On a Problem of Karpilovsky

Let G be a finite elementary group. Let ⁿ (G) denote the nth power of the augmentation ideal (G) of the integral group ring G. In this paper, we give an explicit basis of the quotient group Q_n(G) = ⁿ(G)/ⁿ⁺¹ (G) and compute the order of Qⁿ (G).2000 Mathematics Subject Classification: 16S34, 20C05     

Rigidity theorem for harmonic maps from Riemannian manifold to Grassmannian manifold

We first study the Grassmannian manifoldG _n (R^n+p)as a submanifold in Euclidean space ⁿ (R ^n+p). Then we give a local expression for each map from Riemannian manifoldM toG ⁿ (R ^n+p) ⁿ (R ^n+p), and use the local expression to establish a formula which is satisfied by any harmonic map fromM toG _n (R ^n+p). As a consequence of this formula we get a rigidity theorem.     

Optimal strong parity edge-coloring of complete graphs

A parity walk in an edge-coloring of a graph is a walk along which each color is used an even number of times. Let p(G) be the least number of colors in an edge-coloring of G having no parity path (a parity edge-coloring). Let (G) be the least number of colors in an edge-coloring of G having no open parity walk (a strong parity edge-coloring). Always (G) ≥ p(G) ≥ χ′(G). We prove that (K _n ) = 2^⌈lgn⌉ − 1 for all n. The optimal strong parity edge-coloring of K _n is unique when n is a power of 2, and the optimal colorings are completely described for all n. Partially supported by NSF grant CCR 0093348. Work supported in part by the NSA under Award No. MDA904-03-1-0037.     

On Some Sets of Group Functions

Let G be a group, let A be an Abelian group, and let n be an integer such that n –1. In the paper, the sets _n (G,A) of functions from G into A of degree not greater than n are studied. In essence, these sets were introduced by Logachev, Sal'nikov, and Yashchenko. We describe all cases in which any function from G into A is of bounded (or not necessarily bounded) finite degree. Moreover, it is shown that if G is finite, then the study of the set _n (G,A) is reduced to that of the set n(G/O ^p (G),A _p ) for primes p dividing G/G. Here O ^p (G) stands for the p-coradical of the group G, A _p for the p-component of A, and G for the commutator subgroup of G.     

Random graphs and covering graphs of posets

For a graph G, we define c(G) to be the minimal number of edges we must delete in order to make G into a covering graph of some poset. We prove that, if p=n ^-1+(n) ,where (n) is bounded away from 0, then there is a constant k ₀>0 such that, for a.e. G _p , c(G _p )k ₀ n ¹⁺⁽ⁿ⁾ .In other words, to make G _p into a covering graph, we must almost surely delete a positive constant proportion of the edges. On the other hand, if p=n ^-1+(n) , where (n)0, thenc(G _p )=o(n ¹⁺⁽ⁿ⁾ ), almost surely.Partially supported by MCS Grant 8104854.     

p-Group actions on homology spheres

When an arbitraryp-groupG acts on a _n -homologyn-sphereX, it is proved here that the dimension functionn:S(G)(S(G) is the set of subgroups ofG), defined byn(H)=dimX ^H, (dim here is cohomological dimension) is realised by a real representation ofG, and that there is an equivariant map fromX to the sphere of this representation. A converse is also established.     

On strong solutions of Poisson's equation in Beppo Levi spaces

LetG sun (n 2) be an unbounded open set having a compact complement and a smooth boundary G of classC ². InG we consider the equations — u=f,u¦G= and prove the existence of a solutionu L ^2,q(G) providedf L ^q(G) and W ^{2 —1/q-q}(G) (1 <q < ). HereL ^2,q(G) is the space of all functionsu L _Ioc ^q (G) having all second order distributional derivatives inL ^q(G). Concerning the uniqueness of this solution we show that the corresponding nullspace has dimensionn + 1 (n 2).

