Some nodes matrices appearing in the numerical analysis for singular integral equations

Let {p _m (w)} be the sequence of Jacobi polynomials corresponding to the weightw(x)=(1–x)(1+x), 0, <1. denote=">x _k (w)=cos _m,k (w),k=1,...,m, the zeros ofp _m (w). If +=0, then the estimates

Rational approximation to Neumann series of Bessel functions

LetJ _n (z) be the Bessel function of the first kind and ordern, and letf(z) be an analytic function in|z|r (r>0); then it is known that the Bessel expansion

Survival time of a random graph

LetV _n ={1, 2, ...,n} ande ₁,e ₂, ...,e _N ,N= be a random permutation ofV _n ⁽²⁾. LetE _t={e ₁,e ₂, ...,e _t} andG _t=(V _n ,E _t ). If is a monotone graph property then the hitting time() for is defined by=()=min {t:G _t }. Suppose now thatG starts to deteriorate i.e. loses edges in order ofage, e ₁,e ₂, .... We introduce the idea of thesurvival time =() defined by t = max {u:(V _n, {e _u,e _u+1, ...,e _T }) }. We study in particular the case where isk-connectivity. We show that

LOWLAD: a locally weightedL 1 smoothing spline algorithm with cross validated choice of smoothing parameters

The computation ofL ₁ smoothing splines on large data sets is often desirable, but computationally infeasible. A locally weighted, LAD smoothing spline based smoother is suggested, and preliminary results will be discussed. Specifically, one can seek smoothing splines in the spacesW _m (D), with [0, 1] ⁿ D. We assume data of the formy _i =f(t _i )+ _i ,i=1,..., N with {t _i } _i=1 ^N D, the _i are errors withE( _i )=0, andf is assumed to be inW _m . An LAD smoothing spline is the solution,s , of the following optimization problem

Multiple Wiener-Ito integrals possessing a continuous extension

LetF(W) be a Wiener functional defined byF(W)=I _n(f) whereI _n(f) denotes the multiple Wiener-Ito integral of ordern of the symmetricL ²([0, 1] ⁿ ) kernelf. We show that a necessary and sufficient condition for the existence of a continuous extension ofF, i.e. the existence of a function ø(·) from the continuous functions on [0, 1] which are zero at zero to which is continuous in the supremum norms and for which ø(W)=F(W) a.s, is that there exists a multimeasure (dt ₁,...,dt _n ) on [0, 1] ⁿ such thatf(t ₁, ...,t _n ) = ((t ₁, 1]), ..., (t _n , 1]) a.e. Lebesgue on [0, 1] ⁿ . Recall that a multimeasure (A ₁,...,A _n ) is for every fixedi and every fixedA _i,...,A_i-1, A_i+1,...,A_n a signed measure inA _i and there exists multimeasures which are not measures. It is, furthermore, shown that iff(t ₁,t ₂, ...,t _n ) = ((t ₁, 1], ..., (t _n , 1]) then all the tracesf ^(k), off exist, eachf(^k) induces ann–2k multimeasure denoted by ^(k), the following relation holds

A note on the behavior of blow-up solutions for one-phase Stefan problems

ＡＮＯＴＥＯＮＴＨＥＢＥＨＡＶＩＯＲＯＦＢＬＯＷ┐ＵＰＳＯＬＵＴＩＯＮＳＦＯＲＯＮＥ┐ＰＨＡＳＥＳＴＥＦＡＮＰＲＯＢＬＥＭＳＺＨＵＮＩＮＧＡｂｓｔｒａｃｔ．Ｉｎｔｈｉｓｐａｐｅｒ,ｔｈｅｆｏｌｏｗｉｎｇｏｎｅ－ｐｈａｓｅＳｔｅｆａｎｐｒｏｂｌｅｍｉｓ...     

Subcoercivity and subelliptic operators on Lie groups II: The general case

Let (,G, U) be a continuous representation of a Lie groupG by bounded operatorsg U (g) on the Banach space and let (, ,dU) denote the representation of the Lie algebra obtained by differentiation. Ifa ₁, ...,a _d is a Lie algebra basis of ,A _i =dU (a _i ) and whenever =(i ₁, ...,i _k ) we reconsider the operators

An extension of the Erdős-Stone theorem

LetK _p(u₁, ..., u_p) be the completep-partite graph whoseith vertex class hasu _i vertices (lip). We show that the theorem of Erds and Stone can be extended as follows. There is an absolute constant >0 such that, for allr1, 0<1 and=">1/r, every graphG=G ⁿ of sufficiently large order |G|=n with at least

A note on regularity for a class of quasilinear elliptic equations

The following regularity of weak solutions of a class of elliptic equations of the form are investigated.     

