L^p -uniqueness for Dirichlet operators with singular potentials

We study the problem of strong uniqueness in L^p for the Dirichlet operator perturbed by a singular complex-valued potential. First we construct the generator -H_p of a C₀-semigroup in L^p, with H_p extending the restriction of the perturbed Dirichlet operator to the set of smooth functions. The corresponding sesquilinear form in L² is not assumed to be sectorial. Then we reveal sufficient conditions on the logarithmic derivative # of the measure rdx \rho dx and the potential q which ensure that -H_p is the only extension of D+b·?-q \upharpoonright_C0^￥ \Delta +\beta \cdot \nabla -q \upharpoonright_{C_0^{\infty}} which generates a C₀-semigroup on L^p. The method of a priori estimates of solutions to corresponding differential equations is employed.     

Concave and positive summing norms for operators on Lp

Let p be a real number such that p ? [1,+ ￥] p \in [1,+ \infty] and its conjugate exponent p' is not an even integer and let T be an operator defined on L^p(l)L^p(\lambda ) with values in a Banach space. We prove that the image of the unit ball determines if T belongs to the space of concave and positive summing operators. We also prove that the image of the unit ball determines the representability of the operator.     

Adapting Some Ideas from Stochastic Control Theory to Studying the Heat Equation in Closed Smooth Domains

Sufficient conditions on bounded domains D ² R ^d={(t,x)} of class C⁴ are given under which solutions of the heat equation u_t=j u+f in D have continuous second-order derivatives with respect to (t,x) in D^- . The equation is supplemented with C⁴ boundary data and it is assumed that f] C² .     

The Cesàro operator on the Bergman space $ A^{2} (\mathbb{D}) $

The Cesàro operator is shown to be subdecomposable on the Bergman spaces A^p (\mathbbD) A^{p} (\mathbb{D}) for p \geqq 2 p \geqq 2 , extending a result of [12] to the case that p < 4. For A² (\mathbbD) A^{2} (\mathbb{D}) , we show that Cesàro operator is in fact subscalar, but in contrast to the situation in the Hardy space, C |_A² C |_{A^2} fails to be hyponormal.     

On the existence of $\lambda $-projective splitters

Let D be an R-module over an arbitrary ring R of projective dimension at most 1. We construct an R-module G containing D such that Ext(D, G) = 0 = Ext(G, G). Moreover, we show that if D is l\lambda -projective over a hereditary ring R, for some infinite cardinal l\lambda , then G is also l\lambda -projective.     

On self-normality and abnormality in the alternating group Ap

An old problem proposed by Huppert, Doerk and Hawkes motivates us to investigate the relationship between an abnormal subgroup and self-normalizing in non-solvable groups. A subgroup H of a group G is called second maximal if H is maximal in all maximal subgroups of G containing H. Our result is that if H is a second maximal subgroup of the alternating group A_p of prime degree, then H is abnormal in A_p if and only if H is self-normalizing.     

Amenable representations and dynamics of the unit sphere in an infinite-dimensional Hilbert space

We establish a close link between the amenability property of a unitary representation p \pi of a group G (in the sense of Bekka) and the concentration property (in the sense of V. Milman) of the corresponding dynamical system (\Bbb S_p, G) ({\Bbb S}_{\pi}, G) , where \Bbb S_H {\Bbb S}_{\cal H} is the unit sphere the Hilbert space of representation. We prove that p \pi is amenable if and only if either p \pi contains a finite-dimensional subrepresentation or the maximal uniform compactification of (\Bbb S_p ({\Bbb S}_{\pi} has a G-fixed point. Equivalently, the latter means that the G-space (\Bbb S_p, G) ({\Bbb S}_{\pi}, G) has the concentration property: every finite cover of the sphere \Bbb S_p {\Bbb S}_{\pi} contains a set A such that for every e > 0 \epsilon > 0 the e \epsilon -neighbourhoods of the translations of A by finitely many elements of G always intersect. As a corollary, amenability of p \pi is equivalent to the existence of a G-invariant mean on the uniformly continuous bounded functions on \Bbb S_p {\Bbb S}_{\pi} . As another corollary, a locally compact group G is amenable if and only if for every strongly continuous unitary representation of G in an infinite-dimensional Hilbert space H {\cal H} the system (\Bbb S_H, G) ({\Bbb S}_{\cal H}, G) has the property of concentration.     