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Zusammenfassung SciG ⁿ (n 2) eine unbeschränkte offene Menge mit kompaktem Komplement und mit glattem Rand G der KlasseC ². InG betrachten wir das Randwertproblem — u=f,u¦g= und beweisen die Existenz einer Lösungu L ^2,q(G) für beliebigef L ^q(G) und Randwerte W ^2-1/q,q(G) (1 <q < ). Dabei istL ^2,q(G) der Raum aller Funktionenu L _Ioc ^q (G), die Distributionsableitungen zweiter Ordnung inL ^q (G) besitzen. Bezüglich der Eindeutigkeit solcher Lösungen zeigen wir, daß der entsprechende Nullraum die Dimensionn + 1 (n 2) besitzt.

     

The Central Approximation Theorems for the Method of Left Gamma Quasi-Interpolants in L p spaces

The optimal degree of approximation of the method of Gammaoperators G _n in L _p spaces is O(n ^-1). In order to obtain much faster convergence, quasi-interpolants G _n ^(k) of G _n in the sense of Sablonnière are considered. We show that for fixed k the operator-norms G _n ^(k) _p are uniformly bounded in n. In addition to this, for the first time in the theory of quasi-interpolants, all central problems for approximation methods (direct theorem, inverse theorem, equivalence theorem) could be solved completely for the L _p metric. Left Gamma quasi-interpolants turn out to be as powerful as linear combinations of Gammaoperators [6].     

Homological Invariants for pro- p Grou ps and Some Finitely Presented pro-${\cal C}$ Grou ps

Let G be a finitely presented pro- group with discrete relations. We prove that the kernel of an epimorphism of G to is topologically finitely generated if G does not contain a free pro- group of rank 2. In the case of pro-p groups the result is due to J. Wilson and E. Zelmanov and does not require that the relations are discrete ([15], [17]).For a pro-p group G of type FP_m we define a homological invariant ^m(G) and prove that this invariant determines when a subgroup H of G that contains the commutator subgroup G is itself of type FP_m. This generalises work of J. King for ¹(G) in the case when G is metabelian [9].Both parts of the paper are linked via two conjectures for finitely presented pro-p groups G without free non-cyclic pro-p subgroups. The conjectures suggest that the above conditions on G impose some restrictions on ¹(G) and on the automorphism group of G.Both authors are partially supported by CNPq, Brazil.     

Covering a Finite Abelian Group by Subset Sums

Let G be an abelian group of order n. The critical number c(G) of G is the smallest s such that the subset sums set (S) covers all G for eachs ubset SG\{0} of cardinality |S|s. It has been recently proved that, if p is the smallest prime dividing n and n/p is composite, then c(G)=|G|/p+p–2, thus establishing a conjecture of Diderrich.We characterize the critical sets with |S|=|G|/p+p–3 and (S)=G, where p3 is the smallest prime dividing n, n/p is composite and n7p²+3p.We also extend a result of Diderrichan d Mann by proving that, for n67, |S|n/3+2 and S=G imply (S)=G. Sets of cardinality for which (S) =G are also characterized when n183, the smallest prime p dividing n is odd and n/p is composite. Finally we obtain a necessary and sufficient condition for the equality (G)=G to hold when |S|n/(p+2)+p, where p5, n/p is composite and n15p².* Work partially supported by the Spanish Research Council under grant TIC2000-1017 Work partially supported by the Catalan Research Council under grant 2000SGR00079     

Incidence Structures of Type (p, n)

Every incidence structure (understood as a triple of sets (G, M, I), I G×M) admits for every positive integer p an incidence structure where G ^p (M ^p) consists of all independent p-element subsets in G (M) and I ^p is determined by some bijections. In the paper such incidence structures are investigated the 's of which have their incidence graphs of the simple join form. Some concrete illustrations are included with small sets G and M.     

