Local currents for the GL(N,C) self-dual Yang-Mills equation

By using a simple Bäcklund-like transformation which linearizes the GL(N, C) self-dual Yang-Mills equation, an infinite number of local conservation laws for this equation are constructed. In the SL(N, C) case, the currents become trivial, which explains why these currents are not found in SU(N) gauge theory.     

M-matrix asymptotics for Sturm–Liouville problems on graphs

We consider a system formulation for Sturm–Liouville operators with formally self-adjoint boundary conditions on a graph. An M-matrix associated with the boundary value problem is defined and related to the matrix Prüfer angle associated with the system boundary value problem, and consequently with the boundary value problem on the graph. Asymptotics for the M-matrix are obtained as the eigenparameter tends to negative infinity. We show that the boundary conditions may be recovered, up to a unitary equivalence, from the M-matrix and that the M-matrix is a Herglotz function. This is the first in a series of papers devoted to the reconstruction of the Sturm–Liouville problem on a graph from its M-matrix.     

Discrete Galerkin method for Fredholm integro-differential equations with weakly singular kernels

Approximations to a solution and its derivatives of a boundary value problem of an nth order linear Fredholm integro-differential equation with weakly singular or other nonsmooth kernels are determined. These approximations are piecewise polynomial functions on special graded grids. For their finding a discrete Galerkin method and an integral equation reformulation of the boundary value problem are used. Optimal global convergence estimates are derived and an improvement of the convergence rate of the method for a special choice of parameters is obtained. To illustrate the theoretical results a collection of numerical results of a test problem is presented.     

Rédei-Funktionen für Zweierpotenzen

L. Rédei has introduced in 1946 a class of rational functions over finite fields of orderp ^t –p being an odd prime — and has proved some interesting results on permutations which are induced by these functions. Recently, similar results have been found for factor rings of the integers of orderp ^t ; moreover, these functions have been applied to construct public key cryptosystems. In this paper we consider functions of Rédei type over finite fields and factor rings of the integers of order 2 ^t . We show that most of the results for oddp also hold in this case.

     

Admissibility and minimaxity of generalized Bayes estimators for spherically symmetric family

We give a sufficient condition for admissibility of generalized Bayes estimators of the location vector of spherically symmetric distribution under squared error loss. Compared to the known results for the multivariate normal case, our sufficient condition is very tight and is close to being a necessary condition. In particular, we establish the admissibility of generalized Bayes estimators with respect to the harmonic prior and priors with slightly heavier tail than the harmonic prior. We use the theory of regularly varying functions to construct a sequence of smooth proper priors approaching an improper prior fast enough for establishing the admissibility. We also discuss conditions of minimaxity of the generalized Bayes estimator with respect to the harmonic prior.     

Unification of lower-bound analyses of the lift-and-project rank of combinatorial optimization polyhedra

We present a unifying framework to establish a lower bound on the number of semidefinite-programming-based lift-and-project iterations (rank) for computing the convex hull of the feasible solutions of various combinatorial optimization problems. This framework is based on the maps which are commutative with the lift-and-project operators. Some special commutative maps were originally observed by Lovász and Schrijver and have been used usually implicitly in the previous lower-bound analyses. In this paper, we formalize the lift-and-project commutative maps and propose a general framework for lower-bound analysis, in which we can recapture many of the previous lower-bound results on the lift-and-project ranks.     

Analysis of variance and linear contrasts in experimental design with generalized secant hyperbolic distribution

We consider one-way classification model in experimental design when the errors have generalized secant hyperbolic distribution. We obtain efficient and robust estimators for block effects by using the modified maximum likelihood estimation (MML) methodology. A test statistic analogous to the normal-theory F statistic is defined to test block effects. We also define a test statistic for testing linear contrasts. It is shown that test statistics based on MML estimators are efficient and robust. The methodology readily extends to unbalanced designs.     

Frequency stabilization and transverse mode discrimination in injection-seeded unstable resonator TEA CO2 lasers

Longitudinal mode selection by injection has been demonstrated as a viable technique for TEA-CO₂ lasers with pulse energies of a Joule or greater. Once reliable generation of single-longitudinal-mode (SLM) pulses is obtained, one can study the characteristics and the causes of intrapulse frequency variation. These include the effect of the decaying plasma, the thermal gradient due to the energy dissipation associated with the laser mechanism itself, and the pressure shift of the center frequency of the laser transition. The use of the positive-branch unstable resonator as an efficient means of coupling a discharge with large spatial dimensions to an optical cavity mode introduces another concern: namely, what can be done to emphasize transverse mode discrimination in an unstable resonator cavity while maintaining high coupling efficiency. These issues are discussed in this paper, and relevant experimental results are included.