Local Stereology of Tensors of Convex Bodies

In this paper, we present local stereological estimators of Minkowski tensors defined on convex bodies in ? ^d . Special cases cover a number of well-known local stereological estimators of volume and surface area in ?³, but the general set-up also provides new local stereological estimators of various types of centres of gravity and tensors of rank two. Rank two tensors can be represented as ellipsoids and contain information about shape and orientation. The performance of some of the estimators of centres of gravity and volume tensors of rank two is investigated by simulation.     

Convergence to steady states of solutions of semilinear evolutionary integral equations

We study the convergence and decay rate to a steady state of bounded solutions of the nonlinear evolutionary integral equation and we apply our abstract results to the viscoelastic Euler-Bernoulli beam and to Kelvin-Voigt solids.Received: 25 March 2003, Accepted: 5 April 2004, Published online: 16 July 2004This work was partially supported by the DFG project Regularität und Asymptotik für elliptische und parabolische Probleme and by the grants GAR 201/01/D094, MSM 113200007.     

Some identities on the Catalan, Motzkin and Schröder numbers

In this paper, some identities between the Catalan, Motzkin and Schröder numbers are obtained by using the Riordan group. We also present two combinatorial proofs for an identity related to the Catalan numbers with the Motzkin numbers and an identity related to the Schröder numbers with the Motzkin numbers, respectively.     

String cosmology: a review

We give an overview of the status of string cosmology. We explain the motivation for the subject, outline the main problems, and assess some of the proposed solutions. Our focus is on those aspects of cosmology that benefit from the data of an ultraviolet-complete theory.     

A genetic algorithm for the unrelated parallel machine scheduling problem with sequence dependent setup times

In this work a genetic algorithm is presented for the unrelated parallel machine scheduling problem in which machine and job sequence dependent setup times are considered. The proposed genetic algorithm includes a fast local search and a local search enhanced crossover operator. Two versions of the algorithm are obtained after extensive calibrations using the Design of Experiments (DOE) approach. We review, evaluate and compare the proposed algorithm against the best methods known from the literature. We also develop a benchmark of small and large instances to carry out the computational experiments. After an exhaustive computational and statistical analysis we can conclude that the proposed method shows an excellent performance overcoming the rest of the evaluated methods in a comprehensive benchmark set of instances.     

Ekedahl–Oort and Newton strata for Shimura varieties of PEL type

We study the Ekedahl–Oort stratification for good reductions of Shimura varieties of PEL type. These generalize the Ekedahl–Oort strata defined and studied by Oort for the moduli space of principally polarized abelian varieties (the “Siegel case”). They are parameterized by certain elements $w$ in the Weyl group of the reductive group of the Shimura datum. We show that for every such $w$ the corresponding Ekedahl–Oort stratum is smooth, quasi-affine, and of dimension $\ell (w)$ (and in particular non-empty). Some of these results have previously been obtained by Moonen, Vasiu, and the second author using different methods. We determine the closure relations of the strata. We give a group-theoretical definition of minimal Ekedahl–Oort strata generalizing Oort’s definition in the Siegel case and study the question whether each Newton stratum contains a minimal Ekedahl–Oort stratum. As an interesting application we determine which Newton strata are non-empty. This criterion proves conjectures by Fargues and by Rapoport generalizing a conjecture by Manin for the Siegel case. We give a necessary criterion when a given Ekedahl–Oort stratum and a given Newton stratum meet.     

Uniformly defining valuation rings in Henselian valued fields with finite or pseudo-finite residue fields

We give a definition, in the ring language, of _Zp${\mathbb{Z}}_p$ inside _Qp${\mathbb{Q}}_p$ and of _Fp[[t]]${\mathbb{F}}_p[[t]]$ inside _Fp((t))${\mathbb{F}}_p((t))$, which works uniformly for all p   and all finite field extensions of these fields, and in many other Henselian valued fields as well. The formula can be taken existential-universal in the ring language, and in fact existential in a modification of the language of Macintyre. Furthermore, we show the negative result that in the language of rings there does not exist a uniform definition by an existential formula and neither by a universal formula for the valuation rings of all the finite extensions of a given Henselian valued field. We also show that there is no existential formula of the ring language defining _Zp${\mathbb{Z}}_p$ inside _Qp${\mathbb{Q}}_p$ uniformly for all p  . For any fixed finite extension of _Qp${\mathbb{Q}}_p$, we give an existential formula and a universal formula in the ring language which define the valuation ring.     

Hilbert, Dirichlet and Fejér families of operators arising from C 0-groups, cosine functions and holomorphic semigroups

The main aim of this paper is to extend definitions of Hilbert transform, Dirichlet and Fejér operators (defined by convolution with suitable kernels in Lebesgue spaces) in arbitrary Banach spaces. We present a self-contained theory which includes different approaches of other authors whose starting points were usually C ₀-groups or cosine functions. We present relations with holomorphic semigroups. We characterize the geometric property of UMD spaces in terms of the Dirichlet and Fejér operators. To end the paper, we give examples to illustrate our results.     

Function Spaces and Contractive Extensions in Approach Theory: The Role of Regularity

Two classical results characterizing regularity of a convergence space in terms of continuous extensions of maps on one hand, and in terms of continuity of limits for the continuous convergence on the other, are extended to convergence-approach spaces. Characterizations are obtained for two alternative extensions of regularity to convergence-approach spaces: regularity and strong regularity. The results improve upon what is known even in the convergence case. On the way, a new notion of strictness for convergence-approach spaces is introduced.