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18
Representation of spatial orientation by the intrinsic dynamics of the headdirection cell ensemble: A theory
 J. Neurosci
, 1996
"... The headdirection (HD) cells found in the limbic system in freely moving rats represent the instantaneous head direction of the animal in the horizontal plane regardless of the location of the animal. The internal direction represented by these cells uses both selfmotion information for inettiall ..."
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Cited by 130 (4 self)
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The headdirection (HD) cells found in the limbic system in freely moving rats represent the instantaneous head direction of the animal in the horizontal plane regardless of the location of the animal. The internal direction represented by these cells uses both selfmotion information for inettially based updating and familiar visual landmarks for calibration. Here, a model of the dynamics of the HD cell ensemble is presented. The stability of a localized static activity profile in the network and a dynamic shift mechanism are explained naturally by synaptic weight distribution components with even and odd symmetry, respectively. Under symmetric weights or symmetric reciprocal connections, a stable activity profile close to the known directional tuning curves will emerge. By adding a slight asymmetry to the weights, the activity profile will shift continuously without 1
Transformations of quadrilateral lattices
 J. Math. Phys
"... Abstract. We investigate the τfunction of the quadrilateral lattice using the nonlocal ¯ ∂dressing method, and we show that it can be identified with the Fredholm determinant of the integral equation which naturally appears within that approach. ..."
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Cited by 24 (10 self)
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Abstract. We investigate the τfunction of the quadrilateral lattice using the nonlocal ¯ ∂dressing method, and we show that it can be identified with the Fredholm determinant of the integral equation which naturally appears within that approach.
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
Spectral Properties of Totally Positive Kernels and Matrices
 Total Positivity and its Applications, volume 359 of Mathematics and its Applications
"... . We detail the history and present complete proofs of the spectral properties of totally positive kernels and matrices. x1. Introduction There seem to have been four central topics which historically led to the development of the theory of total positivity. These included the variation diminishing ..."
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Cited by 7 (0 self)
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. We detail the history and present complete proofs of the spectral properties of totally positive kernels and matrices. x1. Introduction There seem to have been four central topics which historically led to the development of the theory of total positivity. These included the variation diminishing (V D) properties, the study of which was initiated by I. J. Schoenberg in 1930 (see [30]), and the study of P'olya frequency functions also initiated by I. J. Schoenberg in the late 1940's and early 1950's (see the relevant papers in [31]). These two topics are extensively investigated and expanded upon in Karlin [20]. In addition we have research connected with ordinary differential equations whose Green's function is totally positive (work started by M. G. Krein and some of his colleagues in the mid 1930's), and finally the study of the spectral properties of totally positive kernels and matrices. This last topic was actually the first studied. There are various misconceptions relating t...
Numerical Solution of a Cauchy Problem for the Laplace Equation
"... We consider a two dimensional steady state heat conduction problem. ..."
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Cited by 5 (1 self)
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We consider a two dimensional steady state heat conduction problem.
On Gaussian Processes Equivalent In Law To Fractional Brownian Motion
 J. Theoret. Probab
, 2004
"... We consider Gaussian processes that are equivalent in law to the fractional Brownian motion and their canonical representations. We prove a Hitsuda type representation theorem for the fractional Brownian motion with Hurst index H 2 we show that such a representation cannot hold. We also consider bri ..."
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Cited by 1 (1 self)
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We consider Gaussian processes that are equivalent in law to the fractional Brownian motion and their canonical representations. We prove a Hitsuda type representation theorem for the fractional Brownian motion with Hurst index H 2 we show that such a representation cannot hold. We also consider briefly the connection between Hitsuda and Girsanov representations. Using the Hitsuda representation we consider a certain special kind Gaussian stochastic equation with fractional Brownian motion as noise.
On τfunction of conjugate nets
, 2004
"... We study a potential introduced by Darboux to describe conjugate nets, which within the modern theory of integrable systems can be interpreted as a τfunction. We investigate the potential using the nonlocal ¯ ∂ dressing method of Manakov and Zakharov, and we show that it can be interpreted as the ..."
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Cited by 1 (1 self)
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We study a potential introduced by Darboux to describe conjugate nets, which within the modern theory of integrable systems can be interpreted as a τfunction. We investigate the potential using the nonlocal ¯ ∂ dressing method of Manakov and Zakharov, and we show that it can be interpreted as the Fredholm determinant of an integral equation which naturally appears within that approach. Finally, we give some arguments extending that interpretation to multicomponent Kadomtsev–Petviashvili hierarchy. 1
DESARGUES MAPS AND THE HIROTA–MIWA EQUATION
"... Abstract. We study the Desargues maps φ: Z N → P M, which generate lattices whose points are collinear with all their nearest (in positive directions) neighbours. The multidimensional consistency of the map is equivalent to the Desargues theorem and its higherdimensional generalizations. The nonlin ..."
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Cited by 1 (1 self)
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Abstract. We study the Desargues maps φ: Z N → P M, which generate lattices whose points are collinear with all their nearest (in positive directions) neighbours. The multidimensional consistency of the map is equivalent to the Desargues theorem and its higherdimensional generalizations. The nonlinear counterpart of the map is the noncommutative (in general) Hirota–Miwa system. In the commutative case of the complex field we apply the nonlocal ¯ ∂dressing method to construct Desargues maps and the corresponding solutions of the equation. In particular, we identify the Fredholm determinant of the integral equation inverting the nonlocal ¯ ∂dressing problem with the τfunction. Finally, we establish equivalence between the Desargues maps and quadrilateral lattices provided we take into consideration also their Laplace transforms.
Linear Estimation of Boundary Value Stochastic ProcessesPart I: The Role and Construction of Complementary Models
"... Abstract'Ibis paper presents a substantial extension of the method of complementary modek for minimum variance linear estimation introduced by Weinert and Desai in their important paper [l]. Specifically, the method of complementary models is extended to solve estimation problems for both discrete ..."
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Abstract'Ibis paper presents a substantial extension of the method of complementary modek for minimum variance linear estimation introduced by Weinert and Desai in their important paper [l]. Specifically, the method of complementary models is extended to solve estimation problems for both discrete and continuous parameter linear boundary value stochastic processes in one and higher dimensions. A major contribution of this paper is an application of Green's identity in denting a differential operator representation of the estimator. To clarify the development and to illustrate the range of applications of our approach, two brief examples are provided: one is a ID discrete twopoint boundaq value process and the other is a 2D process governed by Poisson's equation on the unit disk. I.