Cyclic Ostrom Spreads

We call a transitive permutation group G stable if every non-trivial orbit B, of a point stabilizerG _O , is such that some element of G-G _O leaves invariant B {O}. We characterize all finite affine planes of order n that admit a collineation group G that acts, on the points of , as a stable 3/2-transitive permutation group of rank n+2 . The planes obtained are precisely what we have called cyclic Ostrom planes; these are translation planes whose associated spreads are obtained from the Desarguesian spread , of order n=p^r , by replacing a partial subspread of by another consisting of GF(p) subspaces that are unions of kern orbits. All André spreads, as well as many other spreads, are examples of cyclic Ostrom spreads.     

Linearity of Zp[[t]]-Perfect Groups

Let G be a _p[[t]]-standard group of level 1. Then G is _p[[t]]-perfect if its lower central series is given by powers of the maximal ideal (p, t), i.e. if _n(G) = G((p,t)ⁿ). We prove that a _p[[t]]-perfect group is linear by imitating the proof that a _p[[t]]-standard group is linear.     

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.     

<Emphasis Type="Italic">L</Emphasis>-functions of symmetric products of the Kloosterman sheaf over Z

The classical n-variable Kloosterman sums over the finite field F _p give rise to a lisse -sheaf Kl _n+1 on , which we call the Kloosterman sheaf. Let L _p (G _{m, F p} , Sym ^k Kl _n+1, s) be the L-function of the k-fold symmetric product of Kl _n+1. We construct an explicit virtual scheme X of finite type over Spec Z such that the p-Euler factor of the zeta function of X coincides with L _p (G _{m, F p} , Sym ^k Kl _n+1, s). We also prove similar results for and . The research of L. Fu is supported by the NSFC (10525107).     

Compact groups having almost discrete orbit hypergroups

LetG be a compact group of automorphism acting continuously on a compact groupH. Then the orbit spaceH ^G is a compact hypergroup. We characterize, all solvable groupsH and compact automorphism groupsG for whichH ^G is almost discrete, i.e.,H ^G is homeomorphic to the one-point-compactification of . It turns out that thenH is isomorphic either to the infinite direct product (p) of the cyclic groups (p) or to _p ⁿ ( _p the group of allp-adic numbers) for some primep and some . The almost discrete orbit hypergroupsH ^G are determined explicitly for some examples.     

A sharp form of Poincaré type inequalities on balls and spheres

Summary Letu be a real valued function on ann-dimensional Riemannian manifoldM ⁿ. We consider an inequality between theL ^q-norm ofu minus its mean value overM ⁿ and theL ^p-norm of the gradient ofu.The best constant in such inequality is exhibited in the following cases: i)M ⁿ is an open ball inIR ⁿ andp=1, 0<qn/(n–1); ii)M ⁿ is a sphere inIR ⁿ +1 and eitherp=1, 0<qn/(n–1) orp>n,q=.

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Sunto Siau una funzione a valori reali dafinita su una varietà riemannianan-dimensionaleM ⁿ. Si considera una disuguaglianza tra la normaL ^q diu meno il suo valor medio suM ⁿ e la normaL ^p del gradiente diu.Si determina la costante ottimale in tale disuguaglianza nei seguenti casi: i)M ⁿ è un disco aperto inIR ⁿ ep=1, 0<qn/(n–1); ii)M ⁿ è una sfera inIR ⁿ +1 ep=1, 0<qn/(n–1) oppurep>n,q=.

     

On boundary properties of the components of polyharmonic functions

The following two classes of functions are introduced forp0: the classCU ^p (G) of uniformly continuous functions of orderp in a domainG , and the classM ^p (G) of functions of the boundedness of orderp inG. Criterions are established for ann-analytic function to belong to each of these classes.Translated fromMatematicheskie Zametki, Vol. 63, No. 6, pp. 821–834, June, 1998.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-013766     

Collineation groups preserving a unital in a projective plane of even order

We investigate the structure of a collineation group G leaving invariant a unital U in a finite projective plane of even order n=m ². When G is transitive on the points of U and the socle of G has even order, then must be a Desarguesian plane, U a classical unital and PSU(3,m ²)GPU(3,m ²) — for m>2. The primitive case follows as an easy corollary.This research was supported by a grant from the M.P.I.