The classification of <Emphasis Type="Italic">p</Emphasis>-local finite groups over the extraspecial group of order <Emphasis Type="Italic">p</Emphasis><Superscript>3</Superscript> and exponent <Emphasis Type="Italic">p</Emphasis>

In this paper we classify the p-local finite groups over p¹⁺²₊, the extraspecial group of order p³ and exponent p for odd p. This study reduces to the classification of the saturated fusion systems over p¹⁺²₊, which will be characterized by the outer automorphism group, the number of -radical subgroups and the automorphism group of each nontrivial -radical subgroup. As part of this classification, we obtain three new exotic 7-local finite groups.Partially supported by MCYT grant BFM2001-2035.Partially supported by MCYT grant BFM2001-1825.Both authors have been supported by the EU grant nr HPRN-CT-1999-00119.in final form: 1 October 2003     

On the <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-conjecture for locally compact groups

Let G be a locally compact group. For 1 < p < ∞, it is well-known that f * g exists and belongs to L^p(G) for all f, g L^p (G) if and only if G is compact. Here, for 2 < p < ∞, we show that f * g exists for all f, g L^p(G) if and only if G is compact. We also show that this result does not remain true for 1 < p ≤ 2. Received: 23 April 2006     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> approximation capability of RBF neural networks

L ^p approximation capability of radial basis function (RBF) neural networks is investigated. If g: R ₊¹ → R ¹ and ∈ L _loc ^p (R ⁿ ) with 1 ≤ p < ∞, then the RBF neural networks with g as the activation function can approximate any given function in L ^p (K) with any accuracy for any compact set K in R ⁿ , if and only if g(x) is not an even polynomial. Partly supported by the National Natural Science Foundation of China (10471017)     

Sobolev Inequalities on Forms and <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis>,<Emphasis Type="Italic">q</Emphasis></Superscript>-Cohomology on Complete Riemannian Manifolds

We prove some Sobolev inequalities on differential forms over a class of complete non-compact Riemannian manifolds with suitable geometric conditions. Moreover, we establish some L ^p,q -estimates and existence theorems of the Cartan-De Rham equation and the Hodge systems. As applications, we prove some vanishing theorems of the L ^p,q -cohomology and prove the L ^q -solvability of the nonlinear p-Laplace equation on forms on complete non-compact Riemannian manifolds with suitable geometric conditions.     

A Characterization of PSL(3, <Emphasis Type="Italic">q</Emphasis>) for <Emphasis Type="Italic">q</Emphasis>=2<Superscript><Emphasis Type="Italic">m</Emphasis></Superscript>

The order components of a finite group are introduced in [12]. In [9], it is proved that the group PSL(3,q), where q is an odd prime power, is uniquely determined by its order components. In this paper, we show that the group PSL(3, q), where q=2 ^m , is also uniquely determined by its order components. Received December 15, 2000, Revised August 15, 2001, Accepted November 13, 2001     

Finite solvable groups in which the Sylow <Emphasis Type="Italic">p</Emphasis>-subgroups are either bicyclic or of order <Emphasis Type="Italic">p</Emphasis><Superscript>3</Superscript>

All groups considered in this paper will be finite. Our main result here is the following theorem. Let G be a solvable group in which the Sylow p-subgroups are either bicyclic or of order p ³ for any p ∈ π(G). Then the derived length of G is at most 6. In particular, if G is an A₄-free group, then the following statements are true: (1) G is a dispersive group; (2) if no prime q ∈ π(G) divides p ² + p + 1 for any prime p ∈ π(G), then G is Ore dispersive; (3) the derived length of G is at most 4.     

