a fast conformal mapping algorithm with no fft abstract an algorithm is presented for the computation of a conformal mapping discretized on a non uniformly spaced point set useful for the numerical solution of many problems of fluid dynamics most existing iterative techniques both those having a linear and those having a quadratic type of convergence rely on the fast fourier transform fft algorithm for calculating a convolution integral which represents the most time consuming phase of the computation the fft however definitely cannot be applied to a non uniform spacing the algorithm presented in this paper has been made possible by the construction of a calculation method for convolution integrals which despite not using an fft maintains a computation time of the same order as that of the fft the new technique is successfully applied to the problem of conformally mapping a closely spaced cascade of airfoils onto a circle which requires an exceedingly large number of points if it is solved with uniform spacing

numerical conformal mapping based on the generalised conjugation operator an iterative procedure for numerical conformal mapping is presented which imposes no restriction on the boundary complexity the formulation involves two analytically equivalent boundary integral equations established by applying the conjugation operator to the real and the imaginary parts of an analytical function the conventional approach is to use only one and ignore the other equation however the discrete version of the operator using the boundary element method bem leads to two non equivalent sets of linear equations forming an over determined system the generalised conjugation operator is introduced so that both sets of equations can be utilised and their least square solution determined without any additional computational cost a strategy largely responsible for the stability and efficiency of the proposed method numerical tests on various samples including problems with cracked domains suggest global convergence although this cannot be proved theoretically the computational efficiency appears significantly higher than that reported earlier by other investigators

an iterative method for conformal mapping abstract in this paper we discuss an iterative method for the calculation of the boundary values of the conformal mapping of a simply connected region g onto a region h with smooth boundary the method is based on a certain riemann hilbert problem it turns out that this problem is the linearized version of a singular integral equation of the second kind hence the method is a newton method whenever the boundaries of g and h are sufficiently smooth its convergence is locally quadratic if g is the unit circle the solution of the linearized problem can be represented explicitly in terms of integral transforms from this one derives a quadratically convergent newton like numerical method that avoids the numerical solution of systems of linear equations and therefore in comparison with other methods based on integral equations is quite economical in terms of computer time and storage requirements

conformal grid generation for high aspect ratio simply and doubly connected regions conformal co ordinate transformations are used to map rectangular computational domains onto arbitrary simply and doubly connected regions with smooth boundaries the efficient numerical schemes of wegmann involving the solution of the inverse boundary correspondence function problems associated with the mapping of the unit disc or circular annulus onto simply or doubly connected domains respectively are employed the numerical implementation of these schemes is emphasized examples are generated for regions with elliptic inner and outer boundaries additional examples are used to demonstrate the accuracy and convergence of the schemes and their practical limitations the techniques are found to converge well if holomorphic functions are used to describe the boundaries the use of preconditioning maps is also discussed

portfolio selection under possibilistic mean variance utility and a smo algorithm in this paper we propose a new portfolio selection model with the maximum utility based on the interval valued possibilistic mean and possibilistic variance which is a two parameter quadratic programming problem we also present a sequential minimal optimization smo algorithm to obtain the optimal portfolio the remarkable feature of the algorithm is that it is extremely easy to implement and it can be extended to any size of portfolio selection problems for finding an exact optimal solution

flow around simply and multiply connected bodies a new iterative scheme for conformal mapping discussion des techniques de representation conforme numerique des profils aerodynamiques des cascades soit pour le calcul des champs d ecoulement irrotationnel soit pour la construction de grilles de calcul numerique presentation d un algorithme iteratif a convergence quadratique

a general framework for solving riemann hilbert problems numerically a new numerical framework for the approximation of solutions to matrix valued riemann hilbert problems is developed based on a recent method for the homogeneous painleve ii riemann hilbert problem we demonstrate its effectiveness by computing solutions to other painleve transcendents an implementation in mathematica is made available online

numerical conformal mapping methods for simply and doubly connected regions methods are presented and analyzed for approximating the conformal map from the interior exterior of the disk to the interior exterior of a smooth simple closed curve and from an annulus to a bounded doubly connected region with smooth boundaries the methods are newton like methods for computing the boundary correspondences and conformal moduli similar to fornberg s method for the interior of the disk we show that the linear systems are discretizations of the identity plus a compact operator and hence that the conjugate gradient method converges superlinearly

