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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
Billiard systems in three dimensions: The boundary integral equation and the trace formula
 Nonlinearity
, 1998
"... We derive semiclassical contributions of periodic orbits from a boundary integral equation for threedimensional billiard systems. We use an iterative method that keeps track of the composition of the stability matrix and the Maslov index as an orbit is traversed. Results are given for isolated peri ..."
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Cited by 5 (0 self)
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We derive semiclassical contributions of periodic orbits from a boundary integral equation for threedimensional billiard systems. We use an iterative method that keeps track of the composition of the stability matrix and the Maslov index as an orbit is traversed. Results are given for isolated periodic orbits and rotationally invariant families of periodic orbits in axially symmetric billiard systems. A practical method for determining the stability matrix and the Maslov index is described. PACS numbers: 03.65.Sq Semiclassical theories and applications. 05.45.+b Theory and models of chaotic systems. 1 1
Transposition hypergroups of Fredholm integral operators and related hyperstructures Part I
"... In this contribution we construct noncommutative transposition hypergroups of integral operators on spaces of continuous functions which are created by Fredholm integral equations of the first and second kinds. Moreover, we investigate the obtained hyperstructures as transposition hypergroups and al ..."
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In this contribution we construct noncommutative transposition hypergroups of integral operators on spaces of continuous functions which are created by Fredholm integral equations of the first and second kinds. Moreover, we investigate the obtained hyperstructures as transposition hypergroups and also related quasihypergroups of blocks of equivalence of integral operators. Theory of linear integral equations which are a certain continuous analogue of systems of linear algebraic equations belongs to classical parts of contemporary pure and applied mathematics and plays an important role from the point of view of technical sciences. The basic constructed structures are ordered groups of integral operators. Moreover, we use also the object function (where the corresponding binary hyperoperation on an ordered group is defined as principal end generated by products of pairs of elements of the considered group) of a functor enabling the transfer from the category of ordered groups and their isotone homomorphisms into the category of hypergroups and their inclusion homomorphisms. The basic group of integral operators contains an invariant subgroup. Using another binary operation on the set of suitable Fredholm integral operators of the second kind we get a group with a significant noninvariant subgroup of operators of the first kind enabling to construct a quasihypergroup of decomposition classes of operators, structure of which is also clarified.
Parallel Number Theoretical Numerics for Solving sdimensional Integral Equations of the Convolution Type
, 1994
"... This paper is devoted to show that parallel number theoretical methods for solving special types of the integral equation \Phi(T ) = F (T ) + (K \Phi)(T ) 2 IR for higher dimensions s can be very efficient with a view to accuracy and computation time. To enable a numerical solution of such integr ..."
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Cited by 2 (1 self)
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This paper is devoted to show that parallel number theoretical methods for solving special types of the integral equation \Phi(T ) = F (T ) + (K \Phi)(T ) 2 IR for higher dimensions s can be very efficient with a view to accuracy and computation time. To enable a numerical solution of such integral equations we will follow a classical way which will lead us to the problem of the inversion of multivariate Laplace transforms. The inherent parallelism of the number theoretical methods to compute the inversion of the sdimensional Laplace transform has been exploited.
On the Solution of the Generalized Airfoil Equation
, 1996
"... this paper let be the Lebesgue measure in the open interval\Omega = (\Gamma1; 1). Those functions on\Omega which coincide outside a Lebesgue null set will be identified as usual. Define functions % and oe ..."
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Cited by 1 (0 self)
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this paper let be the Lebesgue measure in the open interval\Omega = (\Gamma1; 1). Those functions on\Omega which coincide outside a Lebesgue null set will be identified as usual. Define functions % and oe
Boltzmann Collision Kernels and Velocity Saturation in Semiconductors
, 1997
"... For different models of the electronphonon interaction, the asymptotic behaviour of the moments of the stationary homogeneous solution of the linear Boltzmann equation is determined in the limit of a high external field. For HilbertSchmidt kernels of a finite rank, a result recently proven for ker ..."
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For different models of the electronphonon interaction, the asymptotic behaviour of the moments of the stationary homogeneous solution of the linear Boltzmann equation is determined in the limit of a high external field. For HilbertSchmidt kernels of a finite rank, a result recently proven for kernels of rank one is found generally valid; as a consequence velocity saturation is excluded for these collision models. For a class of singular collision kernels in contrast, velocity saturation is generally obtained. 1
unknown title
"... Bounds of high quality for first kind Volterra integral equations HANSJORGEN DOBNER EMethnds for.,aflving linear Volterra integral equatitms of the first kind with smot~th kernels are considered. EMethotL ~ are a new type of numerical algorithms computing numerical approximations together with m ..."
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Bounds of high quality for first kind Volterra integral equations HANSJORGEN DOBNER EMethnds for.,aflving linear Volterra integral equatitms of the first kind with smot~th kernels are considered. EMethotL ~ are a new type of numerical algorithms computing numerical approximations together with mathematically guaranteed do. ~ error I~amds, The basic concepts from verifi~ttion thet~ry are sketched and such ~lfvalidating numerics derived. O~mputati~mal experiments show the effidency t)f these prtn:edures being an advance in numerical methods. TeCHBIe rpaHmI~i peTneHm ~ HHTerpaABHHX ypaBHeHHfi BOABTeppa nepBoro poAa