A Telescoping method for Double Summations

We present a method to prove hypergeometric double summation identities. Given a hypergeometric term $F(n,i,j)$, we aim to find a difference operator $L=a_0(n) N^0 + a_1(n) N^1 +...+a_r(n) N^r$ and rational functions $R_1(n,i,j),R_2(n,i,j)$ such that $L F = \Delta_i (R_1 F) + \Delta_j (R_2 F)$. Based on simple divisibility considerations, we show that the denominators of $R_1$ and $R_2$ must possess certain factors which can be computed from $F(n, i,j)$. Using these factors as estimates, we may find the numerators of $R_1$ and $R_2$ by guessing the upper bounds of the degrees and solving systems of linear equations. Our method is valid for the Andrews-Paule identity, Carlitz's identities, the Ap\'ery-Schmidt-Strehl identity, the Graham-Knuth-Patashnik identity, and the Petkov\v{s}ek-Wilf-Zeilberger identity.Comment: 22 pages. to appear in J. Computational and Applied Mathematic

Applicability of the qq-Analogue of Zeilberger's Algorithm

The applicability or terminating condition for the ordinary case of Zeilberger's algorithm was recently obtained by Abramov. For the $q$-analogue, the question of whether a bivariate $q$-hypergeometric term has a $qZ$-pair remains open. Le has found a solution to this problem when the given bivariate $q$-hypergeometric term is a rational function in certain powers of $q$. We solve the problem for the general case by giving a characterization of bivariate $q$-hypergeometric terms for which the $q$-analogue of Zeilberger's algorithm terminates. Moreover, we give an algorithm to determine whether a bivariate $q$-hypergeometric term has a $qZ$-pair.Comment: 15 page

In an Attempt to Introduce Long-range Interactions into Small-world Networks

Distinguishing the long-range bonds with the regular ones, the critical temperature of the spin-lattice Guassian model built on two typical Small-world Networks (SWNs) is studied. The results show much difference from the classical case, and thus may induce some more accurate discussion on the critical properties of the spin-lattice systems combined with the SWNs.Comment: 4 pages, 3 figures, 18 referenc

The co-creation of social ventures through bricolage, for the displaced, by the displaced

Purpose Although scholars have investigated how social entrepreneurs create and develop social enterprises in the penurious stable environment, how they are created in the penurious unstable environment has yet been overlooked. The purpose of this paper is to address this research gap by exploring how internally displaced individuals, despite the lack of resources, create and develop a social enterprise to serve the other displaced population in the war and conflict zones. Design/methodology/approach Underpinned by a biographical research design, in-depth interviews with internally displaced individuals who have created social enterprises in the war and conflict zones were undertaken. Three social entrepreneurs were chosen for this study from three different social enterprises that are created by internally displaced individuals to serve the other internally displaced people of three different countries, namely, Pakistan, Afghanistan and Syria. Findings The single and cross-case analysis found that internally displaced individuals deploy bricolage strategy, for example, reconfiguration of pre-existing resources and competencies (both internal and external), to start up a social venture in the war and conflict zones. They utilise pre-existing internal resources, mainly human capital, and external resources, through a frugal approach towards resources acquisitions. The authors also found that the displaced social entrepreneurs utilise resources of other displaced individuals, for example, networks, volunteers, local knowledge and financial supports mainly from older arrivals, and develop their own enterprise ecosystem within the host location to co-create and co-develop social enterprise and social values for all of them. Research limitations/implications The findings show that internally displaced individuals utilise bricolage strategies to create and develop socially entrepreneurial venture to serve other internally displaced individuals in the war and conflict zones. As the findings are based on three case studies, for confirmatory approach, a quantitative study with a large sample size is necessary. Furthermore, as the differences in economic, cultural and linguistic in between the home and host locations can have impact on the creation and the development of a social venture, they should be considered in the future studies. Originality/value This study contributes to the limited literature and studies on social entrepreneurship, specifically, to the context of unstable penurious environment. It also contributes to the literature on bricolage by extending its application from penurious stable environment to the penurious unstable environment. By exploring what and how internal and external resources are utilised to create and develop a socially entrepreneurial venture in a war and conflict zones, this study has added value to the literature on not only bricolage but also entrepreneurship in war and conflict zones

Applicative Bidirectional Programming with Lenses

A bidirectional transformation is a pair of mappings between source and view data objects, one in each direction. When the view is modified, the source is updated accordingly with respect to some laws. One way to reduce the development and maintenance effort of bidirectional transformations is to have specialized languages in which the resulting programs are bidirectional by construction---giving rise to the paradigm of bidirectional programming. In this paper, we develop a framework for applicative-style and higher-order bidirectional programming, in which we can write bidirectional transformations as unidirectional programs in standard functional languages, opening up access to the bundle of language features previously only available to conventional unidirectional languages. Our framework essentially bridges two very different approaches of bidirectional programming, namely the lens framework and Voigtlander’s semantic bidirectionalization, creating a new programming style that is able to bag benefits from both

Entropy and Its Quantum Thermodynamical Implication for Anomalous Spectral Systems

The state function entropy and its quantum thermodynamical implication for two typical dissipative systems with anomalous spectral densities are studied by investigating on their low-temperature quantum behavior. In all cases it is found that the entropy decays quickly and vanishes as the temperature approaches zero. This reveals a good conformity with the third law of thermodynamics and provides another evidence for the validity of fundamental thermodynamical laws in the quantum dissipative region.Comment: 10 pages, 3 figure