Commuting Rank 2 Differential Operators Corresponding to a Curve of Genus 2

We construct commuting rank 2 formally self-adjoint ordinary differential operators corresponding to a curve of genus 2.     

Positive set‐operators of low complexity

The powerset operator, ??, is compared with other operators of similar type and logical complexity. Namely we examine positive operators whose defining formula has a canonical form containing at most a string of universal quantifiers. We call them ?‐operators. The question we address in this paper is: How is the class of ?‐operators generated ? It is shown that every positive ?‐operator Γ such that Γ(??) ≠ ??, is finitely generated from ??, the identity operator Id, constant operators and certain trivial ones by composition, ∪ and ∩. This extends results of [3] concerning bounded positive operators.     

Discrete duality finite volume schemes for Leray−Lions−type elliptic problems on general 2D meshes

Discrete duality finite volume schemes on general meshes, introduced by Hermeline and Domelevo and Omnès for the Laplace equation, are proposed for nonlinear diffusion problems in 2D with nonhomogeneous Dirichlet boundary condition. This approach allows the discretization of non linear fluxes in such a way that the discrete operator inherits the key properties of the continuous one. Furthermore, it is well adapted to very general meshes including the case of nonconformal locally refined meshes. We show that the approximate solution exists and is unique, which is not obvious since the scheme is nonlinear. We prove that, for general W^?1,p′(Ω) source term and W^1‐(1/p),p(?Ω) boundary data, the approximate solution and its discrete gradient converge strongly towards the exact solution and its gradient, respectively, in appropriate Lebesgue spaces. Finally, error estimates are given in the case where the solution is assumed to be in W^2,p(Ω). Numerical examples are given, including those on locally refined meshes. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

The completely positive and doubly nonnegative completion problems

We prove the following. Let G be an undirected graph. Every partially specified symmetric matrix, the graph of whose specified entries is G and each of whose fully specified submatrices is completely positive (equal to BB^T for some entrywise nonnegative matrix B), may be completed to a completely positive matrix if and only if G is a block-clique graph (a chordal graph in which distinct maximal cliques overlap in at most one vertex). The same result holds for matrices that are doubly nonnegative (entrywise nonnegative and positive semidefinite).     

Behaviors of the positive temperature coefficient of resistance of poly(styrene‐co‐n‐butylacrylate) filled with Ni‐plated core‐shell polymeric particles

Conductive composite films of poly(styrene‐co‐n‐butylacrylate) copolymers filled with low‐density, Ni‐plated core‐shell polymeric particles were prepared and their behaviors of positive temperature coefficient of resistance (PTCR) were investigated. When the conductive fillers in the composite film were loaded beyond the critical volume, 10 up to 25 vol %, composite films exhibited a unique electrical resistant transition behavior, which the electrical resistance rapidly increased by several orders of magnitude at the critical temperature. The PTCR transition temperature, in general, occurred before the glass transition temperature of polymer matrix. Further increased the conductive filler loading to 30 vol %, the overpacked conduction paths were formed in the entire composite and the PTCR effects became blurred. While the composite film treated with thermal cycle several times from room temperature up to 120 °C, the electrical resistivity increased accompanied with the shift of the PTCR transition to lower temperature. The reason might have been caused by the formed interfacial cracks within the composite film. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 322–329, 2007     

The complete symmetrization of quantum operators: new thoughts on an old problem

The complete symmetrization with respect to x, p _x,... of the operators associated with dynamical properties can sometimes lead to results different from those obtained by the conventional quantum formalism based on the rule op (A ²)=(op A)². For example, angular momentum operators M _z ² and M ² are modified by the additive constants ²/2 and 3 ²/2 respectively (M ²0 for electron in the ground state of H atom, rotator never at rest, but spectra unchanged); the average quadratic dispersion of energy is different from zero. These results can be interpreted by assuming that the system is never strictly isolated but communicates with the other systems of the universe by means of electromagnetic interactions. Quantum mechanics would give only average values over a sufficiently long time and would exhibit a quasi-ergodic character. Examples supporting this possibility are given, in particular that of arsines for which quantum forecasts correspond to average values over one year.Dedicated to Professor J. Koutecký on the occasion of his 65th birthday     

Mass spectrometer-wall probe diagnostic of Ar discharges containing SF6 and/or O2: Reactive ions in etching plasmas

Positive and negative ions of Ar/SF₆ and Ar/SF₆/O₂ plasmas (etching plasmas) and of Ar/O₂ plasmas (cleaning plasmas) in Pyrex tubes have been investigated using a mass spectrometer-wall probe diagnostic technique. The measurement of negative ions proved to be a very sensitive method for the detection of wall material. In etching plasmas with small admixtures of SF₆, oxygen was found as the only representative of wall material. At larger amounts of SF₆, silicon could be detected. In cleaning plasmas with small admixtures of O₂ applied to a previously etched Pyrex surface, fluorine was found, indicating the reversal of fluoridation by oxygenation.     

Integral Operators Between Fock Spaces∗

In this paper, the authors study the integral operator Sφf(z) = Z C φ(z, w)f(w)dλα(w) induced by a kernel function φ(z, ·) ∈ F ∞α between Fock spaces. For 1 ≤ p ≤ ∞, they prove that Sφ : F 1 α → F p α is bounded if and only if sup a∈C kSφkakp,α < ∞, (?) where ka is the normalized reproducing kernel of F 2 α; and, Sφ : F 1 α → F p α is compact if and only if lim |a|→∞ kSφkakp,α = 0. When 1 < q ≤ ∞, it is also proved that the condition (?) is not sufficient for boundedness of Sφ : F q α → F p α . In the particular case φ(z, w) = eαzw?(z ? w) with ? ∈ F 2 α, for 1 ≤ q < p < ∞, they show that Sφ : F p α → F q α is bounded if and only if ? = 0; for 1 < p ≤ q < ∞, they give sufficient conditions for the boundedness or compactness of the operator Sφ : F p α → F q α.     

Absence of absolutely continuous spectrum of Floquet operators

The spectrum of the Floquet operator associated with time-periodic perturbations of discrete Hamiltonians is considered. If the gap between successive eigenvalues _j of the unperturbed Hamiltonian grows as _j - _j-1 j and the multiplicity of _j grows asj with >0 asj tends to infinity, then the corresponding Floquet operator possesses no absolutely continuous spectrum provided the perturbation is smooth enough.     

Positive solutions of singular boundary value problem of negative exponent Emden—Fowler equation

This paper investigates the existence of positive solutions of a singular boundary value problem with negative exponent similar to standard Emden-Fowler equation. A necessary and sufficient condition for the existence of C[0, 1] positive solutions as well as C¹[0, 1] positive solutions is given by means of the method of lower and upper solutions with the Schauder fixed point theorem.