We prove that the class number of the imaginary quadratic field is divisible by n for any positive integers k and n with 22k < 3n, by using Y. Bugeaud and T. N. Shorey's result on Diophantine equations.

In this paper, we study Lp-boundedness properties for higher order Littlewood-Paley g-functions in the Bessel setting. We use the Calderón-Zygmund theory in a homogeneous-type space (in the sense of Coifman and Weiss) ((0, ∞), d, γα), where d represents the usual metric on (0, ∞) and γα denotes the doubling measure on (0, ∞) with respect to d defined by dγα(x) = x2α+1dx, with α > −1/2.

Let n be the n-dimensional hyperbolic space with n ≥ 2. Suppose that G is a discrete, sense-preserving subgroup of Isomn, the isometry group of n. Let p be the projection map from n to the quotient space M = n/G. The first goal of this paper is to prove that for any a ∈ ∂n (the sphere at infinity of n), there exists an open neighbourhood U of a in n ∪ ∂ n such that p is an isometry on U ∩ n if and only if a ∈ oΩ(G) (the domain of proper discontinuity of G). This is a generalization of the main result discussed in the work by Y. D. Kim (A theorem on discrete, torsion free subgroups of Isomn, Geometriae Dedicata109 (2004), 51–57). The second goal is to obtain a new characterization for the elements of Isomn by using a class of hyperbolic geometric objects: hyperbolic isosceles right triangles. The proof is based on a geometric approach.

We study the duality problem for order weakly compact operators by giving sufficient and necessary conditions under which the order weak compactness of an operator implies the order weak compactness of its adjoint and conversely.

This paper deals with the global existence and blow-up properties of the following non-Newton polytropic filtration system coupled with local source: ut − Δm,pu = avα, vt − Δn,qv = buβ. Under appropriate hypotheses, we prove that the solution either exists globally or blows up in finite time depending on the initial data and the relations between αβ and mn(p − 1)(q − 1).

In this paper, we prove a theorem concerning asymptotic estimates of the eigenvalues of certain positive integral operators with Laplace transform type kernels.

In this paper, we are computing asymptotic formulas for a base of solutions of the second-order difference equations in the double root case. Two methods are presented.

The c-dimension of a group is the maximum length of a chain of nested centralizers. It is proved that a periodic locally soluble group of finite c-dimension k is soluble of derived length bounded in terms of k, and the rank of its quotient by the Hirsch–Plotkin radical is bounded in terms of k. Corollary: a pseudo-(finite soluble) group of finite c-dimension k is soluble of derived length bounded in terms of k.

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.

We investigate the spectral properties of matrices associated with comb graphs. We show that the adjacency matrices and adjacency matrix Laplacians of the sequences of graphs show a spectral similarity relationship in the sense of work by L. Malozemov and A. Teplyaev (Self-similarity, operators and dynamics, Math. Phys. Anal. Geometry6 (2003), 201–218), and hence these sequences of graphs show a spectral decimation property similar to that of the Laplacians of the Sierpiński gasket graph and other fractal graphs.

We denote the numerical range of the normal operator T by W(T). A characterization is given to the points in W(T) that lie on the boundary. The collection of such boundary points together with the interior of the the convex hull of the spectrum of T will then be the set W(T). Moreover, it is shown that such boundary points reveal a lot of information about the normal operator. For instance, such a boundary point always associates with an invariant (reducing) subspace of the normal operator. It follows that a normal operator acting on a separable Hilbert space cannot have a closed strictly convex set as its numerical range. Similar results are obtained for the Davis-Wielandt shell of a normal operator. One can deduce additional information of the normal operator by studying the boundary of its Davis-Wielandt shell. Further extension of the result to the joint numerical range of commuting operators is discussed.

Let k be a finite field with q elements and characteristic coprime with 6. Our main result is: Every polynomial P ∈ k[T] is a strict sum of three cubes and two squares.

We show that there is an infinite set of natural numbers with the property that is square-free for every finite subset ⊆ .

Vincent Lafforgue's bivariant K-theory for Banach algebras is invariant in the second variable under a rather general notion of Morita equivalence. In particular, the ordinary topological K-theory for Banach algebras is invariant under Morita equivalences.

Let (X, cX) be a convex projective surface equipped with a real structure. The space of stable maps carries different real structures induced by cX and any order two element τ of permutation group Sk acting on marked points. Each corresponding real part ℝτ is a real normal projective variety. As the singular locus is of codimension bigger than two, these spaces thus carry a first Stiefel–Whitney class for which we determine a representative in the case k = c1(X)d − 1 where c1(X) is the first Chern class of X. Namely, we give a homological description of these classes in term of the real part of boundary divisors of the space of stable maps.

The Hecke algebra of a Hecke pair (G, H) is studied using the Schlichting completion (Ḡ, ), which is a Hecke pair whose Hecke algebra is isomorphic to and which is topologized so that is a compact open subgroup of Ḡ. In particular, the representation theory and C*-completions of are addressed in terms of the projection using both Fell's and Rieffel's imprimitivity theorems and the identity . An extended analysis of the case where H is contained in a normal subgroup of G (and in particular the case where G is a semi-direct product) is carried out, and several specific examples are analysed using this approach.

Let be the self-adjoint operator associated with the Dirichlet form

where ϕ is a positive C2 function, dλϕ = ϕdλ and λ denotes Lebesgue measure on ℝd. We study the boundedness on Lp(λϕ) of spectral multipliers of . We prove that if ϕ grows or decays at most exponentially at infinity and satisfies a suitable ‘curvature condition’, then functions which are bounded and holomorphic in the intersection of a parabolic region and a sector and satisfy Mihlin-type conditions at infinity are spectral multipliers of Lp(λϕ). The parabolic region depends on ϕ, on p and on the infimum of the essential spectrum of the operator on L2(λϕ). The sector depends on the angle of holomorphy of the semigroup generated by on Lp(λϕ).

We show that if G is an infinitely generated locally (polycyclic-by-finite) group with cohomology almost-everywhere finitary, then every finite subgroup of G acts freely and orthogonally on some sphere.

In this paper the spectral properties of the abstract Klein–Gordon equation are studied. The main tool is an indefinite inner product known as the charge inner product. Under certain assumptions on the potential V, two operators are associated with the Klein–Gordon equation and studied in Krein spaces generated by the charge inner product. It is shown that the operators are self-adjoint and definitizable in these Krein spaces. As a consequence, they possess spectral functions with singularities, their essential spectra are real with a gap around 0 and their non-real spectra consist of finitely many eigenvalues of finite algebraic multiplicity which are symmetric to the real axis. One of these operators generates a strongly continuous group of unitary operators in the Krein space; the other one gives rise to two bounded semi-groups. Finally, the results are applied to the Klein–Gordon equation in ℝn.