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

characterization of quantum chaos by the autocorrelation function of spectral determinants the autocorrelation function of spectral determinants is proposed as a convenient tool for the characterization of spectral statistics in general and for the study of the intimate link between quantum chaos and the random matrix theory in particular for this purpose the correlation functions of spectral determinants are evaluated for various random matrix ensembles and are compared with the corresponding semiclassical expressions the method is demonstrated by applying it to the spectra of the quantized sinai billiards in two and three dimensions

the loewner equation maps and shapes an approach called schramm loewner evolution sle provides a new method for dealing with a wide variety of scale invariant problems in two dimensions this approach is based upon an older method called loewner evolution le which connects analytic and geometrical constructions in the complex plane in this paper the bases of le and sle are described and some simple applications are discussed in relatively non technical form a bibliography of the subject is presented

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 conjugate function method and conformal mappings in multiply connected domains the conjugate function method is an algorithm for numerical computation of conformal mappings for simply and doubly connected domains in this paper the conjugate function method is generalized for

stable and unstable manifolds of the henon mapping by using a parametric representation of the stable and unstable manifolds we prove that for some given values of the parameter in particular in the case first investigated by henon the henon mapping has a transversal homoclinic orbit

radial and circular slit maps of unbounded multiply connected circle domains infinite product formulae for conformally mapping an unbounded multiply connected circle domain to an unbounded canonical radial or circular slit domain or to domains with both radial and circular

application of the schwarz christoffel map to the laplacian growth of needles and fingers a numerical procedure based on the schwarz christoffel map suitable for the study of the laplacian growth of thin two dimensional protrusions is presented the protrusions take the form of either straight needles or curved fingers satisfying loewner s equation and are represented by slits in the complex plane particular use is made of driscoll s numerical procedure the sc toolbox for computing the schwarz christoffel map from a half plane to a slit half plane since the schwarz christoffel map applies only to polygonal regions the growth of curved fingers is approximated by an increasing number of short straight line segments the growth rate is given by a fixed power of the harmonic measure at the finger or needle tips and so includes the possibility of screening as the needles of fingers interact with themselves and with boundaries the method is illustrated with examples of multiple needle and finger growth in half plane and channel geometries the effect of on the trajectories of asymmetric bifurcating fingers is also studied

fixed equilibria of point vortices in symmetric multiply connected domains abstract the paper provides symmetric fixed configurations of point vortices in multiply connected domains in the unit circle with many circular obstacles when the circular domain is invariant with respect to rotation around the origin by a degree of 2 m a regular m polygonal ring configuration of identical point vortices becomes a fixed equilibrium on the other hand when we assume a special symmetry called the folding symmetry on the circular domain we find a fixed equilibrium in which m point vortices with the positive unit strength and m point vortices with the negative unit strength are arranged alternately at the vertices of a 2 m polygon we also investigate the stability of these fixed equilibria and their bifurcation for a special circular domain with the rotational symmetry as well as the folding symmetry furthermore we discuss fixed equilibria in non circular multiply connected domains with the same symmetries we give sufficient conditions for the conformal mappings by which fixed equilibria in the circular domains are mapped to those in the general multiply connected domains some examples of such conformal mappings are also provided

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

invariant curves attractors and phase diagram of a piecewise linear map with chaos we introduce equations describing the invariant curves associated with periodic points in a wide class of two dimensional invertible maps which in the special case of the mapt x z 1 a x bz x can be solved by analytical methods in the dissipative case several branches of the separatrices of the fixed points as well as of the period 2 and 4 points are constructed the regions of the parameter space where a given type of strange attractor exists are located we point out that the disappearance of homoclinic intersections between the separatrices of the fixed point and that of heteroclinic intersections between the unstable manifolds of the period 2 points and the stable manifold of the fixed point may occur separately and the latter leads already to the appearance of a two piece strange attractor this phenomenon may happen at weak dissipation in other maps too in the conservative caseb 1 separatrices and certain invariant tori are calculated

on capacity computation for symmetric polygonal condensers abstract making use of two different analytical numerical methods for capacity computation we obtain matching to a very high precision numerical values for capacities of a wide family of planar condensers these two methods are based respectively on the use of the lauricella function bezrodnykh and vlasov 2002 bezrodnykh 2016 64 65 and riemann theta functions bogatyrev 2012 grigoriev 2013 bogatyrev and grigor ev 2017 bogatyrev and grigor ev 2018 we apply these results to benchmark the performance of numerical algorithms which are based on adaptive h p finite element method hakula et al 2011 2013 2018 and boundary integral method tsuji 1959 jaswon and symm 1977 albrecht and collatz 1980

