periodic schwarz christoffel mappings with multiple boundaries per period we present an extension to the theory of schwarzchristoffel sc mappings by permitting the target domain to be a single period window of a periodic configuration having multiple polygonal straigh

two dimensional integrable potentials with quartic invariants for autonomous two dimensional conservative dynamical systems the author derives four necessary and sufficient conditions which the potential function u x y has to satisfy in order that it is integrable with the second constant of motion quartic in the velocity components the author also develops the method by means of which he finds the quartic invariant for a given potential satisfying these conditions certain degenerate cases leading to pseudo quartic integrals are discussed two examples and a counter example are presented

configurational invariants of hamiltonian systems in this paper we explore the general conditions in order that a two dimensional natural hamiltonian system possess a second invariant which is a polynomial in the momenta and is therefore liouville integrable we examine the possibility that the invariant is preserved by the hamiltonian flow on a given energy hypersurface only weak integrability and derive the additional requirement necessary to have conservation at arbitrary energy strong integrability using null complex coordinates we show that the leading order coefficient of the polynomial is an arbitrary holomorphic function in the case of weak integrability and a polynomial in the coordinates in the strongly integrable one we review the results obtained so far with strong invariants up to degree four and provide some new examples of weakly integrable systems with linear and quadratic invariants

equilibria for the n vortex problem in a general bounded domain abstract this article is concerned with the study of existence of stationary solutions for the dynamics of n point vortices in an idealised fluid constrained to a bounded two dimensional domain which is governed by a hamiltonian system i d x i d t h y i z 1 z n i d y i d t h x i z 1 z n where z i x i y i i 1 n under selected conditions on the vorticities i and various topological and geometrical assumptions on here h z j 1 n j 2 h z j i j 1 i j n i j g z i z j is the so called kirchhoff routh path function in particular we will prove that h has a critical point if it is possible to align the vortices along a line such that the signs of the i are alternating and i is non increasing we are also able to derive a critical point of h if j j j 2 i j j i j i j for all j 1 n j 2 and is not simply connected

relatively robust representations of symmetric tridiagonals let ldlt be the triangular factorization of an unreduced symmetric tridiagonal matrix t i small relative changes in the nontrivial entries of l and d may be represented by diagonal scaling matrices 1 and 2 ldlt 2l 1d 1lt 2 the effect of 2 on the eigenvalues i is benign in this paper we study the inner perturbations induced by 1 suitable condition numbers govern the relative changes in the eigenvalues i we show that when j is an eigenvalue then the relative condition number of m j m j is the same for all n twisted factorizations one of which is ldlt that could be used to represent t i see section 2 we prove that as j the smallest eigenvalue has relative condition number relcond 1 o j each relcond is a rational function of we identify the poles and then use orthogonal polynomial theory to develop upper bounds on the sum of the relconds of all the eigenvalues these bounds require o n operations for an n n matrix we show that the sum of all the relconds is bounded by trace l d lt and conjecture that n ldlt the quantity trace l d lt ldlt is a natural measure of element growth in the context of this paper an algorithm for computing numerically orthogonal eigenvectors without recourse to the gram schmidt process is sketched it requires that there exist values of close to each cluster of close eigenvalues such that all the relconds belonging to the cluster are modest say 10 the sensitivity of the other eigenvalues is not important for this reason we develop o n bounds on the sum of the relconds associated with a cluster none of our bounds makes reference to the nature of the distribution of the eigenvalues within a cluster which can be very complicated

critical points of the n vortex hamiltonian in bounded planar domains and steady state solutions of the incompressible euler equations we prove the existence of critical points of the n vortex hamiltonian h kr x 1 ldots x n sum n i 1 gamma 2 i h x i sum genfrac 0pt i j 1 j ne k n gamma i gamma jg x i x j 2 sum i 1 n gamma i psi 0 x i in a bounded domain omega subset mathbb r 2 which may be simply or multiply connected here g denotes the green function for the dirichlet laplace operator in omega more generally a hydrodynamic green function and h the robin function moreover psi 0 in mathcal c 1 overline omega is a harmonic function on omega we obtain new critical points x x 1 dots x n for n 3 or n 4 under conditions on the vorticities gamma i in mathbb r setminus 0 these critical points correspond to point vortex equilibria of the euler equation in vorticity form the case gamma i 1 i of counterrotating vortices with identical vortex strength is included the point vortex equilibria can be desingularized to obtain smooth steady sta

