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600
New results in linear filtering and prediction theory
 Trans. ASME, Ser. D, J. Basic Eng
, 1961
"... A nonlinear differential equation of the Riccati type is derived for the covariance matrix of the optimal filtering error. The solution of this "variance equation " completely specifies the optimal filter for either finite or infinite smoothing intervals and stationary or nonstationary statistics. T ..."
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Cited by 322 (0 self)
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A nonlinear differential equation of the Riccati type is derived for the covariance matrix of the optimal filtering error. The solution of this "variance equation " completely specifies the optimal filter for either finite or infinite smoothing intervals and stationary or nonstationary statistics. The variance equation is closely related to the Hamiltonian (canonical) differential equations of the calculus of variations. Analytic solutions are available in some cases. The significance of the variance equation is illustrated by examples which duplicate, simplify, or extend earlier results in this field. The Duality Principle relating stochastic estimation and deterministic control problems plays an important role in the proof of theoretical results. In several examples, the estimation problem and its dual are discussed sidebyside. Properties of the variance equation are of great interest in the theory of adaptive systems. Some aspects of this are considered briefly. 1
Optimal paths for a car that goes both forwards and backwards
 Pacific Journal of Mathematics
, 1990
"... The path taken by a car with a given minimum turning radius has a lower bound on its radius of curvature at each point, but the path has cusps if the car shifts into or out of reverse gear. What is the shortest such path a car can travel between two points if its starting and ending directions are s ..."
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Cited by 190 (0 self)
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The path taken by a car with a given minimum turning radius has a lower bound on its radius of curvature at each point, but the path has cusps if the car shifts into or out of reverse gear. What is the shortest such path a car can travel between two points if its starting and ending directions are specified? One need consider only paths with at most 2 cusps or reversals. We give a set of paths which is sufficient in the sense that it always contains a shortest path and small in the sense that there are at most 68, but usually many fewer paths in the set for any pair of endpoints and directions. We give these paths by explicit formula. Calculating the length of each of these paths and selecting the (not necessarily unique) path with smallest length yields a simple algorithm for a shortest path in each case. These optimal paths or geodesies may be described as follows: If C is an arc of a circle of the minimal turning radius and S is a line segment, then it is sufficient to consider only certain paths of the form CCSCC where arcs and segments fit smoothly, one or more of the arcs or segments may vanish, and where reversals, or equivalently cusps, between arcs or segments are allowed. This contrasts with the case when cusps are not allowed, where Dubins (1957) has shown that paths of the form CCC and CSC suffice. 1. Introduction. We
Pricing the risks of default
 Review of Derivatives Research
, 1998
"... the problems and opportunities facing the financial services industry in its search for competitive excellence. The Center's research focuses on the issues related to managing risk at the firm level as well as ways to improve productivity and performance. The Center fosters the development of a comm ..."
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Cited by 120 (6 self)
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the problems and opportunities facing the financial services industry in its search for competitive excellence. The Center's research focuses on the issues related to managing risk at the firm level as well as ways to improve productivity and performance. The Center fosters the development of a community of faculty, visiting scholars and Ph.D. candidates whose research interests complement and support the mission of the Center. The Center works closely with industry executives and practitioners to ensure that its research is informed by the operating realities and competitive demands facing industry participants as they pursue competitive excellence. Copies of the working papers summarized here are available from the Center. If you would like to learn more about the Center or become a member of our research community, please let us know of your interest.
Dynamics of encoding in a population of neurons
 J. Gen. Physiol
, 1972
"... ABSTRACT A simple encoder model, which is a reasonable idealization from known electrophysiological properties, yields a population in which the variation of the firing rate with time is a perfect replica of the shape of the input stimulus. A population of noisefree encoders which depart even sligh ..."
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Cited by 93 (7 self)
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ABSTRACT A simple encoder model, which is a reasonable idealization from known electrophysiological properties, yields a population in which the variation of the firing rate with time is a perfect replica of the shape of the input stimulus. A population of noisefree encoders which depart even slightly from the simple model yield a very much degraded copy of the input stimulus. The presence of noise improves the performance of such a population. The firing rate of a population of neurons is related to the firing rate of a single member in a subtle way. 1.
Equations and Mirror Maps For Hypersurfaces
 in Essays on Mirror Manifolds, Ed. S.T.Yau, International Press
, 1992
"... Abstract. We describe a strategy for computing Yukawa couplings and the mirror map, based on the PicardFuchs equation. (Our strategy is a variant of the method used by Candelas, de la Ossa, Green, and Parkes [5] in the case of quintic hypersurfaces.) We then explain a technique of Griffiths [14] wh ..."
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Cited by 77 (4 self)
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Abstract. We describe a strategy for computing Yukawa couplings and the mirror map, based on the PicardFuchs equation. (Our strategy is a variant of the method used by Candelas, de la Ossa, Green, and Parkes [5] in the case of quintic hypersurfaces.) We then explain a technique of Griffiths [14] which can be used to compute the PicardFuchs equations of hypersurfaces. Finally, we carry out the computation for four specific examples (including quintic hypersurfaces, previously done by Candelas et al. [5]). This yields predictions for the number of rational curves of various degrees on certain hypersurfaces in weighted projective spaces. Some of these predictions have been confirmed by classical techniques in algebraic geometry.
