Let G be a group which has a faithful representation as a transitive permutation group on m letters in which no permutation other than the identity leaves two letters unaltered, and there is at least one permutation leaving exactly one letter fixed. It is easily seen that if G has order mh, a necessary and sufficient condition for G to have such a representation is that G contains a subgroup H of order h which is its own normalizer in G and is disjoint from all its conjugates. Such a group G is called a Frobenius group of type (h, m).

We consider the Neumann problem for the Schrödinger equations $-\Delta u\,+\,Vu\,=\,0$, with singular nonnegative potentials $V$ belonging to the reverse Hölder class ${{\mathcal{B}}_{n}}$ , in a connected Lipschitz domain $\Omega \,\subset \,{{\text{R}}^{n}}$ . Given boundary data $g$ in ${{H}^{p}}\text{or}\,{{L}^{p}}\,\text{for}\,\text{1}-\in \,<\,p\,\le \,2,\text{where}\,\text{0}<\in <\frac{1}{n}$ , it is shown that there is a unique solution, $u$, that solves the Neumann problem for the given data and such that the nontangential maximal function of $\nabla u$ is in ${{L}^{p}}(\partial \Omega )$. Moreover, the uniform estimates are found.

In recent years there has been considerable interest in the conjecture that |G| divides |Aut G| for all finite non-cyclic p-groups G of order greater than p2. In particular, the conjecture has been established for a considerable number of (not necessarily distinct) classes of finite p-groups ([6], [7], [8], [9], [15], [16]); additionally, results have been obtained, often using homological methods, which permit reductions in any attempt to establish the overall conjecture ([5], [10], [13], [15]). In the former case, the p-groups G have generally been regular p-groups (see, for example, [6]) and the prime p = 2 has either been excluded (see, for example, [8]) or treated as a special case (as in [9]).It is the purpose of this paper to establish the conjecture for the class of all p-groups G where |G: Z(G)| ≦ p4 with no restrictions on the prime p.

We establish the asymptotic behaviour of the partition function, the heat content, the integrated eigenvalue counting function, and, for certain points, the on-diagonal heat kernel of generalized Sierpinski carpets. For all these functions the leading term is of the form $ {{x}^{\text{ }\!\!\gamma\!\!\text{ }}}\phi \left( \log x \right)$ for a suitable exponent $\text{ }\!\!\gamma\!\!\text{ }$ and $\phi $ a periodic function. We also discuss similar results for the heat content of affine nested fractals.

This paper introduces a class of abstract linear representations on Banach convolution function algebras over homogeneous spaces of compact groups. Let $G$ be a compact group and $H$ a closed subgroup of $G$. Let $\mu $ be the normalized $G$-invariant measure over the compact homogeneous space $G/H$ associated with Weil's formula and $1\,\le \,p\,<\,\infty $. We then present a structured class of abstract linear representations of the Banach convolution function algebras ${{L}^{p}}\left( G/H,\,\mu \right)$.

The dual of Hp of the unit disk for 0 < p < 1 has been characterized by Duren, Romberg and Shields (see [3]). The present paper is concerned with the analogous result for Hp(R+n+1) in the sense of Stein and Weiss (see [11]). In this connection it may be recalled that the dual of H1 has been characterized by Fefferman (see [4]).

Let be the Fourier transform of a Borel measure of finite total variation. The formula

can be justified if the integral on the right converges absolutely. For

where

Now let h → 0 in both sides of this equation and use the Lebesgue dominated convergence theorem.

