In this article, we explore the problem of determining isomorphisms between the twisted complex group algebras of finite $p$-groups. This problem bears similarity to the classical group algebra isomorphism problem and has been recently examined by Margolis-Schnabel. Our focus lies on a specific invariant, referred to as the generalized corank, which relates to the twisted complex group algebra isomorphism problem. We provide a solution for non-abelian $p$-groups with generalized corank at most three.

G is a graph on n nodes with q edges, without loops or multiple edges. We write α = q/n and β for the maximum degree of any node of G. We write

and H for the number of Hamiltonian circuits (H.c.) in the complement of G, or, what is the same thing, the number of those H.c. in the complete graph Kn which have no edge in common with G. Our object here is to prove the following theorem.

This paper proves the existence of potentials of the first and second kind of a Frobenius like structure in a frame, which encompasses families of arrangements. The frame uses the notion of matroids. For the proof of the existence of the potentials, a power series ansatz is made. The proof that it works requires that certain decompositions of tuples of coordinate vector fields are related by certain elementary transformations. This is shown with a nontrivial result on matroid partition.

Piochi in [10] gives a description of the least commutative congruence γ of an inverse semigroup in terms of congruence pairs and generalizes to inverse semigroups the notion of solvability. The object of this paper is to give an explicit construction of λ for simple regular ω-semigroups exploiting the work of Baird on congruences on such semigroups. Moreover the connection between the solvability classes of simple regular ω-semigroups and those of their subgroups is studied.

A maximal symmetry group is a group of isomorphisms of a three-dimensional hyperbolic manifold of maximal order in relation to the volume of the manifold. In this paper we determine all maximal symmetry groups of the types PSL(2, q) and PGL(2, q). Depending on the prime p there are one or two such groups with q=pk and k always equals 1, 2 or 4.

Let C be the Cayley algebra denned over the real field. If, for given elements α, β, and γ of a quaternion subalgebra of C, α = βγ, it follows, by associativity, that for any nonzero element δ of the same quaternion subalgebra, α = (βδ)(δ-1γ). For Cayley numbers ζ ξ, and η with ζ = ξη, the relation ζ = (ξδ)(δ-1η) in general only holds when δ is a nonzero real number. Because of the existence of factorization results [1, 2] in the orders of C, the question naturally arises of whether it is possible to choose one-to-one mappings, θ and φ, of C onto itself such that ζ = θξ. φη whenever ζ = ξη. To discuss this question, we make the following definition.

We consider a class of operators that contains the strictly singular operators and it is contained in the perturbation class of the upper semi-Fredholm operators PΦ+. We show that this class is strictly contained in PΦ+, solving a question of Friedman. We obtain similar results for the strictly cosingular operators and the perturbation class of the lower semi-Fredholm operators PΦ−. We also characterize in terms of PΦ+ and in terms of PΦ−. As a consequence, we show that and are the biggest operator ideals contained in PΦ+ and PΦ−, respectively.

The development of the theory of absolute integrals derives from certain key facts. Among them are:

(I) An integral is a positive linear functional on a vector lattice, which is continuous in a certain sense.

(II) A function equal almost everywhere to a summable function is itself summable.

(III) Every measurable function is the pointwise limit of a sequence of elementary step functions.

A device that often plays an important role in measure theory, but which has not beenfully exploited in the theory of abstract integrals is that of

(IV) the smallest class containing a given class and having a certain property

(such as being a σ-ring of sets). It is our purpose in this paper to examine the theory of abstract real-valued absolute integrals axiomatically, in such a way as to isolate and clarify the roles of (I) through (IV).

Let Ω be a smooth bounded domain in RN, with N ≥ 5. We provide existence and bifurcation results for the elliptic fourth-order equation Δ2u − Δpu = f(λ, x, u) in Ω, under the Dirichlet boundary conditions u = 0 and ∇u = 0. Here λ is a positive real number, 1 < p ≤ 2# and f(.,., u) has a subcritical or a critical growth s, 1 < s ≤ 2*, where and . Our approach is variational, and it is based on the mountain-pass theorem, the Ekeland variational principle and the concentration-compactness principle.

