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45
A Review of Kernel Methods in Machine Learning
, 2006
"... We review recent methods for learning with positive definite kernels. All these methods formulate learning and estimation problems as linear tasks in a reproducing kernel Hilbert space (RKHS) associated with a kernel. We cover a wide range of methods, ranging from simple classifiers to sophisticate ..."
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Cited by 35 (3 self)
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We review recent methods for learning with positive definite kernels. All these methods formulate learning and estimation problems as linear tasks in a reproducing kernel Hilbert space (RKHS) associated with a kernel. We cover a wide range of methods, ranging from simple classifiers to sophisticated methods for estimation with structured data.
Spectral gap and coercivity estimates for linearized Boltzmann collision operators without angular cutoff
 J. Math. Pures Appl
"... Abstract. In this paper we prove new constructive coercivity estimates for the Boltzmann collision operator without cutoff, that is for longrange interactions. In particular we give a generalized sufficient condition for the existence of a spectral gap which involves both the growth behavior of the ..."
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Cited by 16 (2 self)
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Abstract. In this paper we prove new constructive coercivity estimates for the Boltzmann collision operator without cutoff, that is for longrange interactions. In particular we give a generalized sufficient condition for the existence of a spectral gap which involves both the growth behavior of the collision kernel at large relative velocities and its singular behavior at grazing and frontal collisions. It provides in particular existence of a spectral gap and estimates on it for interactions deriving from the hard potentials φ(r) = r −(s−1) , s ≥ 5 or the socalled moderately soft potentials φ(r) = r −(s−1) , 3 < s < 5, (without angular cutoff). In particular this paper recovers (by constructive means), improves and extends previous results of Pao [46]. We also obtain constructive coercivity estimates for the Landau collision operator for the optimal coercivity norm pointed out in [34] and we formulate a conjecture about a unified necessary and sufficient condition for the existence of a spectral gap for Boltzmann and Landau linearized collision operators.
The linearized Boltzmann equation: Concise and accurate solutions to basic flow problems
 ZEITSCHRIFT FÜR ANGEWANDTE MATHEMATIK UND PHYSIK ZAMP
, 2003
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Localized boundarydomain integral formulations for problems with variable coefficients
"... Specially constructed localized parametrixes are used in this paper instead of a fundamental solution to reduce a Boundary Value Problem with variable coefficients to a Localized BoundaryDomain Integral or IntegroDifferential Equation (LBDIE or LBDIDE). After discretization, this results in a spar ..."
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Cited by 13 (9 self)
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Specially constructed localized parametrixes are used in this paper instead of a fundamental solution to reduce a Boundary Value Problem with variable coefficients to a Localized BoundaryDomain Integral or IntegroDifferential Equation (LBDIE or LBDIDE). After discretization, this results in a sparsely populated system of linear algebraic equations, which can be solved by wellknown efficient methods. This make the method competitive with the Finite Element Method for such problems. Some methods of the parametrix localization are discussed and the corresponding LBDIEs and LBDIDEs are introduced. Both meshbased and meshless algorithms for the localized equations discretization are described.
ON THE NUMERICAL EVALUATION OF FREDHOLM DETERMINANTS
, 804
"... Abstract. Some significant quantities in mathematics and physics are most naturally expressed as the Fredholm determinant of an integral operator, most notably many of the distribution functions in random matrix theory. Though their numerical values are of interest, there is no systematic numerical ..."
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Cited by 10 (5 self)
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Abstract. Some significant quantities in mathematics and physics are most naturally expressed as the Fredholm determinant of an integral operator, most notably many of the distribution functions in random matrix theory. Though their numerical values are of interest, there is no systematic numerical treatment of Fredholm determinants to be found in the literature. Instead, the few numerical evaluations that are available rely on eigenfunction expansions of the operator, if expressible in terms of special functions, or on alternative, numerically more straightforwardly accessible analytic expressions, e.g., in terms of Painlevé transcendents, that have masterfully been derived in some cases. In this paper we close the gap in the literature by studying projection methods and, above all, a simple, easily implementable, general method for the numerical evaluation of Fredholm determinants that is derived from the classical Nyström method for the solution of Fredholm equations of the second kind. Using Gauss–Legendre or Clenshaw– Curtis as the underlying quadrature rule, we prove that the approximation error essentially behaves like the quadrature error for the sections of the kernel. In particular, we get exponential convergence for analytic kernels, which are typical in random matrix theory. The application of the method to the distribution functions of the Gaussian unitary ensemble (GUE), in the bulk and the edge scaling limit, is discussed in detail. After extending the method to systems of integral operators, we evaluate the twopoint correlation functions of the more recently studied Airy and Airy 1 processes. Key words. Fredholm determinant, Nyström’s method, projection method, trace class operators, random
Hydrodynamic Limit Of The Stationary Boltzmann Equation In A Slab
, 1994
"... . We study the stationary solution of the Boltzmann equation in a slab with a constant external force parallel to the boundary and complete accommodation condition on the walls at a specified temperature. We prove that when the force is sufficiently small there exists a solution which converges, in ..."