the master integrable two dimensional system with a quartic second integral in this paper we construct a new integrable natural conservative mechanical system admitting a second invariant of the fourth degree in velocities this system is quite general and involves 21 parameters we also show that all systems with a quartic integral known up to date can be obtained from it as special cases by a relevant choice of the values of parameters the results are applied to problems of particle and rigid body dynamics new integrable cases are obtained as special versions of the new system these cases include motions in a plane lobachevsky plane sphere and surfaces of variable curvature they also include generalizations of the classical cases of kovalevskaya chaplygin and goriatchev with the addition of certain singular terms to the potential

analytical formulae for the kirchhoff routh path function in multiply connected domains explicit formulae for the kirchhoff routh path functions or hamiltonians governing the motion of n point vortices in multiply connected domains are derived when all circulations around the holes in the domain are zero the method uses the schottky klein prime function to find representations of the hydrodynamic green s function in multiply connected circular domains the green s function is then used to construct the associated kirchhoff routh path function the path function in more general multiply connected domains then follows from a transformation property of the path function under conformal mapping of the canonical circular domains illustrative examples are presented for the case of single vortex motion in multiply connected domains

on hamiltonian systems in two degrees of freedom with invariants quartic in the momenta of form p21p22 a search is made for autonomous hamiltonian systems in two degrees of freedom which admit a second invariant quartic in the momenta with leading term p21p22 2 a sufficient condition for the resulting functional equation to possess solutions is deduced and a family of integrable systems is identified which under the equivalence class of linear transformations reduce to a simpler integrable system found originally by bozis the method of lax pairs is used to find further solutions to the functional equation and give new classes of integrable but nonseparable hamiltonians

analytical solutions for an escape problem in a disc with an arbitrary distribution of exit holes along its boundary we analytically construct solutions for the mean first passage time and splitting probabilities for the escape problem of a particle moving with continuous brownian motion in a confining planar disc with an arbitrary distribution i e of any number size and spacing of exit holes absorbing sections along its boundary the governing equations for these quantities are poisson s equation with a non zero constant forcing term and laplace s equation respectively and both are subject to a mixture of homogeneous neumann and dirichlet boundary conditions our solutions are expressed as explicit closed formulae written in terms of a parameterising variable via a conformal map using special transcendental functions that are defined in terms of an associated schottky group they are derived by exploiting recent results for a related problem of fluid mechanics that describes a unidirectional flow over no slip no shear surfaces as well as results from potential theory all of which were themselves derived using the same theory of schottky groups they are exact up to the determination of a finite set of mapping parameters which is performed numerically their evaluation also requires the numerical inversion of the parameterising conformal map computations for a series of illustrative examples are also presented

accurate singular values and differential qd algorithms in september 1991 j w demmel and w m kahan were awarded the second siam prize in numerical linear algebra for their paper accurate singular values of bidiagonal matrices 1 referred to as dk hereafter among several valuable results was the observation that the standard bidiagonal qr algorithm used in linpack 2 and in many other svd programs can be simplified when the shift is zero and of greater importance no subtractions occur the last feature permits very small singular values to be found with almost all the accuracy permitted by the data and at no extra cost

calculating the lift on a finite stack of cylindrical aerofoils the classic exact solution due to lagally lagally m 1929 die reibungslose stromung im aussengebiet zweier kreise z angew math mech 9 299 305 for streaming flow past two cylindrical aerofoils or obstacles is generalized to the case of an arbitrary finite number of cylindrical aerofoils given the geometry of the aerofoils the speed and direction of the oncoming uniform flow and the individual round aerofoil circulations the complex potential associated with the flow is found in analytical form in a parametric pre image region that can be conformally mapped to the fluid region a complete determination of the flow then follows from knowledge of the conformal mapping between the two regions in the special case where the aerofoils are all circular the conformal mapping from the parametric pre image region to the fluid domain is a mobius mapping the solution for the complex potential in such a case can then be used in combination with the blasius theorem to compute the distribution of hydrodynamic forces on the multi aerofoil configuration

