for various pairs of weight functions w1 : X → R + and w2 : Y → R . Roughly speaking, these weighted estimates are a variant of the usual L estimates which assert that if f avoids the regions where w1 is large, then Tf avoids the regions where w2 is large. If X = Y and one can establish the above type of estimate for many pairs w1, w2, with w1 “looking similar to” w2, then these estimates begin...

Let p be a prime number. Let w2 and w ∗ 2 denote the exponents of approximation defined by Mahler and Koksma, respectively, in their classifications of p-adic numbers. It is well-known that every p-adic number ξ satisfies w∗ 2(ξ) ≤ w2(ξ) ≤ w∗ 2(ξ) + 1, with w∗ 2(ξ) = w2(ξ) = 2 for almost all ξ. By means of Schneider’s continued fractions, we give explicit examples of p-adic numbers ξ for which ...

The grading Becchi-Rouet-Stora-Tyutin (BRST) method gives a way to construct the integer W2,s strings, where the BRST charge is written as QB = Q0+Q1. In this paper, using this method, we reconstruct the nilpotent BRST charges Q0 for the integer W2,s strings and the half-integer W2,s strings. Then we construct the exact grading BRST charge with spinor fields and give the new realizations of the...

Let R −∞,∞ , and let Q ∈ C2 : R → 0,∞ be an even function. In this paper, we consider the exponential-type weights wρ x |x| exp −Q x , ρ > −1/2, x ∈ R, and the orthonormal polynomials pn w2 ρ;x of degree n with respect to wρ x . So, we obtain a certain differential equation of higher order with respect to pn w2 ρ;x and we estimate the higher-order derivatives of pn w2 ρ;x and the coefficients o...

We make a detailed study of norm retrieval. We give several classification theorems for norm retrieval and give a large number of examples to go with the theory. One consequence is a new result about Parseval frames: If a Parseval frame is divided into two subsets with spans W1, W2 and W1 ∩W2 = {0}, then W1 ⊥W2.

Lower bound Here we describe the densities on the two manifolds M1 and M2. There are two sets of interest to us: W1 = M1 \M2 which corresponds to the two “holes” of radius 4τ in the annulus, and W2 = M2\M1 which corresponds to the d-dimensional piece added to smoothly join the inner pieces of the two annuli in M2. By construction, vol(W1) = 2vd(4τ) d where vd is the volume of the unit d-ball. v...

This is a companion paper of [Rou05]. It concentrates on asymptotic properties of determinants of some random matrices in the Jacobi ensemble. Let M ∈ Mn1+n2,r(R) (with r ≤ n1 + n2) be a random matrix whose entries are standard i.i.d. Gaussian. We can decompose MT = (MT 1 ,M T 2 ) with M1 ∈ Mn1,r and M2 ∈ Mn2,r. Then, W1 := MT 1 M1 and W2 := MT 2 M2 are independent r× r Wishart matrices with pa...

Cross swtions for threshold electron scattering from the proton have been measured in the missing-mass squared region M2 < W2 < 2 (GeV) 2 and the four-momentum transfer squared region 6< Q2 <30 (GeV/c)2. Scaling of the extracted values of the structure function F2 = VW2 is examined in the variables z, ~, and W2. The best scaling is found for the quantity Q6F2, which is found to be linearly prop...

The fact that certain non-linear W2,s algebras can be linearized by the inclusion of a spin-1 current can provides a simple way to realize W2,s algebras from linear W1,2,s algebras. In this paper, we first construct the explicit field realizations of linear W1,2,s algebras with double scalar fields and spinor fields respectively. Then, after a basis inverting, the realizations of W2,s algebras ...