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Plane Partition Polynomial Asymptotics  [PDF]
Robert Boyer,Daniel Parry
Mathematics , 2014,
Abstract: The plane partition polynomial $Q_n(x)$ is the polynomial of degree $n$ whose coefficients count the number of plane partitions of $n$ indexed by their trace. Extending classical work of E.M. Wright, we develop the asymptotics of these polynomials inside the unit disk using the circle method.
Geometry of the reduced quantum plane  [PDF]
R. Coquereaux,A. O. Garcia,R. Trinchero
Physics , 1998,
Abstract: We consider the space M of NxN matrices as a reduced quantum plane and discuss its geometry under the action and coaction of finite dimensional quantum groups (a quotient of U_q(SL(2)), q being an N-th root of unity, and its dual). We also introduce a differential calculus for M: a quotient of the Wess Zumino complex. We shall restrict ourselves to the case N odd and often choose the particular value N=3. The present paper (to appear in the proceedings of the conference "Quantum Groups and Fundamental Physical Applications", Palerme, December 1997) is essentially a short version of math-ph/9807012.
The Limiting Distribution of the Trace of a Random Plane Partition  [PDF]
Ljuben Mutafchiev,Emil Kamenov
Mathematics , 2004,
Abstract: We study the asymptotic behaviour of the trace (the sum of the diagonal parts) of a plane partition of the positive integer n, assuming that this parfition is chosen uniformly at random from the set of all such partitions.
Boxed Skew Plane Partition and Integrable Phase Model  [PDF]
Keiichi Shigechi,Masaru Uchiyama
Mathematics , 2005, DOI: 10.1088/0305-4470/38/48/003
Abstract: We study the relation between the boxed skew plane partition and the integrable phase model. We introduce a generalization of a scalar product of the phase model and calculate it in two ways; the first one in terms of the skew Schur functions, and another one by use of the commutation relations of operators. In both cases, a generalized scalar product is expressed as a determinant. We show that a special choice of the spectral parameters of a generalized scalar product gives the generating function of the boxed skew plane partition.
The shape of a typical boxed plane partition  [PDF]
Henry Cohn,Michael Larsen,James Propp
Mathematics , 1998,
Abstract: Using a calculus of variations approach, we determine the shape of a typical plane partition in a large box (i.e., a plane partition chosen at random according to the uniform distribution on all plane partitions whose solid Young diagrams fit inside the box). Equivalently, we describe the distribution of the three different orientations of lozenges in a random lozenge tiling of a large hexagon. We prove a generalization of the classical formula of MacMahon for the number of plane partitions in a box; for each of the possible ways in which the tilings of a region can behave when restricted to certain lines, our formula tells the number of tilings that behave in that way. When we take a suitable limit, this formula gives us a functional which we must maximize to determine the asymptotic behavior of a plane partition in a box. Once the variational problem has been set up, we analyze it using a modification of the methods employed by Logan and Shepp and by Vershik and Kerov in their studies of random Young tableaux.
A periodic approach to plane partition congruences  [PDF]
Matthew S. Mizuhara,James A. Sellers,Holly Swisher
Mathematics , 2015,
Abstract: Ramanujan's celebrated congruences of the partition function $p(n)$ have inspired a vast amount of results on various partition functions. Kwong's work on periodicity of rational polynomial functions yields a general theorem used to establish congruences for restricted plane partitions. This theorem provides a novel proof of several classical congruences and establishes two new congruences. We additionally prove several new congruences which do not fit the scope of the theorem, using only elementary techniques, or a relationship to existing multipartition congruences.
One Loop Partition Function in Plane Waves R-R Background  [PDF]
Amine B. Hammou
Physics , 2002, DOI: 10.1088/1126-6708/2002/11/028
Abstract: We compute the one loop partition function of type IIB string in plane wave R-R 5-form background $F^5$ using both path integral and operator formalisms and show that the two results agree perfectly. The result turns out to be equal to the partition function in the flat background. We also study the Tadpole cancellation for the unoriented closed and open string model in plane wave R-R 5-form background studied in hep-th/0203249 and find that the cancellation of the Tadpole requires the gauge group to be SO(8).
Mathematical aspects of decentralized control of formations in the plane  [PDF]
M. -A. Belabbas
Mathematics , 2011,
Abstract: In formation control, an ensemble of autonomous agents is required to stabilize at a given configuration in the plane, doing so while agents are allowed to observe only a subset of the ensemble. As such, formation control provides a rich class of problems for decentralized control methods and techniques. Additionally, it can be used to model a wide variety of scenarios where decentralization is a main characteristic. We introduce here some mathematical background necessary to address questions of stability in decentralized control in general and formation control in particular. This background includes an extension of the notion of global stability to systems evolving on manifolds and a notion of robustness of feedback control for nonlinear systems. We then formally introduce the class of formation control problems, and summarize known results.
The mathematical theory of reduced MHD models for fusion plasmas  [PDF]
Hervé Guillard
Physics , 2015,
Abstract: The derivation of reduced MHD models for fusion plasma is here formulated as a special instance of the general theory of singular limit of hyperbolic system of PDEs with large operator. This formulation allows to use the general results of this theory and to prove rigorously that reduced MHD models are valid approximations of the full MHD equations. In particular, it is proven that the solutions of the full MHD system converge to the solutions of an appropriate reduced model.
On some mathematical identities resulting from evaluation of the partition function for an electron moving in a periodic lattice  [PDF]
Jakub J?drak
Physics , 2007,
Abstract: We consider a simple model of the dynamics of a single electron in a crystal lattice. Although this is a standard problem in condensed matter physics, alternative ways of evaluating a partition function for such a system lead to equalities, that may be interesting from the point of view of mathematical analysis, combinatorics and graph theory.
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