portfolio optimization with an envelope based multi objective evolutionary algorithm the problem of portfolio selection is a standard problem in financial engineering and has received a lot of attention in recent decades classical mean variance portfolio selection aims at simultaneously maximizing the expected return of the portfolio and minimizing portfolio variance in the case of linear constraints the problem can be solved efficiently by parametric quadratic programming i e variants of markowitz critical line algorithm however there are many real world constraints that lead to a non convex search space e g cardinality constraints which limit the number of different assets in a portfolio or minimum buy in thresholds as a consequence the efficient approaches for the convex problem can no longer be applied and new solutions are needed in this paper we propose to integrate an active set algorithm optimized for portfolio selection into a multi objective evolutionary algorithm moea the idea is to let the moea come up with some convex subsets of the set of all feasible portfolios solve a critical line algorithm for each subset and then merge the partial solutions to form the solution of the original non convex problem we show that the resulting envelope based moea significantly outperforms existing moeas

randomly generating portfolio selection covariance matrices with specified distributional characteristics abstract in portfolio selection there is often the need for procedures to generate realistic covariance matrices for security returns for example to test and benchmark optimization algorithms for application in portfolio optimization such a procedure should allow the entries in the matrices to have distributional characteristics which we would consider realistic for security returns deriving motivation from the fact that a covariance matrix can be viewed as stemming from a matrix of factor loadings a procedure is developed for the random generation of covariance matrices a whose off diagonal covariance entries possess a pre specified expected value and standard deviation and b whose main diagonal variance entries possess a likely different pre specified expected value and standard deviation the paper concludes with a discussion about the futility one would likely encounter if one simply tried to invent a valid covariance matrix in the absence of a procedure such as in this paper

frontiers of stochastically nondominated portfolios we consider the problem of constructing a portfolio of finitely many assets whose returns are described by a discrete joint distribution we propose mean risk models that are solvable by linear programming and generate portfolios whose returns are nondomi nated in the sense of second order stochastic dominance next we develop a specialized parametric method for recovering the entire mean risk efficient frontiers of these models and we illustrate its operation on a large data set involving thousands of assets and real izations the problem of optimizing a portfolio of finitely many assets is a classical problem in theoretical and computational finance since the seminal work of markowitz 1952 it is generally agreed that portfolio performance should be measured in two distinct dimen sions the mean describing the expected return and the risk which measures the uncer tainty of the return in the mean risk approach we select from the universe of all possible portfolios those that are efficient for a given value of the mean they minimize the risk or equivalently for a given value of risk they maximize the mean such an approach has many advantages it allows one to formulate the problem as a parametric optimization problem and it facilitates the trade off analysis between mean and risk markowitz used the variance of the return as the measure of the risk it is easy to compute and it reduces the portfolio selection problem to a parametric quadratic pro gramming problem one can however construct simple counterexamples that show the imperfection of the variance as the risk measure it treats over performance equally as under performance and more importantly its use may suggest a portfolio that is always outperformed by another portfolio the use of the semivariance rather than the variance was already recommended by markowitz 1959 but even in this case significant deficien cies remain as we shall explain another theoretical approach to the portfolio selection problem is that of stochastic dominance see whitmore and findlay 1978 levy 1992 the usual first order defi nition of stochastic dominance gives a partial order in the space of real random variables more important from the portfolio point of view is the notion of second order dominance which is also defined as a partial order but which is equivalent to this statement a random variable y dominates the random variable z if e u y e u z for all nondecreasing concave functions u for which these expected values are finite thus no risk averse decision marker will prefer a portfolio with return z over a portfolio with return y while

the structure of the tautological ring in genus one we prove getzler s claims about the cohomology of the moduli space of stable curves of genus one that is that the even cohomology ring is spanned by the strata classes and that all relations between these classes follow from the witten dijkgraaf verlinde verlinde wdvv relation and getzler s relation in particular the even cohomology ring is isomorphic to the tautological ring

large scale mv efficient frontier computation via a procedure of parametric quadratic programming abstract despite the volume of research conducted on efficient frontiers in many cases it is still not the easiest thing to compute a mean variance mv efficient frontier even when all constraints are linear this is particularly true of large scale problems having dense covariance matrices and hence they are the focus in this paper because standard approaches for constructing an efficient frontier one point at a time tend to bog down on dense covariance matrix problems with many more than about 500 securities we propose as an alternative a procedure of parametric quadratic programming for more effective usage on large scale applications with the proposed procedure we demonstrate through computational results on problems in the 1000 3000 security range that the efficient frontiers of dense covariance matrix problems in this range are now not only solvable but can actually be computed in quite reasonable time

