numerical solution of nonlinear quasi stationary magnetic fields the problem of quasi stationary nonlinear magnetic fields is formulated in a variational form a galerkin finite element solution is constructed discretization in time is carried out by means of one and two step linear a stable schemes the resulting system of nonlinear equations is solved by newton s method a numerical example is given

adaptive weighted learning for unbalanced multicategory classification summary in multicategory classification standard techniques typically treat all classes equally this treatment can be problematic when the dataset is unbalanced in the sense that certain classes have very small class proportions compared to others the minority classes may be ignored or discounted during the classification process due to their small proportions this can be a serious problem if those minority classes are important in this article we study the problem of unbalanced classification and propose new criteria to measure classification accuracy moreover we propose three different weighted learning procedures two one step weighted procedures as well as one adaptive weighted procedure we demonstrate the advantages of the new procedures using multicategory support vector machines through simulated and real datasets our results indicate that the proposed methodology can handle unbalanced classification problems effectively

growth and integrability in the dynamics of mappings the growth of some numerical characteristics of the mappings under their iterations in the context of the general problem of integrability is discussed in the general case such characteristics as complexity by arnold or the number of the different images for the multiple valued mappings are growing exponentially it is shown that the integrability is closely related with thepolynomial growth the analogies with quantum integrable systems are discussed

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

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

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

diophantine non integrability of a third order recurrence with the laurent property we consider a one parameter family of third order nonlinear recurrence relations each member of this family satisfies the singularity confinement test has a conserved quantity and moreover has the laurent property all of the iterates are laurent polynomials in the initial data however we show that these recurrences are not diophantine integrable according to the definition proposed by halburd 2005 j phys a math gen 38 l1 explicit bounds on the asymptotic growth of the heights of iterates are obtained for a special choice of initial data as a by product of our analysis infinitely many solutions are found for a certain family of diophantine equations studied by mordell that includes markoff s equation

multicategory support vector machines two category support vector machines svm have been very popular in the machine learning community for classification problems solving multicategory problems by a series of binary classifiers is quite common in the svm paradigm however this approach may fail under various circumstances we propose the multicategory support vector machine msvm which extends the binary svm to the multicategory case and has good theoretical properties the proposed method provides a unifying framework when there are either equal or unequal misclassification costs as a tuning criterion for the msvm an approximate leave one out cross validation function called generalized approximate cross validation is derived analogous to the binary case the effectiveness of the msvm is demonstrated through the applications to cancer classification using microarray data and cloud classification with satellite radiance profiles

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

multicategory ψ learning in binary classification margin based techniques usually deliver high performance as a result a multicategory problem is often treated as a sequence of binary classifications in the absence of a dominating class this treatment may be suboptimal and may yield poor performance such as for support vector machines svms we propose a novel multicategory generalization of learning that treats all classes simultaneously the new generalization eliminates this potential problem while at the same time retaining the desirable properties of its binary counterpart we develop a statistical learning theory for the proposed methodology and obtain fast convergence rates for both linear and nonlinear learning examples we demonstrate the operational characteristics of this method through a simulation our results indicate that the proposed methodology can deliver accurate class prediction and is more robust against extreme observations than its svm counterpart

algebraic entropy for algebraic maps we propose an extension of the concept of algebraic entropy as introduced by bellon and viallet for rational maps to algebraic maps or correspondences of a certain kind the corresponding entropy is an index of the complexity of the map the definition inherits the basic properties from the definition of entropy for rational maps we give an example with positive entropy as well as two examples taken from the theory of backlund transformations

comparisons of predictive values of binary medical diagnostic tests for paired designs positive and negative predictive values of a diagnostic test are key clinically relevant measures of test accuracy surprisingly statistical methods for comparing tests with regard to these parameters have not been available for the most common study design in which each test is applied to each study individual in this paper we propose a statistic for comparing the predictive values of two diagnostic tests using this paired study design the proposed statistic is a score statistic derived from a marginal regression model and bears some relation to mcnemar s statistic as mcnemar s statistic can be used to compare sensitivities and specificities of diagnostic tests parameters that condition on disease status our statistic can be considered as an analog of mcnemar s test for the problem of comparing predictive values parameters that condition on test outcome we report on the results of a simulation study designed to examine the properties of this test under a variety of conditions the method is illustrated with data from a study of methods for diagnosis of coronary artery disease

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

bias corrected diagonal discriminant rules for high dimensional classification diagonal discriminant rules have been successfully used for high dimensional classification problems but suffer from the serious drawback of biased discriminant scores in this article we propose improved diagonal discriminant rules with bias corrected discriminant scores for high dimensional classification we show that the proposed discriminant scores dominate the standard ones under the quadratic loss function analytical results on why the bias corrected rules can potentially improve the predication accuracy are also provided finally we demonstrate the improvement of the proposed rules over the original ones through extensive simulation studies and real case studies

