Refined matrix models from BPS counting

We construct a free fermion and matrix model representation of refined BPS generating functions of D2 and D0 branes bound to a single D6 brane, in a class of toric manifolds without compact four-cycles. In appropriate limit we obtain a matrix model representation of refined topological string amplitudes. We consider a few explicit examples which include a matrix model for the refined resolved conifold, or equivalently five-dimensional U(1) gauge theory, as well as a matrix representation of the refined MacMahon function. Matrix models which we construct have ordinary unitary measure, while their potentials are modified to incorporate the effect of the refinement.Comment: 27 pages, 4 figures, published versio

BPS counting for knots and combinatorics on words

We discuss relations between quantum BPS invariants defined in terms of a product decomposition of certain series, and difference equations (quantum A-polynomials) that annihilate such series. We construct combinatorial models whose structure is encoded in the form of such difference equations, and whose generating functions (Hilbert-Poincar\'e series) are solutions to those equations and reproduce generating series that encode BPS invariants. Furthermore, BPS invariants in question are expressed in terms of Lyndon words in an appropriate language, thereby relating counting of BPS states to the branch of mathematics referred to as combinatorics on words. We illustrate these results in the framework of colored extremal knot polynomials: among others we determine dual quantum extremal A-polynomials for various knots, present associated combinatorial models, find corresponding BPS invariants (extremal Labastida-Mari\~no-Ooguri-Vafa invariants) and discuss their integrality.Comment: 41 pages, 1 figure, a supplementary Mathematica file attache

Topological strings, strips and quivers

We find a direct relation between quiver representation theory and open topological string theory on a class of toric Calabi-Yau manifolds without compact four-cycles, also referred to as strip geometries. We show that various quantities that characterize open topological string theory on these manifolds, such as partition functions, Gromov-Witten invariants, or open BPS invariants, can be expressed in terms of characteristics of the moduli space of representations of the corresponding quiver. This has various deep consequences; in particular, expressing open BPS invariants in terms of motivic Donaldson-Thomas invariants, immediately proves integrality of the former ones. Taking advantage of the relation to quivers we also derive explicit expressions for classical open BPS invariants for an arbitrary strip geometry, which lead to a large set of number theoretic integrality statements. Furthermore, for a specific framing, open topological string partition functions for strip geometries take form of generalized $q$-hypergeometric functions, which leads to a novel representation of these functions in terms of quantum dilogarithms and integral invariants. We also study quantum curves and A-polynomials associated to quivers, various limits thereof, and their specializations relevant for strip geometries. The relation between toric manifolds and quivers can be regarded as a generalization of the knots-quivers correspondence to more general Calabi-Yau geometries.Comment: 47 pages, 9 figure

Topological recursion and mirror curves

We study the constant contributions to the free energies obtained through the topological recursion applied to the complex curves mirror to toric Calabi-Yau threefolds. We show that the recursion reproduces precisely the corresponding Gromov-Witten invariants, which can be encoded in powers of the MacMahon function. As a result, we extend the scope of the "remodeling conjecture" to the full free energies, including the constant contributions. In the process we study how the pair of pants decomposition of the mirror curves plays an important role in the topological recursion. We also show that the free energies are not, strictly speaking, symplectic invariants, and that the recursive construction of the free energies does not commute with certain limits of mirror curves.Comment: 37 pages, 4 figure

Trust-based quality culture conceptual model for higher education institutions

Higher Education Institutions (HEIs) play a crucial role in societies as they enhance the sustainable development of nations. In a context of increasing competition and financial difficulties in higher education institutions, the loyalty of students, faculty and administration staff as well as institutional reputation are key factors for survival and success. They are built upon trust and high quality of services rendered by HEIs. The intentional development of trust serves the purpose of enhancing the quality culture in higher education. The concept of quality culture has become a natural successor of quality management and quality assurance in universities presenting a new perspective for viewing quality at HEIs - as a combination of structural and managerial with cultural and psychological components. This paper provides an elaboration of a novel Trust-Based Quality Culture Conceptual Model for Higher Education Institutions which presents the perceived interconnections between trust and quality culture at HEIs. It can form a source for an inquiry process at HEIs, thus contributing to better contextual diagnosis of the stage where HEI is in the process of building the quality culture based on trust. The findings of this study are important in better understanding the quality culture development in HEIs that is based on trust, loyalty and reputation. It may have an impact on the decision-making processes concerning HEIs’ management. The proposed model contributes to the need for greater clarity, ordering and systematization of the role of trust in the processes of quality culture development

Medical family businesses in Poland : model and managerial challanges

W Polsce występuje deficyt danych o cechach przedsiębiorstw rodzinnych świadczących usługi medyczne. Sektor medyczny w Polsce stoi w obliczu szybkiego rozwoju firm rodzinnych i jest zróżnicowany, ponieważ obejmuje różne wielkości podmiotów gospodarczych, które specjalizują się w wielu możliwych aspektach branży medycznej. Artykuł dotyczy cech przedsiębiorstw rodzinnych świadczących usługi medyczne oraz zakresu, w jakim przypominają one firmy rodzinne i do których wywodzą się z działalności usług medycznych. Artykuł ma charakter teoretyczny i jego celem jest zaprezentowanie modelu funkcjonowania rodzinnych firm medycznych, biorąc pod uwagę wpływ rodzaju działalności i organizacji rodzinnej. Pierwsza część artykułu koncentruje się na charakterystyce firm rodzinnych, stosunkowo mało jest reprezentatywnych badań analizujących udział przedsiębiorstw rodzinnych w polskiej gospodarce i opisujących ich charakter. Druga część artykułu to problem etosu zawodów medycznych w odniesieniu do logiki biznesowej i ekonomicznej organizacji. W trzeciej części artykułu znajduje się propozycja modelu łączącego oba aspekty funkcjonowania tego typu podmiotów gospodarczych.There is a deficit of data in Poland about characteristics of family enterprises providing medical services. The medical sector in Poland faces a rapid development of family businesses and is diverse because it encompasses various size business entities that specialize in many possible aspects of the medical business. The article is about the characteristics of family enterprises providing medical services and extent to which they resemble family businesses, and to which they are derived from medical service activities. The article is of theoretical nature and its aim is to propose the model for the functioning of family-owned medical businesses, taking into account the impact of the type of activity and the family organization. The first part of the article is focuses on characteristics of family businesses, there is relatively little representative research analyzing the share of family enterprises in the Polish economy and describing their character. The second part of the article is the problem of ethos of medical professions in relation to business and economic logics of an organization. In the third of the article part there is a proposal for a model combining both aspects of the functioning of this type of economic entities

Knots, BPS states, and algebraic curves

We analyze relations between BPS degeneracies related to Labastida-Marino-Ooguri-Vafa (LMOV) invariants, and algebraic curves associated to knots. We introduce a new class of such curves that we call extremal A-polynomials, discuss their special properties, and determine exact and asymptotic formulas for the corresponding (extremal) BPS degeneracies. These formulas lead to nontrivial integrality statements in number theory, as well as to an improved integrality conjecture stronger than the known M-theory integrality predictions. Furthermore we determine the BPS degeneracies encoded in augmentation polynomials and show their consistency with known colored HOMFLY polynomials. Finally we consider refined BPS degeneracies for knots, determine them from the knowledge of super-A-polynomials, and verify their integrality. We illustrate our results with twist knots, torus knots, and various other knots with up to 10 crossings.Comment: 43 pages, 6 figure