On (p a,p, p a,p a−1 )-relative difference sets

Abelian relative difference sets of parameters (m, n, k, )=(p ^a , p, p ^a , p ^a–1 )are studied in this paper. In particular, we show that for an abelian groupG of orderp ^2c+1 and a subgroupN ofG of orderp, a (p ^2c , p, p ^2c , p ^2c–1 )-relative difference set exists inG relative toN if and only if exp (G)p ^c+1 .Furthermore, we have some structural results on (p ^2c p, p ^2c , p ^2c–1 )-relative difference sets in abelian groups of exponentp ^c+1 . We also show that for an abelian groupG of order 2^2c+2 and a subgroupN ofG of order 2, a (2^2c+1, 2, 2^2c+1, 2^2c )-relative difference set exists inG relative toN if and only if exp(G)2 ^c+2 andN is contained in a cyclic subgroup ofG of order 4. New constructions of (p ^2c+1 , p, p ^2c+1 , p ^2c )-relative difference sets, wherep is an odd prime, are given. However, we cannot find the necessary and sufficient condition for this case.     

The Automorphism Groups of Steiner Triple Systems Obtained by the Bose Construction

The automorphism group of the Steiner triple system of order v 3 (mod 6), obtained from the Bose construction using any Abelian Group G of order 2s + 1, is determined. The main result is that if G is not isomorphic to Z ₃ ⁿ × Z ₉ ^m , n 0, m 0, the full automorphism group is isomorphic to Hol(G) × Z ₃, where Hol(G) is the Holomorph of G. If G is isomorphic to Z ₃ ⁿ × Z ₉ ^m , further automorphisms occur, and these are described in full.     

Characterization,structure and analysis on abelian ℒ∞ groups

An abelian topological group is an group if and only if it is a locally -compactk-space and every compact subset in it is contained in a compactly generated locally compact subgroup. Every abelian groupG is topologically isomorphic to G ₀ where ₀ andG ₀ is an abelian group where every compact subset is contained in a compact subgroup. Intrinsic definitions of measures, convolution of measures, measure algebra,L ₁-algebra, Fourier transforms of abelian groups are given and their properties are studied.     

Elementary abelian p-subgroups of algebraic groups

Let be an algebraically closed field and let G be a finite-dimensional algebraic group over which is nearly simple, i.e. the connected component of the identity G ⁰ is perfect, C _G(G ⁰)=Z(G ⁰) and G ⁰/Z(G ⁰) is simple. We classify maximal elementary abelian p-subgroups of G which consist of semisimple elements, i.e. for all primes p char .Call a group quasisimple if it is perfect and is simple modulo the center. Call a subset of an algebraic group toral if it is in a torus; otherwise nontoral. For several quasisimple algebraic groups and p=2, we define complexity, and give local criteria for whether an elementary abelian 2-subgroup of G is toral.For all primes, we analyze the nontoral examples, include a classification of all the maximal elementary abelian p-groups, many of the nonmaximal ones, discuss their normalizers and fusion (i.e. how conjugacy classes of the ambient algebraic group meet the subgroup). For some cases, we give a very detailed discussion, e.g. p=3 and G of type E ₆, E ₇ and E ₈. We explain how the presence of spin up and spin down elements influences the structure of projectively elementary abelian 2-groups in Spin(2n, ). Examples of an elementary abelian group which is nontoral in one algebraic group but toral in a larger one are noted.Two subsets of a maximal torus are conjugate in G iff they are conjugate in the normalizer of the torus; this observation, with our discussion of the nontoral cases, gives a detailed guide to the possibilities for the embedding of an elementary abelian p-group in G. To give an application of our methods, we study extraspecial p-groups in E ₈( ).Dedicated to Jacques Tits for his sixtieth birthday     

On Groups with Finite Involution and Locally Finite 2-Isolated Subgroup of Even Period