fast conformal mapping of multiply connected regions abstract an iterative method is presented which constructs for an unbounded region g with m holes and sufficiently smooth boundary a circular region h and a conformal mapping from h to g with the usual normalization both h and are uniquely determined by g with a few modifications the method can also be applied to a bounded region g with m holes the canonical region h is then the unit disc with m circular holes the proposed method also determines the centers and radii of the boundary circles of h and requires at each iterative step the solution of a riemann hilbert rh problem which has a unique solution numerically the rh problem can be treated efficiently by the method of successive conjugation using the fast fourier transform fft the iteration for the solution of the rh problem converges linearly the conformal mapping method converges quadratically the results of some test calculations exemplify the performance of the method

a simplified fornberg like method for the conformal mapping of multiply connected regions comparisons and crowding we present a new fornberg like method for the numerical conformal mapping of multiply connected regions exterior to circles to multiply connected regions exterior to smooth curves the method is based on new symmetric conditions for analytic extension of functions given on circular boundaries we also briefly discuss a similar method due to wegmann and compare some computations with both methods some examples of regions which exhibit crowding of the circles are also presented

numerical conformal mapping methods based on function conjugation abstract a unifying treatment of methods for computing conformal maps from the unit disk onto a jordan region is presented integral and integro differential equations involving the conjugation operator for the boundary correspondence function are first derived using an arbitrary auxiliary function having certain properties then various iteration methods for solving these equations are described in this generality so that the basic ideas become manifest specific methods are then treated as examples of the general theory among them are in particular the successive conjugation methods of theodorsen melentiev and kulisch timman and friberg the projection method of bergstrom and the newton methods of vertgeim wegmann and hubner which make use of the easy construction of the solutions of riemann hilbert problems many of these methods are treated in greater generality than in the literature the connections with the methods of fornberg menikoff zemach chakravarthy anderson and challis burley are also outlined

mean absolute deviation portfolio optimization model and its applications to tokyo stock market the purpose of this paper is to demonstrate that a portfolio optimization model using the l1 risk mean absolute deviation risk function can remove most of the difficulties associated with the classical markowitz s model while maintaining its advantages over equilibrium models in particular the l1 risk model leads to a linear program instead of a quadratic program so that a large scale optimization problem consisting of more than 1 000 stocks may be solved on a real time basis numerical experiments using the historical data of nikkei 225 stocks show that the l1 risk model generates a portfolio quite similar to that of the markowitz s model within a fraction of time required to solve the latter

an iterative method for the conformal mapping of doubly connected regions an iterative method is described for the determination of the conformal mapping of a circular annulus rq z q z 1 onto a doubly connected region g with smooth boundary curves the method is analogous to the method of 7 for simply connected regions it requires in each step the solution of a riemann hilbert problem on rq this problem can be solved explicitly in terms of a generalized conjugation operator k a degeneracy of this problem leads in a natural way to an adjustment of the parameter q in each step if the boundary curves of g admit parametrizations by functions with second derivatives which are holder continuous with exponent 0 1 then the method converges locally in the sobolev space w of 2 periodic absolutely continuous functions with square integrable derivatives the order of convergence is at least 1 the convergence is quadratic if the have lipschitz continuous second derivatives for the numerical implementation of the method the operator k can be approximated by a wittich type method which can be performed very effectively using fft a calculation with n 2m grid points on the computer requires storage of the order o n and computing time o mn the results of some test calculations are reported and compared with results of calculations with the method of theodorsen garrick

recursive numerical algorithm for conformal mapping in two dimensional hydrodynamics an original recursive numerical method rt algorithm for conformal mapping is presented based on the lavrentiev variational principle it is a powerful means for the solution of various two dimensional boundary value problems the rt algorithm can be used to solve internal and external hydrodynamics problems occurring in spacecraft research namely the calculation of fluid motions with local vorticity regions for external problems conformal mapping of the fluid flow area onto the outside of the unit circle is constructed for internal problems the area is mapped onto one of standard domains the unit circle or circular annulus fluid flows in a cylindrical tank with inner radial damping ribs and nearby an air brake with sharp edges with or without free eddies are discussed as examples

constructive solution of a certain class of riemann hilbert problems on multiply connected circular regions abstract a class of riemann hilbert rh problems on multiply connected circular regions is treated iteratively by a method of successive conjugation sc which solves the problem on a single boundary circle each time running through all circles in each iterative step it is shown that the method converges convergence is linear with a rate which can be calculated with an eigenvalue problem the rh problems considered have negative index in order to fulfill the solvability conditions free parameters are introduced in form of polynomials the proof of convergence of the sc method shows at the same time that the rh problems with this specification of the free parameters have unique solutions rh problems of this kind are of practical use in conformal mapping problems of multiply connected regions to circular regions they play a role in the mapping of nearly circular regions lavrentev s principle there is also a close relationship to koebe s iterative method for the calculation of conformal mappings the properties of rh problems are different for unbounded and bounded regions the sc iteration must be adapted accordingly the performance of the sc iteration is illustrated with several examples

