Let $G$ be a finite group; there exists a uniquely determined Dirichlet polynomial $P_G(s)$ such that if $t \in \mathbb N$, then $P_G(t)$ gives the probability of generating $G$ with $t$ randomly chosen elements. We show that if $P_G(s)=P_{\text{Alt}(n)}(s)$, then $G/\text{Frat}\, G\cong \text{Alt}(n).$

Let A be an n × n complex matrix and c = (c1… cn) єℂn. Define the c-numerical range of A to be the set is an orthonormal set in , where * denotes the conjugate transpose. Westwick [8[ proved that if c … cn are collinear, then Wc(A) is convex. (Poon [6] gave another proof.) But in general for n ≧3, Wc(A) may fail to be convex even for normal A (for example, see Marcus [4] or Lemma 3 in this note) though it is star-shaped (Tsing [7]). In the following, we shall assume that A is normal. Let W(A) = {diag UAU*: U is unitary}. Horn [3] proved that if the eigenvalues of A are collinear, then W(A) is convex. Au-Yeung and Sing [2] showed that the converse is also true. Marcus [4] further conjectured (and proved for n = 3) that if Wc(A) is convex for all cєℂn then the eigenvalues of A are collinear. Let λ = (λ1, …, λn єℂn. We denote by the vector λ1, …, λn and by [λ] the diagonal matrix with λ1, …, λn lying on its diagonal. Since, for any unitary matrix U,. Wc(A) = Wc (UAU*), the Marcus conjecture reduces to: if Wc([λ]) is convex for all c єℂn then λ1, … λn are collinear. For the case n = 3, Au-Yeung and Poon [1] gave a complete characterization on the convexity of the set Wc([λ]) in terms of the relative position of the points , where σ є S3 the permutation group of order 3. As an example they showed that if λ1, λ2, λ3 are not collinear, then is not convex (Lemma 3 in this note gives another proof). We shall show that for the case n = 4, is not convex if λ1, λ2. λ3. λ4 are not collinear. Thus for n = 3, 4 the Marcus conjecture is answered and improved.

The problem of computing the integral cohomology ring of the symmetric square of a topological space has long been of interest, but limited progress has been made on the general case until recently. We offer a solution for the complex and quaternionic projective spaces $\mathbb{K}$Pn, by utilising their rich geometrical structure. Our description involves generators and relations, and our methods entail ideas from the literature of quantum chemistry, theoretical physics, and combinatorics. We begin with the case $\mathbb{K}$P∞, and then identify the truncation required for passage to finite n. The calculations rely upon a ladder of long exact cohomology sequences, which compares cofibrations associated to the diagonals of the symmetric square and the corresponding Borel construction. These incorporate the one-point compactifications of classic configuration spaces of unordered pairs of points in $\mathbb{K}$Pn, which are identified as Thom spaces by combining Löwdin's symmetric orthogonalisation (and its quaternionic analogue) with a dash of Pin geometry. The relations in the ensuing cohomology rings are conveniently expressed using generalised Fibonacci polynomials. Our conclusions are compatible with those of Gugnin mod torsion and Nakaoka mod 2, and with homological results of Milgram.

The trace on matrix rings, along with the augmentation map and Kaplansky trace on group rings, are some of the many examples of linear functions on algebras that vanish on all commutators. We generalize and unify these examples by studying traces on (contracted) semigroup rings over commutative rings. We show that every such ring admits a minimal trace (i.e., one that vanishes only on sums of commutators), classify all minimal traces on these rings, and give applications to various classes of semigroup rings and quotients thereof. We then study traces on Leavitt path algebras (which are quotients of contracted semigroup rings), where we describe all linear traces in terms of central maps on graph inverse semigroups and, under mild assumptions, those Leavitt path algebras that admit faithful traces.

We show that the commutative V*-algebras with relatively weakly compact unit spheres are those that are representable by means of hermitian spectral measures. This provides a more unified approach to the results of [15], and allows us to generalise some of them.

Motivated by a problem in mathematical economics [4] Gretsky and Ostroy have shown [5] that every positive operator T:L1[0, 1] → c0 is a Dunford-Pettis operator (i.e. T maps weakly convergent sequences to norm convergent ones), and hence that the same is true for every regular operator from L1[0, 1] to c0. In a recent paper [6] we showed the converse also holds, thereby characterizing the D–P operators by this condition. In each case the proof depends (as do so many concerning D–P operators on Ll[0, 1]) on the following well-known result (see, e.g., [2]): If μ is a finite measure, an operator T:L1(μ) → E is a D–P operator is compact, where i:L∞(μ) → L1(μ) is the canonical injection of L∞(μ) into L1(μ). If μ is not a finite measure this characterization of D–P operators is no longer available, and hence results based on its use (e.g. [5], [6]) do not always have straightforward extensions to the case of operators on more general L1(μ) spaces.

