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160
An adaptive multielement generalized polynomial chaos method for stochastic differential equations
 J. COMPUT. PHYS
, 2005
"... We formulate a MultiElement generalized Polynomial Chaos (MEgPC) method to deal with longterm integration and discontinuities in stochastic differential equations. We first present this method for Legendrechaos corresponding to uniform random inputs, and subsequently we generalize it to other ra ..."
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Cited by 79 (11 self)
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We formulate a MultiElement generalized Polynomial Chaos (MEgPC) method to deal with longterm integration and discontinuities in stochastic differential equations. We first present this method for Legendrechaos corresponding to uniform random inputs, and subsequently we generalize it to other random inputs. The main idea of MEgPC is to decompose the space of random inputs when the relative error in variance becomes greater than a threshold value. In each subdomain or random element, we then employ a generalized Polynomial Chaos expansion. We develop a criterion to perform such a decomposition adaptively, and demonstrate its effectiveness for ODEs, including the KraichnanOrszag threemode problem, as well as advectiondiffusion problems. The new method is similar to spectral element method for deterministic problems but with hp discretization of the random space.
New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function
, 1996
"... Dedicated to the memory of GianCarlo Rota who encouraged me to write this paper in the present style Abstract. In this paper we derive many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi’s 4 and 8 squares identities to 4n ..."
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Cited by 45 (1 self)
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Dedicated to the memory of GianCarlo Rota who encouraged me to write this paper in the present style Abstract. In this paper we derive many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi’s 4 and 8 squares identities to 4n 2 or 4n(n + 1) squares, respectively, without using cusp forms. In fact, we similarly generalize to infinite families all of Jacobi’s explicitly stated degree 2, 4, 6, 8 Lambert series expansions of classical theta functions. In addition, we extend Jacobi’s special analysis of 2 squares, 2 triangles, 6 squares, 6 triangles to 12 squares, 12 triangles, 20 squares, 20 triangles, respectively. Our 24 squares identity leads to a different formula for Ramanujan’s tau function τ(n), when n is odd. These results, depending on new expansions for powers of various products of classical theta functions, arise in the setting of Jacobi elliptic functions, associated continued fractions, regular Cfractions, Hankel or Turánian determinants, Fourier series, Lambert series, inclusion/exclusion, Laplace expansion formula for determinants, and Schur functions. The Schur function form of these infinite families of identities are analogous to the ηfunction identities of Macdonald. Moreover, the powers 4n(n + 1), 2n 2 + n, 2n 2 − n that appear in Macdonald’s work also arise at appropriate places in our analysis. A special case of our general methods yields a proof of the two Kac–Wakimoto conjectured identities involving representing
Relation between Kinematic Boundaries, Stirring, and Barriers for the Antarctic Polar Vortex
 J. Atmos. Sci
"... Maximum stretching lines in the lower stratosphere around the Antarctic polar vortex are diagnosed using a method based on finitesize Lyapunov exponents. By analogy with the mathematical results known for simple dynamical systems, these curves are identified as stable and unstable manifolds of the ..."
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Cited by 33 (0 self)
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Maximum stretching lines in the lower stratosphere around the Antarctic polar vortex are diagnosed using a method based on finitesize Lyapunov exponents. By analogy with the mathematical results known for simple dynamical systems, these curves are identified as stable and unstable manifolds of the underlying hyperbolic structure of the flow. For the first time, the exchange mechanism associated with lobe dynamics is characterized using atmospheric analyzed winds. The tangling manifolds form a stochastic layer around the vortex. It is found that fluid is not only expelled from this layer toward the surf zone but also is injected inward from the surf zone, through a process similar to the turnstile mechanism in lobe dynamics. The vortex edge, defined as the location of the maximum gradient in potential vorticity or tracer, is found to be the southward (poleward) envelope of this stochastic layer. Exchanges with the inside of the vortex are therefore largely decoupled from those, possibly intense, exchanges between the stochastic layer and the surf zone. It is stressed that using the kinematic boundary defined by the hyperbolic points and the manifolds as an operational definition of vortex boundary is not only unpractical but also leads to spurious estimates of exchanges. The authors anticipate that more accurate dynamical systems tools are needed to analyze stratospheric transport in terms of lobe dynamics. 1.
Numerical solution of the small dispersion limit of Kortewegde Vries and Witham equations
 Comm. Pure Appl. Math
"... Abstract. The Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion of order ǫ2, ǫ ≪ 1, is characterized by the appearance of a zone of rapid modulated oscillations of wavelength of order ǫ. These oscillations are approximately described by the elliptic solution of KdV where ..."
