homological and monodromy representations of framed braid groups in this paper we introduce two new classes of representations of the framed braid groups one is the homological representation constructed as the action of a mapping class group on a certain homology group the other is the monodromy representation of the confluent kz equation which is a generalization of the kz equation to have irregular singularities we also give a conjectural equivalence between these two classes of representations

einstein maxwell solitons the application of the inverse scattering method to the einstein maxwell equations for stationary axisymmetric exterior fields leads the authors to an explicit formula for the ernst and electromagnetic potentials of new exact solutions generated from an arbitrary seed solution

vandermonde like determinants and n fold darboux backlund transformations we define vandermonde like determinants and analyze their structure the resulting scheme is well suited to achieve a remarkable compactness and transparency in n soliton formulas or more generally in formulas for n fold darboux or backlund transformations

a model for stocks dynamics based on a non gaussian path integral abstract we introduce a model for the dynamics of stock prices based on a non quadratic path integral the model is a generalization of ilinski s path integral model more precisely we choose a different action which can be tuned to different time scales the result is a model with a very small number of parameters that provides very good fits of some stock prices and indices fluctuations

creation operators and bethe vectors of the osp 1 2 gaudin model a gaudin model based on the orthosymplectic lie superalgebra osp 1 2 is studied the eigenvectors of the osp 1 2 invariant gaudin hamiltonians are constructed by algebraic bethe ansatz corresponding creation operators are defined by a recurrence relation furthermore explicit solution to this recurrence relation is found the action of the creation operators on the lowest spin vector yields bethe vectors of the model the relation between the bethe vectors and solutions to the knizhnik zamolodchikov equation of the corresponding super conformal field theory is established

husimi transform of an operator product it is shown that the series derived by mizrahi giving the husimi transform or covariant symbol of an operator product is absolutely convergent for a large class of operators in particular the generalized liouville equation describing the time evolution of the husimi function is absolutely convergent for a large class of hamiltonians in contrast the series derived by groenewold giving the weyl transform of an operator product is often only asymptotic or even undefined the result is used to derive an alternative way of expressing expectation values in terms of the husimi function the advantage of this formula is that it applies in many of the cases where the anti husimi transform or contravariant symbol is so highly singular that it fails to exist as a tempered distribution

confluent kz equations for mathfrak sl _n with poincare rank 2 at infinity we construct confluent kz equations with poincar rank 2 at infinity for the case of and the integral representation for the solutions hamiltonians of these confluent kz equations are derived from suitable quantization of dlog constructed in the theory of monodromy preserving deformation mpd in jimbo et al 1981 physica d 2 306 52 our confluent kz equations can be viewed as a quantization of mpd with poincar rank 2 at infinity

quantum mechanics in the gaussian wave packet phase space representation abstract we present here some results related to an alternative mapping formulation to the well known wigner weyl transform which makes use of the gaussian wave packet or coherent states representation pq the pair p q which labels this state defines a phase space in which the abstract operators p q of momentum and position are represented as differential operators in the mapped expression of an operator a p q the quantum effects appear when they are absent in the corresponding wigner weyl transform

arbitrage free self organizing markets with garch properties generating them in the lab with a lattice model abstract we extend our studies of a quantum field model defined on a lattice having the dilation group as a local gauge symmetry the model is relevant in the cross disciplinary area of econophysics a corresponding proposal by ilinski aimed at gauge modeling in non equilibrium pricing is realized as a numerical simulation of the one asset version the gauge field background enforces minimal arbitrage yet allows for statistical fluctuations the new feature added to the model is an updating prescription for the simulation that drives the model market into a self organized critical state taking advantage of some flexibility of the updating prescription stylized features and dynamical behaviors of real world markets are reproduced in some detail

a method of linearization for painleve equations painleve iv v abstract a method to linearize the initial value problem of the painleve equations iv v is given the procedure involves formulating a riemann hilbert boundary value problem on intersecting lines for the inverse monodromy problem this boundary value problem is reduced to a sequence of standard problems on single lines in a certain range of parameter space schlesinger transformations allow one to completely cover the parameter space special solutions are constructed from special cases of the riemann problem as well