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

sigma function solution of the initial value problem for somos 5 sequences the somos 5 sequences are a family of sequences defined by a fifth order bilinear recurrence relation with constant coefficients for particular choices of coefficients and initial data integer sequences arise by making the connection with a second order nonlinear mapping with a first integral we prove that the two subsequences of odd even index terms each satisfy a somos 4 fourth order recurrence this leads directly to the explicit solution of the initial value problem for the somos 5 sequences in terms of the weierstrass sigma function for an associated elliptic curve

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

finite element variational crimes in parabolic elliptic problems the study of a completely discretized variational problem corresponding to a nonlinear second order parabolic elliptic initial boundary value problem with an initial value only inl 2 is presented the discretization in time is done by the euler backward method the discretization in space by the finite element method with linear functions on triangular elements the changes in domain are taken into account the convergence and the unconditional stability of the method is proved under the assumption that the boundary is piecewise of classc 3 and the forma v w has a potential and is strongly monotone and lipschitz continuous as a by product the existence and uniqueness theorem is obtained

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

integrable mappings and soliton equations ii abstract some simple solutions of discrete soliton equations are shown to satisfy 2d mappings we show that these belong to a recently introduced 18 parameter family of integrable reversible mappings of the plane thus lending weight to a previous conjecture we also give an example of an integrable mapping occuring in an exactly solvable model in statistical mechanics finally we discuss the notion of generalized reversibility

a conjecture based on somos 4 sequence and its extension abstract in two recent papers by barry 2010 29 and 2011 30 it is conjectured that somos 4 admits a solution expressed in terms of hankel determinant with its elements satisfying a convolution recursion relation in this paper barry s conjecture on somos 4 is firstly confirmed actually we present a more generalized result the proof is mainly based on new findings on properties for so called block hankel determinants the method can also be used to prove another conjecture proposed by michael somos which has been solved by guoce xin

real arnold complexity versus real topological entropy for birational transformations we consider a family of birational measure preserving transformations of two variables depending on one parameter for which simple rational expressions with integer coefficients for the exact expression of the dynamical zeta function have been conjectured together with an equality between the multiplicative rate of growth of the arnold complexity and the exponential of the topological entropy this identification takes place for the birational mapping seen as a mapping bearing on two complex variables acting in a complex projective space we revisit this identification between these two quite universal complexities by considering now the mapping as a mapping bearing on two real variables the definitions of the two previous topological complexities arnold complexity and topological entropy are modified according to this real variables point of view most of the universality is lost however the results presented here are again in agreement with an identification between the multiplicative rate of growth of some real arnold complexity and the exponential of some real topological entropy a detailed analysis of this real arnold complexity as a function of the parameter of this family of birational transformations of two variables is given one can also slightly modify the definition of the dynamical zeta function introducing a real dynamical zeta function associated with the counting of the real cycles only similarly one can also introduce some real arnold complexity generating functions we show that several of these two real generating functions seem to have the same singularities furthermore we actually conjecture several simple rational expressions for them yielding again algebraic values for the exponential of the real topological entropy in particular when the parameter of our family of birational transformations becomes large we obtain two interesting compatible nontrivial rational expressions these rational results for real mappings cannot be understood in terms of any obvious markov s partition or symbolic dynamics hyperbolic systems interpretation the birational transformation is far from being hyperbolic it is measure preserving it would be useful to know whether this relation between the arnold complexity and the topological entropy as well as the rationality of the degree generating functions and dynamical zeta functions are a consequence of the measure preserving property of the mapping

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

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

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

bilinear discrete painleve equations based on the results of singularity confinement we derive bilinear expressions for the discrete painleve equations in these cases where a bilinear expression is not sufficient we obtain trilinear or higher multilinear expressions we show that the bilinear approach provides a natural framework for the derivation of backlund and schlesinger transforms for the discrete painleve equations

a generalized fellegi sunter framework for multiple record linkage with application to homicide record systems we present a probabilistic method for linking multiple datafiles this task is not trivial in the absence of unique identifiers for the individuals recorded this is a common scenario when linking census data to coverage measurement surveys for census coverage evaluation and in general when multiple record systems need to be integrated for posterior analysis our method generalizes the fellegi sunter theory for linking records from two datafiles and its modern implementations the goal of multiple record linkage is to classify the record k tuples coming from k datafiles according to the different matching patterns our method incorporates the transitivity of agreement in the computation of the data used to model matching probabilities we use a mixture model to fit matching probabilities via maximum likelihood using the expectation maximization algorithm we present a method to decide the record k tuples membership to the subsets of matching patterns and we prove its optimality we apply our method to th

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

weighted distance weighted discrimination and its asymptotic properties while distance weighted discrimination dwd is an appealing approach to classification in high dimensions it was designed for balanced datasets in the case of unequal costs biased sampling or unbalanced data there are major improvements available using appropriately weighted versions of dwd wdwd a major contribution of this paper is the development of optimal weighting schemes for various nonstandard classification problems in addition we discuss several alternative criteria and propose an adaptive weighting scheme awdwd and demonstrate its advantages over nonadaptive weighting schemes under some situations the second major contribution is a theoretical study of weighted dwd both high dimensional low sample size asymptotics and fisher consistency of dwd are studied the performance of weighted dwd is evaluated using simulated examples and two real data examples the theoretical results are also confirmed by simulations

