(Homo) Flatness of posets on poideal extensions

We consider pomonoids , where G is a pogroup and I is a poideal of S and show that if an S-poset is principally weakly flat, (weakly) flat, po-flat, (principally) weakly po-flat, (po-) torsion free or satisfies Conditions (P), (P _E ), (P _w ), (PWP), (PWP) _w , (WP) or (WP) _w as an I ¹-poset, then it has these properties as an S-poset. We also show that an S-poset which is free, projective or strongly flat as an I ¹-poset may not generally have these properties as an S-poset.     

On projectively flat Finsler spaces

First we present a short overview of the long history of projectively flat Finsler spaces. We give a simple and quite elementary proof of the already known condition for the projective flatness, and we give a criterion for the projective flatness of a special Lagrange space (Theorem 1). After this we obtain a second-order PDE system, whose solvability is necessary and sufficient for a Finsler space to be projectively flat (Theorem 2). We also derive a condition in order that an infinitesimal transformation takes geodesics of a Finsler space into geodesics. This yields a Killing type vector field (Theorem 3). In the last section we present a characterization of the Finsler spaces which are projectively flat in a parameter-preserving manner (Theorem 4), and we show that these spaces over ${\mathbb {R}}^{n}$ are exactly the Minkowski spaces (Theorems 5 and 6).     

The Maximal Riesz, Fejér, and Cesàro Operators on Real Hardy Spaces

We prove that the maximal Riesz operator $\sigma^{\alpha,\gamma}_*$ is of strong type from $L^1(\R) \cap H^p$ $ (\R)$ to $L^p (\R)$ for $\alpha, \gamma>0$ and $1/(1+\alpha) < p \le 1$, it is of weak type for $\alpha,\gamma>0$ and $1/(1+\alpha) = p$, and these results are best possible. The proofs are based on sharp estimates of the derivatives of the Riesz kernel. We characterize the real Hardy space $H^p(\R)$ in terms of $\sigma^{\alpha,1}_*$ for $1/(1+ \alpha) < p \le 1$, and draw consequences for real Hardy spaces on $\R^2$, as well. For example, an integrable function $f$ belongs to $H^1(\R)$ if and only if the maximal Fej\er operator $\sigma^{1,1}_*$ applied to $f$ belongs to $L^1(\R)$. We also establish analogous results for real Hardy spaces on $\T$ and $\T^2$.     

Dynamical systems disjoint from any minimal system

Furstenberg showed that if two topological systems and are disjoint, then one of them, say , is minimal. When is nontrivial, we prove that must have dense recurrent points, and there are countably many maximal transitive subsystems of such that their union is dense and each of them is disjoint from . Showing that a weakly mixing system with dense periodic points is in , the collection of all systems disjoint from any minimal system, Furstenberg asked the question to characterize the systems in . We show that a weakly mixing system with dense regular minimal points is in , and each system in has dense minimal points and it is weakly mixing if it is transitive. Transitive systems in and having no periodic points are constructed. Moreover, we show that there is a distal system in .

Recently, Weiss showed that a system is weakly disjoint from all weakly mixing systems iff it is topologically ergodic. We construct an example which is weakly disjoint from all topologically ergodic systems and is not weakly mixing.

     

Principally Weakly and Weakly Coherent Monoids

We shall call a monoid S principally weakly (weakly) left coherent if direct products of nonempty families of principally weakly (weakly) flat right S-acts are principally weakly (weakly) flat. Such monoids have not been studied in general. However, Bulman-Fleming and McDowell proved that a commutative monoid S is (weakly) coherent if and only if the act S ^I is weakly flat for each nonempty set I. In this article we introduce the notion of finite (principal) weak flatness for characterizing (principally) weakly left coherent monoids. Also we investigate monoids over which direct products of acts transfer an arbitrary flatness property to their components.     

Some combinatorics of binomial coefficients and the Bloch-Gieseker property for some homogeneous bundles

A vector bundle has the Bloch-Gieseker property if all its Chern classes are numerically positive. In this paper we show that the non-ample bundle has the Bloch-Gieseker property, except for two cases, in which the top Chern classes are trivial and the other Chern classes are positive. Our method is to reduce the problem to showing, e.g. the positivity of the coefficient of in the rational function (for even).

     

The reactive bargaining set for cooperative games

We introduce a new bargaining set for cooperative games in characteristic function form, and investigate its structure and properties. We prove that the new bargaining set is not empty. In fact, we show that it contains the kernel and is contained in the classical bargaining set ${\mathcal{M}^i_1}$ , and we further prove that it consists of the unique symmetric vector for the class of simple majority games.     

