an interpolating polynomial method for numerical conformal mapping a simple algorithm is presented for numerical approximation of conformal mappings for simply connected planar regions the desired mapping is represented by a normalized polynomial of degree n determined by the boundary values which it is presumed to take at the nth roots of unity then its values at the n intermediate 2nth roots of unity are used to correct the initial guess each iteration which consists essentially of matrix multiplication and boundary projection costs o n log n arithmetic operations numerical results are provided to confirm linear convergence of the algorithm

viscosity versus vorticity stretching global well posedness for a family of navier stokes alpha like models abstract we study global well posedness and regularity of solutions for a family of incompressible three dimensional navier stokes alpha like models that employ fractional laplacian operators this family of equations depends on two parameters 1 and 2 which affect the strength of non linearity vorticity stretching and the degree of viscous smoothing varying 1 and 2 interpolates between the incompressible navier stokes equations and the incompressible lagrangian averaged navier stokes model our main result which contains previously established results of j l lions and others provides a relationship between 1 and 2 that is sufficient to guarantee global existence uniqueness and regularity of solutions

accessory parameters in conformal mapping exploiting the isomonodromic tau function for painleve vi we present a novel method to solve the accessory parameter problem arising in constructing conformal maps from a canonical simply connected planar region to the interior of a circular arc quadrilateral the schwarz christoffel accessory parameter problem relevant when all sides have zero curvature is also captured within our approach the method exploits the isomonodromic tau function associated with the painleve vi equation recently these tau functions have been shown to be related to certain correlation functions in conformal field theory and asymptotic expansions have been given in terms of tuples of the young diagrams after showing how to extract the monodromy data associated with the target domain we show how a numerical approach based on the known asymptotic expansions can be used to solve the conformal mapping accessory parameter problem the viability of this new method is demonstrated by explicit examples and we discuss its extension to circular arc polygons with more than four sides

the accuracy of numerical conformal mapping methods a survey of examples and results this paper shows how the geometry of the region affects the conditioning and the accuracy of numerical conformal mapping methods for simply connected regions especially fourier series methods both explicit examples of popular test cases and more general estimates are discussed the severe ill conditioning that is known as the crowding phenomenon is discussed and its effect on a conformally transplanted boundary value problem is illustrated remarks on various numerical methods are included

a nonoverlapping domain decomposition method for symm s equation for conformal mapping symm s equation is a first kind integral equation for computing conformal maps of simply connected regions the package confpack solves symm s equation by an indirect boundary element method using an accurate corner representation this solution technique is extended here to include nonoverlapping domain decomposition degrees of freedom are introduced on one or more interfaces and different unknowns are used leading to a system of second kind equations the corresponding linear system can be expressed in schur complement form the accurate treatment of corners is preserved in the new formulation the results of serial and parallel matlab implementations of the new algorithm show significant speedups as the number of unknowns grows

eigenfrequencies of fractal drums a method for the computation of eigenfrequencies and eigenmodes of fractal drums is presented the approach involves first conformally mapping the unit disk to a polygon approximating the fractal and then solving a weighted eigenvalue problem on the unit disk by a spectral collocation method the numerical computation of the complicated conformal mapping was made feasible by the use of the fast multipole method as described in l banjai l n trefethen a multipole method for schwarz christoffel mapping of polygons with thousands of sides siam j sci comput 25 3 2003 1042 1065 the linear system arising from the spectral discretization is large and dense to circumvent this problem we devise a fast method for the inversion of such a system consequently the eigenvalue problem is solved iteratively we obtain eight digits for the first eigenvalue of the koch snowflake and at least five digits for eigenvalues up to the 20th numerical results for two more fractals are shown

numerical conformal mapping of circular arc polygons abstract howell l h numerical conformal mapping of circular arc polygons journal of computational and applied mathematics 46 1993 7 28 solutions to the schwarzian differential equation are conformal maps from the upper half plane to circular arc polygons plane regions bounded by straight line segments and arbitrary arcs of circles we develop methods for numerically integrating this equation both directly and through the use of a related linear differential equation particular attention is given to the behavior near corner singularities we also derive alternate versions of the transformation which map from the unit disk and from an infinite strip while the former may be of primarily theoretical interest the latter can be used to map highly elongated regions such as channels for internal flow problems such regions are difficult or impossible to map from the disk or the half plane due to the so called crowding phenomenon

