Let the Laurent expansion of the Riemann zeta function ξ(s) about s=1 be written in the form

It has been discovered independently by many authors that, in terms of this notation, the coefficient

The abstract theory of positive compact operators (acting in a partially ordered Banach space) has proved to be particularly useful in the theory of integral equations. In a recent paper (2) it was shown that many of the now classical theorems for positive compact operators can be extended to certain classes of non-compact operators. One result, proved in (2, Theorem 5), was a fixed point theorem for compressive k-set contractions (k<l). The main result of this paper (Theorem 3.3) shows that some of the hypotheses of (2, Theorem 5) are unnecessary. We use techniques based on those used by M. A. Krasnoselskii in the proof of Theorem 4.12 in (4), which is the classical fixed point theorem for compressive compact operators, to obtain a complete generalisation of this classical result to the k-set contractions (k < 1). It should be remarked that J. D. Hamilton has extended the same result to A-proper mappings (3, Theorem 1). However apparently it is not known, even in the case when we are dealing with a Π1-space, whether k-set contractions are A-proper or not.

We consider a class of Jacobi matrices with periodically modulated diagonal in a critical hyperbolic (‘double root’) situation. For the model with ‘non-smooth’ matrix entries we obtain the asymptotics of generalized eigenvectors and analyse the spectrum. In addition, we reformulate a very helpful theorem from a paper by Janas and Moszynski in its full generality in order to serve the needs of our method.

The principal results in this paper are concerned with the description of domains of infinitesimal generators of strongly continuous groups of isometries in non-commutative operator spaces $E(\mathcal{M},\tau)$, which are induced by $\mathbb{R}$-flows on $\mathcal{M}$. In particular, we are concerned with the description of operator functions which leave the domain of such generators invariant in all symmetric operator spaces, associated with a semi-finite von Neumann algebra $\mathcal{M}$ and a separable function space $E$ on $(0,\infty)$. Furthermore, we apply our results to the study of operator functions for which $[D,x]\in E(\mathcal{M},\tau)$ implies that $[D,f(x)]\in E(\mathcal{M},\tau)$, where $D$ is an unbounded self-adjoint operator. Our methods are partly based on the recently developed theory of double operator integrals in symmetric operator spaces and the theory of adjoint $C_{0}$-semigroups.

In the general theory of relativity, as in many other branches of theoretical physics, the material and energetical content of spacetime is considered, in the first instance, as an extended field, which is specified by means of field quantities (energy-momentum-stress tensor, charge-current density, electromagnetic field strength). From this point of view corpuscles (material particles, photons) are constructs obtained by first considering the field quantities as nonvanishing only within certain world tubes, and then passing by limiting processes to the idealisation in which these world tubes are shrunk into world lines. More precisely, this passage to the corpuscular description may be thought of as accomplished by replacing the original field by successive members of a sequence of field distributions, satisfying the same field laws, which cluster more and more in the neighbourhood of the world lines, and for which in some significant sense the total measure approaches a finite limit. Each such world line, together with the limiting measures of those portions of the field quantities associated therewith, is then a corpuscle; the form of the world line determines the motion of the corpuscle, and the associated “corpuscular quantities” its physical attributes (mass or energy, momentum, charge).

We introduce a new method to study rational conjugacy of torsion units in integral group rings using integral and modular representation theory. Employing this new method, we verify the first Zassenhaus conjecture for the group PSL(2, 19). We also prove the Zassenhaus conjecture for PSL(2, 23). In a second application we show that there are no normalized units of order 6 in the integral group rings of M 10 and PGL(2, 9). This completes the proof of a theorem of Kimmerle and Konovalov that shows that the prime graph question has an affirmative answer for all groups having an order divisible by at most three different primes.

A strongly regular ring R is one in which for all x ∈ R, there is an a ∈ R with x = x2a. Equivalently, for all x there is an a with x = ax2. Such a ring is regular, duo, biregular, and a left and right V-ring. Moreover since R is reduced, all nilpotent elements are central (vacuously) and so all idempotent elements are central.

We obtain some new results about the maximal operator space structure which can be put on a normed space. These results are used to prove some dilation results for contractive linear maps from a normed space into B(H). Finally, we prove CB(MIN(X), MAX(y)) = Γ2(X, Y) and apply this result to prove some new Grothendieck-type inequalities and some new estimates on spans of “free” unitaries.

Let μ λ be the Bernoulli convolution associated with λ ∈ (0, 1). The well-known result of Jorgensen and Pedersen shows that if λ = 1/(2k) for some k ∈ ℕ, then μ 1/(2k) is a spectral measure with spectrum Γ(1/(2k)). The recent research on the spectrality of μ λ shows that μ λ is a spectral measure only if λ = 1/(2k) for some k ∈ ℕ. Moreover, for certain odd integer p, the multiple set pΓ(1/(2k)) is also a spectrum for μ 1/(2k). This is surprising because some spectra for the measure μ 1/(2k) are thinning. In this paper we mainly characterize the number p that has the above property. By applying the properties of congruences and the order of elements in the finite group, we obtain several conditions on p such that pΓ(1/(2k)) is a spectrum for μ 1/(2k).

