Usually in this type of calculations one does not write explicitly the unit vectors ei. This makes the notation more economical but is possible only either if the vectors are decomposed into the Cartesian unit vectors ex, ey, ez, or (for vectors decomposed into unit vectors e1(ξ), e2(ξ), e3(ξ) associated with some curvelinear coordinates ξ, ξ, ξ see below), or if no differentiations are involve...

Definition 1.1. Let A be an involutive algebra over C with involution ξ 7→ ξ (resp. ξ 7→ ξ). We say A is a left (resp. right) Hilbert algebra if A has a inner product (· | ·) satisfying: a. Multiplication on the left (resp. right) is a bounded operator; that is, ∀ξ ∈ A the map πl(ξ) : η 7→ ξη (resp. πr(ξ) : η 7→ ηξ) is bounded on A. b. (ξη | ζ) = (η | ξζ) (resp. (ξη | ζ) = (ξ | ζη)). c. The inv...

We prove that every real number ξ > 1 is the Diophantine exponent of some binary word ω. More precisely, we show that Dio(ω) = ξ for ω = 101102103 · · · , where kn = [ξ] for ξ > 2, kn = [ν] with ν = (−ξ + 1 + √ 6ξ − 3ξ2 + 1)/(4− 2ξ) for 1 < ξ < 2, and kn = n for ξ = 1.

The quantum phase description of Noh, Fougères and Mandel (NFM) leads to a two-mode theory of phase in the strong local oscillator limit. We show that the explicit form of the eigenstates of the sine and the cosine phase operators in two-mode Fock space can be constructed. This brings great convenience to the NFM quantum phase approach. Recently, phase operators and phase measurements in quantu...

Problem 1 Find u(x, y) satisfying u x + u y = u 2 , u(x, −x) = x, −∞ ≤ x ≤ ∞ Sketch the characteristic curves and show the domain of validity of the solution in the (x, y)-plane. In this problem we will examine the solution to a nonhomogeneous first-order PDE with constant coefficients. In order to solve this problem we can follow a typical approach: (1) find an explicit form for the characteri...

(2) |ξ − ξ′| ≥ rm γm(ξ′) . If we assume that ξ′ is also a simple root, then clearly a symmetry argument also proves the inequality (1) and, hence, the present proof remains valid. But this added assumption is unnecessary. To correct the proof of Theorem 3.2 essentially all that is needed is to switch ξ and ξ′; i.e., assume that ξ′ is a simple root while relaxing the simplicity assumption on ξ. ...

We establish boundedness properties on products of weighted Lebesgue, Hardy, and amalgam spaces of certain paraproducts and bilinear pseudodifferential operators with mild regularity. We do so by showing that these operators can be realized as generalized bilinear Calderón-Zygmund operators. 1. Bilinear pseudodifferential operators with mild regularity Let us motivate our main result on bilinea...

For each of the following functions u 0 (x), construct solutions to the initial value problem (1 + ku)u x + u y = 0, y ≥ 0 u(x, 0) = u 0 (x) is a given function for −∞ ≤ x ≤ ∞ Draw the characteristics in the (x, y)-plane. If you find the wave-breaking phenomenon arising, determine the time and place of the initial break appearance and find a solution containing a shockwave. For this first probl...

Let n ≥ 2 and let A : Rn → [0,∞] be any convex function satisfying the following properties: A(0) = 0 and A(ξ) = A(−ξ) for ξ ∈ R; (1.1) for every t > 0, {ξ ∈ R : A(ξ) ≤ t} (1.2) is a compact set whose interior contains 0. Observe that A need not depend on the length |ξ| of ξ nor be the sum of functions of its components ξi, i = 1, . . . , n. The purpose of this note is to exhibit an inequality ...