a critical surface for the solutions to singular elliptic systems abstract in this paper we mainly study the radially symmetric and strictly decreasing solutions to a system of critical elliptic equations in r n which involves strongly coupled critical nonlinearities and different hardy type terms by the odes analysis methods the asymptotic behaviors at the origin and infinity of solutions are proved by the way the asymptotic properties of radial and decreasing minimizers to the related best sobolev constant are established we find the existence of a critical surface above which and below which the asymptotic properties of solutions are deferent

on a reduction of the multi component kp hierarchy we show that the constrained kp hierarchies and their generalizations are natural reductions of the multi component kp hierarchy and that particular solutions of these hierarchies are obtained in a straightforward way from that of the multi component kp hierarchy

dual isomonodromic deformations and moment maps to loop algebras the hamiltonian structure of the monodromy preserving deformation equations of jimboet al jmms is explained in terms of parameter dependent pairs of moment maps from a symplectic vector space to the dual spaces of two different loop algebras the nonautonomous hamiltonian systems generating the deformations are obtained by pulling back spectral invariants on poisson subspaces consisting of elements that are rational in the loop parameter and identifying the deformation parameters with those determining the moment maps this construction is shown to lead to dual pairs of matrix differential operators whose monodromy is preserved under the same family of deformations as illustrative examples involving discrete and continuous reductions a higher rank generalization of the hamiltonian equations governing the correlation functions for an impenetrable bose gas is obtained as well as dual pairs of isomonodromy representations for the equations of the painleve transcendentspv andvi

similarity reduction of the modified yajima oikawa equation we study a similarity reduction of the modified yajima oikawa hierarchy the hierarchy is associated with a non standard heisenberg subalgebra in the affine lie algebra of type a 1 2 the system of equations for self similar solutions is presented as a hamiltonian system of degree of freedom 2 and admits a group of backlund transformations isomorphic to the affine weyl group of type a 1 2 we show that the system is equivalent to a two parameter family of the fifth painleve equation

symmetric solutions for a class of singular biharmonic elliptic systems involving critical exponents this paper deals with a class of singular biharmonic elliptic systems involving critical exponents in a bounded symmetric domain by using the variational method and the symmetric criticality principle of palais we obtain several existence and multiplicity results of g symmetric solutions for the systems

existence and multiplicity of positive solutions to elliptic systems involving critical exponents abstract in this paper a singular elliptic system involving multiple critical exponents and the caffarelli kohn nirenberg inequality is investigated by using the extremals of the best hardy sobolev constants the existence and multiplicity of positive solutions to the system are established

solutions to critical elliptic equations with multi singular inverse square potentials abstract let be an open bounded domain in r n n 3 with smooth boundary we are concerned with the multi singular critical elliptic problem u i 1 k i u x a i 2 q x u 2 2 u u h 0 1 where i r 2 2 n n 2 a i 1 i k and q x is a positive bounded function on using moser iteration we prove the asymptotic behavior of solutions for at points a i 1 i k by exploiting the effect of the coefficient of the critical nonlinearity we by means of a variational method establish the existence of positive and sign changing solutions for problem

inference with minimal gibbs free energy in information field theory non linear and non gaussian signal inference problems are difficult to tackle renormalization techniques permit us to construct good estimators for the posterior signal mean within information field theory ift but the approximations and assumptions made are not very obvious here we introduce the simple concept of minimal gibbs free energy to ift and show that previous renormalization results emerge naturally they can be understood as being the gaussian approximation to the full posterior probability which has maximal cross information with it we derive optimized estimators for three applications to illustrate the usage of the framework i reconstruction of a log normal signal from poissonian data with background counts and point spread function as it is needed for gamma ray astronomy and for cosmography using photometric galaxy redshifts ii inference of a gaussian signal with unknown spectrum and iii inference of a poissonian log normal signal with unknown spectrum the combination of i and ii finally we explain how gaussian knowledge states constructed by the minimal gibbs free energy principle at different temperatures can be combined into a more accurate surrogate of the non gaussian posterior