Formes différentielles abéliennes,bornes de Castelnuovo et géométrie des tissus

A $d$-web ${\Cal W}(d)$ is given by $d$ complex analytic foliations of codimension $n$ in $({\sumbbb C}^N,0)$ such that the leaves are in general position. We are interested in the geometry of such configurations. A complex $({\Cal A}^{\bullet},\delta)$ of ${\sumbbb C}$-vector spaces is defined in which ${\Cal A}^0$ corresponds to functions and ${\Cal A}^p$ to $p$-forms of the web ${\Cal W}(d)$ for $1\leq p\leq n$. If $N=kn$ with $k\geq 2$, it is proved that $r_p:=\dim_{\,\sumbbb C}{\Cal A}^p$ is a finite analytic invariant of ${\Cal W}(d)$ with an optimal upper bound $\pi_{p}(d,k,n)$ for $0\leq p\leq n$. These bounds generalize the Castelnuovos ones for genus of curves in ${\sumbbb P}^{k}$ with degree $d$. Some characterization of the the space $H^0(V_n,\omega^p_{V_n})$ of abelian differentials to an algebraic variety $V_n$ in ${\sumbbb P}^{n+k-1}$ of pure dimension $n$ with degree $d$ is given. Moreover, using duality and Abels theorem, we investigate how for suitable $V_n$ the natural complex $\bigr(H^0(V_n,\omega^{\bullet}_{V_n}),d\,\bigr)$ and the abelian relation complex $({\Cal A}^{\bullet},\delta)$ of the linear web associated to $V_n$ in $({\sumbbb C}^{kn},0)$ are related.

     

Motivic decomposition of projective homogeneous varieties and the Krull-Schmidt theorem

Given an algebraic group G defined over a (not necessarily algebraically closed) field F and a commutative ring R we associate the subcategory of the category of Chow motives with coefficients in R, that is, the Tate pseudo-abelian closure of the category of motives of projective homogeneous G-varieties. We show that is a symmetric tensor category, i.e., the motive of the product of two projective homogeneous G-varieties is a direct sum of twisted motives of projective homogeneous G-varieties. We also study the problem of uniqueness of a direct sum decomposition of objects in We prove that the Krull--Schmidt theorem holds in many cases.     

On flatness of acts

We consider monoids $S=G\dot \cup I$ where G is a group and I is an ideal of S and show that if an S-act is principally weakly flat, (weakly) flat, torsion free or satisfies conditions (P) or (P_E) as an I¹-act, then it has these properties as an S-act. We also show that an S-act which is free, projective or strongly flat as an I¹-act may not generally have these properties as an S-act.     

A Lipschitz estimate for Berezin's operator calculus

F. A. Berezin introduced a general ``symbol calculus" for linear operators on reproducing kernel Hilbert spaces. For the particular Hilbert space of Gaussian square-integrable entire functions on complex -space, , we obtain Lipschitz estimates for the Berezin symbols of arbitrary bounded operators. Additional properties of the Berezin symbol and extensions to more general reproducing kernel Hilbert spaces are discussed.

     

A Helly-type theorem for countable intersections of starshaped sets

Let k and d be fixed integers, 0kd, and let be a collection of sets in If every countable subfamily of has a starshaped intersection, then is (nonempty and) starshaped as well. Moreover, if every countable subfamily of has as its intersection a starshaped set whose kernel is at least k-dimensional, then the kernel of is at least k-dimensional, too. Finally, dual statements hold for unions of sets.Received: 3 April 2004     

Typical <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>-Dimensions of Measures

For a probability measure μ on a subset of , the lower and upper L^q-dimensions of order are defined by We study the typical behaviour (in the sense of Baire’s category) of the L^q-dimensions and . We prove that a typical measure μ is as irregular as possible: for all q ≥ 1, the lower L^q-dimension attains the smallest possible value and the upper L^q-dimension attains the largest possible value.     

