Gronwall's inequality for a nabla fractional difference system with a retarded argument and an application

ABSTRACT

We study the nabla fractional difference system with retarded argument. There are two major ingredients. A Gronwall's inequality for the nabla case is given. This allows us to evaluate the solution of nabla fractional difference system. We shall illustrate the validity of our results by means of examples.     

Synthesis and Spectral Study of Several Solid M3[Eu(2,6-Pyridinedicarboxylate)3] Salts (M=Li+, Na+, K+, Rb+, Cs+, NH4 +, and Pyridinium+)

Abstract

Salts of the [Eu(2,6-pyridinedicarboxylate)₃]^3- complex anion and various monovalent inorganic and organic counterions (Li⁺, Na⁺, K⁺, Rb⁺, Cs⁺, NH₄ ⁺, and pyridinium⁺) have been synthesized and studied by emission spectroscopy. The Eu³⁺ ion emission spectra exhibited by these salts have been observed with high resolution (less than 1.0 cm^?1) and at low temperature (77 K). The emission spectra of these compounds indicate that changing the attached counterion does not affect the site symmetry observed by the europium ion beyond slight distortions indicated by small shifts in the energies of the Eu³⁺ electronic levels.     

Modelling of excited state absorption and upconversion in erbium germanosilicate continuous wave fibre lasers

The effects of upconversion and pump excited state absorption (ESA) on a three-manifold continuous wave (CW) laser are investigated by solving the laser and pump differential equations subject to two-point boundary conditions. This technique is applied to erbium germanosilicate fibre lasers in low-finesse cavities. First, the three-manifold laser problem in a low-finesse cavity is solved analytically, without ESA and upconversion. This shows that the finite value of the upper pump manifold decay rate causes the extracted power to saturate, and it also shows that pump threshold is strongly influenced by the cavity reflectivity. Next, upconversion and ESA are included. This leads to a study of the extracted power and pump threshold as functions of fibre length and outcoupling. This simulation is numerical and is successfully anchored to experiment.     

Comparison theorems and existence theorems for ordinary differential equations

We are concerned with uniqueness and existence theorems for two point boundary value problems for the nonlinear differential equation Ly = f(x, y), where L is the classical nth order linear differential operator. In proving our results interesting comparison theorems are proven for linear differential equations.     

A quark parton model for hadronic fragmentation distributions

The one-particle inclusive distributions in the fragmentation regions in hadronic interactions are described in a parameter free way by a simple additative quark parton model by means of the quark fragmentation functions measured in leptoproduction. We interpret the result so that while the main interaction in hadronic collisions is due to the wee partons almost all the momenta is carried by one of the valence quarks yielding fragmentation region distributions similar to the ones in leptoproduction.     

The complementary unboundedness of dual feasible solution sets in convex programming

F.E. Clark has shown that if at least one of the feasible solution sets for a pair of dual linear programming problems is nonempty then at least one of them is both nonempty and unbounded. Subsequently, M. Avriel and A.C. Williams have obtained the same result in the more general context of (prototype posynomial) geometric programming. In this paper we show that the same result is actually false in the even more general context of convex programming — unless a certain regularity condition is satisfied.We also show that the regularity condition is so weak that it is automatically satisfied in linear programming (prototype posynomial) geometric programming, quadratic programming (with either linear or quadratic constraints),l _p -regression analysis, optimal location, roadway network analysis, and chemical equilibrium analysis. Moreover, we develop an equivalent regularity condition for each of the usual formulations of duality.Research sponsored by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant Number AFOSR-73-2516.     

A study of the reactions KN→Σ(1385)π between 0.6 and 1.2 GeV/c

An energy-dependent partial-wave analysis of the reaction $\text{K}\text{N}\text{→Σ(1385)π}$ has been performed in the energy region lying between 1.65 and 1.85 GeV. The resonant amplitudes of seven well known Y¹ resonances have been determined, and used to compute the corresponding (8) ? (10) SU (3) coupling constants.