double three wave interaction of four waves lax representations and exact solutions the nonlinear resonant interaction of coherent waves is a fundamental process in the study of wave phenomena which has received a great deal of attention in its many aspects in the present article a system of four interacting waves which constitute two resonant triplets is considered the system is described in a simplified model by a hamiltonian system of eight autonomous ordinary differential equations with time as the independent variable both positive and negative energy waves are allowed in the interaction two distinct lax representations for this system two new classes of exact solutions in terms of elliptic functions a solution in the form of a convergent generic laurent series expansion around a movable pole in the independent variable and a stabilization criterion for the explosive instability that may occur when waves of different energy sign interact are obtained herein

equation of motion for point vortices in multiply connected circular domains the paper gives the equation of motion for n point vortices in a bounded planar multiply connected domain inside the unit circle that contains many circular obstacles called the circular domain the velocity field induced by the point vortices is described in terms of the schottky klein prime function associated with the circular domain the explicit representation of the equation enables us not only to solve the euler equations through the point vortex approximation numerically but also to investigate the interactions between localized vortex structures in the circular domain as an application of the equation we consider the motion of two point vortices with unit strength and of opposite signs when the multiply connected domain is symmetric with respect to the real axis the motion of the two point vortices is reduced to that of a single point vortex in a multiply connected semicircle which we investigate in detail

harmonic radius and concentration of energy hyperbolic radius and liouville s equations d u e u and d u u n 2 n 2 the conformal radius of a simply connected planar domain omega is defined as r x 1 f x where f denotes a riemann map from omega to the unit disk mapping the given point x to the origin beyond geometric function theory the conformal radius has various applications in partial differential equations therefore it is desirable to extend it to multiply connected and higher dimensional domains we discuss two natural extensions 1 the harmonic radius is defined in terms of the green s function its maximum points the harmonic centers play an important role in variational problems involving concentration of energy they are of the following form sup left int limits omega f u u in h 0 1 omega int omega nabla u 2 leqslant varepsilon 2 right as varepsilon to 0 the solutions concentrate at a harmonic center our examples include bernoulli s free boundary problem the plasma problem and the elastic membrane with a movable obstacle an efficient method

a note on integrable two degrees of freedom hamiltonian systems with a second integral quartic in the momenta two of the simplest integrable hamiltonians h x y px py px2 py2 2 v x y with a second integral quartic in the momenta are those with potentials v3 x y by 3x2 16y2 d x2 16y2 eta y and v4 x y a x4 6x2y2 8y4 c x2 4y2 vy 2 we show how v3 can be obtained from v4 in the process we obtain a new potential of the class vn that includes both v3 and v4 as particular cases for this potential we give the second integral of motion separating variables a lax representation and a bi hamiltonian structure thus synthesizing the corresponding results for potentials v3 and v4 the integrable extension vn mu x 2 is also discussed

parametric time coherent states for smorodinsky winternitz potentials in this study we construct the coherent states for a particle in the smorodinsky winternitz potentials which are the generalizations of the two dimensional kepler problem in the third case the system is transformed into four ocillators and the parametric time coherent states are constructed in two coordinate frames in the fourth case the system is transformed into two ocillators with the reflection symmetry and the parametric time coherent states are constructed in two coordinate frames

the schwarz christoffel mapping to bounded multiply connected polygonal domains a formula for the generalized schwarz christoffel mapping from a bounded multiply connected circular domain to a bounded multiply connected polygonal domain is derived the theory of classical schottky groups is employed the formula for the derivative of the mapping function contains a product of powers of schottky klein prime functions associated with a schottky group relevant to the circular pre image domain the formula generalizes in a natural way the known mapping formulae for simply and doubly connected polygonal domains