extreme superposition rogue waves of infinite order and the painleve iii hierarchy we study the fundamental rogue wave solutions of the focusing nonlinear schrodinger equation in the limit of large order using a recently proposed riemann hilbert representation of the rogue wave solution of arbitrary order k we establish the existence of a limiting profile of the rogue wave in the large k limit when the solution is viewed in appropriate rescaled variables capturing the near field region where the solution has the largest amplitude the limiting profile is a new particular solution of the focusing nonlinear schrodinger equation in the rescaled variables the rogue wave of infinite order which also satisfies ordinary differential equations with respect to space and time the spatial differential equations are identified with certain members of the painleve iii hierarchy we compute the far field asymptotic behavior of the near field limit solution and compare the asymptotic formulas with the exact solution using numerical methods for solving riemann hilbert problems in a certain transitional region for the asymptotics the near field limit function is described by a specific globally defined tritronquee solution of the painleve ii equation these properties lead us to regard the rogue wave of infinite order as a new special function

two component soliton systems and the painleve equations we give an extension of the two component kp hierarchy by considering additional time variables we obtain the linear 2 2 system by taking into consideration the hierarchy through a reduction procedure the lax pair of the schlesinger system and the sixth painleve equation is given from this linear system a unified approach to treat the other painleve equations from the usual two component kp hierarchy is also considered

existence of solutions for singular elliptic systems with critical exponents abstract let 0 be an open bounded domain in r n n 3 and 2 2 n n 2 we consider the following elliptic system of two equations in h 0 1 h 0 1 u t u x 2 2 u 2 u v u v t v x 2 2 u v 2 v v where 0 and 1 satisfy 2 using moser iteration we prove the asymptotic behavior of solutions for at the origin by exploiting the mountain pass theorem we establish the existence of solutions to we also discuss sign changing solutions for in this paper

operator calculus for information field theory signal inference problems with non gaussian posteriors can be hard to tackle through using the concept of gibbs free energy these posteriors are rephrased as gaussian posteriors for the price of computing various expectation values with respect to a gaussian distribution we present a way of translating these expectation values to a language of operators which is similar to that in quantum mechanics this simplifies many calculations for instance such as those involving log normal priors the operator calculus is illustrated by deriving a self calibrating algorithm which is tested with mock data

elliptic systems involving the critical exponents and potentials abstract in this paper we investigate a singular elliptic system which involves the critical sobolev exponent and multiple hardy type terms by employing variational methods the existence of its positive solutions is established by the moser iteration method some asymptotic properties of its nontrivial solutions at the singular points are verified

elliptic systems involving critical nonlinearities and different hardy type terms abstract in this paper a singular system of elliptic equations is investigated which involves critical nonlinearities and deferent hardy type terms existence of positive solutions to the system is verified by variational methods and asymptotic properties of solutions at the singular point are established by the moser iteration method it is found that the functions in solutions have different singularities at the singular point

positive least energy solutions for a coupled schrodinger system with critical exponent abstract in this paper we consider the following coupled schrodinger system with doubly critical exponents which can be seen as a counterpart of the brezis nirenberg problem u 1 u 1 u 5 u 2 v 3 x v 2 v 2 v 5 v 2 u 3 x u 0 v 0 x u v 0 x where r 3 is a smooth bounded domain 1 2 0 1 2 0 and 0 under certain conditions on 1 2 and we show that this problem has at least one positive least energy solution

semilinear systems involving multiple critical hardy sobolev exponents and three singular points abstract in this paper a semilinear system of elliptic equations is investigated which involves multiple critical exponents and singular points by variational methods and analytic techniques the related best hardy sobolev constants is studied and the existence of positive solutions is established