on the connection between kolmogorov microscales and friction in pipe flows of viscoplastic fluids abstract the present work extends kolmogorov s micro scales to a large family of viscoplastic fluids the new micro scales combined with gioia and chakaborty s 2006 friction phenomenology theory lead to a unified framework for the description of the friction coefficient in turbulent flows a resulting blasius type friction equation is tested against some available experimental data and shows good agreement over a significant range of hedstrom and reynolds numbers the work also comments on the role of the new expression as a possible benchmark test for the convergence of dns simulations the formula also provides limits for the maximum drag reduction of viscoplastic flows

on a leray α model of turbulence in this paper we introduce and study a new model for three dimensional turbulence the leray model this model is inspired by the lagrangian averaged navier stokes model of turbulence also known navier stokes model or the viscous camassa holm equations as in the case of the lagrangian averaged navier stokes model the leray model compares successfully with empirical data from turbulent channel and pipe flows for a wide range of reynolds numbers we establish here an upper bound for the dimension of the global attractor the number of degrees of freedom of the leray model of the order of l l d 12 7 where l is the size of the domain and l d is the dissipation length scale this upper bound is much smaller than what one would expect for three dimensional models i e l l d 3 this remarkable result suggests that the leray model has a great potential to become a good sub grid scale large eddy simulation model of turbulence we support this observation by studying analytically and computationally the energy spectrum and show that in addition to the usual k 5 3 kolmogorov power law the inertial range has a steeper power law spectrum for wavenumbers larger than 1 finally we propose a prandtl like boundary layer model induced by the leray model and show a very good agreement of this model with empirical data for turbulent boundary layers

numerical conformal mapping using cross ratios and delaunay triangulation we propose a new algorithm for computing the riemann mapping of the unit disk to a polygon also known as the schwarz christoffel transformation the new algorithm crdt for cross ratios of the delaunay triangulation is based on cross ratios of the prevertices and also on cross ratios of quadrilaterals in a delaunay triangulation of the polygon the crdt algorithm produces an accurate representation of the riemann mapping even in the presence of arbitrary long thin regions in the polygon unlike any previous conformal mapping algorithm we believe that crdt solves all difficulties with crowding and global convergence although these facts depend on conjectures that we have so far not been able to prove we demonstrate convergence with computational experiments the riemann mapping has applications in two dimensional potential theory and mesh generation we demonstrate crdt on problems in long thin regions in which no other known algorithm can perform comparably

conjugate function method for numerical conformal mappings we present a method for numerical computation of conformal mappings from simply or doubly connected domains onto so called canonical domains which in our case are rectangles or annuli the method is based on conjugate harmonic functions and properties of quadrilaterals several numerical examples are given

generalized intelligent states and squeezing the robertson schrodinger uncertainty relation for two observables a and b is shown to be minimized in the eigenstates of the operator a ib being a complex number such states called generalized intelligent states gis can exhibit arbitrarily strong squeezing of a or b the time evolution of gis is stable for hamiltonians which admit linear in a and b invariants systems of gis for the su 1 1 and su 2 groups are constructed and discussed it is shown that su 1 1 gis contain all the perelomov coherent states cs and the barut and girardello cs while the spin cs are a subset of su 2 gis cs for an arbitrary semisimple lie group can be considered as a gis for the quadratures of the weyl generators

the camassa holm equations and turbulence abstract in this paper we will survey our results on the camassa holm equations and their relation to turbulence as discussed in s chen c foias d d holm e olson e s titi s wynne the camassa holm equations as a closure model for turbulent channel and pipe flow phys rev lett 81 1998 5338 s chen c foias d d holm e olson e s titi s wynne a connection between the camassa holm equations and turbulent flows in channels and pipes phys fluids in press in particular we will provide a more detailed mathematical treatment of those equations for pipe flows which yield accurate predictions of turbulent flow profiles for very large reynolds numbers there are many facts connecting the camassa holm equations to turbulent fluid flows the dimension of the attractor agrees with the heuristic argument based on the kolmogorov statistical theory of turbulence the statistical properties of the energy spectrum agree in numerical simulation with the kolmogorov power law furthermore comparison of mean flow profiles for turbulent flow in channels and pipes given by experimental and numerical data show acceptable agreement with the profile of the corresponding solution of the camassa holm equations

