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...

A computer model was created as a scientific and management tool for understanding the effects of macrophytes on hydrodynamics and water quality. A model was required that could simulate macrophytes in a complex water body and could be coupled to a multicompartment water quality model of phytoplankton, dissolved oxygen, nutrients, pH, and organic matter. This would permit the investigation of w...

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 ...

Bacillus sp. strain W2 is a plant growth-promoting rhizobacterium isolated from saffron fields of Kashmir, India. Here, we report the draft genome sequence (3.9 Mb) of Bacillus amyloliquefaciens strain W2 having 65 contigs (3, 997, 511 bp), 4,163 coding sequences, and an average 46.45% GC content. Despite the 99% identity of the 16S rRNA gene with that of Bacillus amyloliquefaciens subsp. plant...

Diaconis, P. and R. Graham, Binomial coefficient codes over GF(2) Discrete Mathematics 106/107 (1992) 181-188. In this note we study codes over GF(2) which are generated for given d and r by binary vectors of the form ((y), (,‘), , ({), . , (*‘i ‘)) (mod 2), 0 <i =Z d. We describe the weight enumerators of these codes and the numbers of codewords of weights 1 and 2. These results can be used to...