variable length lossy source code using a constrained random number generator a variable length lossy source code is introduced with a rate distortion pair close to the rate distortion function random numbers that satisfy a condition specified by a function and its value are used to construct a stochastic encoder the proof of the theorem is based on the balanced coloring property of an ensemble of functions since an ensemble of systematic sparse matrices has this property we can construct a tractable code for a memoryless source some algorithms for implementing the code are introduced and compared by simulation

special functions arising from discrete painleve equations a survey this article is a survey on recent studies on special solutions of the discrete painleve equations especially on hypergeometric solutions of the q painleve equations the main part of this survey is based on the joint work k kajiwara t masuda m noumi y ohta y yamada hypergeometric solutions to the q painleve equations imrn 2004 47 2004 2497 2521 k kajiwara t masuda m noumi y ohta y yamada construction of hypergeometric solutions to the q painleve equations imrn 2005 24 2005 1439 1463 with kajiwara masuda ohta and yamada after recalling some basic facts concerning painleve equations for comparison we give an overview of the present status of studies on difference discrete painleve equations as a source of special functions

on condition numbers of the spectral projections associated with periodic eigenproblems abstract in this paper we study and analyze absolute and relative condition numbers of the spectral projections for regular periodic eigenproblems the main contribution is to derive explicit expressions of the condition numbers of the j th left and right spectral projections numerical examples are given to illustrate the proposed condition numbers for the spectral projections associated with periodic eigenproblems

degeneration through coalescence of the q painleve vi equation starting from the q discrete form of the painleve vi equation we obtain its degenerate forms by applying the procedure of coalescence of singularities the whole cascade of degenerate forms is thus obtained leading to new forms for the discrete painleve iv and v equations the lax pairs of these discrete painleve equations are explicitly constructed thus confirming their integrability

discretising systematically the painleve equations we apply a method that we have already proposed for the systematic discretisation of differential systems to the construction of discrete analogues of the painleve equations the important point is that in the case of the painleve equations one needs to preserve the integrability we show that this is indeed possible using our systematic procedure applied to the equations obtained from the hamiltonian description of the painleve equations

ultradiscrete painleve ii equation and a special function solution ultradiscretization with parity variables which keeps the information of original variables sign is applied to the q painleve ii equation of type a6 q pii a special solution of the resulting ultradiscrete system which corresponds to the special function solution of q pii is constructed ultradiscrete analogues of the q airy equation and its special solutions are also discussed in the process

painleve equations nonlinear special functions the six painleve equations pi pvi were first discovered about a hundred years ago by painleve and his colleagues in an investigation of nonlinear second order ordinary differential equations recently there has been considerable interest in the painleve equations primarily due to the fact that they arise as reductions of the soliton equations which are solvable by inverse scattering consequently the painleve equations can be regarded as completely integrable equations and possess solutions which can be expressed in terms of solutions of linear integral equations despite being nonlinear equations although first discovered from strictly mathematical considerations the painleve equations have arisen in a variety of important physical applications including statistical mechanics plasma physics nonlinear waves quantum gravity quantum field theory general relativity nonlinear optics and fibre optics the painleve equations may be thought of a nonlinear analogues of the classical special functions they possess hierarchies of rational solutions and one parameter families of solutions expressible in terms of the classical special functions for special values of the parameters further the painleve equations admit symmetries under affine weyl groups which are related to the associated backlund transformations in this paper i discuss some of the remarkable properties which the painleve equations possess including connection formulae backlund transformations associated discrete equations and hierarchies of exact solutions in particular the second painleve equation pii is used to illustrate these properties and some of the applications of pii are also discussed

universal character and q difference painleve equations the universal character is a polynomial attached to a pair of partitions and is a generalization of the schur polynomial in this paper we introduce an integrable system of q difference lattice equations satisfied by the universal character and call it the lattice q uc hierarchy we regard it as generalizing both q kp and q uc hierarchies suitable similarity and periodic reductions of the hierarchy yield the q difference painleve equations of types a 2g 1 a 1 1 g geq 1 d 5 1 and e 6 1 as its consequence a class of algebraic solutions of the q painleve equations is rapidly obtained by means of the universal character in particular we demonstrate explicitly the reduction procedure for the case of type e 6 1 via the framework of based on the geometry of certain rational surfaces

