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.

We investigate in this paper the topological stability of pairs (ω, X), where ω is a germ of an integrable 1-form and X is a germ of a vector field tangent to the foliation determined by ω.

If G is a finite solvable group and p is a prime, then the normalizer of a Sylow p-subgroup has a normal Sylow 2-subgroup if and only if all non-trivial irreducible real 2-Brauer characters of G have degree divisible by p.

We present a systematic method for proving non-terminating basic hypergeometric identities. Assume that k is the summation index. By setting a parameter x to xqn, we may find a recurrence relation of the summation by using the q-Zeilberger algorithm. This method applies to almost all non-terminating basic hypergeometric summation formulae in the work of Gasper and Rahman. Furthermore, by comparing the recursions and the limit values, we may verify many classical transformation formulae, including the Sears–Carlitz transformation, transformations of the very well-poised 8φ7 series, the Rogers–Fine identity and the limiting case of Watson's formula that implies the Rogers–Ramanujan identities.

A result is presented giving conditions on a set of open discs in the complex plane that ensure that a transcendental meromorphic function with Nevanlinna deficient poles omits at most one finite value outside the set of discs. This improves a previous result of Langley, and goes some way towards closing a gap between Langley's result and a theorem of Toppila in which the omitted values considered may include ∞

In this paper, we deal with the non-negative solutions of a degenerate parabolic system with nonlinear coupled boundary conditions and non-negative non-trivial compactly supported initial data. The critical Fujita exponents are given and the blow-up rates of the non-global solution are obtained.

We study the spectrum of periodic Jacobi matrices. We concentrate on the case of slowly oscillating diagonal terms and study the behaviour of the zeros of the associated orthogonal polynomials in the spectral gap. We find precise estimates for the distance from single eigenvalues of truncated matrices in the spectral gap to the diagonal entries of the matrix. We include a brief numerical example to show the exactness of our estimates.

We continue the investigation of the algebraic and topological structure of the algebra of Colombeau generalized functions with the aim of building up the algebraic basis for the theory of these functions. This was started in a previous work of Aragona and Juriaans, where the algebraic and topological structure of the Colombeau generalized numbers were studied. Here, among other important things, we determine completely the minimal primes of and introduce several invariants of the ideals of (Ω). The main tools we use are the algebraic results obtained by Aragona and Juriaans and the theory of differential calculus on generalized manifolds developed by Aragona and co-workers. The main achievement of the differential calculus is that all classical objects, such as distributions, become C∞-functions. Our purpose is to build an independent and intrinsic theory for Colombeau generalized functions and place them in a wider context.

A generalization of the odd Bernoulli polynomials related to the quantum Euler top is introduced and investigated. This is applied in order to compute the coefficients of the spectral polynomials for the classical Lamé operator.

The nonlinear eigenvalue problem

for 0 ≤ x < ∞, fixed p ∈ (1, ∞), and with y′(0)/y(0) specified, is studied under conditions on q related to those of Brinck and Molanov. Topics include Sturmian results, connections between problems on finite intervals and the half-line, and variational principles.