revisiting the crowding phenomenon in schwarz christoffel mapping we address the problem of conformally mapping the unit disk to polygons with elongations the elongations cause the derivative of the conformal map to be exponentially large in some regions this crowding phenomenon creates difficulties in standard numerical methods for the computation of the conformal map we make use of the schwarz christoffel representation of the mapping and show that a simple change to the existing algorithms introduced by trefethen siam j sci statist comput 1 1980 pp 82 102 makes it feasible to accurately compute conformal maps to polygons even in the presence of extreme crowding for an efficient algorithm it is essential that a good initial guess for the parameters of the schwarz christoffel map be available a uniformly close initial guess can be obtained from the cross ratios of certain quadrilaterals as introduced in the crdt algorithm of driscoll and vavasis siam j sci comput 19 1998 pp 1783 1803 we present numerical experiments and compare our algorithms with the crdt which has been particularly designed to combat crowding

algebraic aspects of quantum statistics consistent implementation of the additivity of physical observables in second quantization uniquely establishes quantum statistics in this scheme the usual weyl heisenberg algebra induces maxwell boltzmann distribution while bose einstein statistics are related to su 1 1

representations and properties of generalized ar statistics coherent states and robertson uncertainty relations the generalization of ar statistics including bosonic and fermionic sectors is performed by means of the so called jacobson generators the corresponding fock spaces are constructed the bargmann representations are also considered for the bosonic ar statistics two inequivalent bargmann realizations are developed the first resp second realization induces in a natural way coherent states recognized as gazeau klauder resp klauder perelomov ones in the fermionic case the bargamnn realization leads to the klauder perelomov coherent states for each considered realization the inner product of two analytic functions is defined with respect to a measure explicitly computed the jacobson generators are realized as differential operators it is shown that the obtained coherent states minimize the robertson schr dinger uncertainty relation

representation of conformal maps by rational functions the traditional view in numerical conformal mapping is that once the boundary correspondence function has been found the map and its inverse can be evaluated by contour integrals we propose that it is much simpler and 10 1000 times faster to represent the maps by rational functions computed by the aaa algorithm to justify this claim first we prove a theorem establishing root exponential convergence of rational approximations near corners in a conformal map generalizing a result of d j newman in 1964 this leads to the new algorithm for approximating conformal maps of polygons then we turn to smooth domains and prove a sequence of four theorems establishing that in any conformal map of the unit circle onto a region with a long and slender part there must be a singularity or loss of univalence exponentially close to the boundary and polynomial approximations cannot be accurate unless of exponentially high degree this motivates the application of the new algorithm to smooth domains where it is again found to be highly effective

robertson intelligent states the diagonalization of the uncertainty matrix and the minimization of robertson inequality for n observables are considered it is proved that for even n this relation is minimized in states which are eigenstates of n 2 independent complex linear combinations of the observables in the case of canonical observables this eigenvalue condition is also necessary such minimizing states are called robertson intelligent states ris the group related coherent states cs with maximal symmetry for semisimple lie groups are a particular case of ris for the quadratures of weyl generators explicit constructions of ris are considered for operators of su 1 1 su 2 and sp n r algebras unlike the group related cs ris can exhibit strong squeezing of group generators multimode squared amplitude squeezed states are naturally introduced as sp n r ris it is shown that the uncertainty matrices for quadratures of q deformed boson operators q 0 and of any k power of are positive definite and can be diagonalized by symplectic linear transformations

a dynamical uncertainty principle in von neumann algebras by operator monotone functions suppose that a1 an are observables selfadjoint matrices and is a state density matrix in this case the standard uncertainty principle proved by robertson gives a bound for the quantum generalized variance namely for det cov aj ak using the commutators aj ak this bound is trivial when n is odd recently a dierent inequality of robertson type has been proved by the authors with the help of the theory of operator monotone functions in this case the bound makes use of the commutators a j and is non trivial for any n in the present paper we generalize this new result to the von neumann algebra case nevertheless the proof appears to simplify all the existing ones