achieving the rate distortion bound with low density generator matrix codes it is shown that binary low density generator matrix codes can achieve the rate distortion bound of discrete memoryless sources with general distortion measure via multilevel quantization a practical encoding scheme based on the survey propagation algorithm is proposed the effectiveness of the proposed scheme is verified through simulation

kovalevskaya exponents and the space of initial conditions of a quasi homogeneous vector field formal series solutions and the kovalevskaya exponents of a quasi homogeneous polynomial system of differential equations are studied by means of a weighted projective space and dynamical systems theory a necessary and sufficient condition for the series solution to be a convergent laurent series is given which improves the well known painleve test in particular if a given system has the painleve property an algorithm to construct okamoto s space of initial conditions is given the space of initial conditions is obtained by weighted blow ups of the weighted projective space where the weights for the blow ups are determined by the kovalevskaya exponents the results are applied to the first painleve hierarchy 2m th order first painleve equation

10e9 solution to the elliptic painleve equation a function formalism for sakai s elliptic painleve equation is presented this establishes the equivalence between the two formulations by sakai and by ohta ramani grammaticos we also give a simple geometric description of the elliptic painleve equation as a non autonomous deformation of the addition formula on elliptic curves by using these formulations we construct a particular solution of the elliptic painleve equation expressed in terms of the elliptic hypergeometric function 10e9

isomonodromic deformations of a rational differential system and reconstruction with the topological recursion the sl2 case in this paper we show that it is always possible to deform a differential equation x x l x x with l x sl2 c x by introducing a small formal parameter in such a way that it satisfies the topological type properties of bergere borot and eynard annales henri poincare 16 12 2713 2782 2015 this is obtained by including the former differential equation in an isomonodromic system and using some homogeneity conditions to introduce the topological recursion is then proved to provide a formal series expansion of the corresponding tau function whose coefficients can thus be expressed in terms of intersections of tautological classes in the deligne mumford compactification of the moduli space of surfaces we present a few examples including any fuchsian system of sl2 c x as well as some elements of painleve hierarchies

special solutions of discrete painleve equations through direct linearisation abstract we study the special solutions of discrete painleve equations which can be expressed through the discrete equivalent of quadratures the discrete painleve equations examined are the ones which appear in the classification based on affine weyl groups and which can be written as a system of two first order mappings the solutions obtained are the discrete equivalents of solutions which have been shown to exist for the continuous painleve equations

gravitational instantons with faster than quadratic curvature decay iii this is our third paper in a series on the gravitational instantons in this paper we classify alg and alh gravitational instantons in alg case we extend tian yau hein s construction slightly and show that it is the only alg gravitational instanton in alh case we prove a torelli type theorem

on the lattice geometry and birational group of the six point multi ratio equation the inherent self consistency properties of the six point multi ratio equation allow it to be considered on a domain associated with a t shaped coxeter dynkin diagram this extends the kadomtsev pe

elliptic painleve equations from next nearest neighbor translations on the e_8 1 lattice the well known elliptic discrete painleve equation of sakai is constructed by a standard translation on the lattice given by nearest neighbor vectors in this paper we give a new elliptic discrete painleve equation obtained by translations along next nearest neighbor vectors this equation is a generic 8 parameter version of a 2 parameter elliptic difference equation found by reduction from adler s partial difference equation the so called q4 equation we also provide a projective reduction of the well known equation of sakai

lossy compression via sparse linear regression computationally efficient encoding and decoding we propose computationally efficient encoders and decoders for lossy compression using a sparse regression code the codebook is defined by a design matrix and codewords are structured linear combinations of columns of this matrix the proposed encoding algorithm sequentially chooses columns of the design matrix to successively approximate the source sequence it is shown to achieve the optimal distortion rate function for independent identically distributed i i d gaussian sources under the squared error distortion criterion for a given rate the parameters of the design matrix can be varied to tradeoff distortion performance with encoding complexity an example of such a tradeoff as a function of the block length n is the following with computational resource space or time per source sample of o n log n 2 for a fixed distortion level above the gaussian distortion rate function the probability of excess distortion decays exponentially in n the sparse regression code is robust in the following sense for any ergodic source the proposed encoder achieves the optimal distortion rate function of an i i d gaussian source with the same variance simulations show that the encoder has good empirical performance especially at low and moderate rates

