towards new solutions for the general hurwitz problem a formulation of the generalized hurwitz problem is given the authors shown that the solution of this problem is equivalent to exhibiting a matrix extending yiu s intercalate matrix 1987 consistently signed they give examples of explicit solutions for such a generalized hurwitz problem in low dimensions

analytical positon negaton and complexiton solutions for the coupled modified kdv system new analytical positon negaton and complexiton solutions for the coupled modified kdv system are constructed by means of the darboux transformation with a constant seed solution and given out both analytically and graphically

conformal mappings from annuli to canonical doubly connected bell representations abstract this paper derives formulae expressed in terms of schottky klein prime functions for the conformal mapping between concentric annuli and bell representations which are canonical doubly connected planar domains new formulae relating the conformal moduli associated with the two respective canonical domains are derived and complement recent results by jeong oh and taniguchi m jeong j w oh m taniguchi equivalence problem for annuli and bell representations in the plane j math anal appl 325 2007 1295 1305

random matrices in 2d laplacian growth and operator theory since it was first applied to the study of nuclear interactions by wigner and dyson almost 60 years ago random matrix theory rmt has developed into a field of its own within applied mathematics and is now essential to many parts of theoretical physics from condensed matter to high energy the fundamental results obtained so far rely mostly on the theory of random matrices in one dimension the dimensionality of the spectrum or equilibrium probability density in the last few years this theory has been extended to the case where the spectrum is two dimensional or even fractal with dimensions between 1 and 2 in this paper we review these recent developments and indicate some physical problems where the theory can be applied

constructing multiply connected quadrature domains multiply connected bounded quadrature domains with finite connectivity are re constructed from their quadrature data using conformal mappings that are ratios of products of schottky klein prime functions this method provides the natural generalization of the conformal maps to simply and doubly connected quadrature domains constructed by the first author in a num ber of physical applications the efficacy of the method is demonstrated by the explicit construction of a range of examples as well as by comparison with alternative constructive methods recently in troduced by crowdy r soc lond proc ser a math phys eng sci 457 2001 pp 2337 2359 and richardson european j appl math 12 2001 pp 571 599

an algebraic and geometric approach to non bijective quadratic transformations this work is concerned with an extension of the so called levi civita and kustaanheimo stiefel transformations the extension is achieved along two lines firstly the authors examine the latter transformations in other dimensions than the ones originally considered by on the one hand levi civita and on the other hand by kustaanheimo and stiefel secondly they pass from the compact to the non compact case this leads to quadratic non bijective transformations that they refer to as hurwitz or kustaanheimo stiefel like and quasihurwitz or levi civita like transformations the hurwitz and quasihurwitz transformations are introduced and studied in an algebraic framework which relies on the use of eight dimensional cayley dickson algebras an explicit formulation of the hurwitz and quasihurwitz transformations is also given in terms of clifford algebras the hurwitz transformations are investigated from a geometrical viewpoint indeed they are connected to hopf and pseudohopf fibrations finally some differential aspects of the hurwitz and to a lesser extent quasihurwitz transformations are developed in view of future physical applications

relationship between the prime form and the sigma function for some cyclic r s curves in this paper we study some cyclic r s curves x given by we give an expression for the prime form where p q x in terms of the sigma function for some such curves specifically any hyperelliptic curve r s 2 2g 1 as well as the cyclic trigonal curve r s 3 4 where r is a certain multi index of differentials here u1 and v1 are respectively the first components of u w p and v w q which are given by the abel map where g is the genus of x these explicit formulae are useful in applications for instance to the problem of constructing classes of schwarz christoffel maps to slit domains

real separated algebraic curves quadrature domains ahlfors type functions and operator theory abstract the aim of this paper is to inter relate several algebraic and analytic objects such as real type algebraic curves quadrature domains functions on them and rational matrix functions with special properties and some objects from operator theory such as vector toeplitz operators and subnormal operators our tools come from operator theory but some of our results have purely algebraic formulation we make use of xia s theory of subnormal operators and of the previous results by the author in this direction we also correct in section 5 some inaccuracies in the works of d v yakubovich subnormal operators of finite type i xia s model and real algebraic curves in c 2 rev mat iberoamericana 14 1998 95 115 d v yakubovich subnormal operators of finite type ii structure theorems rev mat iberoamericana 14 1998 623 681 by the author

