Functional form of unitary representations of the quantum "az+b" group

The formula for all unitary representations of the quantum "az+b" group for a real deformation parameter is given. The description involves the quantum exponential function introduced by Woronowicz

Social enterprise-led local development of the circular economy : socio-spatial networks and value-impact scaling pathways

The circular economy (CE) paradigm has emerged to challenge a predominantly linear economic development model by extracting and retaining the highest possible value from existing resources through their recirculation. While CE-related literature and policy discourse continue to grow, there is limited research on socio-spatial mechanisms shaping alternative circular economic development trajectories in the local development context. This thesis considers how the ecological and extra economic premises of CE thinking can be harnessed through mission-driven social enterprises (SEs) aimed at locally tackling poverty, inequality and/or waste. It investigates the extent to which 50 case study SEs operating in three different socio-spatial and institutional contexts (Hull, UK; Santiago, Chile; and Graz, Austria), and across diverse sectors (food, wood, textiles, housing, among others), stimulate and potentially could stimulate the development of a local and socially inclusive CE. In so doing, firstly, it untangles complex socio-material circuits of value and corresponding feedback loops associated with flows of (in)tangible resources across co-existing mainstream and alternative economic spaces of exchange, production and consumption. Secondly, this research adopts a Social Network Analysis approach to map and examine the broader social circular enterprise ecosystem in the City of Hull. It explores how the broader network constellations not only embody, but also could embody symbiotic relationships between environmentally-/CE-, socially- and/or commercially oriented enterprises to foster inclusive CE development. It then offers a heuristic framework illustrating the interplay of factors shaping collaborative ties in the development of inclusive CE. Finally, it explores diverse social-circular impact scaling strategies and develops an Integrated Social-Circular Value-Impact Scaling(ISCIRVIS) framework for academia and context-adaptable toolkit for entrepreneurs. The toolkit is designed to help entrepreneurs to create, deepen and/or broaden the scale and scope of environmental-circular, social and/or economic value outcomes/impacts associated with existing or implementable (circular) activities, yet in the light of potential costs/risks

A Laplace operator and harmonics on the quantum complex vector space

The aim of this paper is to study the q-Laplace operator and q-harmonic polynomials on the quantum complex vector space generated by z_i,w_i, i=1,2,...,n, on which the quantum group GL_q(n) (or U_q(n)) acts. The q-harmonic polynomials are defined as solutions of the equation Delta_qp=0, where p is a polynomial in z_i,w_i, i=1,2,...,n, and the q-Laplace operator Delta_q is determined in terms of q-derivatives. The q-Laplace operator Delta_q commutes with the action of GL_q(n). The projector H_{m,m'}: A_{m,m'} --> H_{m,m'} is constructed, where A_{m,m'} and H_{m,m'} are the spaces of homogeneous (of degree m in z_i and of degree m' in w_i) polynomials and homogeneous q-harmonic polynomials, respectively. By using these projectors, a q-analogue of the classical zonal spherical and associated spherical harmonics are constructed. They constitute an orthogonal basis of H_{m,m'}. A q-analogue of separation of variables is given. The quantum algebra U_q(gl_n), acting on H_{m,m'}, determines an irreducible representation of U_q(gl_n). This action is explicitly constructed. The results of the paper lead to the dual pair (U_q(sl_2), U_q(gl_n)) of quantum algebras.Comment: 26 pages, LaTe

Simultaneous decomposition of two states

Simultaneous decompositions of a pair of states into pure ones are examined. There are privileged decompositions which are distinguished from all the other ones.Comment: 7 page

A mathematical theorem as the basis for the second law: Thomson's formulation applied to equilibrium

There are several formulations of the second law, and they may, in principle, have different domains of validity. Here a simple mathematical theorem is proven which serves as the most general basis for the second law, namely the Thomson formulation (`cyclic changes cost energy'), applied to equilibrium. This formulation of the second law is a property akin to particle conservation (normalization of the wavefunction). It has been stricktly proven for a canonical ensemble, and made plausible for a micro-canonical ensemble. As the derivation does not assume time-inversion-invariance, it is applicable to situations where persistent current occur. This clear-cut derivation allows to revive the ``no perpetuum mobile in equilibrium'' formulation of the second law and to criticize some assumptions which are widespread in literature. The result puts recent results devoted to foundations and limitations of the second law in proper perspective, and structurizes this relatively new field of research.Comment: Revised version. Redundant assumption omitted. Microcanonical ensemble included. Reference added. 7 pages revte

"Partial" Fidelities

For pairs, omega, rho, of density operators on a finite dimensional Hilbert space of dimension d I call k-fidelity the d - k smallest eigenvalues of | omega^1/2 rho^1/2 |. k-fidelities are jointly concave in omega, rho. This follows by representing them as infima over linear functions. For k = 0 known properties of fidelity and transition probability are reproduced. Partial fidelities characterize equivalence classes which are partially ordered in a natural way.Comment: LATEX2e, 14 page

Microfungal diversity of Juncus trifidus L. and Salix herbacea L. at isolated locations in the Sudetes and Carpathian Mountains

During cold periods in the Pleistocene Epoch, many plants known as the “relict species” migrated and inhabited new areas. Together with plants, some microfungi also migrated, remaining present on plants and in plant communities. However, the relationship between fungi and the migrating plants (especially host plants) is not well understood. Therefore, we examined the diversity and distribution of microfungi associated with two migratory relict plants in the Sudetes and Carpathian Mountains: Salix herbacea L. and Juncus trifidus L. In total, we found 17 taxa of fungi that were collected from nine different locations. Nine fungal taxa were collected on S. herbacea, and eight taxa on J. trifidus. Localities richest of fungi on S. herbacea were Mały Śnieżny Kocioł (Karkonosze Mts, Sudetes) and on J. trifidus, the Tatra Mts (Carpathian Mts). This work provides new insights into the distribution of fungi inhabiting S. herbacea and J. trifidus in Poland