Abstract:
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. 1. Introduction This paper (see also preprint [1]), presenting the results of a part of the author's thesis [2], deals with contractions (partial traces) of antisymmetric and symmetric product density operators representing mixed states of systems of identical noninteracting fermions and bosons, respectively. If is a separable Hilbert space of a single fermion (boson), then the space of the -fermion (resp. -boson) system is the antisymmetric (resp. symmetric) subspace (resp. ) of . Density operators of -fermion (resp. -boson) systems are identified with those defined on and concentrated on (resp. ). Recall that the expectation value of an observable represented by a bounded self-adjoint operator on given Hilbert space in a state described by a density operator equals . If is an unbounded self-adjoint operator on a dense subspace of given Hilbert space, instead of one can consider its spectral measure (which is a bounded operator) of a Borel subset of the spectrum of . Then is the probability that the result of the measurement of the observable in question belongs to [3]. -particle observables of -fermion and -boson systems ( ) are represented, respectively, by operators of the form (multiplied by ), where and are projectors of onto and , respectively, is the identity operator on and is a self-adjoint operator on (see [4]). Operators (1.1) are called antisymmetric and symmetric expansions of . In view of the earlier remark it is assumed that is bounded. The expectation values of observables represented by and in states represented by -fermion and -boson density operators and , respectively, can be expressed as (see [4, equations (1.7), (3.19)]), where -particle

Abstract:
'Voicing' in English voiced obstruents has been defined in terms of 'full' vs. 'partial'. When teaching English pronunciation to native speakers of Polish, where voiced sounds can be only fully voiced, it is diffcult to make the students aware of the phonation strategy to be used to obtain 'partially voiced' sounds, especially in plosives. The accessability of digital speech analysis computer software has made it possible to visualize the acoustic properties of speech sounds which can facilitate the teaching of English pronunciation to Poles, providing a visual feedback in class and at home. This is necessary for obtaining the correct phonation control that functions with utmost precision measured in centiseconds. Yet speech visualisation for the purpose of teaching English phonetics in Poland is employed only at the author's institution, and the remaining hundreds of schools and universities do not take advantage of the possibilities modem technology offers. The 'pedagogical perspective' of the paper aims at exerting an encouraging impact both on teachers of phonetics and on students of English. The article also provides a description of Polish voicing rules and a detailed comparison of voicing in English and Polish obstruents based on the concept of Voice Onset Time.

Abstract:
Job within organization can be discussed in the context of its quantity possible to do, quality re-sulting from its difficulty level and effect achieved by the employee that is effects of his work. To measure job from a quantitative point of view we use work norms as a function of time standards, products quantity or service level.It is much more difficult to measure qualitative job parameters than measuring quantity of job and its effects. In the literature we know several methods to job evaluation. However, none of them de-termines precisely the value of individual job evaluation within organization. The paper aims to develop a new method to measure and assess qualitative parameters of job in a simple, transparent, universal and timeless way. When evaluating a given feature, factor, object, subject we weight various quality and quantity criteria relative to an accepted pattern or value in a given organization, society or culture.Weight (priorities) determined based on comparisons designate relative value of a comparative factor. Building a system of job evaluation in the organization 7. synthetic criteria were taken: kno-wledge, experience, wisdom, psychological and physical effort, intellectual effort, responsibility and cooperation. Each synthetic criterion was given a few analytical criteria, which in turn was assigned a verbal, adjective level of intensity. To solve the problem we used a multicriterial problem solution me-thod AHP (Analytic Hierarchy Process). By pairwise comparison of each synthetic criteria (on a verbal scale) in relation to job quality in the Saaty’s fundamental scale we arrived a weight comparison matrix (priorities) within the range [> 0, <1]. A sum of weights from comparisons of all synthetic criteria is equal to one. Next, the same scale was used for comparisons of analytical criteria (sub-criteria) and their values in relation to particular synthetic criteria. Each analytical criterion was given weight (priority) resulting from comparisons, their total sum for each synthetic criterion is also equal to one. Next, each analytical sub-criterion in the system of work quality assessment was assigned adjectival level of intensity, also in a numerical scale calculated from a matrix of verbal adjectival comparison scale. The sum of those weights is also equal to 1.Our method differs form all its predecessors by the fact that a final weight (priority) for a given analytical factor is a multiplicative value transferring values from synthetic criteria onto analytical, those in turn into point assessments. Certain interdependencies be

Abstract:
Study of the η → π+π π0 decay allows for precise tests of the C parity conservation, makes possible to set constraints on the u and d as well as s and d quark mass ratios and can be used for a verification of the predictions of the Chiral Perturbation Theory (ChPT). The WASA-at-COSY collaboration has accumulated the world largest numbers of η mesons produced in p d and p p collisions. This report includes preliminary results of the ongoing analysis of the η → π+π π0 decay from p p data which were collected during 8 weeks of the COSY beam time in 2010 in Forschungszentrum Jülich with more than 108 η mesons produced.

Abstract:
We investigate Laha-Lukacs properties of noncommutative random variables (processes). We prove that some families of free Meixner distributions can be characterized by the conditional moments of polynomial functions of degree 3. We also show that this fact has consequences in describing some free Levy processes. The proof relies on a combinatorial identity. At the end of this paper we show that this result can be extended to a q-Gausian variable.

Abstract:
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].

Abstract:
In this article we give a purely noncommutative criterion for the characterization of two-state normal distribution. We prove that families of two-state normal distribution can be described by relations which is similar to the conditional expectation in free probability, but has no classical analogue. We also show a generalization of Bozejko, Leinert and Speicher's formula (relating moments and noncommutative cumulants).

Abstract:
Complex-valued sparse coding is a data representation which employs a dictionary of two-dimensional subspaces, while imposing a sparse, factorial prior on complex amplitudes. When trained on a dataset of natural image patches, it learns phase invariant features which closely resemble receptive fields of complex cells in the visual cortex. Features trained on natural sounds however, rarely reveal phase invariance and capture other aspects of the data. This observation is a starting point of the present work. As its first contribution, it provides an analysis of natural sound statistics by means of learning sparse, complex representations of short speech intervals. Secondly, it proposes priors over the basis function set, which bias them towards phase-invariant solutions. In this way, a dictionary of complex basis functions can be learned from the data statistics, while preserving the phase invariance property. Finally, representations trained on speech sounds with and without priors are compared. Prior-based basis functions reveal performance comparable to unconstrained sparse coding, while explicitely representing phase as a temporal shift. Such representations can find applications in many perceptual and machine learning tasks.

Abstract:
Scavenging abilities of the ethylenediamine, diethylenetriamine and tris(2-aminoethyl)amine functionalized resins derived from the low cross-linked glycidyl methacrylate-styrene gels in relation to the selected transition metal ions and organic (acid chlorides) electrophiles were explored. The synergetic effects of the solvents were found for the metal ion uptake. In this case a pair of solvents capable simultaneously to swell the hydrophobic core of the resins and interact with the polar amine groups turned out to be a more effective medium for activation of the nucleophilic functionalities of the resins. The resin with ethylenediamine moieties proved to be effective in scavenging acyl chlorides from methylene chloride as well. However, it did not provide complete removing sulphonyl chlorides from the solutions. These organic electrophiles were removed effectively by the resins bearing tris(2-aminoethyl)amine functionalities.