determination of eigenstates via lanczos based forward substitution and filter diagonalization a filter diagonalization scheme based on the lanczos tridiagonalization algorithm is proposed for the determination of highly excited eigenstates in a prespecified energy range filtered states localized in the energy domain can be generated during the lanczos iteration by forward substitution which formally corresponds to a green function filter eigenstates in the energy range of interest are then obtained by diagonalizing a small hamiltonian matrix in a subspace spanned by the filtered states it is shown that the accuracy of this approach is affected neither by the loss of orthogonality among the lanczos states nor by the instability in the forward substitution since the filtered states are assembled while the lanczos states are generated no storage of the lanczos states is needed the computational savings can be significant in calculating high energy eigenstates of large systems the proposed scheme is shown numerically to have comparable accuracy with the basic lanczos method

quantising a generic family of billiards with analytic boundaries a generic family of plane billiards has been discovered recently the shape of the boundary is given by the quadratic conformal image of the unit circle and is thus real analytic for small deformations of the unit disc the billiard is a typical kam system but becomes ergodic or even mixing when the curvature of the boundary vanishes at some point the kolmogorov entropy has been calculated and it increases with the deformation of the boundary the author studies aspects of the quantum chaos for this billiard he solves numerically the eigenvalue problem for the laplace operator with dirichlet s boundary condition he examines the spectrum and inspects the avoided crossings at which mixing of nearby states occurs the variation of the nodal structure and of the localisation properties of the eigenfunctions is studied in analysing the level spacing distribution he finds a continuous transition from the poisson distribution towards the wigner distribution the exponent in the level repulsion law varies continuously along with a generic perturbation for small perturbations it seems to be proportional to the square root of the perturbation parameter

starting methods for two step runge kutta methods of stage order 3 and order 6 jackiewicz and tracogna siam j numer anal 32 1995 1390 1427 proposed a general formulation of two step runge kutta tsrk methods using formulas for two step pairs of tsrk methods constructed in japan jiam 19 2002 227 248 jackiewicz and verner obtain results for order 8 pairs that fail to show this designated order hairer and wanner siam j numer anal 34 1997 2087 2089 identify the problem by using b series to formulate a complete set of order conditions for tsrk methods and emphasize that special starting methods are necessary for the first step of implementation they observe that for methods with stage order at least p 1 and design order p starting methods of order at least p are sufficient in this paper the more general challenge to provide correct starting values for methods of low stage order is met by showing how perturbed starting values should be selected for methods of order 6 and stage order 3 the approach is sufficiently general that it may and later will be provided for such methods of higher orders evidence of the accompanying improvement in the implementation of tsrk methods illustrates that carefully designed starting methods are essential for efficient production codes based on methods of low stage order

a general class of two step runge kutta methods for ordinary differential equations a general class of two step runge kutta methods that depend on stage values at two consecutive steps is studied these methods are special cases of general linear methods introduced by butcher and are quite efficient with respect to the number of function evaluations required for a given order general order conditions are derived using the approach proposed recently by albrecht and examples of methods are given up to the order 5 these methods can be divided into four classes that are appropriate for the numerical solution of nonstiff or stiff differential equations in sequential or parallel computing environments

test of a numerical approach to the quantization of billiards a method for computing large numbers of eigenvalues of self adjoint elliptic operators j comput phys 184 321 2003 is tested in numerical studies of the spectral properties of quantum billiards to this extent we study a time reversal invariant quantum billiard of threefold symmetry that undergoes a transformation in its symmetry properties from c 3v to c3 thereby a transition from gaussian orthogonal to gaussian unitary ensemble statistics is observed verifying earlier experimental indications and theoretical predictions at the same time our numerical ansatz is shown to be applicable to arbitrary billiard shapes

implicit restart lanczos as an eigensolver this paper investigates the efficiency of the implicit restart lanczos and simple without reorthogonalization lanczos algorithms as eigensolvers for large scale computations in molecular and chemical physics using the cardioid billiard and the hydrogen cyanide hydrogen isocyanide hcn hnc molecule as model systems we demonstrate superior efficiency of implicit restart lanczos compared to the simple lanczos algorithm a modified implementation of implicit restart lanczos is also presented which works with a smaller krylov space with associated savings in memory and can handle larger basis sets than the usual implicit restart lanczos it also enables getting all eigenpairs of a matrix or all eigenvalues below a threshold where the number of such is not known before hand which is more difficult with the usual implicit restart algorithm

