flatness Theorem that for an arbitrary A there is an element c?Asuch that D(A[c^-1];ø)?M _m(D(A[c^-1]^ø)). As an application, we consider the structure of D(A;ø)when A is a polynomial or Laurent polynomial ring over k and ø is a diagonalizable linear automorphism.     

On Defining Ideals or Subrings of Hall Algebras

Let A be a finitary algebra over a finite field k, and A- \textmod\text{mod} the category of finite dimensional left A-modules. Let H(A)\mathcal{H}(A) be the corresponding Hall algebra, and for a positive integer r let D _r (A) be the subspace of H(A)\mathcal{H}(A) which has a basis consisting of isomorphism classes of modules in A- \textmod\text{mod} with at least r + 1 indecomposable direct summands. If A is the path algebra of the quiver of type A _n with linear orientation, then D _r (A) is known to be the kernel of the map from the twisted Hall algebra to the quantized Schur algebra indexed by n + 1 and r. For any A, we determine necessary and sufficient conditions for D _r (A) to be an ideal and some conditions for D _r (A) to be a subring of H(A)\mathcal{H}(A). For A the path algebra of a quiver, we also determine necessary and sufficient conditions for D _r (A) to be a subring of H(A)\mathcal{H}(A).     

A Note on Relative Efficiency of Axiom Systems

We introduce a notion of relative efficiency for axiom systems. Given an axiom system A_β for a theory T consistent with S¹₂, we show that the problem of deciding whether an axiom system A_α for the same theory is more efficient than A_β is II₂-hard. Several possibilities of speed-up of proofs are examined in relation to pairs of axiom systems A_α, A_β, with A_α ? A_β, both in the case of A_α, A_β having the same language, and in the case of the language of A_α extending that of A_β: in the latter case, letting Pr_α, Pr_β denote the theories axiomatized by A_α, A_β, respectively, and assuming Pr_α to be a conservative extension of Pr_β, we show that if A_α — A_β contains no nonlogical axioms, then A_α can only be a linear speed-up of A_β; otherwise, given any recursive function g and any A_β, there exists a finite extension A_α of A_β such that A_α is a speed-up of A_β with respect to g. Mathematics Subject Classification: 03F20, 03F30.     

Novikov superalgebras with <Emphasis Type="Italic">A</Emphasis><Subscript>0</Subscript> = <Emphasis Type="Italic">A</Emphasis><Subscript>1</Subscript><Emphasis Type="Italic">A</Emphasis><Subscript>1</Subscript>

Novikov superalgebras are related to quadratic conformal superalgebras which correspond to the Hamiltonian pairs and play a fundamental role in completely integrable systems. In this note we show that the Novikov superalgebras with A ₀ = A ₁ A ₁ and dim A ₁ = 2 are of type N and give a class of Novikov superalgebras of type S with A ₀ = A ₁ A ₁.     

Transitioning Between Tableaux and Spider Bases for Specht Modules

Regarding the Specht modules associated to the two-row partition (n, n), we provide a combinatorial path model to study the transitioning matrix from the tableau basis to the A₁-web basis (i.e. cup diagrams), and prove that the entries in this matrix are positive in the upper-triangular portion with respect to a certain partial order.

     

Generation of groups by rank-2 tori

Let k be a field of characteristic different from 2 and 3. In this paper, we study the number of rank-2 k-tori required (in fact exhibit such tori explicitly) for the generation of groups of type A₂, G₂ and F₄ arising from division algebras and subgroups of type D₄ of Aut(A), for A an Albert division algebra, over infinite perfect fields.     

On generalized stable rings

In this paper, we introduce the class of generalized stable rings and investigate equivalent characterizations of such rings. We show that End_R R is a generalized stable ring if and only if any right H-module decompositions M = A ₁ + B ₁ = A ₂ + B ₂ with A ₁ ? A ? A ₂ implies that there exist C,D,E ≤ M such that M = C + D B ₁ = c + E + B ₂ with.C ? A. Also we show that every generalized stable ring is a GE-ring and matrices over generalized stable regular ring can be diagonalized by some weakly invertible matrices.     