A proper subgroup H of a group G is said to be strongly isolated if it contains the centralizer of any nonidentity element of H and 2-isolated if the conditions >C _G(g) H 1 and 2(C_G(g)) imply that C_G(g)H. An involution i in a group G is said to be finite if |ii ^g| < (for any g G). In the paper we study a group G with finite involution i and with a 2-isolated locally finite subgroup H containing an involution. It is proved that at least one of the following assertions holds:1) all 2-elements of the group G belong to H;2) (G,H) is a Frobenius pair, H coincides with the centralizer of the only involution in H, and all involutions in G are conjugate;3) G=FFC_G(i) is a locally finite Frobenius group with Abelian kernel F;4) H=V D is a Frobenius group with locally cyclic noninvariant factor D and a strongly isolated kernel V, U=O₂(V) is a Sylow 2-subgroup of the group G, and G is a Z-group of permutations of the set =U ^g g G.     

Difference Sets Corresponding to a Class of Symmetric Designs

We study difference sets with parameters(v, k, ) = (p ^s(r ^2m - 1)/(r - 1), p ^s-1 r ^2m-2 r - 1)r ^{2m -2}, where r = r ^s - 1)/(p - 1) and p is a prime. Examples for such difference sets are known from a construction of McFarland which works for m = 1 and all p,s. We will prove a structural theorem on difference sets with the above parameters; it will include the result, that under the self-conjugacy assumption McFarland's construction yields all difference sets in the underlying groups. We also show that no abelian .160; 54; 18/-difference set exists. Finally, we give a new nonexistence prove of (189, 48, 12)-difference sets in Z ₃ × Z ₉ × Z ₇.     

On the Coates–Sinnott Conjecture

Let E signify a totally real Abelian number field with a prime power conductor and ring of pintegers R _E for a prime p. Let G denote the Galois group of E over the rationals, and let be a padic character of G of order prime to p. Theorem A calculates, under a minor restriction on , the Fitting ideals of H ² _ét(R _E;Z _p (n/2+1))() over Z _p [G](). Here we require that n2 mod 4. These Fitting ideals are principal and generated by a Stickelberger element. This gives a partial verification and also a strong indication of the Coates–Sinnott conjecture.     

Elementary Abelian Covers of Graphs

Let _G (X) be the set of all (equivalence classes of) regular covering projections of a given connected graph X along which a given group G Aut X of automorphisms lifts. There is a natural lattice structure on _G (X), where _{1 2} whenever ₂ factors through ₁. The sublattice _G () of coverings which are below a given covering : X^~ X naturally corresponds to a lattice _G () of certain subgroups of the group of covering transformations. In order to study this correspondence, some general theorems regarding morphisms and decomposition of regular covering projections are proved. All theorems are stated and proved combinatorially in terms of voltage assignments, in order to facilitate computation in concrete applications.For a given prime p, let _G ^p (X) _G (X) denote the sublattice of all regular covering projections with an elementary abelian p-group of covering transformations. There is an algorithm which explicitly constructs _G ^p (X) in the sense that, for each member of _G ^p (X), a concrete voltage assignment on X which determines this covering up to equivalence, is generated. The algorithm uses the well known algebraic tools for finding invariant subspaces of a given linear representation of a group. To illustrate the method two nontrival examples are included.     

L p Spectral Independence of Elliptic Operators via Commutator Estimates

Let {T _p:q ₁ p q ₂} be a family of consistent C ₀ semigroups on L ^p(), with q ₁,q ₂ [1,) and open. We show that certain commutator conditions on T _p and on the resolvent of its generator A _p ensure the p independence of the spectrum of A _p for p [q ₁,q ₂.Applications include the case of Petrovskij correct systems with Hölder continuous coefficients, Schrödinger operators, and certain elliptic operators in divergence form with real, but not necessarily symmetric, or complex coefficients.     