on fornberg s numerical method for conformal mapping it is shown that fornberg s method 1 is a discretized version of an iterative process which is essentially the same we introduced and studied in 7 8 the main difference consists in the determination of the shift u k in each iterative step while we give an explicit representation of u k fornberg determines u k from a linear equation which involves a certain integral operator r we prove some properties and estimates for this operator which explain several experimental results reported in 1 especially the fact that the conjugate gradient method is so efficient the results of 8 concerning the convergence can be applied also for fornberg s method finally we give estimates for the numerical error

representation and properties of para bose oscillator operators ii coherent states and the minimum uncertainty states the energy position and momentum eigenstates of a para bose oscillator system were considered in paper i here we consider the bargmann or the analytic function description of the para bose system this brings in in a natural way the coherent states z defined as the eigenstates of the annihilation operator the transformation functions relating this description to the energy position and momentum eigenstates are explicitly obtained possible resolution of the identity operator using coherent states is examined a particular resolution contains two integrals one containing the diagonal basis z z and the other containing the pseudodiagonal basis z z we briefly consider the normal and antinormal ordering of the operators and their diagonal and discrete diagonal coherent state approximations the problem of constructing states with a minimum value of the product of the position and momentum uncertainties and the possible dependence of this minimum value is considered

spectral scaling of the leray α model for two dimensional turbulence we present data from high resolution numerical simulations of the navier stokes and the leray models for two dimensional turbulence it was shown previously lunasin et al 2007 j turbul 8 30 that for wavenumbers k such that k 1 the energy spectrum of the smoothed velocity field for the two dimensional navier stokes ns model scales as k 7 t his result is in agreement with the scaling deduced by dimensional analysis of the flux of the conserved enstrophy using its characteristic time scale we therefore hypothesize that the spectral scaling of any model in the sub spatial scales must depend only on the characteristic time scale and dynamics of the dominant cascading quantity in that regime of scales the data presented here from simulations of the two dimensional leray model confirm our hypothesis we show that for k 1 the energy spectrum for the twodimensional leray scales as k 5 as expected by the characteristic time scale for the flux of the conserved enstrophy of the leray model these results lead to our conclusion that the dominant directly cascading quantity of the model equations must determine the scaling of the energy spectrum

solving theodorsen s integral equation for conformal maps with the fast fourier transform and various nonlinear iterative methods we investigate several iterative methods for the numerical solution of theodorsen s integral equation the discretization of which is either based on trigonometric polynomials or function families with known attenuation factors all our methods require simultaneous evaluations of a conjugate periodic function at each step and allow us to apply the fast fourier transform for this in particular we discuss the nonlinear jor iteration the nonlinear sor iteration a nonlinear second order euler iteration the nonlinear chebyshev semi iterative method and its cyclic variant under special symmetry conditions for the region to be mapped onto we establish local convergence in the case of discretization by trigonometric interpolation and give simple formulas for the optimal parameters e g the underrelaxation factor and the asymptotic convergence factor weaker related results for the general non symmetric case are presented too practically our methods extend the range of application of theodorsen s method and improve its effectiveness strikingly

coherent states for systems with discrete and continuous spectrum for a system with one degree of freedom coherent states that are parametrized by classical canonical action angle variables are introduced these states also possess continuity of labelling a resolution of unity and temporal stability the insistence on canonical action angle variables strongly restricts any remaining arbitrariness in the coherent state definition such states are introduced for semibounded hamiltonian operators having either a discrete or a continuous spectrum hamiltonians that have both discrete and continuous parts in their spectrum are also discussed

