On The Postulation of The Multiples of The Nodes for Generic Nodal Curves on Generic Rational Surfaces

Fix integers x > 0, m₁ ≥ … ≥ ^m _x > 0 and P₁,…,P_x ∈ P² such that no 3 of them are collinear. Let C ? P² a “ general ” degree d plane curve with an ordinary point with multiplicity m _i at each P _i and y further singularities which are ordinary nodes. Fix any A ? Sing(C){P₁,…, P_x} and any integer m > 0. Here we study the postulation of the fat points m A ? ∪_Q∈AmQ.     

On distinct distances among points in general position and other related problems

A set of points in the plane is said to be in general position if no three of them are collinear and no four of them are cocircular. If a point set determines only distinct vectors, it is called parallelogram free. We show that there exist n-element point sets in the plane in general position, and parallelogram free, that determine only O(n ²/√log n) distinct distances. This answers a question of Erd?s, Hickerson and Pach. We then revisit an old problem of Erd?s: given any n points in the plane (or in d dimensions), how many of them can one select so that the distances which are determined are all distinct? — and provide (make explicit) some new bounds in one and two dimensions. Other related distance problems are also discussed.     

On the irreducibility of multivariate subresultants

Let P₁,…,P_n be generic homogeneous polynomials in n variables of degrees d₁,…,d_n respectively. We prove that if ν is an integer satisfying ∑_i=1ⁿd_i?n+1?min{d_i}<ν, then all multivariate subresultants associated to the family P₁,…,P_n in degree ν are irreducible. We show that the lower bound is sharp. As a byproduct, we get a formula for computing the residual resultant of $\text{ρ?ν+n?1}\text{n?1}$ smooth isolated points in $\text{P}{}^{\mathrm{<mspace\; sp="0.2"\; width="2px"\; linebreak="nobreak"\; is="true"></mspace><mtext>n?1</mtext>}}\text{.}$To cite this article: L. Busé, C. D'Andrea, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Inversions of Infinitely Divisible Distributions and Conjugates of Stochastic Integral Mappings

The dual of an infinitely divisible distribution on ? ^d without Gaussian part defined in Sato (ALEA Lat. Am. J. Probab. Math. Statist. 3:67–110, 2007) is renamed to the inversion. Properties and characterization of the inversion are given. A stochastic integral mapping is a mapping μ=Φ _f ? ρ of ρ to μ in the class of infinitely divisible distributions on ? ^d , where μ is the distribution of an improper stochastic integral of a nonrandom function f with respect to a Lévy process on ? ^d with distribution ρ at time 1. The concept of the conjugate is introduced for a class of stochastic integral mappings and its close connection with the inversion is shown. The domains and ranges of the conjugates of three two-parameter families of stochastic integral mappings are described. Applications to the study of the limits of the ranges of iterations of stochastic integral mappings are made.     

Tractability of Multivariate Integration for Periodic Functions

We study multivariate integration in the worst case setting for weighted Korobov spaces of smooth periodic functions of d variables. We wish to reduce the initial error by a factor ε for functions from the unit ball of the weighted Korobov space. Tractability means that the minimal number of function samples needed to solve the problem is polynomial in ε⁻¹ and d. Strong tractability means that we have only a polynomial dependence in ε⁻¹. This problem has been recently studied for quasi-Monte Carlo quadrature rules and for quadrature rules with non-negative coefficients. In this paper we study arbitrary quadrature rules. We show that tractability and strong tractability in the worst case setting hold under the same assumptions on the weights of the Korobov space as for the restricted classes of quadrature rules. More precisely, let γ_j moderate the behavior of functions with respect to the jth variable in the weighted Korobov space. Then strong tractability holds iff ∑^∞_j=1 γ_j<∞, whereas tractability holds iff lim sup_d→∞ ∑^d_j=1 γ_j/ln d<∞. We obtain necessary conditions on tractability and strong tractability by showing that multivariate integration for the weighted Korobov space is no easier than multivariate integration for the corresponding weighted Sobolev space of smooth functions with boundary conditions. For the weighted Sobolev space we apply general results from E. Novak and H. Woźniakowski (J. Complexity17 (2001), 388–441) concerning decomposable kernels.     

Derivative Formula and Gradient Estimates for Gruschin Type Semigroups

By solving a control problem and using Malliavin calculus, an explicit derivative formula is derived for the semigroup P _t generated by the Gruschin type operator on ? ^m ×? ^d : where σ∈C ¹(? ^m ;? ^d ?? ^d ) might be degenerate. In particular, if σ(x) is comparable with |x| ^l I _d×d for some l≥1 in the sense of (1.5), then for any p>1 there exists a constant C _p >0 such that which implies a new Harnack type inequality for the semigroup. A more general model is also investigated.     