the fixed energy problem for a class of nonconvex singular hamiltonian systems we consider a noncompact hypersurface h in r2n which is the energy level of a singular hamiltonian of strong force type under global geometric assumptions on h we prove that it carries a closed characteristic as a consequence of a result by hofer and viterbo on the weinstein conjecture in cotangent bundles of compact manifolds our theorem contains as particular cases earlier results on the fixed energy problem for singular lagrangian systems of strong force type

the generalized mic kepler system this paper deals with the dynamical system that generalizes the mic kepler system it is shown that the schrodinger equation for this generalized mic kepler system can be separated in spherical and parabolic coordinates the spectral problem in spherical and parabolic coordinates is solved

gluing methods for vortex dynamics in euler flows a classical problem for the two dimensional euler flow for an incompressible fluid confined to a smooth domain is that of finding regular solutions with highly concentrated vorticities around n moving vortices the formal dynamic law for such objects was first derived in the 19th century by kirkhoff and routh in this paper we devise a gluing approach for the construction of smooth n vortex solutions we capture in high precision the core of each vortex as a scaled finite mass solution of liouville s equation plus small more regular terms gluing methods have been a powerful tool in geometric constructions by desingularization we succeed in applying those ideas in this highly challenging setting

algebraic treatment of super integrable potentials the so 2 1 lie algebra is applied to three classes of two and three dimensional smorodinsky winternitz super integrable potentials for which the path integral discussion has been recently presented in the literature we have constructed the green s functions for two important super integrable potentials in r2 among the super integrable potentials in r3 we have considered two examples one is maximally super integrable and another one minimally super integrable the discussion is made in various coordinate systems the energy spectrum and the suitably normalized wave functions of bound and continuous states are then deduced

fernando s solution to wilkinson s problem an application of double factorization abstract suppose that one knows a very accurate approximation to an eigenvalue of a symmetric tridiagonal matrix t a good way to approximate the eigenvector x is to discard an appropriate equation say the rth from the system t i x 0 and then to solve the resulting underdetermined system in any of several stable ways however the output x can be completely inaccurate if r is chosen poorly and in the absence of a quick and reliable way to choose r this method has lain neglected for over 35 years experts in boundary value problems have known about the special structure of the inverse of a tridiagonal matrix since the 1960s and their double triangular factorization technique down and up gives directly the redundancy of each equation and so reveals the set of good choices for r the relation of double factorization to the eigenvector algorithm of godunov and his collaborators is described the results extend to band matrices and to zero entries in eigenvectors and have uses beyond eigenvector computation

convexity of solutions of semilinear elliptic equations on considere le probleme elliptique suivant u f u dans u m sur ou est un domaine convexe de r 2 et m est soit une constante reelle soit on etablit que pour une bonne fonction monotone g t pout toute solution u on a g u est strictement convexe dans

periodic solutions of the n vortex hamiltonian system in planar domains abstract we investigate the existence of collision free nonconstant periodic solutions of the n vortex problem in domains c these are solutions z t z 1 t z n t n of the first order hamiltonian system z k t i z k h z t k 1 n where the hamiltonian h has the form h z 1 z n 1 2 j k 1 j k n log 1 z j z k f z the function f n r depends on the regular part of the hydrodynamic green s function and is unbounded from above f z if z k for some k the hamiltonian is unbounded from above and below it is singular not integrable energy surfaces are not compact and not known to be of contact type using singular perturbation techniques and conley index theory we prove the existence of a family of periodic solutions z r t 0 r r 0 with arbitrarily small minimal period t r 0 as r 0 the solutions are close to the singular set of h our result applies in particular to generic bounded domains which may be simply or multiply connected it also applies to certain unbounded domains depending on the domain there are multiple such families

a fast symmetric svd algorithm for square hankel matrices abstract this paper presents an o n 2 log n algorithm for computing the symmetric singular value decomposition of square hankel matrices of order n in contrast with existing o n 3 svd algorithms the algorithm consists of two stages first a complex square hankel matrix is reduced to a complex symmetric tridiagonal matrix using the block lanczos method in o n 2 log n flops second the singular values and singular vectors of the symmetric tridiagonal matrix resulted from the first stage are computed in o n 2 flops the singular vector matrix is given in the form of a product of three or two unitary matrices the performance of our algorithm is demonstrated by comparing it with the svd subroutines in matlab and lapack