Lebesgue weighted <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-algebra on locally compact groups

Let G be a locally compact group, ω a weight function on G, and 1<p<∞. We introduce the Lebesgue weighted L ^p -space \({\mathcal{L}}_{\omega}^{1,p}(G)= L^{p}(G,\omega)\cap L^{1}(G)\) as a Banach space and introduce its dual. Furthermore, we consider this space as a Banach algebra with respect to the usual convolution and show that \({\mathcal{L}}_{\omega}^{1,p}(G)\) admits a bounded approximate identity if and only if G is discrete. In addition, we prove that amenability of this algebra implies that G is discrete and amenable. Moreover, we discuss the converse of this result.     

Kernels and <Emphasis Type="Italic">p</Emphasis>-Kernels of <Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">r</Emphasis></Superscript>-ary 1-Perfect Codes

The rank of a q-ary code C is the dimension of the subspace spanned by C. The kernel of a q-ary code C of length n can be defined as the set of all translations leaving C invariant. Some relations between the rank and the dimension of the kernel of q-ary 1-perfect codes, over as well as over the prime field , are established. Q-ary 1-perfect codes of length n=(q^m − 1)/(q − 1) with different kernel dimensions using switching constructions are constructed and some upper and lower bounds for the dimension of the kernel, once the rank is given, are established.Communicated by: I.F. Blake     

Weighted <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-conjecture for locally compact groups

Let G be a locally compact group, ω be a weight function on G and 1 < p < ∞. Here, we give a sufficient condition for that the weighted L ^p -space L ^p (G, ω) is a Banach algebra. Also, we get some necessary conditions on G and the weight function ω for L ^p (G, ω) to be a Banach algebra. As a consequence, we show that if G is abelian and L ^p (G, ω) is a Banach algebra, then G is σ-compact.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-maximal regularity for second order Cauchy problems

We introduce the concept of L^p-maximal regularity for second order Cauchy problems. We prove L^p-maximal regularity for an abstract model problem and we apply the abstract results to prove existence, uniqueness and regularity of solutions for nonlinear wave equations. The author acknowledges with thanks the support provided by the Department ofApplied Analysis, University of Ulm, and the travel grants provided by NBMH India and MSF Delhi, India.     

On the <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-consistency of wavelet estimators

This paper deals with the L ^p -consistency of wavelet estimators for a density function based on size-biased random samples. More precisely, we firstly show the L ^p -consistency of wavelet estimators for independent and identically distributed random vectors in R ^d . Then a similar result is obtained for negatively associated samples under the additional assumptions d = 1 and the monotonicity of the weight function.     

LP（Rn）-Boundedness of certain commutators

In this paper, the Lp(Rn)-boundedness of the commutators generalized by BMO(Rn) function and the singular integral operator T with rough kernel Ω∈ Llog+ L(Sn-1) is proved by using the Bony's formula for the paraproduct of two functions.     

Typical <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>-Dimensions of Measures

For a probability measure μ on a subset of , the lower and upper L^q-dimensions of order are defined by We study the typical behaviour (in the sense of Baire’s category) of the L^q-dimensions and . We prove that a typical measure μ is as irregular as possible: for all q ≥ 1, the lower L^q-dimension attains the smallest possible value and the upper L^q-dimension attains the largest possible value.     

On <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis>,<Emphasis Type="Italic">q</Emphasis></Subscript>-factorization of complete bipartite multigraphs

Let λK _m,n be a complete bipartite multigraph with two partite sets having m and n vertices, respectively. A K _p,q -factorization of λK _m,n is a set of edge-disjoint K _p,q -factors of λK _m,n which partition the set of edges of λK _m,n . When p = 1 and q is a prime number, Wang, in his paper [On K _1,q -factorization of complete bipartite graph, Discrete Math., 126: (1994), 359-364], investigated the K _1,q -factorization of K _m,n and gave a sufficient condition for such a factorization to exist. In papers [K _1,k -factorization of complete bipartite graphs, Discrete Math., 259: 301-306 (2002),; K _p,q -factorization of complete bipartite graphs, Sci. China Ser. A-Math., 47: (2004), 473-479], Du and Wang extended Wang’s result to the case that p and q are any positive integers. In this paper, we give a sufficient condition for λK _m,n to have a K _p,q -factorization. As a special case, it is shown that the necessary condition for the K _p,q -factorization of λK _m,n is always sufficient when p : q = k : (k + 1) for any positive integer k.     