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Cited by 32 (12 self)
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Abstract. The Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion of order ǫ2, ǫ ≪ 1, is characterized by the appearance of a zone of rapid modulated oscillations of wavelength of order ǫ. These oscillations are approximately described by the elliptic solution of KdV where the amplitude, wavenumber and frequency are not constant but evolve according to the Whitham equations. In this manuscript we give a quantitative analysis of the discrepancy between the numerical solution of the KdV equation in the small dispersion limit and the corresponding approximate solution for values of ǫ between 10−1 and 10−3. The numerical results are compatible with a difference of order ǫ within the ‘interior ’ of the Whitham oscillatory zone, of order ǫ 1 3 at the left boundary outside the Whitham zone and of order √ ǫ at the right boundary outside the Whitham zone. 1.
Special Lagrangian cones
"... Abstract. We study homogeneous special Lagrangian cones in C n with isolated singularities. Our main result constructs an infinite family of special Lagrangian cones in C 3 each of which has a toroidal link. We obtain a detailed geometric description of these tori. We prove a regularity result for s ..."
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Cited by 28 (2 self)
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Abstract. We study homogeneous special Lagrangian cones in C n with isolated singularities. Our main result constructs an infinite family of special Lagrangian cones in C 3 each of which has a toroidal link. We obtain a detailed geometric description of these tori. We prove a regularity result for special Lagrangian cones in C 3 with a spherical link – any such cone must be a plane. We also construct a oneparameter family of asymptotically conical special Lagrangian submanifolds from any special Lagrangian cone. 1.
Lagrangian surfaces in the complex Euclidean plane with conformal Maslov form
 565–582, MR 1245723 (94j:53064), Zbl 0792.53050
, 1993
"... Abstract. We present a method to construct a large family of Lagrangian surfaces in complex Euclidean plane C 2 by using Legendre curves in the 3sphere and in the anti de Sitter 3space or, equivalently, by using spherical and hyperbolic curves, respectively. Among this family, we characterize mini ..."
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Cited by 26 (10 self)
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Abstract. We present a method to construct a large family of Lagrangian surfaces in complex Euclidean plane C 2 by using Legendre curves in the 3sphere and in the anti de Sitter 3space or, equivalently, by using spherical and hyperbolic curves, respectively. Among this family, we characterize minimal, constant mean curvature, Hamiltonianminimal and Willmore surfaces in terms of simple properties of the curvature of the generating curves. As applications, we provide explicitly conformal parametrizations of known and new examples of minimal, constant mean curvature, Hamiltonianminimal and Willmore surfaces in C 2. 1.
strata in Euler’s elastic problem
 Journal of Dynamical and Control Systems
, 2007
"... Abstract. The classical Euler problem on stationary configurations of elastic rod in the plane is studied in detail by geometric control techniques as a leftinvariant optimal control problem on the group of motions of a twodimensional plane E(2). The attainable set is described, the existence and ..."
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Cited by 20 (3 self)
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Abstract. The classical Euler problem on stationary configurations of elastic rod in the plane is studied in detail by geometric control techniques as a leftinvariant optimal control problem on the group of motions of a twodimensional plane E(2). The attainable set is described, the existence and boundedness of optimal controls are proved. Extremals are parametrized by the Jacobi elliptic functions of natural coordinates induced by the flow of the mathematical pendulum on fibers of the cotangent bundle of E(2). The group of discrete symmetries of the Euler problem generated by reflections in the phase space of the pendulum is studied. The corresponding Maxwell points are completely described via the study of fixed points of this group. As a consequence, an upper bound on cut points in the Euler problem is obtained. 1.
Special Lagrangian cones with higher genus links, Preprint: math.DG/0512178
"... Abstract. For every odd natural number g = 2d+1 we prove the existence of a countably infinite family of special Lagrangian cones in C 3 over a closed Riemann surface of genus g, using a geometric PDE gluing method. 1. ..."
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Cited by 19 (5 self)
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Abstract. For every odd natural number g = 2d+1 we prove the existence of a countably infinite family of special Lagrangian cones in C 3 over a closed Riemann surface of genus g, using a geometric PDE gluing method. 1.
Determinantal processes with number variance saturation
 Comm. Math. Phys
, 2004
"... Abstract. Consider Dyson’s Hermitian Brownian motion model after a finite time S, where the process is started at N equidistant points on the real line. These N points after time S form a determinantal process and has a limit as N → ∞. This limting determinantal proceess has the interesting feature ..."
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Cited by 18 (0 self)
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Abstract. Consider Dyson’s Hermitian Brownian motion model after a finite time S, where the process is started at N equidistant points on the real line. These N points after time S form a determinantal process and has a limit as N → ∞. This limting determinantal proceess has the interesting feature that it shows number variance saturation. The variance of the number of particles in an interval converges to a limiting value as the length of the interval goes to infinity. Number variance saturation is also seen for example in the zeros of the Riemann ζfunction, [20], [2]. The process can also be constructed using nonintersecting paths and we consider several variants of this construction. One construction leads to a model which shows a transition from a nonuniversal behaviour with number variance saturation to a universal sinekernel behaviour as we go up the line. 1.