topology of quadrature domains we address the problem of topology of quadrature domains namely we give upper bounds on the connectivity of the domain in terms of the number of nodes and their multiplicities in the quadrature identity

uniformizing the boundaries of multiply connected quadrature domains using fuchsian groups abstract this paper proposes a new method for uniformizing the boundaries of multiply connected quadrature domains by means of conformal maps that are automorphic functions invariant under the action of a schottky group that is a fuchsian group of the second kind the conformal maps are expressed as ratios of schottky klein prime functions the result is significant because a classical infinite product formula for the schottky klein prime function associated with fuchsian groups of the second kind is known to be convergent

multipolar vortices and algebraic curves this paper demonstrates that analytical solutions to the steady two dimensional euler equations possessing all the qualitative properties of multipolar vortices observed experimentally and numerically can be constructed using the theory of algebraic curves and quadrature domains the solutions consist of a finite set of line vortices superposed on finite area patches of uniform vorticity by way of example new solutions are presented in which the support of the vorticity is a quintuply connected vortex patch with five vorticity maxima modelling a symmetric pentapolar vortex consisting of a central core vortex four satellite vortices and four enclosed zones of irrotational fluid between the core and the satellite vortices the method of construction is amenable to the derivation of exact solutions of the two dimensional euler equations having vorticity distributions of even greater geometrical complexity

vortex patch equilibria of the euler equation and random normal matrices a new analogy is drawn between vortex patch or v state equilibria of the euler equations for ideal fluids and the planar limit of random normal matrix models physically the former is a quite different fluid dynamical problem to hele shaw flow or laplacian growth to which an analogy with matrix models has become well known in recent years the connection of random matrices with vortex dynamics is made via the so called modified schwarz potential this theoretical link while interesting in itself has immediate ramifications for random matrix theory by virtue of a transfer of mathematical technology already well developed for vortex dynamics hence for multi support matrix models in the planar limit we describe a constructive approach to the inverse problem of finding the shapes of the eigenvalue supports for a given potential using automorphic conformal mappings within a schottky model of the underlying algebraic curves and use of the schottky klein prime function two support eigenvalue distributions in a quartic potential are given as an illustrative example

multiple steady bubbles in a hele shaw cell this paper presents new solutions in analytical form for the shapes of an assembly of steady co travelling bubbles in a hele shaw cell the associated velocity field is also derived the assembly can consist of any finite number of bubbles the solutions are expressed in terms of schottky klein prime functions and are derived by exploiting results on multiply connected conformal slit mappings

pseudo euclidean hurwitz pairs and the kaluza klein theories following the geometrical concept of pseudo euclidean hurwitz pairs the authors given their systematic classification according to the relationship with real clifford algebras next a generalisation of the pairs to the curvilinear case is given and it is shown that they provide a convenient framework for the kaluza klein theories

the effect of surface tension on steadily translating bubbles in an unbounded hele shaw cell new numerical solutions to the so called selection problem for one and two steadily translating bubbles in an unbounded hele shaw cell are presented our approach relies on conformal mapping which for the two bubble problem involves the schottky klein prime function associated with an annulus we show that a countably infinite number of solutions exist for each fixed value of dimensionless surface tension with the bubble shapes becoming more exotic as the solution branch number increases our numerical results suggest that a single solution is selected in the limit that surface tension vanishes with the scaling between the bubble velocity and surface tension being different to the well studied problems for a bubble or a finger propagating in a channel geometry

strongly φ like functions of order α in two dimensional free boundary problems abstract in this paper we apply certain results in the theory of univalent functions to investigate the time evolution of the free boundary of a viscous fluid for a planar flow problem in the hele shaw cell model under injection to this end we prove that the property of strongly likeness of order 0 1 is preserved in time for both inner and outer problems under the assumption of nonzero small surface tension

pseudo euclidean hurwitz pairs in any signature the author considers the pseudo euclidean hurwitz problem as formulated in a recent study by lawrynowicz and rembielinsky 1987 however he adopts a more coordinate free approach and employs the spinorial formalism developed by crumeyrolle 1989 to solve this problem completely his results in part disagree with those of lawrynowicz and rembielinsky in particular he uncovers some extra possibilities he also gives a link between pseudo euclidean hurwitz pairs and z2 graded lie algebras and discusses the introduction of some fibre bundle with pseudo euclidean hurwitz pairs