why the general zakharov shabat equations form a hierarchy the totality of all zakharov shabat equations zs i e the zero curvature equations with rational dependence on a spectral parameter if properly defined can be considered as a hierarchy the latter means a collection of commuting vector fields in the same phase space further properties of the hierarchy are discussed such as additional symmetries an analogue to the string equation a grassmannian related to the zs hierarchy and a grassmannian definition of soliton solutions

on the singular elliptic systems involving multiple critical sobolev exponents in this paper a singular elliptic system is investigated which involves multiple critical sobolev exponents and hardy type terms by using variational methods and analytical techniques the existence of positive and sign changing solutions to the system is established

towards information optimal simulation of partial differential equations most simulation schemes for partial differential equations pdes focus on minimizing a simple error norm of a discretized version of a field this paper takes a fundamentally different approach the discretized field is interpreted as data providing information about a real physical field that is unknown this information is sought to be conserved by the scheme as the field evolves in time such an information theoretic approach to simulation was pursued before by information field dynamics ifd in this paper we work out the theory of ifd for nonlinear pdes in a noiseless gaussian approximation the result is an action that can be minimized to obtain an information optimal simulation scheme it can be brought into a closed form using field operators to calculate the appearing gaussian integrals the resulting simulation schemes are tested numerically in two instances for the burgers equation their accuracy surpasses finite difference schemes on the same resolution the ifd scheme however has to be correctly informed on the subgrid correlation structure in certain limiting cases we recover well known simulation schemes like spectral fourier galerkin methods we discuss implications of the approximations made

random words toeplitz determinants and integrable systems ii this paper connects the analysis of the length of the longest weakly increasing subsequence of inhomogeneous random words to a riemnn hilbert problem and an associated system of integrable partial differential equations in particular we show that the poissonization of the distribution function of this length can be identified as the jimbo miwa ueno tau function

an area preserving action of the modular group on cubic surfaces and the painleve vi equation we construct an area preserving action of the modular group on a general 4 parameter family of affine cubic surfaces we present a geometrical background behind this construction that is a natural symplectic structure on a moduli space of rank two linear monodromy representations over the 2 dimensional sphere with four punctures and a natural symplectic action upon it of the braid group on three strings studying this action as a discrete dynamical system will be important in discussing the monodromy of the painleve vi equation

the partition function of the two matrix model as an isomonodromic τ function we consider the itzykson zuber eynard mehta two matrix model and prove that the partition function is an isomonodromic function in a sense that generalizes that of jimbo et al monodromy preserving deformation of linear ordinary differential equations with rational coefficients physica d 2 306 1981 in order to achieve the generalization we need to define a notion of function for isomonodromic systems where the adregularity of the leading coefficient is not a necessary requirement

singularities of solutions to elliptic systems involving different hardy type terms abstract in this paper a system of elliptic equations is studied which involves multiple critical nonlinearities and deferent hardy type terms asymptotic properties at the singular point of positive solutions are proved by the moser iteration method under certain conditions it is found that the functions in solutions have different singularities at the singular point

signal inference with unknown response calibration uncertainty renormalized estimator the calibration of a measurement device is crucial for every scientific experiment where a signal has to be inferred from data we present cure the calibration uncertainty renormalized estimator to reconstruct a signal and simultaneously the instrument s calibration from the same data without knowing the exact calibration but its covariance structure the idea of the cure method developed in the framework of information field theory is to start with an assumed calibration to successively include more and more portions of calibration uncertainty into the signal inference equations and to absorb the resulting corrections into renormalized signal and calibration solutions thereby the signal inference and calibration problem turns into a problem of solving a single system of ordinary differential equations and can be identified with common resummation techniques used in field theories we verify the cure method by applying it to a simplistic toy example and compare it against existent self calibration schemes wiener filter solutions and markov chain monte carlo sampling we conclude that the method is able to keep up in accuracy with the best self calibration methods and serves as a noniterative alternative to them

