On modular inequalities in variable L p spaces

We show that the Hardy-Littlewood maximal operator and a class of Calderón-Zygmund singular integrals satisfy the strong type modular inequality in variable L^p spaces if and only if the variable exponent p(x) ∼ const. Received: 15 September 2004     

On the spectral norm of algebraic numbers

In this paper we continue to study the spectral norms and their completions ([4]) in the case of the algebraic closure $ \overline {\mathbb Q} $ of ? in ?. Let $ \widetilde{\overline{\mathbb{Q}}} $ be the completion of $ \overline {\mathbb Q} $ relative to the spectral norm. We prove that $ \widetilde{\overline{\mathbb{Q}}} $ can be identified with the R‐subalgebra of all symmetric functions of C(G), where C(G) denotes the ?‐Banach algebra of all continuous functions defined on the absolute Galois group G = Gal$ {\overline {\mathbb Q}} / {\mathbb Q} $. We prove that any compact, closed to conjugation subset of ? is the pseudo‐orbit of a suitable element of $ \widetilde{\overline{\mathbb{Q}}} $. We also prove that the topological closure of any algebraic number field in $ \widetilde{\overline{\mathbb{Q}}} $ is of the form $\widetilde{\mathbb{Q}[x]}$ with x in $ \widetilde{\overline{\mathbb{Q}}} $.     

Modulation of the distance dependence of paramagnetic relaxation enhancements by CSA × DSA cross-correlation

Paramagnetic metal ions with fast-relaxing electronic spin and anisotropic susceptibility tensor provide a rich source of structural information that can be derived from pseudo-contact shifts, residual dipolar couplings, dipole-dipole Curie spin cross-correlation, and paramagnetic relaxation enhancements. The present study draws attention to a cross-correlation effect between nuclear relaxation due to anisotropic chemical shielding (CSA) and due to the anisotropic dipolar shielding (DSA) caused by the electronic Curie spin. This CSA x DSA cross-correlation contribution seems to have been overlooked in previous interpretations of paramagnetic relaxation enhancements. It is shown to be sufficiently large to compromise the 1/r6 distance dependence usually assumed. The effect cannot experimentally be separated from auto-correlated DSA relaxation. It can increase or decrease the observed paramagnetic relaxation enhancement. Under certain conditions, the effect can dominate the entire paramagnetic relaxation, resulting in nuclear resonances narrower than in the absence of the paramagnetic center. CSAxDSA cross-correlation becomes important when paramagnetic relaxation is predominantly due to the Curie rather than the Solomon mechanism. Therefore the effect is most pronounced for relaxation by metal ions with large magnetic susceptibility and fast-relaxing electron spin. It most strongly affects paramagnetic enhancements of transverse relaxation in macromolecules and of longitudinal relaxation in small molecules.     

Astrogeometry: Toward mathematical foundations

Inimage processing (e.g., inastronomy), the desired black-and-white image is, from the mathematical viewpoint, aset. Hence, to process images, we need to process sets. To define a generic set, we need infinitely many parameters; therefore, if we want to represent and process sets in the computer, we must restrict ourselves to finite-parameter families of sets that will be used to approximate the desired sets. The wrong choice of a family can lead to longer computations and worse approximation. Hence, it is desirable to find the family that it isthe best in some reasonable sense. In this paper, we show how the problems of choosing the optimal family of sets can be formalized and solved. As a result of the described general methodology, forastronomical images, we get exactly the geometric shapes that have been empirically used by astronomers and astrophysicists; thus, we have atheoretical explanation for these shapes.     

Balance constraints reduction of large-scale linear programming problems

This paper describes a method and the corresponding algorithms for simplification of large-scale linear programming models. It consists of the elimination of the balance constraints (i.e. constraints with zero RHS term). The idea is to apply some linear transformations to the original problem in order to nullify the balance constraints. These transformations are able to simultaneously eliminate more balance rows. The core of this contribution is the introduction of the reduction matrix and the associated theorems on the equivalent linear programs (original and reduced). The numerical experiments with this method of simplification proved this approach to be beneficial for a large class of LP problems.This research work was done while the first author was at Duisburg University, Mess-, Steuer und Regelungstechnik, Germany, under the greatly appreciated financial assistance given by the Alexander-von-Humboldt Foundation.     

Rank, term rank, and chromatic number

Let G be a graph with a nonempty edge set, and with rank rk(G), term rank Rk(G), and chromatic number χ(G). We characterize Rk(G) as being the maximum number of colors in certain proper colorings of G. In particular, we observe that χ(G)Rk(G), with equality holding if and only if (besides isolated vertices) G is either complete or a star. For a twin-free graph G, we observe the bound and we show that this bound is sharp.