Sharp Error Terms for Return Time Statistics under Mixing Conditions

We describe the statistics of repetition times of a string of symbols in a stochastic process. Denote by τ _A the time elapsed until the process spells a finite string A and by S _A the number of consecutive repetitions of A. We prove that, if the length of the string grows unboundedly, (1) the distribution of τ _A , when the process starts with A, is well approximated by a certain mixture of the point measure at the origin and an exponential law, and (2) S _A is approximately geometrically distributed. We provide sharp error terms for each of these approximations. The errors we obtain are point-wise and also allow us to get approximations for all the moments of τ _A and S _A . To obtain (1) we assume that the process is φ-mixing, while to obtain (2) we assume the convergence of certain conditional probabilities.      

Isomorphism and elementary equivalence of multilinear maps

In [14], we proved that two finitely generated finite-by-nilpotent groups G,H are elementarily equivalent if and only if Z×G and Z×H are isomorphic. In the present paper, we obtain similar characterizations of elementary equivalence for the following classes of structures:

1. the (n+2)-tuples (A ₁…,A _n+1,f),where n≥2 is an integerA ₁…,A _n+1 are disjoint finitely generated abelian groups and f A ₁×…×A _n →A _n+1: is a n-linear map;

2. the triples (A,B f), where n≥2 is an integerA,B are disjoint finitely generated abelian groups and f : A ⁿ →B is a n-linear map;

3. the couples (A,f), where n≥2 is an integerA is a finitely generated abelian group and f:A ⁿ →A is a n-linear map.

For each class, we show that elementary equivalence does not imply isomorphism. In particular, we give an example of two nonisomorphic finitely generated torsion-free Lie rings which are elementarily equivalent.     

KK ‐theory of amalgamated free products of C *‐algebras

J. Cuntz has conjectured the existence of two cyclic six terms exact sequences relating the KK ‐groups of the amalgamated free product A _{1 ?? B} A ₂ to the KK ‐groups of A ₁, A ₂ and B. First we establish automatic existence of strict and absorbing homomorphisms. Then we use this result to verify the conjecture when B is a countable direct sum of matrix algebras and the embeddings of B into A ₁ and A ₂ are quasiunital. Inspired by the proof we achieve the following nice classification result: A separable C *‐algebra B is a countable direct sum of matrix algebras if and only if the unitary group of the multiplier algebra U M (B) is compact in the strict topology. Finally we prove the conjecture when the amalgamated free product has the property that any asymptotically split extension of A _{1 ?? B} A ₂ is split. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some Results on Systems of Finite Sets That Satisfy a Certain Intersection Condition

Let S be a finite set, and fix K>2. Let F be a family of subsets of S with the property that whenever A₁,...,A_k are sets in F, not necessarily distinct, and A₁ ? ? ? A_k = ?, then A₁ ? ? ? A_k = S. We prove here that the maximum size of such a family is 2^|S|?1 + 1. If we require that the sets A₁,...,A_k be distinct, then the maximum size of F is again 2^|S|?1 + 1, provided that |S| ≥ log₂(K?2)+3.     

Hopf-Type Algebras

In this article, we provide an alternative approach to the definition of a weak Hopf algebra (WHA). For an associative unital algebra A with a coassociative comultiplication Δ ∈Alg ^u (A, A ? A), the set of homomorphisms from A to A ? A, which do not preserve the units. If the linear maps Ξ₁, Ξ₂ ∈ End(A ? A), defined by Ξ₁(a ? b) = Δ(a)(1 ? b), Ξ₂(a ? b) = (a ? 1)Δ(b), are von Neumann regular elements in the ring End(A ? A) of endomorphisms of A ? A satisfying some appropriate assumptions, we call the A a Hopf-type algebra. We show the existence of a target, a source, a counit, and an antipode of A as in the usual WHA.     

Classification of Finite Completions of a Class Locally D 8 Amalgams

Let 𝒜 = (A ₁, A ₁₂, A ₂) be a locally D ₈ amalgam with a finite completion G. Suppose that A ₁ ∈ Syl ₂(G). We show that under these conditions |A ₁| ≤2⁵, or N _{A 1} (A ₁₂) is Abelian. As applications of our results, we determine all the finite completions G, up to O(G), in the case where N _{A 1} (A ₁₂) is non-Abelian.