complexiton solutions to integrable equations abstract complexiton solutions or complexitons for short are exact solutions newly introduced to integrable equations are newly introduced starting with the solution classification for a linear differential equation the korteweg de vries equation and the toda lattice equation are considered as examples to exhibit complexiton structures of nonlinear integrable equations the crucial step in the solution process is to apply the wronskian and casoratian techniques for hirota s bilinear equations correspondence between complexitons of the korteweg de vries equation and complexitons of the toda lattice equation is provided

complexiton solutions of the toda lattice equation a set of coupled conditions consisting of differential difference equations is presented for casorati determinants to solve the toda lattice equation one class of the resulting conditions leads to an approach for constructing complexiton solutions to the toda lattice equation through the casoratian formulation an analysis is made for solving the resulting system of differential difference equations thereby providing the general solution yielding eigenfunctions required for forming complexitons moreover a feasible way is presented to compute the required eigenfunctions along with examples of real complexitons of lower order

interactions among periodic waves and solitary waves of the n 1 dimensional sine gordon field exact solutions of the n 1 dimensional sine gordon field equation are studied with help of those of the cubic nonlinear klein gordon fields the mapping relations among the sine gordon field equation and the cubic nonlinear klein gordon fields are pure algebraic by solving the cubic nonlinear klein gordon equations many new types of exact explicit solutions such as the periodic periodic interaction waves periodic kink interaction waves periodic perturbed snake shape solitary waves etc are displayed both analytically and graphically

localized excitations in 2 1 dimensional systems by means of a special variable separation approach a common formula with some arbitrary functions has been obtained for some suitable physical quantities of various 2 1 dimensional models such as the davey stewartson ds model the nizhnik novikov veselov nnv system asymmetric nnv equation asymmetric ds equation dispersive long wave equation broer kaup kupershmidt system long wave short wave interaction model maccari system and a general n m component ablowitz kaup newell segur akns system selecting the arbitrary functions appropriately one may obtain abundant stable localized interesting excitations such as the multidromions lumps ring soliton solutions breathers instantons etc it is shown that some types of lower dimensional chaotic patterns such as the chaotic chaotic patterns periodic chaotic patterns chaotic line soliton patterns chaotic dromion patterns fractal lump patterns and fractal dromion patterns may be found in higher dimensional soliton systems the interactions between the traveling ring type soliton solutions are completely elastic the traveling ring solitons pass through each other and preserve their shapes velocities and phases some types of localized weak solutions peakons are also discussed especially the interactions between two peakons are not completely elastic after the interactions the traveling peakons also pass through each other and preserve their velocities and phases however they completely exchange their shapes

genus n algebraic reductions of the benney hierarchy within a schottky model by exploiting a new theoretical connection between reductions of the benney hierarchy and the dirichlet problem for laplace s equation the solution to a spectral problem associated with n parameter algebraic reductions of the benney hierarchy is found explicitly the solutions can be written in terms of the modified green s function associated with reflectionally symmetric n connected planar domains whose holes are all centred on the symmetry axis explicit formulae for the modified green s function in a canonical class of circular domains are constructed using a schottky model of the schottky double of these domains uniformizations of the spectral problem associated with two different types of reductions then follow from these formulae

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

laplacian growth and whitham equations of soliton theory the laplacian growth the hele shaw problem of multiply connected domains in the case of zero surface tension is proven to be equivalent to an integrable system of whitham equations known in soliton theory the whitham equations describe slowly modulated periodic solutions of integrable hierarchies of nonlinear differential equations through this connection the laplacian growth is understood as a flow in the moduli space of riemann surfaces

new analytical positon negaton and complexiton solutions of a coupled kdv mkdv system new positon negaton and complexiton solutions of a coupled kdv mkdv system are constructed directly from zero seed solution by means of darboux transformation these new exact solutions are all analytical with different spectral parameters and constants selection

streamer branching rationalized by conformal mapping techniques spontaneous branching of discharge channels is frequently observed but not well understood we recently proposed a new branching mechanism based on simulations of a simple continuous discharge model in high fields we here present analytical results for such streamers in the lozansky firsov limit where they can be modeled as moving equipotential ionization fronts this model can be analyzed by conformal mapping techniques which allow the reduction of the dynamical problem to finite sets of nonlinear ordinary differential equations our solutions illustrate that branching is generic for the intricate head dynamics of streamers in the lozansky firsov limit