Multiple Blocking Sets in PG(n, q), n > 3

This article discusses minimal s-fold blocking sets B in PG (n, q), q = p^h, p prime, q > 661, n > 3, of size |B| > sq + c ^p q ^2/3 - (s - 1) (s - 2)/2 (s > min (c ^p q ^1/6, q ^1/4/2)). It is shown that these s-fold blocking sets contain the disjoint union of a collection of s lines and/or Baer subplanes. To obtain these results, we extend results of Blokhuis–Storme–Szönyi on s-fold blocking sets in PG(2, q) to s-fold blocking sets having points to which a multiplicity is given. Then the results in PG(n, q), n 3, are obtained using projection arguments. The results of this article also improve results of Hamada and Helleseth on codes meeting the Griesmer bound.     

Descriptive properties of the set of exposed points of compact convex sets in

We construct a compact convex subset of such that the set of its exposed points is not the intersection of an set and a set. The existence of such a set answers a question posed by G. Choquet, H.H. Corson and V.L. Klee.

     

On Diophantine definability and decidability in some infinite totally real extensions of

Let be a number field, and a set of its non-Archimedean primes. Then let . Let be a finite set of prime numbers. Let be the field generated by all the -th roots of unity as and . Let be the largest totally real subfield of . Then for any 0$">, there exist a number field , and a set of non-Archimedean primes of such that has density greater than , and has a Diophantine definition over the integral closure of in .

     

Analytic -adic cell decomposition and integrals

We prove a conjecture of Denef on parameterized -adic analytic integrals using an analytic cell decomposition theorem, which we also prove in this paper. This cell decomposition theorem describes piecewise the valuation of analytic functions (and more generally of subanalytic functions), the pieces being geometrically simple sets, called cells. We also classify subanalytic sets up to subanalytic bijection.

     

On a Class of Uniformly Distributed Sequences of Point Sets

In this note we construct certain sequences of finite point sets in [0, 1)^s(s ≥ 1) and give the upper bounds of their discrepancy. Furthermore we prove that these sequences are uniformly distrbuted in [0, 1)^s.     

Software news and updates. Basis-set completeness profiles in two dimensions

A two-electron basis-set completeness profile is proposed by analogy with the one-electron profile introduced by D. P. Chong (Can J Chem 1995, 73, 79). It is defined as Y(alpha, beta) = sigmam sigman (Galpha(1)Gbeta(2)/(1/r12)/ psim(1)psin(2)) (psim(1)psin(2)/r12/Galpha(1)Gp(2)) and motivated by the expression for the basis-set truncation correction that occurs in the framework of explicitly correlated methods (Galpha is a scanning Gaussian-type orbital of exponent alpha and [psim] is the orthonormalized one-electron basis under study). The two-electron basis-set profiles provide a visual assessment of the suitability of basis sets to describe electron-correlation effects. Furthermore, they provide the opportunity to assess the quality of the basis set as a whole--not only of the individual angular momentum subspaces, as is the case for the one-electron basis-set profiles. The two-electron completeness profiles of the cc-pVXZ (X = D, T, Q), aug-cc-pVTZ, cc-pCVTZ, and SVP-auxiliary basis sets for the carbon atom are presented as illustrative examples.     

Linear Point Sets and Rédei Type k-blocking Sets in PG(n, q)

In this paper, k-blocking sets in PG(n, q), being of Rédei type, are investigated. A standard method to construct Rédei type k-blocking sets in PG(n, q) is to construct a cone having as base a Rédei type k-blocking set in a subspace of PG(n, q). But also other Rédei type k-blocking sets in PG(n, q), which are not cones, exist. We give in this article a condition on the parameters of a Rédei type k-blocking set of PG(n, q = p ^h ), p a prime power, which guarantees that the Rédei type k-blocking set is a cone. This condition is sharp. We also show that small Rédei type k-blocking sets are linear.     

Simplicial Properties of the Set of Planar Binary Trees

Planar binary trees appear as the the main ingredient of a new homology theory related to dialgebras, cf.(J.-L. Loday, C.R. Acad. Sci. Paris 321 (1995), 141–146.) Here I investigate the simplicial properties of the set of these trees, which are independent of the dialgebra context though they are reflected in the dialgebra homology.The set of planar binary trees is endowed with a natural (almost) simplicial structure which gives rise to a chain complex. The main new idea consists in decomposing the set of trees into classes, by exploiting the orientation of their leaves. (This trick has subsequently found an application in quantum electrodynamics, c.f. (C. Brouder, On the Trees of Quantum Fields, Eur. Phys. J. C12, 535–549 (2000).) This decomposition yields a chain bicomplex whose total chain complex is that of binary trees. The main theorem of the paper concerns a further decomposition of this bicomplex. Each vertical complex is the direct sum of subcomplexes which are in bijection with the planar binary trees. This decomposition is used in the computation of dialgebra homology as a derived functor, cf. (A. Frabetti, Dialgebra (co) Homology with Coefficients, Springer L.N.M., to appear).     

A universal coanalytic linear ordering

We construct a linear ordering in which every (coanalytic) linear ordering can be order embedded.