The head-direction (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 self-motion information for inet-tially based updating and familiar visual landmarks for calibration. Here, a model of the dynamics of the HD cell ensemble is presented. The sta-bility 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 direc-tional tuning curves will emerge. By adding a slight asymmetry to the weights, the activity profile will shift continuously without 1

Abstract. We investigate the τ-function 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.

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

. 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...

We consider a two dimensional steady state heat conduction problem.

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.

We study a potential introduced by Darboux to describe conjugate nets, which within the modern theory of integrable systems can be interpreted as a τ-function. We investigate the potential using the non-local ¯ ∂ 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

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 higher-dimensional generalizations. The nonlinear counterpart of the map is the non-commutative (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 τ-function. Finally, we establish equivalence between the Desargues maps and quadrilateral lattices provided we take into consideration also their Laplace transforms.

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 I-D discrete two-point boundaq value process and the other is a 2-D process governed by Poisson's equation on the unit disk. I.

contain these Terms of use.