ldgm based multiple description coding for finite alphabet sources this work presents an ldgm based practical successive coding scheme for the multiple description md problem for finite alphabet sources the scheme which targets the zhang berger zb rate distortion region is shown to be asymptotically optimal with joint typicality encoding while as a practical encoding solution a message passing algorithm is adopted we further discuss in more detail the application of the coding scheme in three cases of the md problem with the hamming distortion measure 1 no excess sum rate for binary sources 2 successive refinement and 3 no excess marginal rate for the uniform binary source in the no excess sum rate case some progress is made in the characterization of fundamental limits by deriving the analytical expression of the distortion region for general binary sources and of the auxiliary variables needed to achieve its boundary the exact expression of the zhang berger upper bound to the central distortion is also provided for the case of no excess marginal rate for the uniform binary source the proposed ldgm based coding scheme is tested in practice for all three aforementioned cases the experimental results show very good performance demonstrating its ability to approach the theoretical rate distortion limits or the available upper bounds

a geometrical description of the discrete painleve vi and v equations we present a geometrical approach for the discrete painleve equations based on weyl groups the method relies on the bilinear formalism and assumes that the multidimensional function lives on the weight lattice of the appropriate affine weyl group the equations for the function a system of nonautonomous hirota miwa equations govern the evolution along the independent variable and the parameters of the equation the latter evolution induced by the schlesinger transformations in the present paper we analyse the case of the e 1 7 group using the geometrical description we derive the nonlinear discrete equations we find that in the case of the e 1 7 group these are the asymmetric q pvi and d pv that were recently proposed

singularity confinement and full deautonomisation a discrete integrability criterion abstract we present a new approach to singularity confinement which makes it an efficient and reliable discrete integrability detector our method is based on the full deautonomisation procedure which consists in analysing non autonomous extensions of a given discrete system obtained by adding terms that are initially absent but whose presence does not alter the singularity pattern a justification for this approach is given through an algebro geometric analysis we also introduce the notions of early and late confinement while the former is a confinement that may exist already for the autonomous system the latter corresponds to a singularity pattern longer than that of the autonomous case late confinement will be shown to play an important role in the singularity analysis of systems with non trivial gauge freedom for which the existence of an undetected gauge in conjunction with a sketchy analysis might lead to erroneous conclusions as to their integrability an algebro geometric analysis of the role of late confinement in this context is also offered this novel type of singularity confinement analysis will be shown to allow for the exact calculation of the algebraic entropy of a given mapping

tropical geometric interpretation of ultradiscrete singularity confinement using the interpretation of the ultradiscretization procedure as a non archimedean valuation we use results of tropical geometry to show how roots and poles manifest themselves in piece wise linear systems as points of non differentiability this will allow us to demonstrate a correspondence between singularity confinement for discrete integrable systems and ultradiscrete singularity confinement for ultradiscrete integrable systems

two variations on a3 a1 a1 1 type discrete painlev equations by considering the normalizers of reflection subgroups of types a 1 1 and a 1 3 in w d5 1 two subgroups w a3 a1 1 w a1 1 and w a1 a1 1 w a3 1 can be constructed from a a3a1a1 1 ty

a classification of two dimensional integrable mappings and rational elliptic surfaces we classify two dimensional integrable mappings by investigating the actions on the fiber space of rational elliptic surfaces while the qrt mappings can be restricted on each fiber there exist several classes of integrable mappings which exchange fibers we also show an equivalent condition when a generalized halphen surface becomes a halphen surface of index m

singularity analysis for difference painleve equations associated with the affine weyl group e 8 we propose a new expression for the generic additive discrete painleve equation associated with the affine weyl group this expression makes the singularity analysis of the mapping perfectly straightforward and allows the implementation of the singularity confinement criterion in an extremely simple way using this new approach we derive new classes of discrete painleve equations related to

