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82
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 35 (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
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 14 (6 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.
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 12 (5 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.
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 11 (2 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.
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 11 (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.
Measures Of Simultaneous Approximation For QuasiPeriods Of Abelian Varieties
 J. Number Theory, 94 (2002), N
, 2002
"... this paper, the functions # i will be assumed to be normalized as above, i.e. so that all secondorder derivatives of # 0 vanish at 0. 3.4. Conclusion ..."
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Cited by 9 (0 self)
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this paper, the functions # i will be assumed to be normalized as above, i.e. so that all secondorder derivatives of # 0 vanish at 0. 3.4. Conclusion
Conjugate points in Euler’s elastic problem
 Journal of Dynamical and Control Systems (accepted), available at: arXiv:0705.1003
"... Abstract. For the classical Euler elastic problem, conjugate points are described. Inflexional elasticas admit the first conjugate point between the first and third inflexion points. All other elasticas do not have conjugate points. As a result, the problem of stability of Euler elasticas is solved. ..."
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Cited by 8 (4 self)
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Abstract. For the classical Euler elastic problem, conjugate points are described. Inflexional elasticas admit the first conjugate point between the first and third inflexion points. All other elasticas do not have conjugate points. As a result, the problem of stability of Euler elasticas is solved. 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 6 (1 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.
The Basic Attractor of the Viscous MooreGreitzer Equation
 J. Nonlinear Sci
, 1998
"... The basic attractor of nonlinear partial differential equations is the part of the global attractor that attracts a prevalent set in phase space. We give a qualitative description of the basic attractor of a model for axial compression systems and show that it consists of three types of components, ..."
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Cited by 5 (4 self)
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The basic attractor of nonlinear partial differential equations is the part of the global attractor that attracts a prevalent set in phase space. We give a qualitative description of the basic attractor of a model for axial compression systems and show that it consists of three types of components, uniform (design) flow, surge and stall. The basic attractor can contain more than one stall component. The existence of these components is proven and their stability explored. Numerical results are presented, showing the shape and evolution of stall cells over large parameter regions. 1 Introduction In recent years a lot of attention has been devoted to the study of air flow through turbomachines. The main reason for this interest is that when a turbomachine, such as a jet engine, operates close to its optimal operating parameter values, the flow can become unstable. These instabilities put a large stress on the engine and in some cases the engine needs to be turned off in order to recover...