Contractions of product density operators of systems of identical fermions and bosons

Recurrence and explicit formulae for contractions (partial traces) of antisymmetric and symmetric products of identical trace class operators are derived. Contractions of product density operators of systems of identical fermions and bosons are proved to be asymptotically equivalent to, respectively, antisymmetric and symmetric products of density operators of a single particle, multiplied by a normalization integer. The asymptotic equivalence relation is defined in terms of the thermodynamic limit of expectation values of observables in the states represented by given density operators. For some weaker relation of asymptotic equivalence, concerning the thermodynamic limit of expectation values of product observables, normalized antisymmetric and symmetric products of density operators of a single particle are shown to be equivalent to tensor products of density operators of a single particle. This paper presents the results of a part of the author's thesis [W. Radzki, "Kummer contractions of product density matrices of systems of $n$ fermions and $n$ bosons" (Polish), MS thesis, Institute of Physics, Nicolaus Copernicus University, Toru\'{n}, 1999].Comment: 20 pages. The manuscript has been shortened. A few typos correcte

On the structure of the set of bifurcation points of periodic solutions for multiparameter Hamiltonian systems

This paper deals with periodic solutions of the Hamilton equation with many parameters. Theorems on global bifurcation of solutions with periods $2\pi/j,$ $j\in\mathbb{N},$ from a stationary point are proved. The Hessian matrix of the Hamiltonian at the stationary point can be singular. However, it is assumed that the local topological degree of the gradient of the Hamiltonian at the stationary point is nonzero. It is shown that (global) bifurcation points of solutions with given periods can be identified with zeros of appropriate continuous functions on the space of parameters. Explicit formulae for such functions are given in the case when the Hessian matrix of the Hamiltonian at the stationary point is block-diagonal. Symmetry breaking results concerning bifurcation of solutions with different minimal periods are obtained. A geometric description of the set of bifurcation points is given. Examples of constructive application of the theorems proved to analytical and numerical investigation and visualization of the set of all bifurcation points in given domain are provided. This paper is based on a part of the author's thesis [W. Radzki, ``Branching points of periodic solutions of autonomous Hamiltonian systems'' (Polish), PhD thesis, Nicolaus Copernicus University, Faculty of Mathematics and Computer Science, Toru\'{n}, 2005].Comment: 35 pages, 4 figures, PDFLaTe

UAVS FLIGHT ROUTES OPTIMIZATION IN CHANGING WEATHER CONDITIONS – CONSTRAINT PROGRAMMING APPROACH

The problem of delivering goods in a distribution network is considered in which a fleet of Unmanned Aerial Vehicles (UAV) carries out transport operations. The changing weather conditions in which the transport operations take place and the UAVs energy capacity levels influenced by the weather conditions are taken into account as factors that affect the determination of a collision-free route. The goods must be delivered to the customers in a given time window. Establishing the routes are the focus of this study. Solutions maximizing the level of customer satisfaction are focused and the computational experiments presented in the study show the impact of weather conditions on route determination

Periodic solutions of second order Hamiltonian systems bifurcating from infinity

The goal of this article is to study closed connected sets of periodic solutions, of autonomous second order Hamiltonian systems, emanating from infinity. The main idea is to apply the degree for SO(2)-equivariant gradient operators defined by the second author. Using the results due to Rabier we show that we cannot apply the Leray-Schauder degree to prove the main results of this article. It is worth pointing out that since we study connected sets of solutions, we also cannot use the Conley index technique and the Morse theory.Comment: 24 page

Effect of Strontium Ranelate on Femur Densitometry and Antioxidative/Oxidative Status in Castrated Male Rats

The studies were aimed at determinatning of the effect of strontium ranelate (SR) on the mineralization  processes and selected parameters of oxidative stress in orchidectomized rats during the development of  osteopenia. Male Wistar rats were sham-operated (SHO) and orchidectomized (ORX). ORX animals were  divided into control (ORX-C) and gavaged with SR (ORX-SR), at a dose of 900mg/kg/b.w. After 60 days  the animals were scanned for determination of bone mineral density (BMD) of the whole skeleton. Isolated  femora were examined by DEXA and pQCT. Tomographic measurements were performed for a total  slice and separately for the cortical and trabecular parts of the distal end of the femora. The intensity of  lipid peroxidation (ILP) and total antioxidant capacity (TAC) in blood serum were measured. SR treatment  increased vBMD and BMC of total, trabecular and cortical bone in ORX rats compared to ORX-C and  SHO rats. ORX significantly increased TAC in control animals, and SR limited this increase. ILP in SHO  and ORX-C rats which on a similar level. SR increased ILP by 21.3%, as compared to SHO. SR improved  densitometric and geometric parameters of femora by orchidectomized rats what prevented degradation of  bone tissue. Beneficial effects of SR were also demonstrated in stabilization of TAC in ORX rats at the level  noted in SHO rats.