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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?
Most tensor problems are NP hard
 CORR
, 2009
"... The idea that one might extend numerical linear algebra, the collection of matrix computational methods that form the workhorse of scientific and engineering computing, to numerical multilinear algebra, an analogous collection of tools involving hypermatrices/tensors, appears very promising and has ..."
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Cited by 44 (6 self)
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The idea that one might extend numerical linear algebra, the collection of matrix computational methods that form the workhorse of scientific and engineering computing, to numerical multilinear algebra, an analogous collection of tools involving hypermatrices/tensors, appears very promising and has attracted a lot of attention recently. We examine here the computational tractability of some core problems in numerical multilinear algebra. We show that tensor analogues of several standard problems that are readily computable in the matrix (i.e. 2tensor) case are NP hard. Our list here includes: determining the feasibility of a system of bilinear equations, determining an eigenvalue, a singular value, or the spectral norm of a 3tensor, determining a best rank1 approximation to a 3tensor, determining the rank of a 3tensor over R or C. Hence making tensor computations feasible is likely to be a challenge.
ILP: A Short Look Back and a Longer Look Forward
 Journal of Machine Learning Research
, 2003
"... Inductive logic programming (ILP) is built on a foundation laid by research in other areas of machine learning and computational logic. But in spite of this strong foundation, at just over 10 years of age ILP now faces a number of new challenges brought on by exciting areas of application. Research ..."
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Cited by 27 (0 self)
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Inductive logic programming (ILP) is built on a foundation laid by research in other areas of machine learning and computational logic. But in spite of this strong foundation, at just over 10 years of age ILP now faces a number of new challenges brought on by exciting areas of application. Research in other areas of machine learning and computational logic can contribute much to help ILP meet these challenges. After a brief review, the paper presents ve future research directions for ILP and points to initial approaches or results where they exist. It is hoped that the paper will motivate research workers in machine learning and computational logic to invest some time into ILP.
Approximability and proof complexity
, 2012
"... This work is concerned with the proofcomplexity of certifying that optimization problems do not have good solutions. Specifically we consider boundeddegree “Sum of Squares ” (SOS) proofs, a powerful algebraic proof system introduced in 1999 by Grigoriev and Vorobjov. Work of Shor, Lasserre, and Pa ..."
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Cited by 19 (6 self)
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This work is concerned with the proofcomplexity of certifying that optimization problems do not have good solutions. Specifically we consider boundeddegree “Sum of Squares ” (SOS) proofs, a powerful algebraic proof system introduced in 1999 by Grigoriev and Vorobjov. Work of Shor, Lasserre, and Parrilo shows that this proof system is automatizable using semidefinite programming (SDP), meaning that any nvariable degreed proof can be found in time n O(d). Furthermore, the SDP is dual to the wellknown Lasserre SDP hierarchy, meaning that the “d/2round Lasserre value ” of an optimization problem is equal to the best bound provable using a degreed SOS proof. These ideas were exploited in a recent paper by Barak et al. (STOC 2012) which shows that the known “hard instances ” for the UniqueGames problem are in fact solved close to optimally by a constant level of the Lasserre SDP hierarchy. We continue the study of the power of SOS proofs in the context of difficult optimization problems. In particular, we show that the BalancedSeparator integrality gap instances proposed by Devanur et al. can have their optimal value certified by a degree4 SOS proof. The key ingredient is an SOS proof of the KKL Theorem. We also investigate the extent to which the Khot–Vishnoi MaxCut integrality gap instances can have their optimum value certified by an SOS proof. We show they can be certified to within a factor.952 (>.878) using a constantdegree proof. These investigations also raise an interesting mathematical question: is there a constantdegree SOS proof of the Central Limit Theorem?
The Sources of Kolmogorov’s Grundbegriffe
, 2006
"... Andrei Kolmogorov’s Grundbegriffe Wahrscheinlichkeitsrechnung put probability’s modern mathematical formalism in place. It also provided a philosophy of probability—an explanation of how the formalism can be connected to the world of experience. In this article, we examine the sources of these two a ..."