Some explicit formulae in simultaneous padé approximation

Using old results on the explicit calculation of determinants, formulae are given for the coefficients of P₀(z) and P₀(z)f_i(z) ? P_i(z), where P_i(z) are polynomials of degree σ ? ρ_i (i=0,1,…,n), P₀(z)f_i(z) ? P_i(z) are power series in which the terms with z^k, 0?k?σ, vanish (i=1,2,…,n), (ρ₀,ρ₁,…,ρ_n) is an (n+1)-tuple of nonnegative integers, σ=ρ₀+ρ₁+?+ρ_n, and {f_i}ⁿ_i=1 is the set of hypergeometric functions {₁F₁(1;c_i;z)}ⁿ_i=1$\text{(c}{}_{\mathrm{i}}\text{?}\text{Z}\text{z.drule;}\text{N}\text{, c}{}_{\mathrm{i}}\text{? c}{}_{\mathrm{j}}\text{?}\text{Z}\text{)}$ or {₂F₀(a_i,1;z)}ⁿ_i=1$\text{(a}{}_{\mathrm{i}}\text{?}\text{Z}\text{?}\text{N}\text{, a}{}_{\mathrm{i}}\text{? a}{}_{\mathrm{j}}\text{?}\text{Z}\text{)}$ under the condition ρ₀?ρ_i ? 1 (i=1,2,…,n).     

Vanishing power values of commutators with derivations

Let R be a prime ring of char R ≠ 2, let d be a nonzero derivation of R, and let ρ be a nonzero right ideal of R such that [[d(x)x ⁿ , d(y)] _m , [y, x] _s ] ^t = 0 for all x, y ? ρ, where n ≥ 1, m ≥ 0, s ≥ 0, and t ≥ 1 are fixed integers. If [ρ, ρ]ρ ≠ 0 then d(ρ)ρ = 0.     

A strong law of large numbers for vector gaussian martingales and a statistical application in linear regression

For a d-dimensional gaussian martingae M with tensor increasingprocess we prove that <M>⁺_tM_t converges in R^d with probability 1 as t → ∞ and the limit is zero a.s. iff tr <M>⁺_t tends to zero. We apply this result to study the strong consistency of estimates in a linear regression model.     

Equivalence of measures for some class of Gaussian random fields

We consider two Gaussian measures P₁ and P₂ on (C(G), $\text{B}$) with zero expectations and covariance functions R₁(x, y) and R₂(x, y) respectively, where R_ν(x, y) is the Green's function of the Dirichlet problem for some uniformly strongly elliptic differential operator A^(ν) of order $\text{2m, m ≥ [}\text{d}\text{2}\text{] + 1}$, on a bounded domain G in $\text{R}$^d (ν = 1, 2). It is shown that if the order of A⁽²⁾ ? A⁽¹⁾ is at most $\text{2m ? [}\text{d}\text{2}\text{] ? 1}$, then P₁ and P₂ are equivalent, while if the order is greater than $\text{2m ? [}\text{d}\text{2}\text{] ? 1}$, then P₁ and P₂ are not always equivalent.     

Patterson-Sullivan-type measures on Riccati foliations with quasi-Fuchsian holonomy

We consider Riccati foliations ?_ρ with hyperbolic leaves, over a finite hyperbolic Riemann Surface S, constructed by suspending a representation ρ: π ₁(S) → PSL(2,?) in a quasi-Fuchsian group. The foliated geodesic flow has a repeller-attractor dynamic with generic statistics µ⁺ and µ^? for positive and negative times, respectively. These measures have a common projection to a harmonic measure μ_ρ for the Riccati foliation. We describe μ _ρ ⁺ , μ _ρ ^- and μ_ρ in terms of the Patterson-Sullivan construction, and we show that the measures μ_ρ provide examples of the conformal harmonic measures introduced by M. Brunella.     

BOUNDEDNESS OF FRACTIONAL INTEGRALS IN HARDY SPACES WITH NON-DOUBLING MEASURE - In Memory of Professor Sun Yongsheng

Let μ be a Borel measure on Rd which may be non doubling. The only condition that μ must satisfy is μ(Q) ≤ col(Q)n for any cube Q () Rd with sides parallel to the coordinate axes and for some fixed n with 0 ＜ n ≤ d. The purpose of this paper is to obtain a boundedness property of fractional integrals in Hardy spaces H1 (μ).     

