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166
A Rigorous ODE Solver and Smale’s 14th Problem
 Found. Comp. Math
"... Abstract. We present an algorithm for computing rigorous solutions to a large class of ordinary differential equations. The main algorithm is based on a partitioning process and the use of interval arithmetic with directed rounding. As an application, we prove that the Lorenz equations support a str ..."
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Cited by 104 (10 self)
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Abstract. We present an algorithm for computing rigorous solutions to a large class of ordinary differential equations. The main algorithm is based on a partitioning process and the use of interval arithmetic with directed rounding. As an application, we prove that the Lorenz equations support a strange attractor, as conjectured by Edward Lorenz in 1963. This conjecture was recently listed by Steven Smale as one of several challenging problems for the twentyfirst century. We also prove that the attractor is robust, i.e., it persists under small perturbations of the coefficients in the underlying differential equations. Furthermore, the flow of the equations admits a unique SRB measure, whose support coincides with the attractor. The proof is based on a combination of normal form theory and rigorous computations. 1.
Discretizing manifolds via minimum energy points
 NOTICES OF THE AMS
, 2004
"... There are a variety of needs for the discretization of a manifold—statistical sampling, quadrature rules, starting points for Newton’s method, computeraided design, interpolation schemes, finite element tessellations—to name but a few. So let us assume we are given a ddimensional manifold A in the ..."
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Cited by 67 (13 self)
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There are a variety of needs for the discretization of a manifold—statistical sampling, quadrature rules, starting points for Newton’s method, computeraided design, interpolation schemes, finite element tessellations—to name but a few. So let us assume we are given a ddimensional manifold A in the Euclidean space R d′ and wish to determine, say, 5000 points that “represent A”. How can we go about this if A is described by some geometric property or by some parametrization of the unit cube U d: = [0, 1] d in R d? Naturally, we must be guided by the particular application in mind. For an historical perspective as well as a brief motivational journey, let’s look at the simple case when A is the interval [−1, 1] ⊂ R. One obvious choice for N points that discretize A is the set of equally spaced points xk,N = −1 +
HILBERT’S 16TH PROBLEM AND BIFURCATIONS OF PLANAR POLYNOMIAL VECTOR FIELDS
, 2003
"... The original Hilbert’s 16th problem can be split into four parts consisting of Problems A{D. In this paper, the progress of study on Hilbert’s 16th problem is presented, and the relationship between Hilbert’s 16th problem and bifurcations of planar vector elds is discussed. The material is presented ..."
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Cited by 46 (6 self)
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The original Hilbert’s 16th problem can be split into four parts consisting of Problems A{D. In this paper, the progress of study on Hilbert’s 16th problem is presented, and the relationship between Hilbert’s 16th problem and bifurcations of planar vector elds is discussed. The material is presented in eight sections. Section 1: Introduction: what is Hilbert’s 16th problem?
A counterexample to the Hirsch conjecture
, 2011
"... The Hirsch Conjecture (1957) stated that the graph of a ddimensional polytope with n facets cannot have (combinatorial) diameter greater than n − d. That is, that any two vertices of the polytope can be connected by a path of at most n − d edges. This paper presents the first counterexample to the ..."
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Cited by 41 (3 self)
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The Hirsch Conjecture (1957) stated that the graph of a ddimensional polytope with n facets cannot have (combinatorial) diameter greater than n − d. That is, that any two vertices of the polytope can be connected by a path of at most n − d edges. This paper presents the first counterexample to the conjecture. Our polytope has dimension 43 and 86 facets. It is obtained from a 5dimensional polytope with 48 facets which violates a certain generalization of the dstep conjecture of Klee and Walkup.
Relative equilibria of the fourbody problem
 Erg. Th. & Dyn. Sys
, 1985
"... Abstract. We show that the number of relative equilibria of the Newtonian fourbody problem is finite, up to symmetry. In fact, we show that this number is always between 32 and 8472. The proof is based on symbolic and exact integer computations which are carried out by computer. 1. ..."
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Cited by 33 (3 self)
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Abstract. We show that the number of relative equilibria of the Newtonian fourbody problem is finite, up to symmetry. In fact, we show that this number is always between 32 and 8472. The proof is based on symbolic and exact integer computations which are carried out by computer. 1.
THE nBODY PROBLEM IN SPACES OF CONSTANT CURVATURE
, 2008
"... Abstract. We generalize the Newtonian nbody problem to spaces of curvature κ = constant, and study the motion in the 2dimensional case. For κ> 0, the equations of motion encounter noncollision singularities, which occur when two bodies are antipodal. These singularities of the equations are re ..."
