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Articles 1 - 10 of 10
Full-Text Articles in Physical Sciences and Mathematics
An Excursion From Enumerative Geometry To Solving Systems Of Polynomial Equations With Macaulay 2, Frank Sottile
An Excursion From Enumerative Geometry To Solving Systems Of Polynomial Equations With Macaulay 2, Frank Sottile
Mathematics and Statistics Department Faculty Publication Series
Solving a system of polynomial equations is a ubiquitous problem in the applications of mathematics. Until recently, it has been hopeless to find explicit solutions to such systems, and mathematics has instead developed deep and powerful theories about the solutions to polynomial equations. Enumerative Geometry is concerned with counting the number of solutions when the polynomials come from a geometric situation and Intersection Theory gives methods to accomplish the enumeration. We use Macaulay 2 to investigate some problems from enumerative geometry, illustrating some applications of symbolic computation to this important problem of solving systems of polynomial equations. Besides enumerating solutions …
Large Deviation Principles And Complete Equivalence And Nonequivalence Results For Pure And Mixed Ensembles, Rs Ellis, K Haven, B Turkington
Large Deviation Principles And Complete Equivalence And Nonequivalence Results For Pure And Mixed Ensembles, Rs Ellis, K Haven, B Turkington
Mathematics and Statistics Department Faculty Publication Series
We consider a general class of statistical mechanical models of coherent structures in turbulence, which includes models of two-dimensional fluid motion, quasi-geostrophic flows, and dispersive waves. First, large deviation principles are proved for the canonical ensemble and the microcanonical ensemble. For each ensemble the set of equilibrium macrostates is defined as the set on which the corresponding rate function attains its minimum of 0. We then present complete equivalence and nonequivalence results at the level of equilibrium macrostates for the two ensembles. Microcanonical equilibrium macrostates are characterized as the solutions of a certain constrained minimization problem, while canonical equilibrium macrostates …
Statistical Mechanics Of A Discrete Nonlinear System, K Rasmussen, T Cretegny, Pg Kevrekidis
Statistical Mechanics Of A Discrete Nonlinear System, K Rasmussen, T Cretegny, Pg Kevrekidis
Mathematics and Statistics Department Faculty Publication Series
Statistical mechanics of the discrete nonlinear Schrödinger equation is studied by means of analytical and numerical techniques. The lower bound of the Hamiltonian permits the construction of standard Gibbsian equilibrium measures for positive temperatures. Beyond the line of T = ∞, we identify a phase transition through a discontinuity in the partition function. The phase transition is demonstrated to manifest itself in the creation of breatherlike localized excitations. Interrelation between the statistical mechanics and the nonlinear dynamics of the system is explored numerically in both regimes.
Some Real And Unreal Enumerative Geometry For Flag Manifolds, Frank Sottile
Some Real And Unreal Enumerative Geometry For Flag Manifolds, Frank Sottile
Mathematics and Statistics Department Faculty Publication Series
We present a general method for constructing real solutions to some problems in enumerative geometry which gives lower bounds on the maximum number of real solutions. We apply this method to show that two new classes of enumerative geometric problems on flag manifolds may have all their solutions be real and modify this method to show that another class may have no real solutions, which is a new phenomenon. This method originated in a numerical homotopy continuation algorithm adapted to the special Schubert calculus on Grassmannians and in principle gives optimal numerical homotopy algorithms for finding explicit solutions to these …
An Excursion From Enumerative Geometry To Solving Systems Of Polynomial Equations With Macaulay 2, Frank Sottile
An Excursion From Enumerative Geometry To Solving Systems Of Polynomial Equations With Macaulay 2, Frank Sottile
Mathematics and Statistics Department Faculty Publication Series
Solving a system of polynomial equations is a ubiquitous problem in the applications of mathematics. Until recently, it has been hopeless to find explicit solutions to such systems, and mathematics has instead developed deep and powerful theories about the solutions to polynomial equations. Enumerative Geometry is concerned with counting the number of solutions when the polynomials come from a geometric situation and Intersection Theory gives methods to accomplish the enumeration. We use Macaulay 2 to investigate some problems from enumerative geometry, illustrating some applications of symbolic computation to this important problem of solving systems of polynomial equations. Besides enumerating solutions …
Four Entries For Kluwer Encyclopaedia Of Mathematics, Frank Sottile
Four Entries For Kluwer Encyclopaedia Of Mathematics, Frank Sottile
Mathematics and Statistics Department Faculty Publication Series
No abstract provided.
Perverse Sheaves On Affine Grassmannians And Langlands Duality, I Mirkovic, K Vilonen
Perverse Sheaves On Affine Grassmannians And Langlands Duality, I Mirkovic, K Vilonen
Mathematics and Statistics Department Faculty Publication Series
No abstract provided.
Boundedness Of Bilinear Operators With Nonsmooth Symbols, John Gilbert, Andrea Nahmod
Boundedness Of Bilinear Operators With Nonsmooth Symbols, John Gilbert, Andrea Nahmod
Mathematics and Statistics Department Faculty Publication Series
We announce the Lp-boundedness of general bilinear operators associated to a symbol or multiplier which need not be smooth. We establish a general result for multipliers that are allowed to have singularities along the edges of a cone as well as possibly at its vertex. It thus unifies ealier results of CoifmanMeyer for smooth multipliers and ones, such the Bilinear Hilbert transform of Lacey-Thiele, where the multiplier is not smooth.
Parametric Quantum Resonances For Bose–Einstein Condensates, Pg Kevrekidis
Parametric Quantum Resonances For Bose–Einstein Condensates, Pg Kevrekidis
Mathematics and Statistics Department Faculty Publication Series
We generalize recent work on parametric resonances for nonlinear Schrödinger (NLS) type equations to the case of three dimensional Bose–Einstein condensates at zero temperatures. We show the possibility of such resonances in the three-dimensional case, using a moment method and numerical simulations.
Noncommutative Pieri Operators On Posets, Nantel Bergeron, Stefan Mykytuik, Frank Sottile, Stephanie Van Willigenburg
Noncommutative Pieri Operators On Posets, Nantel Bergeron, Stefan Mykytuik, Frank Sottile, Stephanie Van Willigenburg
Mathematics and Statistics Department Faculty Publication Series
We consider graded representations of the algebra NC of noncommutative symmetric functions on the Z-linear span of a graded poset P. The matrix coefficients of such a representation give a Hopf morphism from a Hopf algebra HP generated by the intervals of P to the Hopf algebra of quasi-symmetric functions. This provides a unified construction of quasi-symmetric generating functions from different branches of algebraic combinatorics, and this construction is useful for transferring techniques and ideas between these branches. In particular we show that the (Hopf) algebra of Billera and Liu related to Eulerian posets is dual to the peak (Hopf) …