connection matrices for ultradiscrete linear problems we present theory outlining associated linear problems for ultradiscrete equations the appropriate domain for these problems is the max plus semiring our main result is that despite the restrictive nature of the max plus semiring it is still possible to define a theory of connection matrices analogous to that of birkhoff and his school for systems of linear difference equations we use such theory to provide evidence for the integrability of an ultradiscrete difference equation

transformations of elliptic hypergeometric integrals we prove a pair of transformations relating elliptic hypergeometric integrals of different dimensions corresponding to the root systems bc n and a n as a special case we recover some integral identities conjectured by van diejen and spiridonov for bc n we also consider their type ii integral their proof of that integral together with our transformation gives rise to pairs of adjoint integral operators a different proof gives rise to pairs of adjoint difference operators these allow us to construct a family of biorthogonal abelian functions generalizing the koornwinder polynomials and satisfying the analogues of the macdonald conjectures finally we discuss some transformations of type ii style integrals in particular we find that adding two parameters to the type ii integral gives an integral invariant under an appropriate action of the weyl group e 7

globally nilpotent differential operators and the square ising model we recall various multiple integrals with one parameter related to the isotropic square ising model and corresponding respectively to the n particle contributions of the magnetic susceptibility to the lattice form factors to the two point correlation functions and to their extensions the univariate analytic functions defined by these integrals are holonomic and even g functions they satisfy fuchsian linear differential equations with polynomial coefficients and have some arithmetic properties we recall the explicit forms found in previous work of these fuchsian equations as well as their russian doll and direct sum structures these differential operators are selected fuchsian linear differential operators and their remarkable properties have a deep geometrical origin they are all globally nilpotent or sometimes even have zero p curvature we also display miscellaneous examples of globally nilpotent operators emerging from enumerative combinatorics problems for which no integral representation is yet known focusing on the factorized parts of all these operators we find out that the global nilpotence of the factors resp p curvature nullity corresponds to a set of selected structures of algebraic geometry elliptic curves modular curves curves of genus five six and even a remarkable weight 1 modular form emerging in the three particle contribution 3 of the magnetic susceptibility of the square ising model noticeably this associated weight 1 modular form is also seen in the factors of the differential operator for another n fold integral of the ising class 3 h for the staircase polygons counting and in ap ry s study of 3 g functions naturally occur as solutions of globally nilpotent operators in the case where we do not have g functions but hamburger functions one irregular singularity at 0 or that correspond to the confluence of singularities in the scaling limit the p curvature is also found to verify new structures associated with simple deformations of the nilpotent property

numerical solution of the discrete time periodic riccati equation in this paper we present a method for the computation of the periodic nonnegative definite stabilizing solution of the periodic riccati equation this method simultaneously triangularizes by orthogonal equivalences a sequence of matrices associated with a cyclic pencil formulation related to the euler lagrange difference equations in doing so it is possible to extract a basis for the stable deflating subspace of the extended pencil from which the riccati solution is obtained this algorithm is an extension of the standard qz algorithm and retains its attractive features such as quadratic convergence and small relative backward error a method to compute the optimal feedback controller gains for linear discrete time periodic systems is dealt with

periodic orbits and chaos in fast slow systems with bogdanov takens type fold points abstract the existence of stable periodic orbits and chaotic invariant sets of singularly perturbed problems of fast slow type having bogdanov takens bifurcation points in its fast subsystem is proved by means of the geometric singular perturbation method and the blow up method in particular the blow up method is effectively used for analyzing the flow near the bogdanov takens type fold point in order to show that a slow manifold near the fold point is extended along the boutroux s tritronquee solution of the first painleve equation in the blow up space

hash property and coding theorems for sparse matrices and maximum likelihood coding the aim of this paper is to prove the achievability of rate regions for several coding problems by using sparse matrices with logarithmic column degree and maximum likelihood ml coding these problems are the gel fand pinsker problem the wyner ziv problem and the one helps one problem source coding with partial side information at the decoder to this end the notion of a hash property for an ensemble of functions is introduced and it is proved that an ensemble of q ary sparse matrices satisfies the hash property based on this property it is proved that the rate of codes using sparse matrices and ml coding can achieve the optimal rate