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Cited by 28 (14 self)
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Abstract. We generalize the Newtonian nbody problem to spaces of curvature κ = constant, and study the motion in the 2dimensional case. For κ> 0, the equations of motion encounter noncollision singularities, which occur when two bodies are antipodal. These singularities of the equations are responsible for the existence of some hybrid solution singularities that end up in finite time in a collisionantipodal configuration. We also point out the existence of several classes of relative equilibria, including those generated by hyperbolic rotations for κ < 0. In the end, we prove Saari’s conjecture when the bodies are on a geodesic that rotates circularly or hyperbolically. Our approach also shows that each of the spaces κ < 0, κ = 0, and κ> 0 is characterized by certain orbits, which don’t occur in the other cases, a fact that might us help determine the nature of the physical space. 1 2 F. Diacu, E. PérezChavela, and M. Santoprete Contents
Cauchy type integrals of algebraic functions
 Israel J. Math
"... Abstract We consider Cauchy type integrals I(t) = 1 2πi γ g(z)dz z−t with g(z) an algebraic function. The main goal is to give constructive (at least, in principle) conditions for I(t) to be an algebraic function, a rational function, and ultimately an identical zero near infinity. This is done by ..."
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Cited by 20 (11 self)
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Abstract We consider Cauchy type integrals I(t) = 1 2πi γ g(z)dz z−t with g(z) an algebraic function. The main goal is to give constructive (at least, in principle) conditions for I(t) to be an algebraic function, a rational function, and ultimately an identical zero near infinity. This is done by relating the Monodromy group of the algebraic function g, the geometry of the integration curve γ, and the analytic properties of the Cauchy type integrals. The motivation for the study of these conditions is provided by the fact that certain Cauchy type integrals of algebraic functions appear in the infinitesimal versions of two classical open questions in Analytic Theory of Differential Equations: the Poincaré CenterFocus problem and the second part of the Hilbert 16th problem. 1.
The P versus NP problem
 Clay Mathematical Institute; The Millennium Prize Problem
, 2000
"... The P versus NP problem is to determine whether every language accepted by some nondeterministic algorithm in polynomial time is also accepted by some (deterministic) algorithm in polynomial time. To define the problem precisely it is necessary to give a formal model of a computer. The standard comp ..."
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Cited by 18 (0 self)
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The P versus NP problem is to determine whether every language accepted by some nondeterministic algorithm in polynomial time is also accepted by some (deterministic) algorithm in polynomial time. To define the problem precisely it is necessary to give a formal model of a computer. The standard computer model in computability theory is the Turing machine, introduced by Alan Turing in 1936 [37]. Although the model was introduced before physical computers were built, it nevertheless continues to be accepted as the proper computer model for the purpose of defining the notion of computable function. Informally the class P is the class of decision problems solvable by some algorithm within a number of steps bounded by some fixed polynomial in the length of the input. Turing was not concerned with the efficiency of his machines, rather his concern was whether they can simulate arbitrary algorithms given sufficient time. It turns out, however, Turing machines can generally simulate more efficient computer models (for example, machines equipped with many tapes or an unbounded random access memory) by at most squaring or cubing the computation time. Thus P is a
Asymptotics for Discrete Weighted Minimal Riesz Energy Problems on Rectifiable Sets
 Trans. of the Amer. Math. Soc
"... Abstract. Given a closed drectifiable set A embedded in Euclidean space, we investigate minimal weighted Riesz energy points on A; that is, N points constrained to A and interacting via the weighted power law potential V = w(x, y) x − y  −s, where s> 0 is a fixed parameter and w is an admissib ..."
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Cited by 18 (10 self)
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Abstract. Given a closed drectifiable set A embedded in Euclidean space, we investigate minimal weighted Riesz energy points on A; that is, N points constrained to A and interacting via the weighted power law potential V = w(x, y) x − y  −s, where s> 0 is a fixed parameter and w is an admissible weight. (In the unweighted case (w ≡ 1) such points for N fixed tend to the solution of the bestpacking problem on A as the parameter s → ∞.) Our main results concern the asymptotic behavior as N → ∞ of the minimal energies as well as the corresponding equilibrium configurations. Given a distribution ρ(x) with respect to ddimensional Hausdorff measure on A, our results provide a method for generating Npoint configurations on A that are “wellseparated ” and have asymptotic distribution ρ(x) as N → ∞. 1. Introduction. Points on a compact set A that minimize certain energy functions often have desirable properties that reflect special features of A. For A = S 2, the
The center problem for the Abel equations, compositions of functions and moment conditions, Mosc
 Ivanov Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia Evgenii P. Volokitin Sobolev Institute of Mathematics
"... Abstract. An Abel differential equation y ′ = p(x)y2 + q(x)y3 is said to have a center at a pair of complex numbers (a, b) if y(a) = y(b) for every solution y(x) with the initial value y(a) small enough. This notion is closely related to the classical centerfocus problem for plane vector fields. ..."
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Abstract. An Abel differential equation y ′ = p(x)y2 + q(x)y3 is said to have a center at a pair of complex numbers (a, b) if y(a) = y(b) for every solution y(x) with the initial value y(a) small enough. This notion is closely related to the classical centerfocus problem for plane vector fields. Recently, conditions for the Abel equation to have a center have been related to the composition factorization of P = p and Q = q on the one hand and to vanishing conditions for the moments mi,j =∫ P iQjq on the other hand. We give a detailed review of the recent results in each of these directions.