Superlocals in Symmetric and Alternating Groups

On Aschbacher's definition, a subgroup N of a finite group is called a -superlocal for a prime if . We describe the -superlocals in symmetric and alternating groups, thereby resolving part way Problem 11.3 in the Kourovka Notebook [3].     

A Modal Logic Based on Linearly Ordered f-Spaces

A modal logic associated with the -spaces introduced by Ershov is examined. We construct a modal calculus that is complete w.r.t. the class of all strictly linearly ordered -frames, and the class of all strictly linearly ordered -frames.     

Commutativity of the Arens product in lattice ordered algebras

Let be an Abelian Archimedean lattice ordered algebra. The order bidual furnished with the Arens product is again a lattice ordered algebra. We show that the order continuous order bidual is Abelian. This solves an open problem and improves a result of Scheffold, who proved it for the case of normed lattice ordered algebras. The proof is based on the up-down-up approximation of positive elements in the order continuous order bidual by elements in the canonical image of in Components of positive elements in are characterized and the result is applied to the Arens product of -and almost -algebras.     

Domination in generalized Petersen graphs

Generalized Petersen graphs are certain graphs consisting of one quadratic factor. For these graphs some numerical invariants concerning the domination are studied, namely the domatic number , the total domatic number and the -ply domatic number for and . Some exact values and some inequalities are stated.     

Some topological properties of ω-covering sets

We prove the following theorems:1. There exists an -covering with the property s ₀.2. Under cov there exists X such that is not an -covering orX \ B is not an -covering].3. Also we characterize the property of being an -covering.     

Stability of Hypersurfaces with Constant r-Mean Curvature

We deal with compact hypersurfaces immersed in space forms with constant -mean curvature. They are critical points for a variational problem. We show they are stable if and only if they are geodesic spheres, generalizing results on constant curvature hypersurfaces.     

Some New Extrapolation Estimates for the Scale of Spaces

Using new extrapolation estimates for the - and -functionals of couples of limit spaces of the -scale , we introduce a class of extrapolation functors. A characterization of this class via the real interpolation method permits one to obtain new equivalent expressions for the norms in symmetric spaces close to and , which depend only on the -norms of a function.     

A Full Characterization of Multipliers for the Strong ϱ-Integral in the Euclidean Space

We study a generalization of the classical Henstock-Kurzweil integral, known as the strong -integral, introduced by Jarník and Kurzweil. Let be the space of all strongly -integrable functions on a multidimensional compact interval E, equipped with the Alexiewicz norm We show that each element in the dual space of can be represented as a strong -integral. Consequently, we prove that fg is strongly -integrable on E for each strongly -integrable function f if and only if g is almost everywhere equal to a function of bounded variation (in the sense of Hardy-Krause) on E.     

Some remarks on the product of two C α-compact subsets

For a cardinal , we say that a subset B of a space X is C -compact in X if for every continuous function is a compact subset of . If B is a C-compact subset of a space X, then (B, X) denotes the degree of C -compactness of B in X. A space X is called -pseudocompact if X is C -compact into itself. For each cardinal , we give an example of an -pseudocompact space X such that X × X is not pseudocompact: this answers a question posed by T. Retta in Some cardinal generalizations of pseudocompactness Czechoslovak Math. J. 43 (1993), 385–390. The boundedness of the product of two bounded subsets is studied in some particular cases. A version of the classical Glicksberg's Theorem on the pseudocompactness of the product of two spaces is given in the context of boundedness. This theorem is applied to several particular cases.     

On microfunctions at the boundary along CR manifolds

Let X be a complex analytic manifold, a C ² submanifold, an openset with C ² boundary .Denote by (resp. ) the microlocalization along M (resp. ) of the sheaf of holomorphic functions.In the literature (cf. [A-G], [K-S 1,2])one encounters two classical results concerning the vanishing of the cohomology groups .The most general gives the vanishing outside a range of indices j whose length is equal to (with being the number of respectively positive, negative and null eigenvalues for themicrolocal Levi form ).The sharpest result gives the concentration in a single degree, provided that the difference is locally constant for near p (with for z the base point of p).The first result was restated for the complex in [D'A-Z 2], in the case codim We extend it here to any codimension and moreover we also restate for the second vanishing theorem.We also point out that the principle of our proof, related to a criterion for constancy of sheaves due to [K-S 1], is a quite new one.     

Image inverse d'un D-module and Newton Polygon)

We show that, under conditions about the microcharacteristic variety of a coherent -module, the Cauchy problem is well-posed in the spaces of formal power series with Gevrey growth. We deduce that the filtration of the Irregularity Sheaf of a holonomic -module, which we defined in a previous work, is preserved under inverse image if some rather general geometric conditions are fullfilled.     

Uniform and universal Glivenko-Cantelli classes

A class of measurable functions on a probability space is called a Glivenko-Cantelli class if the empirical measuresP _n converge to the trueP uniformly over almost surely. is a universal Glivenko-Cantelli class if it is a Glivenko-Cantelli Cantelli class for all lawsP on a measurable space, and a uniform Glivenko-Cantelli class if the convergence is also uniform inP. We give general sufficient conditions for the Glivenko-Cantelli and universal Glivenko-Cantelli properties and examples to show that some stronger conditions are not necessary. The uniform Glivenko-Cantelli property is characterized, under measurability assumptions, by an entropy condition.     