geometric treatments and a common mechanism in finite time singularities for autonomous odes abstract geometric treatments of blow up solutions for autonomous ordinary differential equations and their blow up rates are concerned our approach focuses on the type of invariant sets at infinity via compactifications of phase spaces and dynamics on their center stable manifolds in particular we show that dynamics on center stable manifolds of invariant sets at infinity with appropriate time scale desingularizations as well as blowing up of singularities characterize dynamics of blow up solutions as well as their rigorous blow up rates similarities for characterizing finite time extinction and asymptotic behavior of compacton traveling waves to blow up solutions are also shown

discrete dynamical systems associated with the configuration space of 8 points in ℙ3 ℂ a 3 dimensional analogue of sakai s theory concerning the relation between rational surfaces and discrete painleve equations is studied for a family of rational varieties obtained by blow ups at 8 points in general position in 3 we define its symmetry group using the inner product that is associated with the intersection numbers and show that the group is isomorphic to the weyl group of type e7 1 by parametrizing the configuration space by means of elliptic curves the action of the weyl group and the dynamical system associated with a translation are explicitly described as a result it is found that the action of the weyl group on 3 preserves a one parameter family of quadratic surfaces and that it can therefore be reduced to the action on 1 1

quantum discrete dubrovin equations the discrete equations of motion for the quantum mappings of kdv type are given in terms of the sklyanin variables which are also known as quantum separated variables both temporal discrete time evolutions and spatial along the lattice at a constant time level evolutions are considered in the classical limit the temporal equations reduce to the classical discrete dubrovin equations as given in a previous publication nijhoff f w 2000 chaos solitons fractals 11 19 28 the reconstruction of the original dynamical variables in terms of the sklyanin variables is also achieved

integrable mappings via rational elliptic surfaces we present a geometric description of the qrt map which is an integrable mapping introduced by quispel roberts and thompson in terms of the addition formula of a rational elliptic surface by this formulation we classify all the cases when the qrt map is periodic and show that its period is 2 3 4 5 or 6 a generalization of the qrt map which acts birationally on a pencil of k3 surfaces or calabi yau manifolds is also presented

isomonodromy transformations of linear systems of difference equations we introduce and study isomonodromy transformations of the matrix linear difference equation y z 1 a z y z with polynomial a z our main result is construction of an isomonodromy action of zm n 1 1 on the space of coefficients a z here m is the size of matrices and n is the degree of a z the birational action of certain rank n subgroups can be described by difference analogs of the classical schlesinger equations and we prove that for generic initial conditions these difference schlesinger equations have a unique solution we also show that both the classical schlesinger equations and the schlesinger transformations known in isomonodromy theory can be obtained as limits of our action in two different limit regimes similarly to the continuous case for m n 2 the difference schlesinger equations and their q analogs yield discrete painlev e equations examples include dpii dpiv dpv and q pvi

finite branch solutions to painleve vi around a fixed singular point every finite branch local solution to the sixth painleve equation around a fixed singular point is an algebraic branch solution in particular a global solution is an algebraic solution if and only if it is finitely many valued globally the proof of this result relies on algebraic geometry of painleve vi riemann hilbert correspondence geometry and dynamics on cubic surfaces resolutions of kleinian singularities and power geometry of algebraic differential equations in the course of the proof we are also able to classify all finite branch solutions up to backlund transformations

the first second and fourth painleve equations on weighted projective spaces abstract the first second and fourth painleve equations are studied by means of dynamical systems theory and three dimensional weighted projective spaces c p 3 p q r s with suitable weights p q r s determined by the newton diagrams of the equations or the versal deformations of vector fields singular normal forms of the equations a simple proof of the painleve property and symplectic atlases of the spaces of initial conditions are given with the aid of the orbifold structure of c p 3 p q r s in particular for the first painleve equation a well known painleve s transformation is geometrically derived which proves to be the darboux coordinates of a certain algebraic surface with a holomorphic symplectic form the affine weyl group dynkin diagram and the boutroux coordinates are also studied from a view point of the weighted projective space