genetic algorithms for portfolio selection problems with minimum transaction lots abstract conventionally portfolio selection problems are solved with quadratic or linear programming models however the solutions obtained by these methods are in real numbers and difficult to implement because each asset usually has its minimum transaction lot methods considering minimum transaction lots were developed based on some linear portfolio optimization models however no study has ever investigated the minimum transaction lot problem in portfolio optimization based on markowitz model which is probably the most well known and widely used based on markowitz model this study presents three possible models for portfolio selection problems with minimum transaction lots and devises corresponding genetic algorithms to obtain the solutions the results of the empirical study show that the portfolios obtained using the proposed algorithms are very close to the efficient frontier indicating that the proposed method can obtain near optimal and also practically feasible solutions to the portfolio selection problem in an acceptable short time one model that is based on a fuzzy multi objective decision making approach is highly recommended because of its adaptability and simplicity

intersection homology ii in 19 20 we introduced topological invariants ih x called intersection homology groups for the study of singular spaces x these groups depend on the choice of a perversity p a perversity is a function from 2 3 to the non negative integers such that both c and c 2 c are positive and increasing functions of c 2 1 the group ihr is defined for spaces x called pseudomanifolds a pseudomanifold of dimension n is a space that admits a stratification x x n x n z x n 3 x 1 x 0

a reference direction approach to multiple objective quadratic linear programming in this paper we propose an interactive procedure for solving multiple criteria problems with one quadratic objective several linear objectives and a set of linear constraints the procedure is based on the use of reference directions and weighted sums reference directions for the linear functions and weighted sums for combining the quadratic function with the linear ones are used as parameters to implement the free search of nondominated solutions the idea leads to the parametric linear complementarity problem formulation an approach to deal with this type of problems is given as well the approach is illustrated with a numerical example

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

motivic random variables and representation stability ii hypersurface sections abstract we prove geometric and cohomological stabilization results for the universal smooth degree d hypersurface section of a fixed smooth projective variety as d goes to infinity we show that relative configuration spaces of the universal smooth hypersurface section stabilize in the completed grothendieck ring of varieties and deduce from this the stabilization of the hodge euler characteristic of natural families of local systems constructed from the vanishing cohomology we prove explicit formulas for the stable values using a probabilistic interpretation along with the natural analogs in point counting over finite fields we explain how these results provide new geometric examples of a weak version of representation stability for symmetric symplectic and orthogonal groups this interpretation of representation stability was studied in the prequel 20 for configuration spaces

numerical conformal mapping onto a rectangle with applications to the solution of laplacian problems abstract let f be the function which maps conformally a simply connected domain onto a rectangle r so that four specified points on are mapped respectively onto the four vertices of r in this paper we consider the problem of approximating the conformal map f and present a survey of the available numerical methods we also illustrate the practical significance of the conformal map by presenting a number of applications involving the solution of laplacian boundary value problems

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

hodge genera of algebraic varieties i the aim of this paper is to study the behavior of hodge theoretic intersection homology genera and their associated characteristic classes under proper morphisms of complex algebraic varieties we obtain formulae that relate parametrized families of global invariants of a complex algebraic variety x to such invariants of singularities of proper algebraic maps defined on x such formulae severely constrain both topologically and analytically the singularities of complex maps even between smooth varieties similar results were announced by the first and third author in 13 32 2007 wiley periodicals inc

kernel search an application to the index tracking problem abstract in this paper we study the problem of replicating the performances of a stock market index i e the so called index tracking problem and the problem of out performing a market index i e the so called enhanced index tracking problem we introduce mixed integer linear programming milp formulations for these two problems furthermore we present a heuristic framework called kernel search we analyze and evaluate the behavior of several implementations of the kernel search framework to the solution of the index tracking problem we show the effectiveness and efficiency of the framework comparing the performances of these heuristics with those of a general purpose solver the computational experiments are carried out using benchmark and newly created instances

a two phase algorithm for the multiparametric linear complementarity problem abstract a new two phase method for solving the multi parametric linear complementarity problem mplcp with sufficient matrices is presented in the first phase an initial feasible solution to mplcp which satisfies certain criteria is determined in the second phase the set of feasible parameters is partitioned into polyhedral regions such that the solution of the mplcp as a function of the parameters is invariant over each region the worst case complexity of the presented algorithms matches that of current methods for nondegenerate problems and is lower than that of current methods for degenerate problems