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

studies on the painlev equations iii second and fourth painlev equations p ii andp iv etude des transformations birationnelles canoniques du systeme de painleve associe a chacune des equations

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

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

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

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

flexible high dimensional classification machines and their asymptotic properties classification is an important topic in statistics and machine learning with great potential in many real applications in this paper we investigate two popular large margin classification methods support vector machine svm and distance weighted discrimination dwd under two contexts the high dimensional low sample size data and the imbalanced data a unified family of classification machines the flexible assortment machine flame is proposed within which dwd and svm are special cases the flame family helps to identify the similarities and differences between svm and dwd it is well known that many classifiers overfit the data in the high dimensional setting and others are sensitive to the imbalanced data that is the class with a larger sample size overly influences the classifier and pushes the decision boundary towards the minority class svm is resistant to the imbalanced data issue but it overfits high dimensional data sets by showing the undesired data piling phenomenon the dwd method was proposed to improve svm in the high dimensional setting but its decision boundary is sensitive to the imbalanced ratio of sample sizes our flame family helps to understand an intrinsic connection between svm and dwd and provides a trade off between sensitivity to the imbalanced data and overfitting the high dimensional data several asymptotic properties of the flame classifiers are studied simulations and real data applications are investigated to illustrate theoretical findings

robust truncated hinge loss support vector machines the support vector machine svm has been widely applied for classification problems in both machine learning and statistics despite its popularity however svm has some drawbacks in certain situations in particular the svm classifier can be very sensitive to outliers in the training sample moreover the number of support vectors svs can be very large in many applications to circumvent these drawbacks we propose the robust truncated hinge loss svm rsvm which uses a truncated hinge loss the rsvm is shown to be more robust to outliers and to deliver more accurate classifiers using a smaller set of svs than the standard svm our theoretical results show that the rsvm is fisher consistent even when there is no dominating class a scenario that is particularly challenging for multicategory classification similar results are obtained for a class of margin based classifiers

singularity confinement and chaos in discrete systems we present a number of second order maps which pass the singularity confinement test commonly used to identify integrable discrete systems but which nevertheless are non integrable as a more sensitive integrability test we propose the analysis of the complexity algebraic entropy of the map using the growth of the degree of its iterates integrability is associated with polynomial growth while the generic growth is exponential for chaotic systems

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

infinitely imbalanced logistic regression in binary classification problems it is common for the two classes to be imbalanced one case is very rare compared to the other in this paper we consider the infinitely imbalanced case where one class has a finite sample size and the other class s sample size grows without bound for logistic regression the infinitely imbalanced case often has a useful solution under mild conditions the intercept diverges as expected but the rest of the coefficient vector approaches a non trivial and useful limit that limit can be expressed in terms of exponential tilting and is the minimum of a convex objective function the limiting form of logistic regression suggests a computational shortcut for fraud detection problems

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

how to detect the integrability of discrete systems several integrability tests for discrete equations will be reviewed all tests considered can be applied directly to a given discrete equation and do not rely on the a priori knowledge of the existence of related structures such as lax pairs specifically singularity confinement algebraic entropy nevanlinna theory diophantine integrability and discrete systems over finite fields will be described

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

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

empirical bayes confidence intervals shrinking both means and variances summary we construct empirical bayes intervals for a large number p of means the existing intervals in the literature assume that variances are either equal or unequal but known when the variances are unequal and unknown the suggestion is typically to replace them by unbiased estimators however when p is large there would be advantage in borrowing strength from each other we derive double shrinkage intervals for means on the basis of our empirical bayes estimators that shrink both the means and the variances analytical and simulation studies and application to a real data set show that compared with the t intervals our intervals have higher coverage probabilities while yielding shorter lengths on average the double shrinkage intervals are on average shorter than the intervals from shrinking the means alone and are always no longer than the intervals from shrinking the variances alone also the intervals are explicitly defined and can be computed immediately

on ψ learning the concept of large margins have been recognized as an important principle in analyzing learning methodologies including boosting neural networks and support vector machines svms however this concept alone is not adequate for learning in nonseparable cases we propose a learning methodology called learning that is derived from a direct consideration of generalization errors we provide a theory for learning and show that it essentially attains the optimal rates of convergence in two learning examples finally results from simulation studies and from breast cancer classification confirm the ability of learning to outperform svm in generalization

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

finite element solution of maxwell s equations for hyperthermia treatment planning abstract a general galerkin formulation is given for the periodic form of maxwell s equations on a bounded inhomogeneous medium the formulation conserves energy exactly and its dispersion relation is studied solutions for test problems representative of hyperthermia applications show good agreement with analytic solutions a representative clinical example is shown

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