a modified schwarz christoffel mapping for regions with piecewise smooth boundaries a method where polygon corners in schwarz christoffel mappings are rounded is used to construct mappings from the upper half plane to regions bounded by arbitrary piecewise smooth curves from a given curve a polygon is constructed by taking tangents to the curve in a number of carefully chosen so called tangent points the schwarz christoffel mapping for that polygon is then constructed and modified to round the corners since such a modification causes effects on the polygon outside the rounded corners the parameters in the mapping have to be redetermined this is done by comparing side lengths in tangent polygons to the given curve and the curve produced by the modified schwarz christoffel mapping the set of equations that this comparison gives can normally be solved using a quasi newton method the resulting function maps the upper half plane on a region bounded by a curve that apart from possible vertices is smooth i e one time continuously differentiable that passes through the tangent points on the given curve has the same direction as the given curve in these points and changes direction monotonically between them furthermore where the original curve has a vertex the constructed curve has a vertex with the same inner angle the method is especially useful for unbounded regions with smooth boundary curves that pass infinity as straight lines such as channels with parallel walls at the ends these properties are kept in the region produced by the constructed mapping

convergence proofs and error estimates for an iterative method for conformal mapping the iterative method as introduced in 8 and 9 for the determination of the conformal mapping of the unit disc onto a domaing is here described explicitly in terms of the operatork which assigns to a periodic functionu its periodic conjugate functionk u it is shown that whenever the boundary curve ofg is parametrized by a function with lipschitz continuous derivative dot eta ne 0 then the method converges locally in the sobolev spacew of 2 periodic absolutely continuous functions with square integrable derivative if is in a holder classc 2 the order of convergence is at least 1 if is inc l 1 withl 1 0 1 then the iteration converges inc l for analytic boundary curves the convergence takes place in a space of analytic functions for the numerical implementation of the method the operatork can be approximated by wittich s method which can be applied very effectively using fast fourier transform the sobolev norm of the numerical error can be estimated in terms of the numbern of grid points it iso n 1 l if is inc l 1 ando exp n 2 if is an analytic curve the number in the latter formula is bounded by logr wherer is the radius of the largest circle into which can be extended analytically such that z 0 for z r the results of some test calculations are reported

conformal mappings to a doubly connected polycircular arc domain the explicit construction of the conformal mapping of a concentric annulus to a doubly connected polygonal domain was first reported by akhiezer in 1928 the construction of an analogous formula for the case of a polycircular arc domain i e for a doubly connected domain whose boundaries are a union of circular arc segments has remained an important open problem in this paper we present this explicit formula we first introduce a new method for deriving the classical formula of akhiezer and then show how to generalize the method to the case of a doubly connected polycircular arc domain as an analytical check of the formula a special exact solution for a doubly connected polycircular arc mapping is derived and compared with that obtained from the more general construction as an illustrative example a doubly connected polycircular arc domain arising in a classic potential flow problem considered in the last century by lord rayleigh is considered in detail

a robertson type uncertainty principle and quantum fisher information let a1 an be complex self adjoint matrices and let be a density matrix the robertson uncertainty principle det cov ah aj det i 2 tr a h aj gives a bound for the quantum generalized variance in terms of the commutators ah aj the right side matrix is antisymmetric and therefore the bound is trivial equal to zero in the odd case n 2m 1 let f be an arbitrary normalized symmetric operator monotone function and let h i f be the associated quantum fisher information we have conjectured the inequality det cov ah aj det f 0 hi a h i a j i f

discrete riemann hilbert problems interpolation of simply closed curves and numerical conformal mapping abstract let g be a simply connected region with 0 g and with a twice lipschitz continuously differentiable boundary curve and let z 1 n be an even number of n 2 n equidistant grid points on the unit circle sfnc z sfnc 1 with z 1 1 then there exists for all sufficiently large n a polynomial p n of degree n 1 normalized by the condition that the coefficient p 0 0 and the coefficients p 1 and p n 1 are real such that p n satisfies the interpolation condition p n z for all 1 n in a neighbourhood of the normalized conformal mapping function there is exactly one such interpolating polynomial the sequence of these p n converges to the conformal mapping function as n if is three times lipschitz continuously differentiable then the p n are also conformal mappings of the unit circle onto regions which approximate g an important tool for the theoretical investigation is a discrete analogon of the riemann hilbert problem we present two fast procedures for the numerical solution of the discrete riemann hilbert problem a conjugate gradient method with computational cost o n log n and a toeplitz matrix method with cost o n log 2 n using this one can calculate the interpolating polynomials by a newton method and in this way obtains very effective methods for the numerical approximation of the conformal mapping function