on the sensitivity of the spectral projection in this paper we apply the theory of condition developed by rice to define condition numbers of the spectral projection explicit expressions of the condition numbers are derived and some relations between the condition numbers of the spectral projection and the condition number of the associated invariant subspace are presented the results are illustrated by a simple numerical example

nonlinear sparse graph codes for lossy compression we propose a scheme for lossy compression of discrete memoryless sources the compressor is the decoder of a nonlinear channel code constructed from a sparse graph we prove asymptotic optimality of the scheme for any separable letter by letter bounded distortion criterion we also present a suboptimal compression algorithm which exhibits near optimal performance for moderate block lengths

an algebraic geometric approach to integrable maps of the plane we show that the dynamics of a birational map on an elliptic curve over a field is typically conjugate to addition by a point under the associated group law when the field is taken to be the function field of rational complex functions of one variable this amounts to an algebraic geometric version of the arnold liouville integrability theorem for planar integrable maps by products of this approach are that birational maps preserving foliations are necessarily the composition of two involutions and that relationships between birational maps preserving the same foliation can be described in terms of the respective points they add on the corresponding weierstrass curves when the result is applied to finite fields it helps explain some universal features of the periodic orbit distribution function for the reductions of integrable maps

folding transformations of the painleve equations new symmetries of the painleve differential equations called folding transformations are determined these transformations are not birational but algebraic transformations of degree 2 3 or 4 these are associated with quotients of the spaces of initial conditions of each painleve equation we make the complete list of such transformations up to birational symmetries we also discuss correspondences of special solutions of painleve equations

new airy type solutions of the ultradiscrete painleve ii equation with parity variables the q difference painleve ii equation admits special solutions written in terms of determinants whose entries are the general solution of the q airy equation an ultradiscrete limit of the special solutions is studied using the procedure of ultradiscretization with parity variables then we obtain new airy type solutions of the ultradiscrete painleve ii equation with parity variables and the solutions have a richer structure than the known solutions

variable length lossy coding using an ldpc code in this paper a new variable length coding scheme using a low density parity check ldpc code is proposed for the lossy compression of general i i d finite sources it is proved that the proposed scheme achieves the rate distortion function asymptotically for an ldpc ensemble for our setting miyake muramatsu already proposed an asymptotically optimal ldpc coding scheme in their scheme a source sequence is first vector quantized using an ldpc matrix and then it is compressed losslessly by fixed length coding with another ldpc matrix however it is not shown whether their scheme can attain good performance practically this is mainly because of the difficulty of lossless coding by a fixed length code in the proposed scheme the lossless compression is performed by arithmetic coding instead of the fixed length code combined with vector quantization using the reinforced belief propagation the proposed scheme attains performance near the rate distortion function practically with time complexity roughly equal to o nlogn for length n source sequences

solution of the equivalence problem for the third painleve equation we find necessary conditions for a second order ordinary differential equation to be equivalent to the painleve iii equation under a general point transformation their sufficiency is established by reduction to known results for the equations of the form y f x y we consider separately the generic case and the case of reducibility to an autonomous equation the results are illustrated by the primary resonance equation

holonomy of the ising model form factors we study the ising model two point diagonal correlation function c n n by presenting an exponential and form factor expansion in an integral representation which differs from the known expansion of wu mccoy tracy and barouch we extend this expansion weighting by powers of a variable the j particle contributions f j n n in the form factor expansion the corresponding extension of the two point diagonal correlation function c n n is shown for arbitrary to be a solution of the sigma form of the painleve vi equation introduced by jimbo and miwa in their isomonodromic approach to the ising model fuchsian linear differential equations for the form factors f j n n are obtained for j 9 and shown to have both a russian doll nesting and a decomposition of the corresponding linear differential operators as a direct sum of operators equivalent to symmetric powers of the second order linear differential operator associated with the elliptic integral e from this we show that each f j n n is unexpectedly simple being expressed polynomially in terms of the elliptic integrals e and k in contrast we exhibit some mathematical objects built from these form factors f j n n which break the direct sum of symmetric powers decomposition with its associated polynomial expressions first we show that the scaling limit of these differential operators and form factors breaks the direct sum structure but not the russian doll structure secondly we show that the previous extension of two point diagonal correlation functions c n n is for singled out values cos m n m n integers also solutions of fuchsian linear differential equations these solutions of painleve vi are not polynomial in e and k but are actually algebraic functions being associated with modular curves

