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Graphical Condensation Generalizations Involving Pfaffians and Determinants  [PDF]
Eric Kuo
Mathematics , 2006,
Abstract: Graphical condensation is a technique used to prove combinatorial identities among numbers of perfect matchings of plane graphs. Propp and Kuo first applied this technique to prove identities for bipartite graphs. Yan, Yeh, and Zhang later applied graphical condensation to nonbipartite graphs to prove more complex identities. Here we generalize some of the identities of Yan, Yeh, and Zhang. We also describe the latest generalization of graphical condensation in which the number of perfect matchings of a plane graph is expressed as a Pfaffian or a determinant where the entries are also numbers of perfect matchings of subgraphs.
Discrete Monodromy, Pentagrams, and the Method of Condensation  [PDF]
Richard Evan Schwartz
Mathematics , 2007,
Abstract: This paper considers a simple geometric construction, called the Pentagram map. The pentagram map, performed on N-gons, gives rise to a birational mapping on the space of all N-gons. This paper finds what conjecturally are all the invariants for this map, and along the way relates the construction to the monodromy of 3rd order differential equations, and also to Dodgson's method of condensation for computing determinants.
Almost Product Evaluation of Hankel Determinants  [PDF]
Omer Egecioglu,Timothy Redmond,Charles Ryavec
Mathematics , 2007,
Abstract: An extensive literature exists describing various techniques for the evaluation of Hankel determinants. The prevailing methods such as Dodgson condensation, continued fraction expansion, LU decomposition, all produce product formulas when they are applicable. We mention the classic case of the Hankel determinants with binomial entries ${3k+2 \choose k}$ and those with entries ${3k \choose k}$; both of these classes of Hankel determinants have product form evaluations. The intermediate case, ${3k+1 \choose k}$ has not been evaluated. There is a good reason for this: these latter determinants do not have product form evaluations. In this paper we evaluate the Hankel determinant of ${3k+1 \choose k}$. The evaluation is a sum of a small number of products, an almost product. The method actually provides more, and as applications, we present the salient points for the evaluation of a number of other Hankel determinants with polynomial entries, along with product and almost product form evaluations at special points.
Heterogeneous Condensation  [PDF]
Sergey P. Fisenko,Manabu Shimada,Kikuo Okuyama
Physics , 2007,
Abstract: Vapor condensation on nanoparticle with radius smaller than the Kelvin radius is considered as fluctuation or as the heterogeneous nucleation. The expression for steady-state heterogeneous nucleation rate is obtained. Nucleation on negatively charged nanoparticles is discussed. The report was made at 17th International Nucleation and Atmospheric Conferences, Galway, Ireland, 2007.
Oddball determinants  [PDF]
J. S. Dowker
Physics , 1995,
Abstract: A simplified direct method is described for obtaining massless scalar functional determinants on the Euclidean ball. The case of odd dimensions is explicitly discussed.
Bose-Einstein condensation  [PDF]
V. I. Yukalov
Physics , 2005,
Abstract: The basic notions and the main historical facts on the Bose-Einstein condensation are surveyed.
TrivializingGeneralizations of some Izergin-Korepin-type Determinants  [cached]
Tewodros Amdeberhan,Doron Zeilberger
Discrete Mathematics & Theoretical Computer Science , 2007,
Abstract: We generalize (and hence trivialize and routinize) numerous explicit evaluations of determinants and pfaffians due to Kuperberg, as well as a determinant of Tsuchiya. The level of generality of our statements render their proofs easy and routine, by using Dodgson Condensation and/or Krattenthaler's factor exhaustion method.
Polariton Condensation and Lasing  [PDF]
David Snoke
Physics , 2012, DOI: 10.1007/978-3-642-24186-4_12
Abstract: The similarities and differences between polariton condensation in microcavities and standard lasing in a semiconductor cavity structure are reviewed. The recent experiments on "photon condensation" are also reviewed.
Quaternionic quasideterminants and determinants  [PDF]
Israel Gelfand,Vladimir Retakh,Robert Lee Wilson
Mathematics , 2002,
Abstract: We compute quaisideterminants and determinants of quaternionic matrices
Quantum determinants  [PDF]
Ulrich Meyer
Physics , 1994,
Abstract: We show how to construct central and grouplike quantum determinants for FRT algebras A(R). As an application of the general construction we give a quantum determinant for the q-Lorentz group.
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