Cyclic Functions in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>(<Emphasis Type="Bold">R</Emphasis>), 1 \le <Emphasis Type="Italic">p</Emphasis> < \infty

Functions whose translates span L ^p (R) are called L ^p-cyclic functions. For a fixed p \memb [1, \infty], we construct Schwartz-class functions which are L ^r -cyclic for r > p and not L ^r - cyclic for r \le p. We then construct Schwartz-class functions which are L ^r -cyclic for r \ge p and not L ^r -cyclic for r < p. The constructions differ for p \memb (1, 2) and p > 2.     

On the <Emphasis Type="Italic">k</Emphasis>-gamma <Emphasis Type="Italic">q</Emphasis>-distribution

We provide combinatorial as well as probabilistic interpretations for the q-analogue of the Pochhammer k-symbol introduced by Díaz and Teruel. We introduce q-analogues of the Mellin transform in order to study the q-analogue of the k-gamma distribution.     

Semisymmetric cubic graphs of order 16<Emphasis Type="Italic">p</Emphasis><Superscript>2</Superscript>

An undirected graph without isolated vertices is said to be semisymmetric if its full automorphism group acts transitively on its edge set but not on its vertex set. In this paper, we inquire the existence of connected semisymmetric cubic graphs of order 16p ². It is shown that for every odd prime p, there exists a semisymmetric cubic graph of order 16p ² and its structure is explicitly specified by giving the corresponding voltage rules generating the covering projections.     

A study of (<Emphasis Type="Italic">x</Emphasis>(<Emphasis Type="Italic">q</Emphasis> + 1), <Emphasis Type="Italic">x</Emphasis>; 2, <Emphasis Type="Italic">q</Emphasis>)-minihypers

In this paper, we study the weighted (x(q + 1), x; 2, q)-minihypers. These are weighted sets of x(q + 1) points in PG(2, q) intersecting every line in at least x points. We investigate the decomposability of these minihypers, and define a switching construction which associates to an (x(q + 1), x; 2, q)-minihyper, with x ≤ q ² − q, not decomposable in the sum of another minihyper and a line, a (j(q + 1), j; 2, q)-minihyper, where j = q ² − q − x, again not decomposable into the sum of another minihyper and a line. We also characterize particular (x(q + 1), x; 2, q)-minihypers, and give new examples. Additionally, we show that (x(q + 1), x; 2, q)-minihypers can be described as rational sums of lines. In this way, this work continues the research on (x(q + 1), x; 2, q)-minihypers by Hill and Ward (Des Codes Cryptogr 44:169–196, 2007), giving further results on these minihypers.     

On the functional equation <Emphasis Type="Italic">Af</Emphasis><Superscript>2</Superscript> + <Emphasis Type="Italic">Bg</Emphasis><Superscript>2</Superscript> = 1 on the field of complex <Emphasis Type="Italic">p</Emphasis>-adic numbers

For a fixed prime p, let C _p denote the complex p-adic numbers. For polynomials A, B ε C p [x] we consider decompositions A (x) f ² (x) + B (x) g ² (x) = 1 of entire functions f, g on C p and try to improve an impossibility result due to A. Boutabaa concerning transcendental f, g. We also provide a new proof of a p-adic diophantic statement due to D. N. Clark, which is an important ingredient of Boutabaa’s method.     

On approximation of meromorphic functions having index-pair (<Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)

The purpose of this paper is to generalize the results obtained by Winiarski (Ann. Polon. Math. 29:259–273, 1970) and Kasana and Kumar (Publ. Mat. 38:255–267, 1994) for the M ₀(C) of all entire functions onto the class M _m (C), m ≥ 0 of all meromorphic functions with exactly m poles on the complex plane C.