improving self calibration response calibration is the process of inferring how much the measured data depend on the signal one is interested in it is essential for any quantitative signal estimation on the basis of the data here we investigate self calibration methods for linear signal measurements and linear dependence of the response on the calibration parameters the common practice is to augment an external calibration solution using a known reference signal with an internal calibration on the unknown measurement signal itself contemporary self calibration schemes try to find a self consistent solution for signal and calibration by exploiting redundancies in the measurements this can be understood in terms of maximizing the joint probability of signal and calibration however the full uncertainty structure of this joint probability around its maximum is thereby not taken into account by these schemes therefore better schemes in sense of minimal square error can be designed by accounting for asymmetries in the uncertainty of signal and calibration we argue that at least a systematic correction of the common self calibration scheme should be applied in many measurement situations in order to properly treat uncertainties of the signal on which one calibrates otherwise the calibration solutions suffer from a systematic bias which consequently distorts the signal reconstruction furthermore we argue that nonparametric signal to noise filtered calibration should provide more accurate reconstructions than the common bin averages and provide a new improved self calibration scheme we illustrate our findings with a simplistic numerical example

the cauchy problem for the second member of a rm p_ iv hierarchy a rigorous method the inverse monodromy transform for studying the riemann hilbert rh problem associated with the classical painleve equations is applied to the second member of a fourth painleve hierarchy we show that the cauchy problem for the second member of this hierarchy admits in general a global meromorphic solution in x moreover for a particular choice of the monodromy data the associated rh problem can be reduced to a set of scalar rh problems and a special solution which can be written in terms of the airy function is obtained

positive minimizers of the best constants and solutions to coupled critical quasilinear systems abstract in this paper systems of quasilinear elliptic equations are investigated which involve critical homogeneous nonlinearities and deferent hardy type terms by variational methods and careful analysis positive minimizers of the related best sobolev constants are found and the existence of positive solutions to the systems is verified the results are new even in the case p 2

transformations rs42 3 of the ranks 4 and algebraic solutions of the sixth painleve equation compositions of rational transformations of independent variables of linear matrix ordinary differential equations odes with the schlesinger transformations rs transformations are used to construct algebraic solutions of the sixth painleve equation rs transformations of the ranks 3 and 4 of 2 2 matrix fuchsian odes with 3 singular points into analogous ode with 4 singular points are classified

monodromy preserving deformation of linear ordinary differential equations with rational coefficients iii abstract a unified treatment of monodromy and spectrum preserving deformations is presented in particular a general procedure is described to reduce the latter into the former consistently the concept of the function previously introduced for the former 2 is extended to the isospectral context it is shown that the general monodromy and spectrum preserving deformation equations can be written as hirota s bilinear differential equations by using the functions as dependent variables

the indefinite anti self dual metrics and the painleve equations we classify the su 2 invariant anti self dual metrics with a signature the metrics are specified by a solution of painleve vi v iii or ii moreover we show the geometric meaning of the metrics specified by each type of painleve function

systems of elliptic equations involving multiple critical nonlinearities and different hardy type terms in rn abstract in this paper we study nonlinear systems of elliptic equations involving different hardy type terms and multiple critical nonlinearities by very technical variational methods local compactness of palais smale sequences is established and values of the best sobolev type constants are compared finally the existence of minimizers to related rayleigh quotient is verified and the existence of positive solutions to elliptic systems is thus proved

the k tacnode process the tacnode process is a universal behavior arising in nonintersecting particle systems and tiling problems for dyson brownian bridges the tacnode process describes the grazing collision of two packets of walkers we consider such a dyson sea on the unit circle with drift for any k in mathbb z we show that an appropriate double scaling of the drift and return time leads to a generalization of the tacnode process in which k particles are expected to wrap around the circle we derive winding number probabilities and an expression for the correlation kernel in terms of functions related to the generalized hastings mcleod solutions to the inhomogeneous painleve ii equation the method of proof is asymptotic analysis of discrete orthogonal polynomials with a complex weight