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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 sta ..."
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Cited by 585 (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.
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 267 (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.
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 ..."
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Cited by 175 (7 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 140 (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.
RigidBody Dynamics With Friction and Impact,”
 SIAM Rev.,
, 2000
"... Abstract. Rigidbody dynamics with unilateral contact is a good approximation for a wide range of everyday phenomena, from the operation of car brakes to walking to rock slides. It is also of vital importance for simulating robots, virtual reality, and realistic animation. However, correctly modeli ..."
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Cited by 131 (1 self)
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Abstract. Rigidbody dynamics with unilateral contact is a good approximation for a wide range of everyday phenomena, from the operation of car brakes to walking to rock slides. It is also of vital importance for simulating robots, virtual reality, and realistic animation. However, correctly modeling rigidbody dynamics with friction is difficult due to a number of discontinuities in the behavior of rigid bodies and the discontinuities inherent in the Coulomb friction law. This is particularly crucial for handling situations with large coefficients of friction, which can result in paradoxical results known at least since Painlevé [C. R. Acad. Sci. Paris, 121 (1895), pp. 112115]. This single example has been a counterexample and cause of controversy ever since, and only recently have there been rigorous mathematical results that show the existence of solutions to his example. The new mathematical developments in rigidbody dynamics have come from several sources: "sweeping processes" and the measure differential inclusions of Moreau in the 1970s and 1980s, the variational inequality approaches of Duvaut and J.L. Lions in the 1970s, and the use of complementarity problems to formulate frictional contact problems by Lötstedt in the early 1980s. However, it wasn't until much more recently that these tools were finally able to produce rigorous results about rigidbody dynamics with Coulomb friction and impulses. Key words. rigidbody dynamics, Coulomb friction, contact mechanics, measuredifferential inclu sions, complementarity problems AMS subject classifications. Primary, 70E55; Secondary, 70F40, 74M PII. S0036144599360110 Rigid Bodies and Friction. Rigid bodies are bodies that cannot deform. They can translate and rotate, but they cannot change their shape. From the outset this must be understood as an approximation to reality, since no bodies are perfectly rigid. However, for a vast number of applications in robotics, manufacturing, biomechanics (such as studying how people walk), and granular materials, this is an excellent approximation. It is also convenient, since it does not require solving large, complex systems of partial differential equations, which is generally difficult to do both analytically and computationally. To see the difference, consider the problem of a bouncing ball. The rigidbody model will assume that the ball does not deform while in flight and that contacts with the ground are instantaneous, at least while the ball is not rolling. On the other hand, a full elastic model will model not only the contacts and the resulting deformation of the entire ball while in contact, but also the elastic oscillations of the ball while it is in flight. Apart from the computational complexity of all this, the analysis of even linearly elastic bodies in contact with a *
PicardFuchs Equations and Mirror Maps For Hypersurfaces
 IN ESSAYS ON MIRROR MANIFOLDS, ED. S.T.YAU, INTERNATIONAL PRESS
, 1992
"... 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 ..."
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Cited by 100 (5 self)
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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.
Resonances, Radiation Damping and Instability in Hamiltonian Nonlinear Wave Equations
, 1998
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On the Absolutely Continuous Spectrum of OneDimensional Schrödinger Operators with Square Summable Potentials
 Comm. Math. Phys
, 1999
"... Abstract: For continuous and discrete onedimensional Schrödinger operators with square summable potentials, the absolutely continuous part of the spectrum is essentially supported by T0;1 / and T−2; 2U respectively. This fact is proved by considering a priori estimates for the transmission coeffici ..."
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Cited by 95 (6 self)
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Abstract: For continuous and discrete onedimensional Schrödinger operators with square summable potentials, the absolutely continuous part of the spectrum is essentially supported by T0;1 / and T−2; 2U respectively. This fact is proved by considering a priori estimates for the transmission coefficient. 1.
A GSMP formalism for discrete event systems
 Proceedings of the IEEE
, 1989
"... We describe here a precise mathematical framework for the study of discrete event systems. The idea i s to define a particular type of stochastic process, called a generalized semiMarkov process (GSMP), which captures the essential dynamical structure of a discrete event system. The paper also att ..."
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Cited by 90 (8 self)
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We describe here a precise mathematical framework for the study of discrete event systems. The idea i s to define a particular type of stochastic process, called a generalized semiMarkov process (GSMP), which captures the essential dynamical structure of a discrete event system. The paper also attempts to give a flavor of the qualitative theory and numerical algorithms that can be obtained as a result of viewing discrete event systems as GSMPs. I.
Stability of Travelling Waves
, 2002
"... An overview of various aspects related to the spectral and nonlinear stability of travellingwave solutions to partial differential equations is given. The point and the essential spectrum of the linearization about a travelling wave are discussed as is the relation between these spectra, Fredholm p ..."
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Cited by 86 (8 self)
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An overview of various aspects related to the spectral and nonlinear stability of travellingwave solutions to partial differential equations is given. The point and the essential spectrum of the linearization about a travelling wave are discussed as is the relation between these spectra, Fredholm properties, and the existence of exponential dichotomies (or Green's functions) for the linear operator. Among the other topics reviewed in this survey are the nonlinear stability of waves, the stability and interaction of wellseparated multibump pulses, the numerical computation of spectra, and the Evans function, which is a tool to locate isolated eigenvalues in the point spectrum and near the essential spectrum. Furthermore, methods for the stability of waves in Hamiltonian and monotone equations as well as for singularly perturbed problems are mentioned. Modulated waves, rotating waves on the plane, and travelling waves on cylindrical domains are also discussed briefly.