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In any infinite dimensional Hilbert space H, a sequence P_n...P_1 x diverges in norm for some x \in H and orthogonal projections P_n \in {Q_1,..., Q_5}.

We investigate the following generalisation of the entropy of quantum measurement. Let H be an infinite-dimensional separable Hilbert space with a 'density' operator {\rho}, tr {\rho}=1. Let I(P)\in R be defined for any partition P = (P_1,...,P_m), P_1+ ... +P_m=1_H, P_i \in proj H$ and let I(P_i Qj, i \leq m, j \leq n) = I(P) + I(Q) for Q =(Q_1,..., Q_n), \sum Q_j = 1_H and P_iQ_j = Q_j P_i, tr {\rho} P_iQ_j = tr {\rho} P_i tr {\rho} Q_j (P, Q are physically independent). Assuming some continuity properties we give a general form of generalised information I., Comment: 20 pages

In global economy, turbulent organization environment strongly influences organization's operation. Organizations must constantly adapt to changing circumstances and search for new possibilities of gaining competitive advantage. To face this challenge, small organizations base their operation on collaboration within Virtual Organizations (VOs). VO operation is based on collaborative processes. Due to dynamism and required flexibility of collaborative processes, existing business information systems are insufficient to efficiently support them. In this paper a novel method for supporting collaborative processes based on process mining techniques is proposed. The method allows activity patterns in various instances of collaborative processes to be identified and used for recommendation of activities. This provides an opportunity for better computer support of collaborative processes leading to more efficient and effective realization of business goals.

We provide, under minimal continuity assumptions, a description of \textsl{additive partition entropies}. They are real functions $I$ on the set of finite partitions that are additive on stochastically independent partitions in a given probability space., Comment: 24 pages

While Enterprise Architecture Modeling (EAM) methodologies become more and more popular, an EAM methodology tailored to the needs of virtual organizations (VO) is still to be developed. Among the most popular EAM methodologies, TOGAF has been chosen as the basis for a new EAM methodology taking into account characteristics of VOs presented in this paper. In this new methodology, referred as Virtual Organization Breeding Methodology (VOBM), concepts developed within the ECOLEAD project, e.g. the concept of Virtual Breeding Environment (VBE) or the VO creation schema, serve as fundamental elements for development of VOBM. VOBM is a generic methodology that should be adapted to a given VBE. VOBM defines the structure of VBE and VO architectures in a service-oriented environment, as well as an architecture development method for virtual organizations (ADM4VO). Finally, a preliminary set of tools and methods for VOBM is given in this paper., Comment: 9 pages, 4 figures

Performance management (PM) is a key function of virtual organization (VO) management. A large set of PM indicators has been proposed and evaluated within the context of virtual breeding environments (VBEs). However, it is currently difficult to describe and select suitable PM indicators because of the lack of a common vocabulary and taxonomies of PM indicators. Therefore, there is a need for a framework unifying concepts in the domain of VO PM. In this paper, a reference model for VO PM is presented in the context of service-oriented VBEs. In the proposed reference model, both a set of terms that could be used to describe key performance indicators, and a set of taxonomies reflecting various aspects of PM are proposed. The proposed reference model is a first attempt and a work in progress that should not be supposed exhaustive., Comment: 10 pages

In Service-Oriented Virtual Organization Breeding Environments (SOVOBEs), services performed by people, organizations and information systems are composed in potentially complex business processes performed by a set of partners. In a SOVOBE, the success of a virtual organization depends largely on the partner and service selection process, which determines the composition of services performed by the VO partners. In this paper requirements for a partner and service selection method for SOVOBEs are defined and a novel Multi-Aspect Partner and Service Selection method, MAPSS, is presented. The MAPSS method allows a VO planner to select appropriate services and partners based on their competences and their relations with other services/partners. The MAPSS method relies on a genetic algorithm to select the most appropriate set of partners and services., Comment: 10 pages, 1 figure