explicit solutions of the four wave mixing model dynamical degenerate four wave mixing is studied analytically in detail by removing the unessential freedom we first characterize this system by a lower dimensional closed subsystem of a deformed maxwell bloch type involving only three physical variables the intensity pattern the dynamical grating amplitude the relative net gain we then classify by the painlev test all the cases when single valued solutions may exist according to the two essential parameters of the system the real relaxation time and the complex response constant in addition to the stationary case the only two integrable cases occur for a purely nonlocal response re 0 these are the complex unpumped maxwell bloch system and another one which is explicitly integrated with elliptic functions for a generic response re 0 we display strong similarities with the cubic complex ginzburg landau equation

trigonometric osp 1 2 gaudin model the problems connected with gaudin models are reviewed by analyzing model related to the trigonometric osp 1 2 classical r matrix the eigenvectors of the trigonometric osp 1 2 gaudin hamiltonians are found using explicitly constructed creation operators the commutation relations between the creation operators and the generators of the trigonometric loop superalgebra are calculated the coordinate representation of the bethe states is presented the relation between the bethe vectors and solutions to the knizhnik zamolodchikov equation yields the norm of the eigenvectors the generalized knizhnik zamolodchikov system is discussed both in the rational and in the trigonometric case

marginal and correlation distribution functions in the squeezed states representation here we consider the husimi function hf p for the squeezed states and calculate the marginal and correlation distribution functions mdfs and cdfs when p is projected onto the photon number states according to the value of the squeezing parameter one verifies the occurence of oscillations and beats as already appointed in the literature we verify that these phenomena are entirely contained in the correlation function in particular we show that since the husimi and its mdfs satisfy partial differential equations where the squeeze parameter plays the role of time the solutions the squeezed functions obtained from initial unsqueezed functions can be expressed by means of kernels responsible for the propagation of squeezing from the calculational point of view this method presents advantages for calculating the mdfs compared with a direct integration over one of the two phase space variables of p since one can use the symmetry properties of the differential equations

pseudo diffusion equation and information entropy of squeezed coherent states abstract as the squeezed coherent states are labeled by 3 tuple the momentum the coordinate and the squeeze parameter a phase space probability distribution function associated with a generic wavefunction can be introduced we verify that this probability function is the solution of a partial differential equation the pseudo diffusion equation hence permitting to extend the concept of information entropy as introduced by wehrl for the coherent states to the squeezed coherent states it is shown that the entropy functionals can be used as a measure of quantum correlations between the phase space variables for a given wavefunction we also present several properties and discuss the use of the entropic inequality as a concept that complements the heisenberg uncertainty relationship

new similarity reductions of the boussinesq equation some new similarity reductions of the boussinesq equation which arises in several physical applications including shallow water waves and also is of considerable mathematical interest because it is a soliton equation solvable by inverse scattering are presented these new similarity reductions including some new reductions to the first second and fourth painleve equations cannot be obtained using the standard lie group method for finding group invariant solutions of partial differential equations they are determined using a new and direct method that involves no group theoretical techniques

quantum mechanics in the gaussian wave packet phase space representation iii from phase space probability functions to wave functions abstract using a mapping formalism the wave packet phase space representation we show that for phase space probability distributions that do not violate the uncertainty principle one can associate a density matrix x x when some conditions are fulfilled furthermore if condition tr 2 1 is satisfied it becomes possible to find a wave function associated to the probability distribution

path integral approach to asian options in the black scholes model abstract we derive a closed form solution for the price of an average strike as well as an average price geometric asian option by making use of the path integral formulation our results are compared to a numerical monte carlo simulation we also develop a pricing formula for an asian option with a barrier on a control process combining the method of images with a partitioning of the set of paths according to the average along the path this formula is exact when the correlation is zero and is approximate when the correlation increases