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Cited by 15 (9 self)
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Andrei Kolmogorov’s Grundbegriffe Wahrscheinlichkeitsrechnung put probability’s modern mathematical formalism in place. It also provided a philosophy of probability—an explanation of how the formalism can be connected to the world of experience. In this article, we examine the sources of these two aspects of the Grundbegriffe—the work of the earlier scholars whose ideas Kolmogorov synthesized.
Small limit cycles bifurcating from fine focus points in cubic order Z2equivariant vector fields
, 2005
"... ..."
LMI approximations for cones of positive semidefinite forms
 Fachbereich Mathematik, Universität Konstanz, 78457
"... An interesting recent trend in optimization is the application of semidefinite programming techniques to new classes of optimization problems. In particular, this trend has been successful in showing that under suitable circumstances, polynomial optimization problems can be approximated via a sequen ..."
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Cited by 11 (4 self)
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An interesting recent trend in optimization is the application of semidefinite programming techniques to new classes of optimization problems. In particular, this trend has been successful in showing that under suitable circumstances, polynomial optimization problems can be approximated via a sequence of semidefinite programs. Similar ideas apply to conic optimization over the cone of copositive matrices, and to certain optimization problems involving random variables with some known moment information. We bring together several of these approximation results by studying the approximability of cones of positive semidefinite forms (homogeneous polynomials). Our approach enables us to extend the existing methodology to new approximation schemes. In particular, we derive a novel approximation to the cone of copositive forms, that is, the cone of forms that are positive semidefinite over the nonnegative orthant. The format of our construction can be extended to forms that are positive semidefinite over more general conic domains. We also construct polyhedral approximations to cones of positive semidefinite forms over a polyhedral domain. This opens the possibility of using linear programming technology in optimization problems over these cones.
Combinatorics of Boolean automata circuits dynamics, submitted to Discrete applied mathematics
, 2010
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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Cited by 9 (5 self)
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Phase portraits of planar vector fields: computer proofs
 J. Exp. Math
, 1995
"... This paper presents an algorithm for computer verification of the global structure of structurally stable planar vector fields. Constructing analytical proofs for the qualitative properties of phase portraits has been difficult. We try to avoid this barrier by augmenting numerical computations of tr ..."
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Cited by 6 (2 self)
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This paper presents an algorithm for computer verification of the global structure of structurally stable planar vector fields. Constructing analytical proofs for the qualitative properties of phase portraits has been difficult. We try to avoid this barrier by augmenting numerical computations of trajectories of dynamical systems with error estimates that yield rigorous proofs. Our approach is one that lends itself to high precision estimates, because the proofs are broken into independent calculations whose length in floating point operations does not increase with increasing precision. The algorithm that we present is tested on a system that arises in the study of Hopf bifurcation of periodic orbits with 1:4 resonance. 1
HILBERT’S 6TH PROBLEM: EXACT AND APPROXIMATE HYDRODYNAMIC MANIFOLDS FOR KINETIC EQUATIONS
"... Abstract. The problem of the derivation of hydrodynamics from the Boltzmann equation and related dissipative systems is formulated as the problem of slow invariant manifold in the space of distributions. We review a few instances where such hydrodynamic manifolds were found analytically both as the ..."
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Cited by 5 (2 self)
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Abstract. The problem of the derivation of hydrodynamics from the Boltzmann equation and related dissipative systems is formulated as the problem of slow invariant manifold in the space of distributions. We review a few instances where such hydrodynamic manifolds were found analytically both as the result of summation of the Chapman–Enskog asymptotic expansion and by the direct solution of the invariance equation. These model cases, comprising Grad’s moment systems, both linear and nonlinear, are studied in depth in order to gain understanding of what can be expected for the Boltzmann equation. Particularly, the dispersive dominance and saturation of dissipation rate of the exact hydrodynamics in the shortwave limit and the viscosity modification at high divergence of the flow velocity are indicated as severe obstacles to the resolution of Hilbert’s 6th Problem. Furthermore, we review the derivation of the approximate hydrodynamic manifold for the Boltzmann equation using Newton’s iteration and avoiding smallness parameters, and compare this to the exact solutions. Additionally, we discuss the problem of projection of the