Lie p-algebras of finite p-subalgebra rank

We say that a Lie p-algebra L has finite p-subalgebra rank if the minimal number of generators required to generate every finitely generated p-subalgebra is uniformly bounded by some integer r. This paper is concerned with the following problem: does L being of finite p-subalgebra rank force ad(L) to be finite-dimensional? Although this seems unlikely in general, we show that this is indeed the case for Lie p-algebras in a large class including all locally, residually, and virtually soluble Lie p-algebras.     

Non-Commutative Ribbons and Quasi-Differential Operators

This paper is devoted to a non-commutative generalization of a classical result occurring in the context of the modular representation theory of the symmetric group. We prove that a non-commutative Schur ribbon function R_I is annihilated by the quasi-differential operator D_Pk if and only if the composition I is the external border of a k-core.     

Glatte p-Gruppen

A p-group G is called smooth if there is a smooth chief series in G, that is a chief series in which any two factor groups of the same order are isomorphic. It is shown that in most cases non-abelian smooth p-groups are of nilpotency class two and that a smooth p-group is not much larger than the largest abelian group in its smooth chief series.     

On extensions of positive definite integral Fredholm operators

The main result of this paper states that a positive definite Fredholm integral operator acting on L²([0,1]) can be modified on a Lebesque measurable set D\mit\Delta in [0,1]² such that the resulting operator is positive definite and its resolvent kernel is zero on D\mit\Delta . This answers a question raised in [3]. The proof is based on extension results for positive definite operator matrices and their connection to generalized determinants.     

On Ricci curvature of isotropic and Lagrangian submanifolds in complex space forms

Let M ⁿ be a Riemannian n-manifold. Denote by S(p) and [`(Ric)](p)\overline {Ric}(p) the Ricci tensor and the maximum Ricci curvature on M <sup>n</sup> at a point p ? M<sup>n</sup>p\in M^n, respectively. First we show that every isotropic submanifold of a complex space form [(M)\tilde]<sup>m</sup>(4 c)\widetilde M^m(4\,c) satisfies S ￡ ((n-1)c+ [(n<sup>2</sup>)/4] H<sup>2</sup>)gS\leq ((n-1)c+ {n^2 \over 4} H^2)g, where H<sup>2</sup> and g are the squared mean curvature function and the metric tensor on M <sup>n</sup>, respectively. The equality case of the above inequality holds identically if and only if either M <sup>n</sup> is totally geodesic submanifold or n = 2 and M <sup>n</sup> is a totally umbilical submanifold. Then we prove that if a Lagrangian submanifold of a complex space form [(M)\tilde]<sup>m</sup>(4 c)\widetilde M^m(4\,c) satisfies [`(Ric)] = (n-1)c+ [(n²)/4] H²\overline {Ric}= (n-1)c+ {n^2 \over 4} H^2 identically, then it is a minimal submanifold. Finally, we describe the geometry of Lagrangian submanifolds which satisfy the equality under the condition that the dimension of the kernel of second fundamental form is constant.     

Transitivity and full transitivity for p-local modules

A p-local module M is called (fully) transitive if for all x,y ? Mx,y\in M with U_M(x) = U_M(y) ( U_M(x)\leqq U_M(y)U_M(x)\leqq U_M(y)) there exists an automorphism (endomorphism) of M which maps x onto y. In this paper we examine the relationship of these two notions in the case of p-local modules. We show that a module M is fully transitive if and only if M?MM\oplus M is transitive in the case where the divisible part of M/tMM/tM has rank at most one. Moreover, we show that for the same class of modules transitivity implies full transitivity if p > 2. This extends theorems of Files, Goldsmith and of Kaplansky for torsion p-local modules.     