In the context of globalization and dynamic markets, collaboration among organizations is a condition sine qua non for organizations, especially small and medium enterprises, to remain competitive. Virtual organizations have been proposed as an organizational structure adapted to collaboration among organizations. The concept of Virtual Organization Breeding Environment (VOBE) has been proposed as a means to support the creation and operation of virtual organizations. With the rise of the service-oriented architecture (SOA), the concept of service-oriented VOBE (SOVOBE) has been proposed as a VOBE systematically organized around the concept of services. In the context of SOVOBEs, novel competence models supporting both service orientation and collaboration among organizations have to be developed to support efficiently partner selection, a key aspect of VO creation. In this paper, such a competence model is presented. Our competence model consists of a competence description model, a competence verification method, and a competence search method. The competence description model is an information model to describe organizations, their competences, and services they provides. The competence verification method enables the verification of the reliance and relevance of competence descriptions. The competence search method allows a VO planner to select appropriate partners based on VO specifications, encompassing competence requirements. Finally, implementation concerns based on the development of the prototype ErGo system are presented., Comment: 17 pages, 7 figures

Young's and shear moduli and Poisson's ratio of polycrystalline solids consisting of 2D quadratic and 3D cubic randomly oriented grains of the same size and shape is studied. Considered polycrystals are initially unstrained. It is shown that for such polycrystals the division of the mechanical stability regions into areas of various auxeticity properties is different than for monocrystalline solids. In particular the regions of complete auxeticity enlarge., Comment: 6 pages, 7 figures, sent to physica status solidi b

We consider 2- and 3-dimensional cubic monocrystalline and polycrystalline materials. Expressions for Young's and shear moduli and Poisson's ratio are expressed in terms of eigenvalues of the stiffness tensor. Such a form is well suited for studying properties of these mechanical characteristics on sides of the stability triangles. For crystalline high-symmetry directions lines of vanishing Poisson's ratio are found. These lines demarcate regions of the stability triangle into areas of various auxeticity properties. The simplest model of polycrystalline 2D and 3D cubic materials is considered. In polycrystalline phases the region of complete auxetics is larger than for monocrystalline materials., Comment: 9 pages, 3 figures, in proceedings of the Tenth International School on Theoretical Physics, Symmetry and Structural Properties of Condensed Matter, Myczkowce 2009

Blackman's diagram of two dimensionless ratios of elastic constants is frequently used to correlate elastic properties of cubic crystals with interatomic bondings. Every's diagram of a different set of two dimensionless variables was used by us for classification of various properties of such crystals. We compare these two ways of characterization of elastic properties of cubic materials and consider the description of various groups of materials, e.g. simple metals, oxides, and alkali halides. With exception of intermediate valent compounds, the correlation coefficients for Every's diagrams of various groups of materials are greater than for Blackaman's diagrams, revealing the existence of a linear relationship between two dimensionless Every's variables. Alignment of elements and compounds along lines of constant Poisson's ratio $\nu(<100>,\textbf{m})$, ($\textbf{m}$ arbitrary perpendicular to <100>) is observed. Division of the stability region in Blackman's diagram into region of complete auxetics, auxetics and non-auxetics is introduced. Correlations of a scaling and an acoustic anisotropy parameter are considered., Comment: 8 pages, 9 figures, presented on The Ninth International School on Theoretical Physics "Symmetry and Structural Properties of Condensed Matter", 5 - 12 September 2007, Myczkowce, Poland

The maximal value of thermal conductivity \kappa_{max} of the perfect wurzite GaN crystal containing isotopes of natural abundance is estimated. Our upper limit of \kappa=4800 W/Km at T_{max}=32 K is smaller than calculated by Liu and Balandin \kappa=6000 W/Km and higher than obtained by Slack et al \kappa=3750 W/Km. The phenomenological dependence \kappa \propto T^{-1.43} obtained by Mion et al for the temperature interval 300-450 K is extended to 200-300K. For temperatures higher than T_max the best fitting of our experimental data to Callaway's formula is obtained for Grueneisen's constant equal to \gamma = 1.35., Comment: 7 pages, 3 figures

