Dissipative Schrödinger Operators with Matrix Potentials

Maximal dissipative Schrödinger operators are studied in L ²((–,);E) (dimE=n<) that the extensions of a minimal symmetric operator with defect index (n,n) (in limit-circle case at – and limit point-case at ). We construct a selfadjoint dilation of a dissipative operator, carry out spectral analysis of a dilation, use the Lax–Phillips scattering theory, and find the scattering matrix of a dilation. We construct a functional model of the dissipative operator, determine its characteristic function in terms of the Titchmarsh–Weyl function of selfadjoint operator and investigate its analytic properties. Finally, we prove a theorem on completeness of the eigenvectors and associated vectors of a dissipative Schrödinger operators.     

Cross characteristic representations of even characteristic symplectic groups

We classify the small irreducible representations of with even in odd characteristic. This improves even the known results for complex representations. The smallest representation for this group is much larger than in the case when is odd. This makes the problem much more difficult.

     

Conservation laws for a class of third order evolutionary differential systems

Conservation laws of third order quasi-linear scalar evolution equations are studied via exterior differential system and characteristic cohomology. We find a subspace of 2-forms in the infinite prolongation space in which every conservation law has a unique representative. Analysis of the structure of this subspace based upon the symbol of the differential equation leads to a universal integrability condition for an evolution equation to admit any higher order (weight) conservation laws. As an example, we give a complete classification of a class of evolution equations which admit conservation laws of the first three consecutive weights , , . The differential system describing the flow of a curve in the plane by the derivative of its curvature with respect to the arc length is also shown to exhibit the KdV property, i.e., an infinite sequence of conservation laws of distinct weights.

     

An intersection multiplicity in terms of -modules

The main aim of this paper is to discuss the relation between Serre's intersection multiplicity and the Euler form. The Euler form is defined to be an alternating sum of the length of -modules and is used by Mori and Smith to develop intersection theory over noncommutative rings. We show that they differ by a sign and that this relation is closely related to Serre's vanishing theorem.

     

Euler Characteristic and Fixed-Point Theorems

We propose a geometric definition of the Euler characteristic (M) for the class of compact epi-Lipschitzian sets MRⁿ and we provide existence theorems of (generalized) equilibria for set-valued mappings F when the domain M of F is neither assumed to be convex, nor smooth but has a nonzero Euler characteristic.     

Updated Analysis of Gallium Chelate and Vanadium Chelate Solutions with 4‐(2‐Pyridylazo)Resorcinol and Xylenol Orange

The characteristic factors of the gallium and vanadium chelates with the ligands 4(2pyridylazo)resorcinol (PAR) and xylenol orange (XO) have been determined with the aid of the principle of correction, which makes it possible to eliminate the influence of the surplus ligand in the solution of its chelate with methane (which makes calculation easier) and thereby provides a more acceptable method in comparison with others. The real molar coefficients of extinction of the chelates Ga and V with PAR and XO have been determined and a detailed calculation of the stoichiometric ratios of chelates has been made. The results have shown that the chelates formed have the following structure: Ga(PAR), Ga(XO), V(PAR)₂, and V(XO).     

The Characteristic Equation of Corrugated Square Groove Waveguides

When the periodic spacing of corrugated wavesguides is much less than wavelength , the slot width dL, and the space harmonics are neglected, the characteristic equation of corrugated square groove waveguides is formulated based on Maxwell's equations and boundary conditions.     

Trace Formulae of Characteristic Polynomial and Cayley-Hamilton＇s Theorem, and Applications to Chiral Perturbation Theory and General Relativity

By using combinatorics, we give a new proof for the recurrence relations of the characteristic polynomial coefficients, and we further obtain an explicit expression for the generic term of the coefficient sequence, which yields the trace formulae of the Cayley-Hamilton＇s theorem with all coefficients explicitly given. This implies a byproduct, a complete expression for the determinant of any finite-dimensional matrix in terms of the traces of its successive powers. And we discuss some of their applications to ehiral perturbation theory and general relativity.     

Deriving Energy-Gap of Some Nonlinear Hamiltonians by Invariant Eigen-operator Method

By virtue of the invariant eigen-operator method we search for the invariant eigen-operators for some Hamiltonians describing nonlinear processes in particle physics. In this way the energy-gap of the Hamiltonians can be naturally obtained. The characteristic polynomial theory has been fully employed in our derivation.