Prime and Primitive Ideals in Graded Deformations of Algebraic Quantum Groups at Roots of Unity

Let G be a connected, simply connected complex semisimple Lie group of rank n. The deformations employed by Artin, Schelter and Tate, and Hodges, Levasseur and Toro can be applied to the single parameter quantizations, at roots of unity, of the Hopf algebra of regular functions on G. Each of the resulting complex multiparameter quantum groups F _,p [G] depends on both a suitable root of unity and an antisymmetric bicharacter p: Z ⁿ ×Z ⁿ C ^×. These quantizations differ significantly from their single parameter (root-of-unity) counterparts, and, in particular, may have infinite-dimensional irreducible representations. Our approach to F _,p [G] depends on a natural ×-action thereon, where is an n-torus, and our main result offers a classification of the primitive ideals: We use a multiparameter quantum Frobenius map to provide a bijection from (PrimF _,p [G])/× onto G/H×H, where H is a maximal torus of G. In the single parameter case, this bijection is a consequence of work by De Concini and Lyubashenko, and De Concini and Procesi; our results require their analysis. Our methods also exploit earlier work by Moeglin and Rentschler concerning actions of algebraic groups on complex Noetherian algebras. In contrast to generic quantizations of the coordinate ring of G, the primitive spectrum of F _,p [G] is not finitely stratified by the torus action.     

The empirical process of a short-range dependent stationary sequence under Gaussian subordination

Summary Consider the stationary sequenceX ₁=G(Z ₁),X ₂=G(Z ₂),..., whereG(·) is an arbitrary Borel function andZ ₁,Z ₂,... is a mean-zero stationary Gaussian sequence with covariance functionr(k)=E(Z ₁ Z _k+1) satisfyingr(0)=1 and _k=1 |r(k)| ^m < , where, withI{·} denoting the indicator function andF(·) the continuous marginal distribution function of the sequence {X _n }, the integerm is the Hermite rank of the family {I{G(·) x} –F(x):xR}. LetF _n (·) be the empirical distribution function ofX ₁,...,X _n . We prove that, asn, the empirical processn ^1/2{F _n (·)-F(·)} converges in distribution to a Gaussian process in the spaceD[–,].Partially supported by NSF Grant DMS-9208067     

Subsets of a Hilbert space,having finite hausdorff dimension

Let X₂, X₂ be Hilbert spaces, X₂ X₁, X₂ is dense in X₁, the imbedding is compact,m X₂, dim_H ⁱ m and h⁽ⁱ⁾(m) are the Hausdorff dimension and the limit capacity (information dimension) of the setm with respect to the metrics of the spaces X_i (i=1, 2). Two examples are constructed. 1) An example of a setm bounded in X₂, such that: a) h⁽¹⁾(m) < (and, consequently, dim_H ¹ m); b)m cannot be covered by a countable collection of sets, compact in X₂ (and, consequently, dim_H ² m=). 2) an Example of a setm, compact in X₂, such that h⁽¹⁾(m) < and h⁽²⁾(m)=.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 163, pp. 154–165, 1987.     

Williamson Matrices and a Conjecture of Ito's

We point out an interesting connection between Williamson matrices and relative difference sets in nonabelian groups. As a consequence, we are able to show that there are relative (4t, 2, 4t, 2t)-difference sets in the dicyclic groups Q _8t = a, b|a ^4t = b ⁴ = 1, a ^2t = b ², b ^-1ab = a^-1 for all t of the form t = 2^a · 10 ^b · 26 ^c · m with a, b, c 0, m 1\ (mod 2), whenever 2m-1 or 4m-1 is a prime power or there is a Williamson matrix over _m. This gives further support to an important conjecture of Ito IT5 which asserts that there are relative (4t, 2, 4t, 2t)-difference sets in Q _8t for every positive integer t. We also give simpler alternative constructions for relative (4t, 2, 4t, 2t)-difference sets in Q _8t for all t such that 2t - 1 or 4t - 1 is a prime power. Relative difference sets in Q _8t with these parameters had previously been obtained by Ito IT1. Finally, we verify Ito's conjecture for all t 46.     