an implementation of the dqds algorithm positive case abstract the dqds algorithm was introduced in 1994 to compute singular values of bidiagonal matrices to high relative accuracy but it may also be used to compute eigenvalues of tridiagonal matrices this paper discusses in detail the issues that have to be faced when the algorithm is to be realized on a computer criteria for accepting a value for splitting the matrix and for choosing a shift to reduce the number of iterations as well as the relative advantages of using ieee arithmetic when available ways to avoid unnecessary over underflows are described in addition some new formulae are developed to approximate the smallest eigenvalue from a twisted factorization of a matrix the results of extensive testing are presented at the end the list of contents is a valuable guide to the reader interested in specific features of the algorithm

accurate ordering of eigenvectors and singular vectors without eigenvalues and singular values abstract the rank of an eigenvector of an unreduced real symmetric tridiagonal matrix can be determined by just knowing the signs of the elements of the eigenvector and the signs of the off diagonal entries of the tridiagonal matrix surprisingly no arithmetic operations involving real numbers are required to determine this ordinal count the absence of real arithmetic operations guarantees an error free algorithm thus it is possible to rank and order eigenvectors without knowing the corresponding eigenvalues it is known that the singular value decomposition svd of bidiagonal matrices are closely related to three tridiagonal eigenvalue problems using this connection it is possible to order singular vectors of bidiagonal matrices without knowing the singular values again no real arithmetic is needed the ordinal count is a theoretical result which is valid in exact arithmetic if eigenvectors and singular vectors are poorly determined in floating point arithmetic then the ordering procedure can detect faulty eigenpairs three standard symmetric eigenproblem routines and two svd routines from lapack are investigated to verify whether the ordinal counts give valid results for computed eigenvectors for some difficult problems such as the wilkinson w n matrix the ordinal counts disagree with that given by the routines for n 25 in general standard lapack routines appear to be reasonably resilient even though the routines are not particularly designed to be robust in this context for some classes of tridiagonals no failures were detected for n exceeding 3000 however one of the new lapack routines cannot be considered to be robust since it destroys information by thresholding small eigenvector elements to zero an svd routine also gave poor results for graded matrices this could be due to a bug many eigenvalue algorithms do a global sort of the eigenvalues however if the ordinal count is reliable then ranking of eigenvectors across processors can be achieved without global communication

global continua of periodic solutions of singular first order hamiltonian systems of n vortex type the paper deals with singular first order hamiltonian systems of the form begin aligned gamma k dot z k t j nabla z k h z t quad z k t in omega subset mathbb r 2 k 1 dots n end aligned where j in mathbb r 2 times 2 defines the standard symplectic structure in mathbb r 2 gamma 1 dots gamma n in mathbb r setminus 0 are given and the hamiltonian h is of n vortex type begin aligned h z 1 dots z n frac 1 2 pi sum genfrac 0 0pt j k 1 j ne k n gamma j gamma k log left z j z k right f z end aligned this is defined on the configuration space z 1 ldots z n in omega n z j ne z k text for j ne k of n different points in the domain omega subset mathbb r 2 the function f omega n rightarrow mathbb r may have additional singularities near the boundary of omega n we prove the existence of a global continuum of periodic solutions z t z 1 t dots z n t in omega n that emanates after introducing a suitable singular limit scaling from a relative equilibrium z t in mathbb r 2n of the n vortex problem in the whole plane where f 0 examples for z include thomson s vortex configurations or equilateral triangle solutions the domain omega need not be simply connected a special feature is that the associated action integral is not defined on an open subset of the space of 2 pi periodic h 1 2 functions the natural form domain for first order hamiltonian systems this is a consequence of the singular character of the hamiltonian our main tool in the proof is a degree for s 1 equivariant gradient maps that we adapt to this class of potential operators