confluent primary fields in the conformal field theory for any complex simple lie algebra we generalize the primary fields in the wess zumino novikov witten conformal field theory for a case with irregular singularities we refer to these generalized primary fields as confluent primary fields we present the screening currents ward identity a recursion rule for computing the expectation values of the products of confluent primary fields in the case of the expectation values of the products of confluent primary fields are integral formulas of solutions to confluent knizhnik zamolodchikov kz equations given in jimbo et al 2008 j phys a math theor 41 175205 by computing the operator product expansion of the energy momentum tensor t z and the confluent primary fields we obtain new differential operators moreover in the case of these differential operators are the same as those of the confluent kz equations jimbo et al 2008

generalized knizhnik zamolodchikov equations and isomonodromy quantization of the equations integrable via the inverse scattering transform maxwell bloch system with pumping canonical quantization of the isomonodromy solutions of equations integrable via the inverse scattering transform leads to generalized knizhnik zamolodchikov equations one can solve these equations by the off shell bethe ansatz method provided the knizhnik zamolodchikov equations are related with the highest weight representations of the corresponding lie algebras these solutions can be written in terms of multi variable generalizations of special functions of the hypergeometric type in this work we consider a realization of the above scheme for the maxwell bloch system with pumping quantum states for this system are found in terms of the multi variable confluent hypergeometric function

similarity solutions of the deformed maxwell bloch system the authors have found similarity reductions for the deformed maxwell bloch system to the fifth and second painleve equations asymptotics of the solutions of these equations which are relevant to two level atomic systems with pumping have also been derived

application of the darboux transformation to inverse problems with variable spectral parameters it is shown that darboux transformations can be used to find explicit solutions of deformed equations which have recently been found through the inverse method with a variable spectral parameter the author constructs in particular the soliton solution of the deformed maxwell bloch equations and analyses its asymptotic behaviour

an application of aomoto gelfand hypergeometric functions to the su n knizhnik zamolodchikov equation solutions to the knizhnik zamolodchikov equation for verma modules of the lie algebra mathfrak s mathfrak l n 1 mathbb c are explicitly given by certain integrals called aomoto gelfand hypergeometric functions

curve motions from the integrable equations having variable spectral parameters this paper reports on a study of the motion of moving space curves induced from integrable equations having variable spectral parameters we explain the geometric structure of the curves using the associated linear equations of integrable equations a generalized form of the hasimoto transformation is introduced this form relates the curvature and torsion of the curves to the variables of integrable equations some explicit curve motions including the solitonic and kelvin type curves are calculated using the sym tafel formula

gauge invariant lattice quantum field theory implications for statistical properties of high frequency financial markets abstract we report on initial studies of a quantum field theory defined on a lattice with multi ladder geometry and the dilation group as a local gauge symmetry the model is relevant in the cross disciplinary area of econophysics a corresponding proposal by ilinski aimed at gauge modeling in non equilibrium pricing is implemented in a numerical simulation we arrive at a probability distribution of relative gains which matches the high frequency historical data of the nasdaq stock exchange index

nonautonomous integrable nonlinear schrodinger equations with generalized external potentials inhomogeneous nonlinear schrodinger equations are constructed systematically using the covariance with respect to the darboux transformation the darboux invariants and the spectral parameter are functions of time and space which result in equations describing nonautonomous solitons moving in generalized external potentials all the known inhomogeneous nonlinear schrodinger equations as well as new ones of a more generalized form can be derived from the constructed equation one and two soliton solutions are explicitly constructed using the darboux transformation

off shell bethe ansatz equations and n point correlators in the su 2 wznw theory we prove that the wavevectors of the off shell bethe ansatz equation for the inhomogeneous su 2 lattice vertex model render in the quasiclassical limit the solutions of the knizhnik zamolodchikov equation

when does a given function in phase space belong to the class of husimi distributions the problem of finding a density matrix corresponding to a husimi distribution is solved in a simple new way it is shown how the proposed procedure may be effectively used to answer the question raised in the title