Let f: (ℝn, 0)→ (ℝ,0) be a germ of a real analytic function. Let L and F(f) denote the link of f and the Milnor fibre of fc respectively, i. e., L = {x ∈ Sn−1 | f(x) = 0}, , where 0 ≤ ξ ≪ r ≪ 1, . In [2] Szafraniec introduced the notion of an -germ as a generalization of a germ defined by a weighted homogeneous polynomial satisfying some condition concerning the relation between its degree and weights (definition 1). He also proved that if f is an -germ (presumably with nonisolated singularity) then the number χ(F(f)/d mod 2 is a topological invariant of f, where χ(F(f)) is the Euler characterististic of F(f), and gave the formula for χ(L)/2 mod 2 (it is a well-known fact that F(L) is an even number). As a simple consequence he got the fact that χ(F(f)mod 2 is a topological invariant for any f, which is a generalization of Wall's result [3] (he considered only germs with an isolated singularity).

We show that the order of an automorphism of an arithmetic Riemann surface of genus $g$ is not greater than $2g\,{-}\,2$, provided $g$ is large enough. This bound is an arithmetic analog of the classical Wiman bound. We prove that it is sharp and attained for any genus but in contrast to the general case the automorphisms of maximal order act without fixed points. This allows us to consider the automorphisms which act on arithmetic Riemann surfaces and have a given number of fixed points. For these automorphisms we describe the asymptotic behavior of their orders.

Let A be an operator on a Hillbert space with polar decomposition A = |A|, let  = |A|½U|A|½ and let  = V|Â| be the polar decomposition of Â. Write à for the operatorà = |Â|½V|Â|½. If = (A1,…,AN) is a doubly commuting n-tuple of p-hyponormal operators on a Hillbert space with equal defect and nullity, then = (Ã1,…,Ãn) is a doubly commuting n-tuple of hyponormal operators. In this paper we show that

where σ* denotes σTe (Taylor essential spectrum), σTw (Taylor-Weyl spectrum) and σTb (Taylor-Browder spectrum), respectively.

Let E[τ] be a locally convex Hausdorff topological vector space. An extended decomposition of E[τ] is a family {Ea}α∈A of closed subspaces of E such that, for each x in E and each α in A, there exists a unique point xα in Eα, with Here convergence will have the following meaning. Let Ф denote the set of all finite subsets of A. The sum is said to be convergent to x if for each neighbourhood U of 0 in E, there is an element φ0 of Ф such that , for all φ in Ф containing φ0. It follows that is Cauchy if and only if, for each neighbourhood U of 0 in E, there is an element φ0 of Ф such that , for all φ in Ф disjoint from φ0.

En algèbre non-commutative, on dit qu'un anneau noethérien A est local si:

(i) le radical de Jacobson M de A est un idéal maximal,

(ii) ∩ Mn = (0),

(iii) A/M est artinien simple.

Dans [9], Walker definit un anneau local régulier comme un anneau local A dont le radical de Jacobson M est engendré par une A-suite centralisante x1; x2, …, xt, [4], et demontre alors que:

t = cldim A = Kdim A = rgldim A = pdAA/M.

§ 1. Introductory. The formula to be proved is

where n is zero or a positive integer, R(m)>0, R(αs-l)>0, s = 1, 2, …, p, p≧q.

In the theory of self-adjoint operators in Hilbert space and of formally self-adjoint linear differential equations there are many situations involving analytic functions on the complex plane whose singularities are confined to the real axis and where the growth of the function at such singular points is strictly limited.

The cone length Cl(f) of a map f: X → Y is defined to be the least number of attaching maps possible in a conic (or iterated mapping cone) structure for f. Cone length is a homotopy invariant in the sense that if φ: X → X and ρ: Y → Y are homotopy equivalences then Cl (ρ°f°φ) = Cl(f). Furthermore Cl(f) depends only on the homotopy class of f. It was shown by Ganea [8] that the cone length of the map * → X coincides with the strong Lusternik-Schnirelmann category of X as a space (see Proposition 1.6 below). Recent work of Cornea ([3]–[6]) is much concerned with cone length and its role in critical point theory. For example, let f be a smooth real valued function on a manifold triad (M; V0, V1) with V0 ≠ θ. Under certain conditions, if f has only “reasonable” critical points then it must have at least Cl(V0↪M) of them (see [6]).

In this paper we obtain the general solution of the equation

whose kernel is

where 0<α, β<1, μ, η, σ > 0, δ>0, are real parameters, z = max(x, y), φ(t) and g(y) are prescribed functions and ƒ(x) is to be determined.

The aim of this note is to present a new, elementary proof of a result of Baas and Madsen on the mod p cohomology of certain quotients of the spectrum BP.