Dimensions of irreducible modules over twisted group algebras

LetN be a normal subgroup of a finite groupG, letF be an algebraically closed field, letZ ²(G, F ^*) and letV be an irreducible module over the twisted group algebraF . If charF=p>0 divides (GN), assume thatG/N isp-solvable. It is proved that dim _F V divides (GN)d whered is the dimension of an irreducible constituent ofV _N. The special case where=1 andN is abelian yields a well-known theorem of Dade [3]. Another special case, namely whereN is abelian, charF(GN) and the restriction of ofNxN is a coboundary is a generalization of the main result of Ng [5].     

Multivalued Stochastic Differential Equations: Convergence of a Numerical Scheme

In this paper we show the strong mean square convergence of a numerical scheme for a R ^d -multivalued stochastic differential equation: dX _t +A(X _t )dtb(t,X _t )dt+(t,X _t )dW _t and obtain the rate of convergence O(( log(1/)^1/2) when the diffusion coefficient is bounded. By introducing a discrete Skorokhod problem, we establish L ^p -estimates (p2) for the solutions and prove the convergence by using a deterministic result. Numerical experiments for the rate of convergence are presented.     

An extension of the level-2 paramodular group,and the Barth-Nieto quintic

In this note we construct a maximal discrete extension of _1,p(2), the paramodular group with a full level-2 structure. The corresponding Siegel modular variety parametrizes (birationally) the space of Kummer surfaces associated to (1, p)-polarized abelian surfaces with a level-2 structure. In the case p=3 this is related to the Barth-Nieto quintic and in this case we also determine the space of cusp forms of weight 3.Send offprint requests to:M. Friedland, Bockeroder Weg 4a, 31832 Springe, Germany.     

An Investigation of the Fields of Bounded Formal Power Series by Means of Theory of Cuts

We consider a class K of real closed fields F, |F|=|G|=₁, where G is a group of Archimedean classes of F, and cofinality of each symmetric gap of F is ₁. We will show that this class is exactly a class of all bounded formal power series RG,₁, where G is a divisible Abelian group, card(G)=₁, under CH. A nonstandard real line ^*R, which is ₁-set belongs to this class; we will also consider a construction RG(L,P),₁ of fields from this class, where L is a totally ordered set, P is a totally ordered field, G(L,P) is a group of finite words. It will be describes symmetric gaps of such two fields in K, which are not ₁-set. Mathematics Subject Classifications (2000) 03E04, 12J15, 12J25.The work was supported by grant of Ministry of Education PD02-1.1-386.     

On the martingale problem for Banach space valued stochastic differential equations

LetE denote a real separable Banach space and letZ=(Z(t, f) be a family of centered, homogeneous, Gaussian independent increment processes with values inE, indexed by timet0 and the continuous functionsf:[0,t] E. If the dependence ont andf fulfills some additional properties,Z is called a gaussian random field. For continuous, adaptedE-valued processesX a stochastic integral processY = ₀ ^. Z(t, X)(dt) is defined, which is a continuous local martingale with tensor quadratic variation[Y] = ₀ ^. Q(t, X)dt, whereQ(t, f) denotes the covariance operator ofZ(t, f).Y is called a solution of the homogeneous Gaussian martingale problem, ifY = ₀ ^. Z(t, Y)(dt). Such solutions occur naturally in connection with stochastic differential equations of the type (D):dX(t)=G(t, X) dt+Z(t, X)(dt), whereG is anE-valued vector field. It is shown that a solution of (D) can be obtained by a kind of variation of parameter method, first solving a deterministic integral equation only involvingG and then solving an associated homogeneous martingale problem.     

Iterative Pexider equation modulo a subset

Summary LetX, Y, Z be arbitrary nonempty sets,E be a nonempty subset ofZ ^z andK be a groupoid. Assume that {F _t} _{t K} Z ^X, {G _t} _{t K} Y ^X, {H _t} _{t K} Z ^Y are families of functions satisfying the functional equationF _st = k_(s,t) H_s G_t for (s, t) D(K), whereD(K) stands for the domain of the binary operation on the groupoidK andk _(s,t) E for (s, t) D(K). Conditions are established under which the equation can be reduced to the corresponding Cauchy equation. This paper generalizes some results from [4] and [1].