matrix painleve systems we study a class of the isomonodromic deformation equations which we call matrix painleve systems they can be written in hamiltonian form the hamiltonian is written as the trace of the hamiltonian of the sixth painleve equation with canonical variables replaced by matrices we also derive the matrix version of the fifth fourth third and second painleve equation from the sixth matrix painleve system by the degeneration

a dynamical systems approach to the fourth painleve equation we use methods from dynamical systems to study the fourth painleve equation our starting point is the symmetric form of to which the poincare compactification is applied the motion on the sphere at infinity can be completely characterized there are fourteen fixed points which are classified into three different types generic orbits of the full system are curves from one of four asymptotically unstable points to one of four asymptotically stable points with the set of allowed transitions depending on the values of the parameters this allows us to give a qualitative description of a generic real solution of

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

discrete painleve equations with maximal periodicities we present the discrete painleve equations the coefficients of which have the maximal periodicity allowed by the geometric description in terms of affine weyl groups for each equation of the degeneration pattern obtained by sakai and which starts at the group we give the corresponding discrete painleve equation some of these systems were already known but here we proceed in a systematic way filling in all the gaps in particular we derive the equations with maximal periodicity associated to the affine weyl group which is the richest system in the sakai classification and which has yet to be studied in detail

a coalescent diagram of the painleve equations from the viewpoint of isomonodromic deformations we revise garnier okamoto s coalescent diagram of isomonodromic deformations and give a possible coalescent diagram from the viewpoint of isomonodromic deformations we have ten types of isomonodromic deformations and two of them give the same type of painleve equation we can naturally put the 34th painleve equation in our diagram which corresponds to the flaschka newell form of the second painleve equation

affine weyl groups discrete dynamical systems and painleve equations a new class of representations of affine weyl groups on rational functions are constructed in order to formulate discrete dynamical systems associated with affine root systems as an application some examples of difference and differential systems of painleve type are discussed

perturbation analysis for the eigenproblem of periodic matrix pairs this paper is devoted to perturbation analysis for the eigenproblem of periodic matrix pairs a j e j j 1 k we first study perturbation expansions of periodic deflating subspaces and eigenv

lossy source compression using low density generator matrix codes analysis and algorithms we study the use of low density generator matrix ldgm codes for lossy compression of the bernoulli symmetric source first we establish rigorous upper bounds on the average distortion achieved by check regular ensemble of ldgm codes under optimal minimum distance source encoding these bounds establish that the average distortion using such bounded degree families rapidly approaches the shannon limit as the degrees are increased second we propose a family of message passing algorithms ranging from the standard belief propagation algorithm at one extreme to a variant of survey propagation algorithm at the other when combined with a decimation subroutine and applied to ldgm codes with suitably irregular degree distributions we show that such a message passing decimation algorithm yields distortion very close to the shannon rate distortion bound for the binary symmetric source

lossy compression of discrete sources via the viterbi algorithm we present a new lossy compressor for finite alphabet sources for coding a sequence xn the encoder starts by assigning a certain cost to each possible reconstruction sequence it then finds the one that minimizes this cost and describes it losslessly to the decoder via a universal lossless compressor the cost of each sequence is a linear combination of its distance from the sequence xn and a linear function of its kth order empirical distribution the structure of the cost function allows the encoder to employ the viterbi algorithm to find the sequence with minimum cost we identify a choice of the coefficients used in the cost function which ensures that the algorithm universally achieves the optimum rate distortion performance for any stationary ergodic source in the limit of large provided that increases as o log n iterative techniques for approximating the coefficients which alleviate the computational burden of finding the optimal coefficients are proposed and studied

channel coding and lossy source coding using a generator of constrained random numbers stochastic encoders for channel coding and lossy source coding are introduced with a rate close to the fundamental limits where the only restriction is that the channel input alphabet and the reproduction alphabet of the lossy source code are finite random numbers which satisfy a condition specified by a function and its value are used to construct stochastic encoders the proof of the theorems is based on the hash property of an ensemble of functions where the results are extended to general channels sources and alternative formulas are introduced for channel capacity and the rate distortion region since an ensemble of sparse matrices has a hash property we can construct a code by using sparse matrices

