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.

Exercise 1.2. If A,B ⊂ R,m∗(A) = 0, then m∗(A ∪B) = m∗(B) Proof: m∗(A ∪B) ≤ m∗(A) +m∗(B), and m∗(B) ≤ m∗(A ∪B), hence we have m∗(B) ≤ m∗(A ∪B) ≤ m∗(A) +m∗(B) = m∗(B) ∴ m∗(A ∪B) = m∗(B) Exercise 1.3. Prove E ∈M iff ∀ > 0,∃O ⊂ R open, such that E ⊂ O and m∗(O\E) < Proof: (⇒) O\E = E ∩O implies that m∗(O\E) = m∗(Ec ∩O), but we have m∗(O) = m∗(Ec ∩O) +m∗(E ∩O) So suppose m∗(E) < ∞ ⇒ m∗(Ec ∩ O) = m∗...

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...