multimodal method and conformal mapping for the scattering by a rough surface the scattering of a wave from a periodic rough surface in two dimensions is considered the proposed method is based on the use of a conformal mapping and of the multimodal admittance method the use of the conformal mappings induces a spatially varying refractive index and transforms the boundary condition on the rough surface into a boundary condition on a flat surface then the multimodal admittance method reduces this problem to a riccati equation for the modal admittance matrix which is solved numerically with a magnus mobius scheme the method is shown to converge exponentially with the number of fourier modes the method also allows to find geometries having trapped modes or quasi trapped modes at a given frequency by looking at singularities or quasi singularities of this riccati equation besides a simple perturbation expansion based on a small roughness approximation is developed

schwarz christoffel mapping of the annulus a new derivation of the schwarz christoffel s c transformation for doubly connected domains is given the work is based on constructing the pre schwarzian by reflection of singularities a derivation of the simply connected s c map without appeal to liouville s theorem is a byproduct of this work

a piecewise linear invertible noodle map abstract a three dimensional map that is piecewise linear and invertible and cannot be reduced to two dimensional submaps is described it has a constant jacobian determinant and is an analogue to the simple noodle diffeomorphism described previously it stands in the same relationship to this latter diffeomorphism as the piecewise linear two dimensional lozi map does to the henon diffeomorphism the new analytically tractable map covers a fairly wide range of behaviors to be expected from four dimensional flows

numerical conformal mapping of multiply connected domains to regions with circular boundaries we propose a method to map a multiply connected bounded planar region conformally to a bounded region with circular boundaries the norm of the derivative of such a conformal map satisfies the laplace equation with a nonlinear neumann type boundary condition we analyze the singular behavior at corners of the boundary and separate the major singular part the remaining smooth part solves a variational problem which is easy to discretize we use a finite element method and a gradient descent method to find an approximate solution the conformal map is then constructed from this norm function we tested our algorithm on a polygonal region and a curvilinear smooth region

the loewner driving function of trajectory arcs of quadratic differentials we obtain a first order differential equation for the driving function of the chordal loewner differential equation in the case where the domain is slit by a curve which is a trajectory arc of a certain type of quadratic differential in particular this includes the case when the curve is a path on the square triangle or hexagonal lattice in the upper half plane or indeed in any domain with boundary on the lattice we also demonstrate how we use this to calculate the driving function numerically equivalent results for other variants of the loewner differential equation are also obtained

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

fingered growth in channel geometry a loewner equation approach a simple model of laplacian growth is considered in which the growth takes place only at the tips of long thin fingers following carleson and makarov l carleson and n makarov j anal math 87 103 2002 the evolution of the fingers is studied with use of the deterministic loewner equation the method is then extended to study the growth in a linear channel with reflecting sidewalls one and two finger solutions are found and analyzed it turns out that the presence of the walls has a significant influence on the shapes of the fingers and the dynamics of the screening process in which longer fingers suppress the growth of the shorter ones

symmetry of matrix valued stochastic processes and noncolliding diffusion particle systems as an extension of the theory of dyson s brownian motion models for the standard gaussian random matrix ensembles we report a systematic study of hermitian matrix valued processes and their eigenvalue processes associated with the chiral and nonstandard random matrix ensembles in addition to the noncolliding brownian motions we introduce a one parameter family of temporally homogeneous noncolliding systems of the bessel processes and a two parameter family of temporally inhomogeneous noncolliding systems of yor s generalized meanders and show that all of the ten classes of eigenvalue statistics in the altland zirnbauer classification are realized as particle distributions in the special cases of these diffusion particle systems as a corollary of each equivalence in distribution of a temporally inhomogeneous eigenvalue process and a noncolliding diffusion process a stochastic calculus proof of a version of the harish chandra itzykson zuber formula of integral over unitary group is established

on some gaussian ensembles of hermitian matrices a gaussian ensemble of hermitian matrices with an arbitrary ratio of their symmetric and anti symmetric parts is studied similarly a gaussian ensemble of hermitian quaternion matrices with an arbitrary ratio of their self dual and anti self dual parts is studied analytical expressions for the correlation and cluster functions as well as their limits when the order of the matrices is large are derived for both ensembles

spectral properties of random reactance networks and random matrix pencils our goal is to study the statistical properies of dielectric resonances which are poles of conductance of a large random lc network such poles are a particular example of eigenvalues n of matrix pencils h w with w being a positive definite matrix and h a random real symmetric one we first consider spectra of the matrix pencils with independent identically distributed entries of h then we concentrate on an infinite range full connectivity version of a random lc network in all cases we calculate the mean eigenvalue density and the two point correlation function in the framework of efetov s supersymmetry approach fluctuations in spectra turn out to be the same as those provided by the wigner dyson theory of usual random matrices