Evaluating the Frechet derivative of the matrix exponential

LetM _n denote the space ofn×n matrices. GivenX, ZM _n define

The Balian-Low theorem and regularity of Gabor systems

For any positive real numbers A, B, and d satisfying the conditions , d>2, we construct a Gabor orthonormal basis for L²(ℝ), such that the generating function g∈L²(ℝ) satisfies the condition:∫_ℝ|g(x)|²(1+|x| ^A )/log ^d (2+|x|)dx < ∞ and .     

Solutions to deconvolution equations using nonperiodic sampling

Explicit, compactly supported solutions, {v_i, ϕ} _i=1 ^m , to the deconvolution (or Bezout) equation

Bloch-Bergman Pullbacks with Logarithmic Weights

We characterize the composition operators mapping Blochs boundedly into the weighted Bergman spaces of logarithmic weight. For 0 < p < ∞, 1 < α < ∞, let A^{p, log α} denote the space of holomorphic functions F in the unit disc D for which

Dual vectors and lower bounds for the nearest lattice point problem

We prove that given a point outside a given latticeL then there is a dual vector which gives a fairly good estimate for how far from the lattice the vector is. To be more precise, there is a set of translated hyperplanesH _i, such thatL _iH_i andd( _iH_i)(6n ²+1)^–1 d( ,L).Supported by an IBM fellowship.     

On the riemann zeta-function and the divisor problem

Let Δ(x) denote the error term in the Dirichlet divisor problem, and E(T) the error term in the asymptotic formula for the mean square of . If with , then we obtain

Packing-type Measures of the Sample Paths of Fractional Brownian Motion

Abstract   Let Λ = {λ _k } be an infinite increasing sequence of positive integers with λ _k →∞. Let X = {X(t), t ∈? R ^N } be a multi-parameter fractional Brownian motion of index α(0 < α < 1) in R ^d . Subject to certain hypotheses, we prove that if N < αd, then there exist positive finite constants K ₁ and K ₂ such that, with unit probability,

Analysis of FETI methods for multiscale PDEs

In this paper, we study a variant of the finite element tearing and interconnecting (FETI) method which is suitable for elliptic PDEs with highly heterogeneous (multiscale) coefficients α(x); in particular, coefficients with strong variation within subdomains and/or jumps that are not aligned with the subdomain interfaces. Using energy minimisation and cut-off arguments we can show rigorously that for an arbitrary (positive) coefficient function the condition number of the preconditioned FETI system can be bounded by C(α) (1 + log(H/h))² where H is the subdomain diameter and h is the mesh size, and where the function C(α) depends only on the coefficient variation in the vicinity of subdomain interfaces. In particular, if varies only mildly in a layer Ω _i,η of width η near the boundary of each of the subdomains Ω _i , then , independent of the variation of α in the remainder Ω _i \Ω _i,η of each subdomain and independent of any jumps of α across subdomain interfaces. The quadratic dependence of C(α) on H/η can be relaxed to a linear dependence under stronger assumptions on the behaviour of α in the interior of the subdomains. Our theoretical findings are confirmed in numerical tests. C. Pechstein was supported by the Austrian Science Funds (FWF) under grant F1306.     

Some greedyt-intersecting families of finite sequences

Letn, s ₁,s ₂, ... ands _n be positive integers. Assume is an integer for eachi}. For , , and , denotes _p (a)={j|1jn,a _j p}, , and . is called anI _t ^p -intersecting family if, for any a,b ,a _i b _i =min(a _i ,b _i )p for at leastt i's. is called a greedyI _t ^P -intersecting family if is anI _t ^p -intersecting family andW _p (A)W _p (B+A ^c ) for anyAS _p ( ) and any with |B|=t–1.In this paper, we obtain a sharp upper bound of | | for greedyI _t ^p -intersecting families in for the case 2ps _i (1in) ands ₁>s ₂>...>s _n .This project is partially supported by the National Natural Science Foundation of China (No.19401008) and by Postdoctoral Science Foundation of China.     

Travelling front solution for a class of competition-diffusion system with high-order singular point

In this paper, the existence of travelling front solution for a class of competition-diffusion system with high-order singular point