Let $f\,=\,\sum{_{n=1}^{\infty }{{a}_{f}}\left( n \right){{q}^{n}}}$ be a cusp form with integer weight $k\,\ge \,2$ that is not a linear combination of forms with complex multiplication. For $n\,\ge \,1$, let

$${{i}_{f}}\left( n \right)\,=\,\left\{ _{0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{otherwise}\text{.}}^{\max \left\{ i\,:\,{{a}_{f}}\left( n+j \right)\,=\,0\,\text{for}\,\text{all}\,0\,\,\le \,j\,\le \,i \right\}\,\,\,\,\,\,\,\text{if}\,{{\text{a}}_{f}}\left( n \right)\,=\,0,} \right.\,$$

Concerning bounded values of ${{i}_{f}}\left( n \right)$ we prove that for $\in \,>\,0$ there exists $M\,=\,M\left( \in ,\,f \right)$ such that $\#\left\{ n\,\le \,x\,:\,{{i}_{f}}\left( n \right)\,\le \,M \right\}\,\ge \,\left( 1-\in \right)x.$ Using results of Wu, we show that if $f$ is a weight 2 cusp form for an elliptic curve without complex multiplication, then ${{i}_{f}}\left( n \right)\,{{\ll }_{f,\in }}\,{{n}^{\frac{51}{134}+\in }}$. Using a result of David and Pappalardi, we improve the exponent to $\frac{1}{3}$ for almost all newforms associated to elliptic curves without complex multiplication. Inspired by a classical paper of Selberg, we also investigate ${{i}_{f}}\left( n \right)$ on the average using well known bounds on the Riemann Zeta function.

Let M and N be connected Hermitian manifolds of dimensions m and n with Hermitian metrics hM and hN, respectively. Then the space ℓ(M, N) of continuous mappings between M and N endowed with the compact-open topology is second countable so that a metric can be furnished in ℓ(M, N) which induces the compact-open topology. A sequence {fn} in ℓ(M, N) converges to a n f in ℓ(M, N) in this topology if and only if fn converges to f uniformly on compact subsets of M. It is then an easy consequence of the Cauchy integral formula to show that the space ℋ(M, N) of holomorphic mappings f:M → N is a closed subspace of ℓ(M, N).

In this paper, generalizing the classical notions of normal functions, Bloch functions, regular sequences and P-point sequences of one complex variable to the mappings in ℋ(M, N), see also [25], we obtain various relations which exist between these notions.

Let K be a convex body (compact, convex set with interior points) in n-dimensional Euclidean space En, and let V(K) denote the volume of K. Let K′ be a centrally symmetric body of maximum volume contained in K (in fact, K′ is unique; see 2 or 9), and define

c(K) = V(K′)/V(K)

Let

c(n) = inf{c(K) : K ⊂ En}.

Let K be a commutative field, let GL(n, K) be the multiplicative group of all non-singular n × n matrices with elements from K, and let SL(n, K) be the subgroup of GL(n, K) consisting of all matrices in GL(n, K) with determinant one. We denote the determinant of matrix A by |A|, the identity matrix by In, the companion matrix of polynomial p(λ) by C(p(λ)), and the transpose of A by AT. The multiplicative group of nonzero elements in K is denoted by K*. We let GF(pn) denote the finite field having pn elements.

Let $\mathbf{H}$ be the Hilbert function of some set of distinct points in ${{\mathbb{P}}^{n}}$ and let $\alpha \,=\,\alpha (\mathbf{H})$ be the least degree of a hypersurface of ${{\mathbb{P}}^{n}}$ containing these points. Write $\alpha ={{d}_{s}}+{{d}_{s-1}}+\cdot \cdot \cdot +{{d}_{1}}$ (where ${{d}_{i}}>0$ ). We canonically decompose $\mathbf{H}$ into $s$ other Hilbert functions $\text{H}\leftrightarrow \text{(}{{\text{H}'}_{s}}\text{,}...\text{,}{{\text{H}'}_{1}}\text{)}$ and show how to find sets of distinct points ${{\mathbb{Y}}_{s}},...,{{\mathbb{Y}}_{1}}$ , lying on reduced hypersurfaces of degrees ${{d}_{s}},...,{{d}_{1}}$ (respectively) such that the Hilbert function of ${{\mathbb{Y}}_{i}}$ is ${{\text{H'}}_{i}}$ and the Hilbert function of $\mathbb{Y}=\bigcup _{i=1}^{s}\,{{\mathbb{Y}}_{i}}$ is $\mathbf{H}$. Some extremal properties of this canonical decomposition are also explored.