A New Lower Bound Based on Gromov’s Method of Selecting Heavily Covered Points

Boros and Füredi (for d=2) and Bárány (for arbitrary d) proved that there exists a positive real number c _d such that for every set P of n points in R ^d in general position, there exists a point of R ^d contained in at least $c_{d}\binom{n}{d+1}$ d-simplices with vertices at the points of P. Gromov improved the known lower bound on c _d by topological means. Using methods from extremal combinatorics, we improve one of the quantities appearing in Gromov??s approach and thereby provide a new stronger lower bound on c _d for arbitrary d. In particular, we improve the lower bound on c ₃ from 0.06332 to more than 0.07480; the best upper bound known on c ₃ being 0.09375.     

A dimension series for multivariate splines

For a polyhedral subdivision Δ of a region in Euclideand-space, we consider the vector spaceC _k ^r (Δ) consisting of allC ^r piecewise polynomial functions over Δ of degree at mostk. We consider the formal power series ∑ _k≥0 dim_? C _k ^r (Δ)λ^k and show, under mild conditions on Δ, that this always has the formP(λ)/(1?λ) ^d+1, whereP(λ) is a polynomial in λ with integral coefficients which satisfiesP(0)=1,P(1)=f _d (Δ), andP′(1)=(r+1)f _d?1 ⁰ (Δ). We discuss how the polynomialP(λ) and bases for the spacesC _k ^r (Δ) can be effectively calculated by use of Gröbner basis techniques of computational commutative algebra. A further application is given to the theory of hyperplane arrangements.     

Trees with second minimum general Randić index for α>0

The general Randi? index R _α (G) of a graph G is the sum of the weights (d(u)d(v)) ^α of all edges uv of G, where α is a real number(α≠0) and d(u) denotes the degree of the vertex u. We have known that P _n has minimum general Randi? index for α>0 among trees when n≥5. In this paper, we prove that P _n,3 has second minimum general Randi? index for α>0 among trees when n≥7.     

On the stabilization of iterative processes

Processes of the type A^tz, A being a symmetric matrix, are considered. Such a process is called strongly stabilizable iff given an arbitrary sequence of interference times (t_i), it admits a stabilization consisting of the addition of a constant vector y at ”times“ t_i. It is shown (Theorem 3) that A^t is strongly stabilizable iff no eigenvalue of A lies in the interval $\text{( ?}\text{1}\text{2}\text{(1 +}\text{5}\text{), ? 1)}$.     

Flows and generalized coloring theorems in graphs

An (oriented) graph H is said to be F_k(k ≥ 2) iff there exists an integer flow in H with all edge-values in $\text{[1 ? k, ?1] ? [1, k ? 1]}$. It is known that a plane 2-edge-connected graph is face-colorable with k colors (k ≥ 2) iff it is F_k; W. T. Tutte has proposed [1] to seek for extensions to general graphs of coloring results known for planar graphs through the use of the F_k property. In this direction, we prove among other results that every 2-edge-connected graph is F₈.     

On stress matrices of (d + 1)-lateration frameworks in general position

Let (G, P) be a bar framework of n vertices in general position in ${\mathbb{R}^d}$ , for d ≤ n ? 1, where G is a (d + 1)-lateration graph. In this paper, we present a constructive proof that (G, P) admits a positive semidefinite stress matrix with rank (n ? d ? 1). We also prove a similar result for a sensor network, where the graph consists of m(≥ d + 1) anchors.     

On partial polynomial interpolation

The Alexander-Hirschowitz theorem says that a general collection of k double points in Pⁿ imposes independent conditions on homogeneous polynomials of degree d with a well known list of exceptions. We generalize this theorem to arbitrary zero-dimensional schemes contained in a general union of double points. We work in the polynomial interpolation setting. In this framework our main result says that the affine space of polynomials of degree ?d in n variables, with assigned values of any number of general linear combinations of first partial derivatives, has the expected dimension if d≠2 with only five exceptional cases. If d=2 the exceptional cases are fully described.     

Carleson measures and the generalized Campanato spaces of vector-valued martingales

In this paper, the so-called(p, φ)-Carleson measure is introduced and the relationship between vector-valued martingales in the general Campanato spaces Lp,φ(X) and the(p, φ)-Carleson measures is investigated. Specifically, it is proved that for q ∈ [2, ∞), the measure dμ := ||dfk||~qdP ? dm is a(q, φ)-Carleson measure on ? × N for every f ∈ L_q,φ(X)if and only if X has an equivalent norm which is q-uniformly convex; while for p ∈(1, 2], the measure dμ :=||dfk||~pdP ? dm is a(p, φ)-Carleson measure on ? × N implies that f ∈ L_p,φ(X)if and only if X admits an equivalent norm which is p-uniformly smooth. This result extends an earlier result in the literature from BMO spaces to general Campanato spaces.