The fixed points of coprime action

Let P be an odd p\pi -group that acts as a group of automorphisms on the soluble p￠\pi '-group G. We obtain generators for the fixed points of P on [G, P].     

Check digit systems over groups and anti-symmetric mappings

A check digit system over a group which detects all single errors and all adjacent transpositions exists if and only if the group possesses an anti-symmetric mapping. In this article we give a characterisation for (anti-)automorphisms to be anti-symmetric, show how anti-automorphisms are used to construct new anti-symmetric mappings from others and give an upper bound for the number of anti-symmetric mappings of a group. For groups with sign structure, particularly the dihedral group, we present a further construction for anti-symmetric mappings. The fact that groups of order 2(2k + 1) have a non-trivial sign-structure leads to a very short proof that groups of order 2(2k + 1) possess no complete mapping. Finally we show that over the dihedral group D_m, m odd, no check digit system exists, which detects all jump transpositions or all twin errors or all jump twin errors.     

Résonances dans des Petits Domaines près d‚une Énergie Critique

Résumé. On travaille dans le cadre de l‘analyse semi-classique. Considérons p(x, hD_x) p(x, hD_{x}) , une perturbation de -h²D -h^{2}\Delta qui est analytique à l‘infini. On suppose que dans la surface d‘énergie E₀ > 0, les points critiques du symbole p(x, x) p(x, \xi) forment une sous-variété C \mathcal C et que p est non dégénéré dans l‘espace normal à C \mathcal C .¶En utilisant les résultats de [6] et [18], on obtient une majoration du nombre de résonances dans des disques de rayon d \delta centrés en E proche de E₀, où d \delta satisfait Ch < d \delta < 1/C pour une constante C > 0. En généralisant la formule de trace de Sjöstrand qui exprime la trace d‘une différence d‘opérateurs en fonction des résonances, on trouve une minoration du nombre de résonances proches de E₀.     

On permutable subgroups of finite groups

Let \frak Z \frak Z be a complete set of Sylow subgroups of a finite group G, that is, for each prime p dividing the order of G, \frak Z \frak Z contains exactly one and only one Sylow p-subgroup of G. A subgroup H of a finite group G is said to be \frak Z \frak Z -permutable if H permutes with every member of \frak Z \frak Z . The purpose here is to study the influence of \frak Z \frak Z -permutability of some subgroups on the structure of finite groups. Some recent results are generalized.     

The Fredholm determinant of an almost periodic arithmetical function

For functions f in the space of 2-limit-periodic functions on \Bbb Z{\Bbb Z}, the Fredholm determinant D(f; z) is defined as an everywhere convergent power series whose coefficients are mean values of certain determinants involving the values of f. A Weierstraß product representation is given for D(f; z) which shows in particular that the zeros of D(f; z) are the reciprocals of the nonvanishing Fourier coefficients of f.     

Oscillatory radial solutions for subcritical biharmonic equations

It is well known that the biharmonic equation Δ²u=u|u|^p−1 with p∈(1,∞) has positive solutions on Rn if and only if the growth of the nonlinearity is critical or supercritical. We close a gap in the existing literature by proving the existence and uniqueness, up to scaling and symmetry, of oscillatory radial solutions on Rn in the subcritical case. Analyzing the nodal properties of these solutions, we also obtain precise information about sign-changing large radial solutions and radial solutions of the Dirichlet problem on a ball.     

Odd order groups with the rewriting property Q3

Let n be an integer greater than 1, and let G be a group. A subset {x₁, x₂, ..., x_n} of n elements of G is said to be rewritable if there are distinct permutations p \pi and s \sigma of {1, 2, ..., n} such that¶¶x_p(1)x_p(2) ?x_p(n) = x_s(1)x_s(2) ?x_s(n). x_{\pi(1)}x_{\pi(2)} \ldots x_{\pi(n)} = x_{\sigma(1)}x_{\sigma(2)} \ldots x_{\sigma(n)}. ¶¶A group is said to have the rewriting property Q_n if every subset of n elements of the group is rewritable. In this paper we prove that a finite group of odd order has the property Q₃ if and only if its derived subgroup has order not exceeding 5.