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Cited by 10 (8 self)
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. We study the stationary solution of the Boltzmann equation in a slab with a constant external force parallel to the boundary and complete accommodation condition on the walls at a specified temperature. We prove that when the force is sufficiently small there exists a solution which converges, in the hydrodynamic limit, to a local Maxwellian with parameters given by the stationary solution of the corresponding compressible NavierStokes equations with noslip boundary conditions. Corrections to this Maxwellian are obtained in powers of the Knudsen number with a controlled remainder. 1. Introduction. In this paper we continue our study of the derivation of hydrodynamic equations from the Boltzmann equation (BE), a problem which goes back to Hilbert [?]. The BE is believed to accurately describe the time evolution of rarefied gases on a "kinetic"scale intermediate between the microscopic and macroscopic [?]. To go from the BE to the macroscopic (hydrodynamic) descriptions the locally c...
S.H.: Green’s function of Boltzmann equation, 3D waves
 Bull. Inst. Math. Acad. Sin. (N.S
, 2006
"... We study the Green’s function for the linearized Boltzmann equation. For the shorttime period, the Green’s function is dominated by the particlelike waves; and for largetime, by the fluidlike waves exhibiting the weak Huygens principle. The fluidlike waves are constructed by the spectral analysi ..."
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Cited by 6 (5 self)
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We study the Green’s function for the linearized Boltzmann equation. For the shorttime period, the Green’s function is dominated by the particlelike waves; and for largetime, by the fluidlike waves exhibiting the weak Huygens principle. The fluidlike waves are constructed by the spectral analysis and complex analytic techniques, making uses of the rotational symmetry of the equation in the space variables. The particlelike waves are constructed by a Picard iteration, making uses of the exchange of regularity in the microscopic velocity with the regularity in the space variables through a Mixture Lemma. We obtain the pointwise estimates in the space and time variables of the Green’s function through a longshort waves and particlewave decompositions.
Oscillatory climate modes in the Eastern Mediterranean: Synchronization with the NAO and Southern Oscillation
, 2009
"... Oscillatory climatic modes over the North Atlantic, Ethiopian Plateau, Eastern Mediterranean and the Tropical Pacific Ocean were examined in instrumental and proxy records from these regions. The teleconnections between the regions were studied in terms of synchronization of chaotic oscillators. We ..."
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Cited by 6 (2 self)
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Oscillatory climatic modes over the North Atlantic, Ethiopian Plateau, Eastern Mediterranean and the Tropical Pacific Ocean were examined in instrumental and proxy records from these regions. The teleconnections between the regions were studied in terms of synchronization of chaotic oscillators. We modify standard methods for studying synchronization among such oscillators by combining them with advanced spectral methods, including singular spectral analysis. This modification helps test, besides the degree of synchronization, also the strength of the coupling. A prominent oscillatory mode with a 7year period was found in all the climatic regions studied here and is completely synchronized with this mode over the North Atlantic. An energy analysis of the synchronization raises the possibility that this mode originates in fact in the North Atlantic. Evidence is discussed for this mode being induced by the 7–8year oscillation in the position of the Gulf Stream front. A mechanism for the teleconnections between the North Atlantic, Ethiopian Plateau and Eastern Mediterranean is proposed. An
Eigenvalues of elliptic boundary value problems with an indefinite weight function
 TRANS. AMER. MATH. SOC
, 1986
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1 QUANTITATIVE LINEARIZED STUDY OF THE BOLTZMANN COLLISION OPERATOR AND APPLICATIONS †
, 2006
"... Abstract. We present recent results [4, 28, 29] about the quantitative study of the linearized Boltzmann collision operator, and its application to the study of the trend to equilibrium for the spatially homogeneous Boltzmann equation for hard spheres. Key words. Boltzmann equation, spatially homoge ..."
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Cited by 2 (2 self)
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Abstract. We present recent results [4, 28, 29] about the quantitative study of the linearized Boltzmann collision operator, and its application to the study of the trend to equilibrium for the spatially homogeneous Boltzmann equation for hard spheres. Key words. Boltzmann equation, spatially homogeneous, linearized Boltzmann collision operator, spectrum, spectral gap, explicit, trend to equilibrium, rate of convergence. AMS subject classifications. 76P05 Rarefied gas flows, Boltzmann equation [See also 82B40, 82C40, 82D05]. 1.