It is shown that for m = 1,2,3,…, the trigonometric sums and can be represented as integer-valued polynomials in n of degrees 2m – 1 and 2m, respectively. Properties of these polynomials are discussed, and recurrence relations for the coefficients are obtained. The proofs of the results depend on the representations of particular polynomials of degree n – 1 or less as their own Lagrange interpolation polynomials based on the zeros of the nth Chebyshev polynomial Tn(x) = cos(narccos x), -1≤x≤1.

In this paper we investigate totally geodesic surfaces in hyperbolic 3-manifolds. In particular we show that if M is a compact arithmetic hyperbolic 3-manifold containing an immersion of a totally geodesic surface then it contains infinitely many commensurability classes of such surfaces. In addition we show for these M that the Chern-Simons invariant is rational.

We also show, that unlike the figure-eight knot complement in S3, many knot complements in S3 do not contain an immersion of a closed totally geodesic surface.

We discuss the existence of breather solutions for a discrete nonlinear Schrödinger equation in an infinite $N$-dimensional lattice, involving site-dependent anharmonic parameters. We give a simple proof of the existence of (non-trivial) breather solutions based on a variational approach, assuming that the sequence of anharmonic parameters is in an appropriate sequence space (decays with an appropriate rate). We also give a proof of the non-existence of (non-trivial) breather solutions, and discuss a possible physical interpretation of the restrictions, in both the existence and non-existence cases.

We prove that the space $L_\infty(\mu,X)$ has the same numerical index as the Banach space $X$ for every $\sigma$-finite measure $\mu$. We also show that $L_\infty(\mu,X)$ has the Daugavet property if and only if $X$ has or $\mu$ is atomless.

AMS 2000 Mathematics subject classification: Primary 46B20; 47A12

Let A be a matrix over a field Φ partitioned as follows

where A11 is n×n and A22 is m×m. The objective of the present paper is to give further results on the problems mentioned in Section 1 of (3). Concretely we shall consider the following question: “we prescribe the characteristic polynomial f(λ) = λn+m−c1λn+m−1 + … of A and the principal blocks A11, A22. Find a necessary and sufficient condition for the existence of A satisfying these prescribed conditions”.

We deal with existence and uniqueness of the solution to the fully nonlinear equation

−F(D2u) + |u|s−1u = f(x) in ℝn,

where s > 1 and f satisfies only local integrability conditions. This result is well known when, instead of the fully nonlinear elliptic operator F, the Laplacian or a divergence-form operator is considered. Our existence results use the Alexandroff-Bakelman-Pucci inequality since we cannot use any variational formulation. For radially symmetric f, and in the particular case where F is a maximal Pucci operator, we can prove our results under fewer integrability assumptions, taking advantage of an appropriate variational formulation. We also obtain an existence result with boundary blow-up in smooth domains.

It is shown that an invertible disjointness preserving operator from a uniformly complete vector lattice onto a normed vector lattice has a disjointness preserving inverse and is necessarily order bounded.

Consider the Klein-Gordon equation

where q, Aj are real valued functions on Rn, m and e positive constants. Equation (1) describes the motion of a relativistic particle of mass m and charge e in an external field described by the electrostatic potential q and the electromagnetic potential A = (Aj); units are chosen so that the speed of light is one.

In this paper we present the basic tools of a fractional function theory in higher dimensions by means of a fractional correspondence to the Weyl relations via fractional Riemann–Liouville derivatives. A Fischer decomposition, Almansi decomposition, fractional Euler and Gamma operators, monogenic projection, and basic fractional homogeneous powers are constructed. Moreover, we establish the fractional Cauchy–Kovalevskaya extension (FCK extension) theorem for fractional monogenic functions defined on ℝd . Based on this extension principle, fractional Fueter polynomials, forming a basis of the space of fractional spherical monogenics, i.e. fractional homogeneous polynomials, are introduced. We study the connection between the FCK extension of functions of the form xPl and the classical Gegenbauer polynomials. Finally, we present an example of an FCK extension.

A Δ-point x of a Banach space is a norm-one element that is arbitrarily close to convex combinations of elements in the unit ball that are almost at distance 2 from x. If, in addition, every point in the unit ball is arbitrarily close to such convex combinations, x is a Daugavet point. A Banach space X has the Daugavet property if and only if every norm-one element is a Daugavet point. We show that Δ- and Daugavet points are the same in L1-spaces, in L1-preduals, as well as in a big class of Müntz spaces. We also provide an example of a Banach space where all points on the unit sphere are Δ-points, but none of them are Daugavet points. We also study the property that the unit ball is the closed convex hull of its Δ-points. This gives rise to a new diameter-two property that we call the convex diametral diameter-two property. We show that all C(K) spaces, K infinite compact Hausdorff, as well as all Müntz spaces have this property. Moreover, we show that this property is stable under absolute sums.