notes on the local p adic simpson correspondence we complete the theory of the local p adic simpson correspondence developed by faltings for rational coefficients it asserts that there is an equivalence between the category of small mathbb q p generalized representations of the geometric fundamental group and that of small mathbb q p higgs bundles for a certain kind of log smooth affine scheme over a complete discrete valuation ring of mixed characteristic 0 p with perfect residue field the difficulty lies in the construction of the latter from the former and we give it via a generalized sen s theory for the log smooth affine scheme which depends on faltings almost purity theorem inspired by a recent work by r liu and x zhu we also give a formulation of local p adic simpson correspondence for mathbb q p generalized representations of the arithmetic fundamental group and a characterization of hodge tate generalized representations in terms of the correspondence

order conditions for integrators and mappings with applications abstract we regard the elementary weight function as a mapping from input to output so that the mappings allow us to derive the order conditions for a given method more easily and concisely this paper presents the order conditions of one step runge kutta methods runge kutta type methods with derivatives a class of two step runge kutta methods and three step runge kutta methods based on the elementary weight mappings the composition formula of two mappings as well as the mappings i d r e defined on the set t of all rooted trees from an input to an output it is pointed out that the order conditions can be easily achieved for linear multistep methods and second derivative linear multistep methods of order p and it suffices to consider only bushy trees of order up to p together with the empty tree 0 for a linear multistep method or a second derivative linear multistep method of order p

construction of highly stable parallel two step runge kutta methods for delay differential equations it is shown that any a stable two step runge kutta method of order p and stage order q p for ordinary differential equations can be extended to the p stable method of uniform order p p for delay differential equations

purity for hodge tate representations we prove that a kind of purity holds for hodge tate representations of the fundamental group of the generic fiber of a semi stable scheme over a complete discrete valuation ring of mixed characteristic with perfect residue field as an application we see that the relative p adic etale cohomology with proper support of a scheme separated of finite type over the generic fiber is hodge tate if it is locally constant

construction of two step runge kutta methods with large regions of absolute stability we describe the construction of explicit two step runge kutta methods of order p and stage order q p 1 or q p with large regions of absolute stability this process is illustrated for the methods of order p 2 and 3 and leads to new methods which are competitive with explicit runge kutta methods of the same order

stability analysis of two step runge kutta methods for delay differential equations abstract we investigate stability properties of two step runge kutta methods with respect to the linear test equation y t ay t by t t 0 y t g t t 0 where a and b are complex parameters it is known that the solution y t to this equation tends to zero as t if b a we will show that under some conditions this property is inherited by any a stable two step runge kutta method applied on a constrained mesh to delay differential equations i e that the corresponding method is p stable

rigidity and a riemann hilbert correspondence for p adic local systems we construct a functor from the category of p adic etale local systems on a smooth rigid analytic variety x over a p adic field to the category of vector bundles with an integrable connection on its base change to mathrm b text dr which can be regarded as a first step towards the sought after p adic riemann hilbert correspondence as a consequence we obtain the following rigidity theorem for p adic local systems on a connected rigid analytic variety if the stalk of such a local system at one point regarded as a p adic galois representation is de rham in the sense of fontaine then the stalk at every point is de rham along the way we also establish some basic properties of the p adic simpson correspondence finally we give an application of our results to shimura varieties

order conditions for general two step runge kutta methods the use of b series for the derivation of order conditions for general two step runge kutta methods is illustrated

a class of implicit explicit two step runge kutta methods this work develops new implicit explicit time integrators based on two step runge kutta methods the class of schemes of interest is characterized by linear invariant preservation and high stage orders theoretical consistency stability and stiff convergence analyses are performed to reveal the excellent properties of these methods the new framework offers extreme flexibility in the construction of partitioned integrators since no coupling conditions are necessary practical schemes of orders three four and six are constructed and are used to solve several test problems numerical results confirm the theoretical findings

two step almost collocation methods for volterra integral equations in this paper we construct a new class of continuous methods for volterra integral equations these methods are obtained by using a collocation technique and by relaxing some of the collocation conditions in order to obtain good stability properties

exponentially fitted two step runge kutta methods construction and parameter selection abstract we derive exponentially fitted two step runge kutta methods for the numerical solution of y f x y specially tuned to the behaviour of the solution such methods have nonconstant coefficients which depend on a parameter to be suitably estimated the construction of the methods is shown and a strategy of parameter selection is presented some numerical experiments are provided to confirm the theoretical expectations