2 parameter tau τ function for the first painleve equation topological recursion and direct monodromy problem via exact wkb analysis we show that a 2 parameter family of tau functions for the first painleve equation can be constructed by the discrete fourier transform of the topological recursion partition function for a family of elliptic curves we also perform an exact wkb theoretic computation of the stokes multipliers of associated isomonodromy system assuming certain conjectures

geometric reductions of abs equations on an n cube to discrete painleve systems in this paper we show how to relate n dimensional cubes on which abs equations hold to the symmetry groups of discrete painlev equations we here focus on the reduction from the four dimensional cube to the q discrete third painlev equation which is a dynamical system on a rational surface of type with the extended affine weyl group we provide general theorems to show that this reduction also extends to other discrete painlev equations at least of type a

near ideal m ary ldgm quantization with recovery for iterative mean square error mse quantizers with alphabet size m 2k using low density generator matrix ldgm code constructions an efficient recovery algorithm is proposed which adjusts the priors used in belief propagation bp to limit the impact of previous non ideal decimation steps based on an analysis of the bp process under ideal or non ideal decimation the algorithm first estimates the conditional probability distributions describing the effect of non ideal decimation then adjusts the priors to make the distributions match the ideal situation as shown in simulation results the recovery algorithm can improve quantization performance greatly reducing the shaping loss to as low as 0 012 db while the increase in computational complexity is modest thanks to the use of fft techniques

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

computation of pseudospectra via spectral projectors abstract in this note we discuss new techniques for analyzing the pseudospectra of matrices and propose a numerical method for computing the spectral projector associated with a group of eigenvalues enclosed by a polygonal curve numerical tests are reported

bilinear equations and q discrete painleve equations satisfied by variables and coefficients in cluster algebras we construct cluster algebras the variables and coefficients of which satisfy the discrete mkdv equation the discrete toda equation and other integrable bilinear equations several of which lead to q discrete painleve equations these cluster algebras are obtained from quivers with an infinite number of vertices or with the mutation period property we will also show that a suitable transformation of quivers corresponds to a reduction of the difference equation

a geometric approach to singularity confinement and algebraic entropy a geometric approach to the equation found by hietarinta and viallet which satisfies the singularity confinement criterion but exhibits chaotic behaviour is presented it is shown that this equation can be lifted to an automorphism of a certain rational surface and can therefore be considered to be the action of an extended weyl group of indefinite type a method to calculate its algebraic entropy by using the theory of intersection numbers is presented

operational duality between lossy compression and channel coding we explore the duality between lossy compression and channel coding in the operational sense whether a capacity achieving encoder decoder sequence achieves the rate distortion function of the dual problem when the channel decoder encoder is the source compressor decompressor resp and vice versa we show that if used as a lossy compressor the maximum likelihood channel decoder of a randomly chosen capacity achieving codebook achieves the rate distortion function almost surely however operational duality does not hold for every capacity achieving encoder decoder sequence or rate distortion achieving compressor decompressor sequence we show that there exist optimal channel coding lossy compression schemes which fail when used for the dual lossy compression channel coding resp problem

rational surfaces associated with affine root systems and geometry of the painleve equations we present a geometric approach to the theory of painleve equations based on rational surfaces our starting point is a compact smooth rational surface x which has a unique anti canonical divisor d of canonical type we classify all such surfaces x to each x there corresponds a root subsystem of e 1 8 inside the picard lattice of x we realize the action of the corresponding affine weyl group as the cremona action on a family of these surfaces we show that the translation part of the affine weyl group gives rise to discrete painleve equations and that the above action constitutes their group of symmetries by backlund transformations the six painleve differential equations appear as degenerate cases of this construction in the latter context x is okamoto s space of initial conditions and d is the pole divisor of the symplectic form defining the hamiltonian structure