On the Irreducibility of Commuting Varieties Associated with Involutions of Simple Lie Algebras

Let be a reductive Lie algebra over an algebraically closed field of characteristic zero and an arbitrary -grading. We consider the variety , which is called the commuting variety associated with the -grading. Earlier it was proved by the author that is irreducible, if the -grading is of maximal rank. Now we show that is irreducible for and (E₆,F₄). In the case of symmetric pairs of rank one, we show that the number of irreducible components of is equal to that of nonzero non--regular nilpotent G ₀-orbits in . We also discuss a general problem of the irreducibility of commuting varieties.     

The Local Stark Conjecture at a Real Place

A refinement of the rank 1 Abelian Stark conjecture has been formulated by B.Gross. This conjecture predicts some -adic analytic nature of a modification of the Stark unit. The conjecture makes perfect sense even when is an Archimedean place. Here we consider the conjecture when is a real place, and interpret it in terms of 2-adic properties of special values of L-functions. We prove the conjecture for CM extensions; here the original Stark conjecture is uninteresting, but the refined conjecture is nontrivial. In more generality, we show that, under mild hypotheses, if the subgroup of the Galois group generated by complex conjugations has less than full rank, then the refined conjecture implies that the Stark unit should be a square. This phenomenon has been discovered by Dummit and Hayes in a particular type of situation. We show that it should hold in much greater generality.     

Systèmes dynamiques non archimédiens et corps des norms (Non-Archimedean Dynamic Systems and Fields of Norms)

Let _k be the ring of integers of a finite extension k of the field _p of p-adic numbers. The endomorphisms of a formal group law defined over _k provide nontrivial examples of commuting formal series with coefficients in _k . This article deals with the inverse problem formulated by Jonathan Lubin within the context of non-Archimedean dynamical systems. We present a large family of series, with coefficients in _p , which satisfy Lubin's conjecture. These series are constructed with the help of Lubin–Tate formal group laws over _p . We introduce the notion of minimally ramified series which turn out to be modulo p reductions of some series of this family. The commutant monoids of these minimally ramified series are determined by using the Fontaine–Wintenberger theory of the field of norms which allows an interpretation of them as automorphisms of _p -extensions of local fields of characteristic zero. A particularly effective example illustrating the paper is given by a family of series generalizing ebyev polynomials     

Lp Contraction Semigroups for Vector Valued Functions

Let be a contraction semigroup on the space of vector valued functions ( is a Hilbert space). In order to study the extension of to a contaction semigroup on , Shigekawa [Sh] studied recently the domination property where is a symmetric sub-Markovian semigroup on . He gives in the setting of square field operators sufficient conditions for the above inequality. The aim of the present paper is to show that the methods of [12] and [13] can be applied in the present setting and provide two ways for the extension of to We give necessary and sufficient conditions in terms of sesquilinear forms for the contractivity property as well as for the above domination property in a more general situation.     

Erdős–Szekeres-Type Theorems for Segments and Noncrossing Convex Sets

A family of convex sets is said to be in convex position, if none of its members is contained in the convex hull of the others. It is proved that there is a function N(n) with the following property. If is a family of at least N(n) plane convex sets with nonempty interiors, such that any two members of have at most two boundary points in common and any three are in convex position, then has n members in convex position. This result generalizes a theorem of T. Bisztriczky and G. Fejes Tóth. The statement does not remain true, if two members of may share four boundary points. This follows from the fact that there exist infinitely many straight-line segments such that any three are in convex position, but no four are. However, there is a function M(n) such that every family of at least M(n) segments, any four of which are in convex position, has n members in convex position.     

Matroid Shellability, β-Systems,and Affine Hyperplane Arrangements

The broken-circuit complex is fundamental to the shellability and homology of matroids, geometric lattices, and linear hyperplane arrangements. This paper introduces and studies the -system of a matroid, nbc(M), whose cardinality is Crapo's -invariant. In studying the shellability and homology of base-pointed matroids, geometric semilattices, and afflne hyperplane arrangements, it is found that the -system acts as the afflne counterpart to the broken-circuit complex. In particular, it is shown that the -system indexes the homology facets for the lexicographic shelling of the reduced broken-circuit complex , and the basic cycles are explicitly constructed. Similarly, an EL-shelling for the geometric semilattice associated with M is produced,_and it is shown that the -system labels its decreasing chains.Basic cycles can be carried over from The intersection poset of any (real or complex) afflnehyperplane arrangement is a geometric semilattice. Thus the construction yields a set of basic cycles, indexed by nbc(M), for the union of such an arrangement.     

Sur la cohomologie à supports compacts des variétés de Shimura pour GSp(4)_\mathbb{Q}

In this paper we compute the cohomology with compact supports of a Siegelthreefold as a virtual module over the product of the Galois group of over and the Hecke algebra. We use a method which has been developed by Ihara, Langlands and Kottwitz: comparison of the Grothendieck--Lefschetz formula and the Arthur--Selberg trace formula.