stochastic calculus for assets with non gaussian price fluctuations abstract from the path integral formalism for price fluctuations with non gaussian distributions we derive the appropriate stochastic calculus replacing ito s calculus for stochastic fluctuations

option pricing for non gaussian price fluctuations abstract from the path integral description of price fluctuations with non gaussian distributions we derive a stochastic calculus which replaces ito s calculus for harmonic fluctuations we set up a natural martingale for option pricing from the wealth balance of options stocks and bonds and evaluate the resulting formula for truncated levy distributions after this an alternative formula is derived for a model of multivariant gaussian price fluctuations which leads to non gaussian return distributions fitting dow jones data excellently from long to short time scales with a tail behavior e x x 3 2

option pricing from path integral for non gaussian fluctuations natural martingale and application to truncated levy distributions within a path integral formalism for non gaussian price fluctuations we set up a simple stochastic calculus and derive a natural martingale for option pricing from the wealth balance of options stocks and bonds the resulting formula is evaluated for truncated levy distributions

darboux invariants of integrable equations with variable spectral parameters the darboux transformation for integrable equations with variable spectral parameters is introduced darboux invariant quantities are calculated which are used in constructing the lax pair of integrable equations this approach serves as a systematic method for constructing inhomogeneous integrable equations and their soliton solutions the structure functions of variable spectral parameters determine the integrability and nonlinear coupling terms three cases of integrable equations are treated as examples of this approach

squeezed states generalized hermite polynomials and pseudo diffusion equation abstract we show that the wavefunctions pq n of the harmonic oscillator in the squeezed state representation have the generalized hermite polynomials as their natural orthogonal polynomials these wavefunctions lead to generalized poisson distribution pn pq which satisfy an interesting pseudo diffusion equation p n p q 1 4 2 p 2 1 2 2 q 2 p 2 p q in which the squeeze parameter plays the role of time th entropies sn have minima at the unsqueezed states 1 which means that squeezing or stretching decreases the correlation between momentum p and position q

o n symmetries sum rules for generalized hermite polynomials and squeezed states quantum optics has been dealing with coherent states squeezed states and many other non classical states the associated mathematical framework makes use of special functions as hermite polynomials laguerre polynomials and others in this connection we here present some formal results that follow directly from the group o n of complex transformations motivated by the squeezed states structure we introduce the generalized hermite polynomials which include as particular cases the hermite polynomials as well as the heat polynomials using generalized raising operators we derive new sum rules for the which are covariant under o n transformations the and the associated sum rules become useful for evaluating wigner functions in a straightforward manner as a byproduct we use one of these sum rules on the operator level to obtain raising and lowering operators for the laguerre polynomials and show that they generate an sl 2 r su 1 1 algebra

temperature correlators of the impenetrable bose gas as an integrable system it is shown that the temperature equal time correlators of impenetrable bosons in one space dimension are described by a classical integrable system partial differential equations for two point as well as for multipoint correlators are obtained the short distance and low density expansions are constructed

quantum mechanics in terms of non negative distribution functions abstract we construct a formalism which enables us to express quantum mechanics in terms of any non negative smoothed wigner function quantum mechanics in terms of the well known husimi function turns out to be just a special case of this general formalism

a path integral based model for stocks and order dynamics abstract we present a model for the short term dynamics of financial assets based on an application to finance of quantum gauge theory developing ideas of ilinski we present a numerical algorithm for the computation of the probability distribution of prices and compare the results with apple stocks prices and the s p500 index

connection formulae for painleve v functions we study those painleve v equations which admit a one parameter family of solutions analytic at the origin and solve the global problem of connecting the two parameter asymptotic expansion at t with the two parameter expansion about t 0 this connection is the one needed to study the correlation functions of the transverse ising chain at the critical field strength from these connection formulae we derive a one parameter connection formula for the general painleve ii equation and a two parameter connection formula for a class of painleve iii equations

on the equivalence between the wave packet phase space representation wppsr and the phase space generated by the squeezed states departamento de fisica universidade federal de sao carlos rodovia washington luiz km 235 13560 sao carlos sp