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

statistics of wave functions in mesoscopic systems we review the results of a recent study of fluctuations of wave functions in confined chaotic systems the fluctuations can be due to a random potential or be a consequence of a chaotic scattering by the walls the entire distribution function of the local amplitudes of the wave functions f 1 and the joint two point distribution are calculated in various situations the computation is performed using the supersymmetry technique and employs the studies of a reduced version of the non linear supersymmetric model developed especially for investigating the properties of a single eigenstate in a discrete spectrum of a chaotic quantum system for not very large amplitudes the complete description can be achieved using the zero dimensional approximation of the model the distribution function calculated in the limit of various symmetry classes shows the universal behavior known as the porter thomas statistics and fluctuations at distant points do not correlate in the crossover regime between the ensembles the distribution of local amplitudes shows a somewhat more sophisticated behavior the fluctuations in this case are correlated over distances exceeding the mean free path for large amplitudes generated by the states the most affected by the localization we call them prelocalized the zero dimensional approximation is no longer valid instead the statistics of their wave functions is determined by nontrivial vacua of the reduced model which is quite similar to the liouville model known in conformal field theory in particular the vacuum state of the reduced model obeys the liouville equation which indicates that in two dimensions the prelocalized states have nearly critical properties we prove their multifractality and power law statistically averaged envelope r 2 r 2 at the intermediate range of distances below the localization length with a spectrum of exponents 1 as well as obtain a logarithmically normal tail of the distribution functionf 1 we also find an evidence of prelocalized states in quasi one dimensional wires with the length shorter than the localization length their statistically averaged envelope has power law asymptotics x 2 x 2 and the tail of the distribution function is similar to that describing localized states in the infinite wires

green s functions for multiply connected domains via conformal mapping a method is described for the computation of the green s function in the complex plane corresponding to a set of k symmetrically placed polygons along the real axis an important special case is a set of k real intervals the method is based on a schwarz christoffel conformal map of the part of the upper half plane exterior to the problem domain onto a semi infinite strip whose end contains k 1 slits from the green s function one can obtain a great deal of information about polynomial approximations with applications in digital filters and matrix iterations by making the end of the strip jagged the method can be generalized to weighted green s functions and weighted approximations

the squeezing function on doubly connected domains via the loewner differential equation for any bounded domains varomega in mathbb c n deng guan and zhang introduced the squeezing function s varomega z which is a biholomorphic invariant of bounded domains we show that for n 1 the squeezing function on an annulus a r lbrace z in mathbb c r z 1 rbrace is given by s a r z max left z frac r z right for all 0 r 1 this disproves the conjectured formula for the squeezing function proposed by deng guan and zhang and establishes up to biholomorphisms the squeezing function for all doubly connected domains in mathbb c other than the punctured plane it provides the first non trivial formula for the squeezing function for a wide class of plane domains and answers a question of wold our main tools used to prove this result are the schottky klein prime function following the work of crowdy and a version of the loewner differential equation on annuli due to komatu we also show that these results can be used to obtain lower bounds on the squeezing function for certain product domains in mathbb c n

secondary schottky klein prime functions associated with multiply connected planar domains in recent years a general mathematical framework for solving applied problems in multiply connected domains has been developed based on use of the schottky klein s k prime function of an underlying compact riemann surface known as the schottky double of the domain in this paper we describe additional function theoretic objects that are naturally associated with planar multiply connected domains and which we refer to as secondary s k prime functions the basic idea develops and extends an observation of burnside dating back to 1892 applications of the new functions to represent conformal slit maps of mixed type that have been a topic of recent interest in the literature are given other possible applications are also surveyed

time reversal symmetry breaking and the statistical properties of quantum systems the breaking of the generalized time reversal symmetry of a quantum chaotic system corresponds to a transition from orthogonal to unitary ensembles of random matrices investigating this transition for the circular ensembles applicable for time dependent periodic systems the authors demonstrate and explain that there exists a relevant difference of the transition rate in comparison with the gaussian ensembles appropriate for quantum conservative systems the above supposition is supported by a numerical study of the eigenvalues and eigenvectors of the periodically kicked top

some exact correlations in the dyson brownian motion model for transitions to the cue abstract the dyson brownian motion model for transitions to the cue is considered for initial eigenvalue probability density functions corresponding to the coe and cse the density density correlation function between an eigenvalue at position initially 0 and an eigenvalue at position for general is calculated theoretical predictions for the asymptotic behaviour are verified and the distribution for the initial rate of change of the eigenvalues is calculated also for initial conditions corresponding to equispaced eigenvalues the n point equal parameter distribution is computed the extension of the calculation of these quantities to a multiparameter version of the theory is given