Estimating Equations Based on Eigenfunctions for a Discretely Observed Diffusion Process
, 1995
"... : A new type of martingale estimating function is proposed for inference about classes of diffusion processes based on discretetime observations. These estimating functions can be tailored to a particular class of diffusion processes by utilizing a martingale property of the eigenfunctions of the g ..."
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Cited by 58 (13 self)
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: A new type of martingale estimating function is proposed for inference about classes of diffusion processes based on discretetime observations. These estimating functions can be tailored to a particular class of diffusion processes by utilizing a martingale property of the eigenfunctions of the generators of the diffusions. Optimal estimating functions in the sense of Godambe and Heyde are found. Inference based on these is invariant under transformations of data. A result on consistency and asymptotic normality of the estimators is given for ergodic diffusions. The theory is illustrated by several examples and by a simulation study. Keywords: generator, optimal estimating function, stochastic differential equation, quasilikelihood. 1 Introduction Martingale estimating functions have turned out to give good and relatively simple estimators for discretely observed diffusion models, for which the likelihood function is only explicitly known in special cases. These estimators have th...
Numerical Methods For Hyperbolic Conservation Laws With Stiff Relaxation I. Spurious Solutions
 SIAM J. Sci. Comput
, 1992
"... . We consider the numerical solution of hyperbolic systems of conservation laws with relaxation using a shock capturing finite difference scheme on a fixed, uniform spatial grid. We conjecture that certain a priori criteria insure that the numerical method does not produce spurious solutions as the ..."
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Cited by 55 (2 self)
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. We consider the numerical solution of hyperbolic systems of conservation laws with relaxation using a shock capturing finite difference scheme on a fixed, uniform spatial grid. We conjecture that certain a priori criteria insure that the numerical method does not produce spurious solutions as the relaxation time vanishes. One criterion is that the limits of vanishing relaxation time and vanishing viscosity commute for the viscous regularization of the hyperbolic system. A second criterion is that a certain "subcharacteristic" condition be satisfied by the hyperbolic system. We support our conjecture with analytical and numerical results for a specific example, the solution of generalized Riemann problems of a model system of equations with a fractional step scheme in which Godunov's method is coupled with the backward Euler method. We also apply our ideas to the numerical solution of stiff detonation problems. 1. Introduction. Hyperbolic systems of conservation laws with relaxation ...
The absolutely continuous spectrum of onedimensional Schrödinger operators with decaying potentials
, 2008
"... This paper deals with general structural properties of onedimensional Schrödinger operators with some absolutely continuous spectrum. The basic result says that the ω limit points of the potential under the shift map are reflectionless on the support of the absolutely continuous part of the spectr ..."
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Cited by 53 (7 self)
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This paper deals with general structural properties of onedimensional Schrödinger operators with some absolutely continuous spectrum. The basic result says that the ω limit points of the potential under the shift map are reflectionless on the support of the absolutely continuous part of the spectral measure. This implies an Oracle Theorem for such potentials and DenisovRakhmanov type theorems. In the discrete case, for Jacobi operators, these issues were discussed in my recent paper [19]. The treatment of the continuous case in the present paper depends on the same basic ideas.
A new approach to inverse spectral theory, II. General real potentials and the connection to the spectral measure
 Ann. of Math
, 2000
"... Abstract. We continue the study of the Aamplitude associated to a halfline Schrödinger operator, − d2 dx2 + q in L2 ((0, b)), b ≤ ∞. A is related to the WeylTitchmarsh mfunction via m(−κ2) = −κ − ∫ a 0 A(α)e−2ακ dα+O(e −(2a−ε)κ) for all ε> 0. We discuss five issues here. First, we extend the ..."
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Cited by 52 (20 self)
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Abstract. We continue the study of the Aamplitude associated to a halfline Schrödinger operator, − d2 dx2 + q in L2 ((0, b)), b ≤ ∞. A is related to the WeylTitchmarsh mfunction via m(−κ2) = −κ − ∫ a 0 A(α)e−2ακ dα+O(e −(2a−ε)κ) for all ε> 0. We discuss five issues here. First, we extend the theory to general q in L1 ((0, a)) for all a, including q’s which are limit circle at infinity. Second, we prove the following relation between the Aamplitude and the spectral measure ρ: A(α) = −2 ∫ ∞ 1 λ − 2 sin(2α − ∞ √ λ)dρ(λ) (since the integral is divergent, this formula has to be properly interpreted). Third, we provide a Laplace transform representation for m without error term in the case b < ∞. Fourth, we discuss mfunctions associated to other boundary conditions than the Dirichlet boundary conditions associated to the principal WeylTitchmarsh mfunction. Finally, we discuss some examples where one can compute A exactly. 1.