A well known result of B. Osofsky asserts that if R is a left (or right) perfect, left and right selfinjective ring thenR is quasi-Frobenius. It was subsequently conjectured by Carl Faith that every left (or right) perfect, left selfinjective ring is quasi-Frobenius. While several authors have proved the conjecture in the affirmative under some restricted chain conditions, the conjecture remains open even if R is a semiprimary, local, left selfinjective ring withJ(R)^3=0. In this paper we construct a local ring R withJ(R)^3=0 and characterize when R is artinian or selfinjective in terms of conditions on a bilinear mapping from a D-D-bimodule toD , where D is isomorphic to R/J(R). Our work shows that finding a counterexample to the Faith conjecture depends on the existence of aD -D-bimodule over a division ring D satisfying certain topological conditions.

The Arens products are the standard way of extending the product from a Banach algebra to its bidual ″. Ultrapowers provide another method which is more symmetric, but one that in general will only give a bilinear map, which may not be associative. We show that if is Arens regular, then there is at least one way to use an ultrapower to recover the Arens product, a result previously known for C*-algebras. Our main tool is a principle of local reflexivity result for modules and algebras.

We investigate the function R(T,σ), which denotes the error term in the asymptotic formula for \int_0^T|\log\zeta(σ + it)|^2dt. It is shown thatR(T,σ) is uniformly bounded for σ \ge 1 and almost periodic in the sense of Bohr for fixed σ \ge 1; hence R(T,σ) = Ω(1) when T \to \infty. In case {1 \over 2}<σ<1 is fixed we can obtain the bound R(T,σ) \ll_ϵ T\,^{(9-2σ)/8+ϵ}.

Let R be a commutative ring with non-zero identity and M be a unitary R-module. Let (M) be the set of all submodules of M, and φ: (M) → (M) ∪ {∅} be a function. We say that a proper submodule P of M is a prime submodule relative to φ or φ-prime submodule if a ∈ R and x ∈ M, with ax ∈ P ∖ φ(P) implies that a ∈(P :RM) or x ∈ P. So if we take φ(N) = ∅ for each N ∈ (M), then a φ-prime submodule is exactly a prime submodule. Also if we consider φ(N) = {0} for each submodule N of M, then in this case a φ-prime submodule will be called a weak prime submodule. Some of the properties of this concept will be investigated. Some characterisations of φ-prime submodules will be given, and we show that under some assumptions prime submodules and φ1-prime submodules coincide.

If R is a complete discrete valuation ring and M is a reduced, torsion-free R-module of rank \kappa, where \aleph_0 \leq \kappa < 2^ (\aleph_0), we show that M \prop\oplus_(\aleph_0) R \oplus C for some R-module C. As a consequence, it must be the case that M \prop M \oplus (\oplus{_\alpha}R), where \alpha \leq \aleph_0, and {\rm (End)_R}M/\rm (Fin)M has rank at least 2^ (\aleph_0), where Fin M denotes the set of endomorphisms of M with finite rank image.

On présente des exemples de représentations de de dimension 2, de déterminant pair, qui sont de type diédral (I) ou de conducteur premier et de type quelconque (II), en imitant la construction de Tate (Serre [11]) de représentations de déterminant impair.

Recently there have appeared papers ([7], [8]; also see [9]) in which integral equations with kernels involving the confluent hypergeometric function

have been studied. These equations are mainly Volterra equations of the first kind except that they have infinite domain (0, ∞). The rest are of the related type with integrals over (x, ∞) instead of (0, x); and all are convolution equations.

On a semigroup S let the relation ℛ*, sometimes denoted by ℛ, be defined by xℛ*y[(sx = txsy = ty]. A semigroup S is called left type-A, iff the set Es of idempotents of S forms a semilattice under multiplication, each element x of Sis ℛ* related to a (necessarily unique) idempotent x+, and xe = (xe)+x for all x ∈ S, е ∈ Es.

We define a class of equations that are not amenable but are type K and are therefore solvable over torsion-free groups. Moreover, we show that these new equations are solvable over all groups.

We study the possibility of constructing a Frobenius manifold for the standard Landau–Ginzburg model of odd-dimensional quadrics Q2n+1 and matching it with the Frobenius manifold attached to the quantum cohomology of these quadrics. Namely, we show that the initial conditions of the quantum cohomology Frobenius manifold of the quadric can be obtained from the suitably modified standard Landau–Ginzburg model.

We provide corrected versions of Theorems 1 and 2 and Corollaries 3(a) and 4 of the paper mentioned in the title.

In this paper, we study spectral properties and local spectral properties of ∞-complex symmetric operators T. In particular, we prove that if T is an ∞-complex symmetric operator, then T has the decomposition property (δ) if and only if T is decomposable. Moreover, we show that if T and S are ∞-complex symmetric operators, then so is T ⊗ S.

The first discussion of the propagation of elastic waves in a thick plate was given by Lamb [1] for the two-dimensional problem of a harmonic wave travelling in a direction parallel to the medial plane of the plate. Lamb derived equations relating the thickness of the plate to the phase velocities of two types of wave, one symmetric with respect to the medial plane and the other antisymmetric. The symmetric modes of propagation introduced by Lamb have been studied by Holden [2] and the antisymmetric modes have been studied by Osborne and Hart [3]. More recently Pursey [4] has shown how the amplitude of the disturbance is related to a given distribution of stress, varying harmonically with time, applied to the free surfaces of the plate; two types of source are considered by Pursey, one producing a two-dimensional field of the Lamb type, and the other having circular symmetry about an axis normal to the surface of the plate.