the crossover between orthogonal and unitary symmetry in small disordered systems a supersymmetry approach the authors consider the spectral statistics of independent electrons moving at zero temperature in a weakly disordered metallic ring threaded by a magnetic flux the analysis is based on the supersymmetry method involving both commuting and anticommuting variables besides they consider an ensemble of gaussian distributed symmetric random matrices gaussian orthogonal ensemble which are perturbed by a small time reversal symmetry breaking contribution for energies smaller than the inverse diffusion time around the ring ec the spectral correlation functions of both models can be represented in terms of supermatrix integrals of identical structure in conformity with recent numerical results this implies that the spectral properties of the two models coincide these matrix integrals are to a large extent universal i e they depend only on two physical parameters the mean level spacing and a symmetry breaking parameter which is identified as the typical sensitivity of levels to the time reversal symmetry breaking perturbation the authors parametrize the relevant matrix coset space of the nonlinear sigma model in a novel way which is particularly convenient for treating models in the crossover between the two symmetry classes as an example they present a detailed calculation of the level level correlation function the basic formalism however applies quite generally and can be used for the investigation of different types of correlation functions and system geometries as well

determinantal process starting from an orthogonal symmetry is a pfaffian process when the number of particles n is finite the noncolliding brownian motion bm and the noncolliding squared bessel process with index 1 besq are determinantal processes for arbitrary fixed initial configurations in the present paper we prove that if initial configurations are distributed with orthogonal symmetry they are pfaffian processes in the sense that any multitime correlation functions are expressed by pfaffians the 2 2 skew symmetric matrix valued correlation kernels of the pfaffians processes are explicitly obtained by the equivalence between the noncolliding bm and an appropriate dilatation of a time reversal of the temporally inhomogeneous version of noncolliding bm with finite duration in which all particles start from the origin n 0 and by the equivalence between the noncolliding besq and that of the noncolliding squared generalized meander starting from n 0

generalization of the schwarz christoffel mapping to multiply connected polygonal domains a generalization of the schwarz christoffel mapping to multiply connected polygonal domains is obtained by making a combined use of two preimage domains namely a rectilinear slit domain and a bounded circular domain the conformal mapping from the circular domain to the polygonal region is written as an indefinite integral whose integrand consists of a product of powers of the schottky klein prime functions which is the same irrespective of the preimage slit domain and a prefactor function that depends on the choice of the rectilinear slit domain a detailed derivation of the mapping formula is given for the case where the preimage slit domain is the upper half plane with radial slits representation formulae for other canonical slit domains are also obtained but they are more cumbersome in that the prefactor function contains arbitrary parameters in the interior of the circular domain

gaussian ensembles of random hermitian matrices intermediate between orthogonal and unitary ones a gaussian ensemble of hermitian matrices depending on a parameter is considered when 0 the ensemble is gaussian orthogonal and when 1 it is gaussian unitary an analytic expression for then level correlation and cluster functions is given for anyn and 0 1 this ensemble is of relevance in the study of time reversal symmetry breaking of nuclear interactions

laplacian networks growth local symmetry and shape optimization inspired by river networks and other structures formed by laplacian growth we use the loewner equation to investigate the growth of a network of thin fingers in a diffusion field we first review previous contributions to illustrate how this formalism reduces the network s expansion to three rules which respectively govern the velocity the direction and the nucleation of its growing branches this framework allows us to establish the mathematical equivalence between three formulations of the direction rule namely geodesic growth growth that maintains local symmetry and growth that maximizes flux into tips for a given amount of growth surprisingly we find that this growth rule may result in a network different from the static configuration that optimizes flux into tips

fingering in a channel and tripolar loewner evolutions a class of laplacian growth models in the channel geometry is studied using the formalism of tripolar loewner evolutions in which three points namely the channel corners and the point at infinity are kept fixed initially the problem of fingered growth where growth takes place only at the tips of slitlike fingers is revisited and a class of exact solutions of the corresponding loewner equation is presented for the case of stationary driving functions a model for interface growth is then formulated in terms of a generalized tripolar loewner equation and several examples are presented it is shown that the growing interface evolves into a steadily moving finger and that tip competition arises for nonsymmetric initial configurations with multiple tips

interface growth in two dimensions a loewner equation approach the problem of laplacian growth in two dimensions is considered within the loewner equation framework initially the problem of fingered growth is revisited and an exact solution for a three finger configuration is reported then a general class of growth models for an interface growing in the upper half plane is introduced and the corresponding loewner equation for the problem is derived several examples are given including interfaces with one or more tips as well as multiple growing interfaces a generalization of our interface growth model in terms of loewner domains where the growth rule is specified by a time evolving measure is briefly discussed