an o n2 algorithm for the bidiagonal svd the relatively robust representations rrr algorithm computes the eigendecomposition of a symmetric tridiagonal matrix t with an o n2 complexity this article discusses how this method can be extended to the bidiagonal svd b u vt it turns out that using the rrr algorithm as a black box to compute btb v 2vt and bbt u 2ut separately may give poor results for bv u the use of the standard jordan wielandt representation can fail as well if clusters of tiny singular values are present a solution is to work on btb and to keep factorizations of bbt implicitly we introduce a set of coupling transformations which allow us to replace the representation u 1 bv by a more stable representation u xv where x is a diagonal matrix numerical results of our implementation are compared with the lapack routines dstegr dbdsqr and dbdsdc

desingularization of vortices for the euler equation we study the existence of stationary classical solutions of the incompressible euler equation in the planes that approximate singular stationary solutions of this equation the construction is performed by studying the asymptotics of equation varepsilon 2 delta u varepsilon u varepsilon q frac kappa 2 pi log frac 1 varepsilon p with dirichlet boundary conditions and q a given function we also study the desingularization of pairs of vortices by minimal energy nodal solutions and the desingularization of rotating vortices

n vortex equilibria for ideal fluids in bounded planar domains and new nodal solutions of the sinh poisson and the lane emden fowler equations we prove the existence of equilibria of the n vortex hamiltonian in a bounded domain omega subset mathbb r 2 which is not necessarily simply connected on an arbitrary bounded domain we obtain new equilibria for n 3 or n 4 if has an axial symmetry we obtain a symmetric equilibrium for each n in mathbb n we also obtain new stream functions solving the sinh poisson equation delta psi rho sinh psi in with dirichlet boundary conditions for 0 small the stream function psi rho induces a stationary velocity field v rho solving the euler equation in on an arbitrary bounded domain we obtain velocitiy fields having three or four counter rotating vortices if has an axial symmetry we obtain for each n a velocity field v rho that has a chain of n counter rotating vortices analogous to the mallier maslowe row of counter rotating vortices in the plane our methods also yield new nodal solutions for other semilinear dirichlet problems in particular for the lane emden fowler equation delta u u p 1 u in with p large

numerical conformal mapping of multiply connected regions by fornberg like methods we develop a new algorithm for computing conformal maps from regions exterior to non overlapping disks to unbounded multiply connected regions exterior to non overlapping smoothly bounded jordan regions the method is an extension of fornberg s original newton like method for mapping of the disk to simply connected regions a fortran program based on the algorithm has been developed and tested for the 2 and 3 disk case numerical examples are reported

construction of new integrable hamiltonians in two degrees of freedom a new procedure for deriving integrable hamiltonians and their constants of the motion is introduced we term this procedure the truncation program integrable hamiltonians occurring in the truncation program possess constants of the motion which are polynomials in a perturbation parameter e the relationship between this program and the whittaker program in two degrees of freedom is discussed integrable hamiltonians occurring in the whittaker program a generalization of whittaker s work possess constants of the motion which are polynomials in the momentum coordinates many previously known integrable hamiltonians are derived a new family of integrable double resonance hamiltonians and a new family of integrable hamiltonians of the form p21 p22 2 v q1 q2 are derived

force enhancing vortex equilibria for two parallel plates in uniform flow a two dimensional potential flow in an unbounded domain with two parallel plates is considered we examine whether two free point vortices can be trapped near the two plates in the presence of a uniform flow and observe whether these stationary point vortices enhance the force on the plates the present study is an extension of previously published work in which a free point vortex over a single plate is investigated the flow problem is motivated by an airfoil design problem for the double wings moreover it also contributes to a design problem for an efficient wind turbine with vertical blades in order to obtain the point vortex equilibria numerically we make use of a linear algebraic algorithm combined with a stochastic process called the brownian ratchet scheme the ratchet scheme allows us to capture a family of stationary point vortices in multiply connected domains with ease as a result we find that stationary point vortices exist around the two plates and they enhance the downward force and the counter clockwise rotational force acting on the two plates