Anisotropies of Young's modulus E, the shear modulus G, and Poisson's ratio of all 2D symmetry systems are studied. Simple necessary and sufficient conditions on their elastic compliances are derived to identify if any of these crystals are completely auxetic, non-auxetic and auxetic. Particular attention is paid to 2D crystals of quadratic symmetry. All mechanically stable quadratic crystals are characterized by three parameters belonging to a prism with the stability triangle in the base. Regions in the stability triangle in which quadratic materials are completely auxetic, non-auxetic, and auxetic are established. Examples of all types of auxetic properties of crystals of oblique and rectangular symmetry are presented., Comment: 17 pages, 9 figures

Explicit expressions for inverse of Young's modulus E, inverse of shear modulus G, and Poisson's ratio for cubic media are considered. All these characteristics of elastic media depend on three components of the compliance tensor S, and on direction cosines of mutually perpendicular vectors m and n with fourfold symmetry axes. These characteristics are studied for all mechanically stable cubic materials for vectors n belonging to the irreducible body angle subtended by three cubic high symmetry directions [001], [111], and [110]. Regions in the stability triangle of in which cubic elastic materials are completely auxetic, non-auxetic, and auxetic are established. Several intermediate-valence compounds belonging to the region of complete auxecity are indicated. The extreme properties of E^{-1}, G^{-1} and Poisson ratio established by Hayes and Shuvalov are confirmed., Comment: 16 pages, 8 figures, 2nd Workshop on Auxetics and Related Systems Bedlewo (near Poznan), August 19-23, 2005

Fourth-rank tensors of complete Voigt's symmetry, that embody the elastic properties of crystalline anisotropic substances, were constructed for all 2D crystal systems. Using them we obtained explicit expressions for inverse of Young's modulus E, inverse of shear modulus G and Poisson's ratio, which depend on components of the elastic compliances tensor S, on direction cosines of vectors n of uniaxial load and the vector m of lateral strain with crystalline symmetry axes. All 2D crystal systems are considered. Such representation yields decomposition of the above elastic characteristics to isotropic and anisotropic parts. Expressions for Poisson's coefficient are well suited for studying the property of auxeticity and anisotropy 2D crystals. Phase velocities are calculated for all 2D crystal classes., Comment: 13 pages, no figures

Using bases of fourth rank tensorial bases of complete Voigt's symmetry elaborated by Walpole we obtained expressions for inverse of Young's modulus E, inverse of shear modulus G and Poisson's ratio, which depend on components of the stiffness tensor S, on direction cosines of vectors n of uniaxial load and the vector m of lateral strain with crystalline symmetry axes. Crystalline media of high and medium symmetries are considered. Such representation yields decomposition of the above elastic characteristics to isotropic and anisotropic parts. Expressions for Poisson's coefficient are well suited for studying the property of auxeticity., Comment: 14 pages, no figures, 2nd Workshop on Auxetics and Related Systems Bedlewo (near Poznan), August 19-23, 2005

We report on lattice specific heat of bulk hexagonal GaN measured by the heat flow method in the temperature range 20-300 K and by the adiabatic method in the range 5-70 K. We fit the experimental data using two temperatures model. The best fit with the accuracy of 3 % was obtained for the temperature independent Debye's temperature $\theta_{\rm D}=365$ {\rm K} and Einstein's temperature $\theta_{\rm E}=880$ {\rm K}. We relate these temperatures to the function of density of states. Using our results for heat conduction coefficient, we established in temperature range 10-100 K the explicit dependence of the phonon mean free path on temperature $\it{l}_{\rm ph}\propto T^{-2}$. Above 100 K, there is the evidence of contribution of the Umklapp processes which limit phonon free path at high temepratures. For phonons with energy $k_{\rm B}\times 300 $ {\rm K} the mean free path is of the order 100 {\rm nm}, Comment: 5 pages, 4 figures

We characterize sequences of numbers $(a_n)$ such that $\sum_{n\geq 1} a_n\Phi_n$ converges a.e. for any orthonormal system $(\Phi_n)$ in any $L_2$-space. In our criterion, we use the set $B =\{\sum_{m\geq n} |a_m|^2; n\geq 1\}$ and its information function $$h_B(t) = -\log_3(\beta-\alpha)$$ for $t\in (\alpha, \beta]$, $[\alpha, \beta]\cap B =\{\alpha, \beta\}.$, Comment: 44 pages