The Lattice of Cyclic Flats of a Matroid

A flat of a matroid is cyclic if it is a union of circuits. The cyclic flats of a matroid form a lattice under inclusion. We study these lattices and explore matroids from the perspective of cyclic flats. In particular, we show that every lattice is isomorphic to the lattice of cyclic flats of a matroid. We give a necessary and sufficient condition for a lattice of sets and a function to be the lattice of cyclic flats of a matroid and the restriction of the corresponding rank function to . We apply this perspective to give an alternative view of the free product of matroids and we show how to compute the Tutte polynomial of the free product in terms of the Tutte polynomials of the constituent matroids. We define cyclic width and show that this concept gives rise to minor-closed, dual-closed classes of matroids, two of which contain only transversal matroids. Received May 29, 2005     

Hiding Cliques for Cryptographic Security

We demonstrate how a well studied combinatorial optimizationproblem may be used as a new cryptographic primitive. The problemin question is that of finding a "large" clique in a randomgraph. While the largest clique in a random graph with nvertices and edge probability p is very likely tobe of size about , it is widely conjecturedthat no polynomial-time algorithm exists which finds a cliqueof size with significantprobability for any constant > 0. We presenta very simple method of exploiting this conjecture by hidinglarge cliques in random graphs. In particular, we show that ifthe conjecture is true, then when a large clique—of size,say, is randomlyinserted (hidden) in a random graph, finding a clique ofsize remains hard.Our analysis also covers the case of high edge probabilitieswhich allows us to insert cliques of size up to . Our result suggests several cryptographicapplications, such as a simple one-way function.     

Weak type estimates for cone multipliers on

We consider operators associated with the Fourier multipliers and show that is of weak type on , , for the critical value .

     

Maximal, potential and singular operators in vanishing generalized Morrey spaces

We introduce vanishing generalized Morrey spaces ${V\mathcal{L}^{p,\varphi}_\Pi (\Omega), \Omega \subseteq \mathbb{R}^n}$ with a general function ${\varphi(x, r)}$ defining the Morrey-type norm. Here ${\Pi \subseteq \Omega}$ is an arbitrary subset in Ω including the extremal cases ${\Pi = \{x_0\}, x_0 \in \Omega}$ and Π = Ω, which allows to unify vanishing local and global Morrey spaces. In the spaces ${V\mathcal{L}^{p,\varphi}_\Pi (\mathbb{R}^n)}$ we prove the boundedness of a class of sublinear singular operators, which includes Hardy-Littlewood maximal operator and Calderon-Zygmund singular operators with standard kernel. We also prove a Sobolev-Spanne type ${V\mathcal{L}^{p,\varphi}_\Pi (\mathbb{R}^n) \rightarrow V\mathcal{L}^{q,\varphi^\frac{q}{p}}_\Pi (\mathbb{R}^n)}$ -theorem for the potential operator I ^α . The conditions for the boundedness are given in terms of Zygmund-type integral inequalities on ${\varphi(x, r)}$ . No monotonicity type condition is imposed on ${\varphi(x, r)}$ . In case ${\varphi}$ has quasi- monotone properties, as a consequence of the main results, the conditions of the boundedness are also given in terms of the Matuszeska-Orlicz indices of the function ${\varphi}$ . The proofs are based on pointwise estimates of the modulars defining the vanishing spaces     

Approximation on the sphere using radial basis functions plus polynomials

In this paper we analyse a hybrid approximation of functions on the sphere by radial basis functions combined with polynomials, with the radial basis functions assumed to be generated by a (strictly) positive definite kernel. The approximation is determined by interpolation at scattered data points, supplemented by side conditions on the coefficients to ensure a square linear system. The analysis is first carried out in the native space associated with the kernel (with no explicit polynomial component, and no side conditions). A more refined error estimate is obtained for functions in a still smaller space. Numerical calculations support the utility of this hybrid approximation.      

Torus actions on weakly pseudoconvex spaces

We show that the univalent local actions of the complexification of a compact connected Lie group on a weakly pseudoconvex space where is acting holomorphically have a universal orbit convex weakly pseudoconvex complexification. We also show that if is a torus, then every holomorphic action of on a weakly pseudoconvex space extends to a univalent local action of

     

Distributional Point Values and Convergence of Fourier Series and Integrals

In this article we show that the distributional point values of a tempered distribution are characterized by their Fourier transforms in the following way: If and , and is locally integrable, then distributionally if and only if there exists k such that , for each a > 0, and similarly in the case when is a general distribution. Here means in the Cesaro sense. This result generalizes the characterization of Fourier series of distributions with a distributional point value given in [5] by . We also show that under some extra conditions, as if the sequence belongs to the space for some and the tails satisfy the estimate ,\ as , the asymmetric partial sums\ converge to . We give convergence results in other cases and we also consider the convergence of the asymmetric partial integrals. We apply these results to lacunary Fourier series of distributions.