on a regularization of a scalar conservation law with discontinuous coefficients this paper is devoted to a scalar conservation law with a linear flux function involving discontinuous coefficients it is clear that the delta standing wave should be introduced into the riemann solution in some nonclassical situation in order to study the formation of delta standing wave we consider a regularization of the discontinuous coefficient with the helmholtz mollifier and then obtain a regularized system which depends on a regularization parameter 0 the regularization mechanism is a nonlinear bending of characteristic curves that prevents their finite time intersection it is proved rigorously that the solutions of regularized system converge to the delta standing wave solution in the 0 limit compared with the classical method of vanishing viscosity here it is clear to see how the delta standing wave forms naturally along the characteristics

on moduli of rings and quadrilaterals algorithms and experiments moduli of rings and quadrilaterals are frequently applied in geometric function theory see e g the handbook by kuhnau handbook of complex analysis geometric function theory vols 1 and 2 north holland amsterdam 2005 yet their exact values are known only in a few special cases previously the class of planar domains with polygonal boundary has been studied by many authors from the point of view of numerical computation we present here a new hp fem algorithm for the computation of moduli of rings and quadrilaterals and compare its accuracy and performance with previously known methods such as the schwarz christoffel toolbox of driscoll and trefethen we also demonstrate that the hp fem algorithm applies to the case of nonpolygonal boundary and report results with concrete error bounds

schwarz christoffel mappings a general approach abstract a general method is developed for constructing equations of the schwarz christoffel type these equations define conformal transformations between regions of the complex plane each of whose boundaries may consist of straight or continuously curved line segments and corners the more familiar types of schwarz christoffel equations are rederived and some new types are obtained

example of infinite statistics i give a second quantized example of infinite statistics in which any representation of the symmetric group can occur for identical particles and discuss the properties of the example

sub grid effects of the voigt viscoelastic regularization of a singular dyadic model of turbulence in this work we investigate the spectral signature of navier stokes voigt nsv viscoelastic fluid flows by employing numerical simulations of a singular dyadic shell model our results clearly show that as the relaxation time is increased above a threshold the inertial range is reduced conserving part of the large scale statistics these results differ drastically from the two power law scenarios observed in a previous work where the nsv model was studied via sabra shell model simulations instead we also show that the additional elastic term regularizes the singular dyadic model which is the main reason behind this reduction of degrees of freedom the results of this work aim at proposing the nsv regularization as a sub grid model

modified schwarz christoffel mappings using approximate curve factors the schwarz christoffel mapping from the upper half plane to a polygonal region in the complex plane is an integral of a product with several factors where each factor corresponds to a certain vertex in the polygon different modifications of the schwarz christoffel mapping in which factors are replaced with the so called curve factors to achieve polygons with rounded corners are known since long times among other requisites the arguments of a curve factor and its correspondent scl factor must be equal outside some closed interval on the real axis in this paper the term approximate curve factor is defined such that many of the already known curve factors are included as special cases additionally by alleviating the requisite on the argument from exact to asymptotic equality new types of curve factors are introduced while traditional curve factors have a c 1 regularity c regular approximate curve factors can be constructed resulting in smooth boundary curves when used in conformal mappings applications include modelling of wave scattering in waveguides when using approximate curve factors in modified schwarz christoffel mappings numerical conformal mappings can be constructed that preserve two important properties in the waveguides first the direction of the boundary curve can be well controlled especially towards infinity where the application requires two straight parallel walls second a smooth c boundary curve can be achieved

well posedness and a numerical study of a regularization model with adaptive nonlinear filtering for incompressible fluid flow we give a detailed analytical study of a leray model of incompressible flow that uses nonlinear filtering based on indicator functions the indicator functions allow for local regularization instead of global regularization which can over smooth and dampen out important flow structures the key to the analysis is the identification of the indicator function as a nemyskii operator after proving well posedness we provide a numerical study which includes proving optimal convergence of finite element method for the model as well as several numerical experiments