In this paper we shall study the relations between the ranks of g-variate, discrete-parameter, weakly stationary stochastic processes x, y, and z satisfying the condition

1.1

and derive from them a characterization for the Wold decomposition and conditions for the concordance of the Wold and the Lebesgue-Cramér decompositions.

A module M is called a minimax module, if it has a finitely generated submodule U such that M/U is Artinian. This paper investigates minimax modules and some generalized classes over commutative Noetherian rings. One of our main results is: M is minimax iff every decomposition of a homomorphic image of M is finite.

From this we deduce that:

- All couniform modules are minimax.

- All modules of finite codimension are minimax.

- Essential covers of minimax modules are minimax. With the aid of these corollaries we completely determine the structure of couniform modules and modules of finite codimension.

We then examine the following variants of the minimax property:

- replace U “ finitely generated” by U “ coatomic” (i.e. every proper submodule of U is contained in a maximal submodule);

- replace M/U “ Artinian” by M/U “ semi-Artinian” (i.e. every proper submodule of M/U contains a minimal submodule).

We prove a strong variant of the Borwein-Preiss variational principle, and show that on Asplund spaces, Stegall's variational principle follows from it via a generalized Smulyan test. Applications are discussed.

The $q$-semicircular distribution is a probability law that interpolates between the Gaussian law and the semicircular law. There is a combinatorial interpretation of its moments in terms of matchings, where $q$ follows the number of crossings, whereas for the free cumulants one has to restrict the enumeration to connected matchings. The purpose of this article is to describe combinatorial properties of the classical cumulants. We show that like the free cumulants, they are obtained by an enumeration of connected matchings, the weight being now an evaluation of the Tutte polynomial of a so-called crossing graph. The case $q=0$ of these cumulants was studied by Lassalle using symmetric functions and hypergeometric series. We show that the underlying combinatorics is explained through the theory of heaps, which is Viennot's geometric interpretation of the Cartier–Foata monoid. This method also gives a general formula for the cumulants in terms of free cumulants.

In 1951 Fan (1) proved the following interesting extreme value result: Let A1, … , Am be completely continuous operators on a Hilbert space . For be the characteristic roots of Aσ* Aσ. Then, for any positive integer k,

(1),

(2),

where both maxima are taken over all unitary operators U1, … , Um and all sets of k orthonormal (o.n.) vectors.

An integral domain D is said to be an essential domain if D is an intersection of valuation rings that are localizations of D. D is called a v-multiplication ring if the finite divisorial ideals of D form a group. Griffin has shown [2, pp. 717-718] that every v-multiplication ring is an essential domain, and that an essential domain having a defining family of valuation rings {Vα} which is of finite character (i.e., every nonzero element of D is a non-unit in at most finitely many Vα) is necessarily a v-multiplication ring. It is noted in [4, p. 860] that any localization of a v-multiplication ring is again a v-multiplication ring. In this vein, Joe Mott has asked whether a localization of an essential domain must again be an essential domain. An example of an essential domain that is not a v-multiplication ring is given in [4], however it can be seen for this example that each localization is again an essential domain [6].

An invertible operator T on a Hilbert space is a multiplicative commutator if there exist invertible operators A and B on such that T = ABA–1B–1. In this paper we discuss the question of which operators are, and which are not, multiplicative commutators. The analogous question for additive commutators (operators of the form AB — BA) has received considerable attention and has, in fact, been completely settled (2). The present results represent the information we have been able to obtain by carrying over to the multiplicative problem the techniques that proved efficacious in the additive situation. While these results remain incomplete, they suffice, for example, to enable us to determine precisely which normal operators are multiplicative commutators.

In this paper, we revisit Russell-type modular equations, a collection of modular equations first studied systematically by R. Russell in 1887. We give a proof of Russell’s main theorem and indicate the relations between such equations and the constructions of Hilbert class fields of imaginary quadratic fields. Motivated by Russell’s theorem, we state and prove its cubic analogue which allows us to construct Russell-type modular equations in the theory of signature 3.