erratum solution of the one dimensional n body problems with quadratic and or inversely quadratic pair potentials j math phys 12 419 436 1971

identification and estimation of statistical functionals using incomplete data incomplete data due to missing observations or interval measurement of variables usually cause parameters of interest in applications to be unidentified except under untestable and often controversial assumptions however it is often possible to identify sharp bounds on parameters without making untestable assumptions about the process through which data become incomplete the bounds contain all logically possible values of the parameters and can be estimated consistently by replacing the population distribution of the data with the empirical distribution this is straightforward in some circumstances but computationally burdensome in others this paper describes the general problem and presents an empirical illustration

a new class of solvable dynamical systems a new class of dynamical systems are presented together with their solutions some of these models are isochronous namely their generic solutions are all completely periodic with the same period others are characterized by friction all solutions vanishing in the remote future and still others are asymptotically isochronous approaching an isochronous behavior in the remote future

kp hierarchy for the cyclic quiver we introduce a generalisation of the kp hierarchy closely related to the cyclic quiver and the cherednik algebra hk zm this hierarchy depends on m parameters one of which can be eliminated with the usual kp hierarchy corresponding to the m 1 case generalising the result of wilson invent math 133 1 1 41 1998 we show that our hierarchy admits solutions parameterised by suitable quiver varieties the pole dynamics for these solutions is shown to be governed by the classical calogero moser system for the wreath product zm sn and its new spin version these results are further extended to the case of the multi component hierarchy

classification of r operators we classified the r operators which satisfy the quantum yang baxter equation on a function space in this study we gave all the meromorphic solutions of the system of the functional equations which is a necessary and sufficient condition for the r operator to satisfy the yang baxter equation most of the solutions were expressed in terms of the elliptic trigonometric and rational functions

the neatest many body problem amenable to exact treatments a goldfish various formulations findings and conjectures are reviewed which relate to a many body problem that is arguably the neatest nontrivial such model amenable to exact treatments

new approach to the statistical properties of energy levels the joint distribution of energy eigenvalues of a hamiltonian is derived by means of the usual statistical laws of classical many body systems it makes a transition from the poisson type to the gaussian type depending on the value of a single parameter characteristic of the hamiltonian

commutation methods applied to the mkdv equation an explicit construction of solutions of the modified korteweg de vries equation given a solution of the ordinary korteweg de vries equation is provided our theory is based on commutation methods i e n 1 supersymmetry underlying miura s transformation that links solutions of the two evolution equations

a class of integrable hamiltonian systems whose solutions are perhaps all completely periodic we show that the dynamical system characterized by the complex equations of motion qj i qj k 1 k jnqjqkf qj qk j 1 n with f x x x is hamiltonian and integrable and we conjecture that all its solutions qj t j 1 n are completely periodic with a period that is a finite integral multiple of t 2 here n is an arbitrary positive integer is an arbitrary nonvanishing real constant y y is the weierstrass function with arbitrary semiperiods and are two arbitrary constants special cases are f x 2 coth x 1 r2 sinh2 x f x 2 coth x f x 2 sinh x f x 2 x 1 2x2 and of course f x 2 x these findings as well as the conjecture which is shown to be true in some of these special cases are based on the possibility to recast these equations of motion in the modified lax form l i l l m with l and m appropriate n n matrix functions of the n dynamical variables qj and of their time derivatives qj

rigidity functional equations and the calogero moser model suppose we have a natural hamiltonian h of n particles on the line centre of mass momentum p and a further independent quantity q cubic in the momenta if these are each sn invariant and mutually poisson commute we have the calogero moser system with potential v 1 6 i j qi qj const

a technique to identify solvable dynamical systems and a solvable generalization of the goldfish many body problem a simple approach is discussed which associates to solvable matrix equations solvable dynamical systems generally interpretable as interesting many body problems possibly involving auxiliary dependent variables in addition to those identifying the positions of the moving particles we then focus on cases in which the auxiliary variables can be altogether eliminated reobtaining thereby via this unified approach well known solvable many body problems and moreover a solvable extension of the goldfish model

on isochronous bruschi ragnisco ruijsenaars toda lattices equilibrium configurations behaviour in their neighbourhood diophantine relations and conjectures isochronous versions of the bruschi ragnisco ruijsenaars toda lattice and of some of its also integrable variants are introduced their equilibrium configurations are found when they exist and by investigating the motions of these systems near equilibrium some diophantine relations are obtained as well as some insight into the solution of those of these integrable models whose solutions are not yet known

the lax pairs for elliptic cn and bcn ruijsenaars schneider models and their spectral curves we study the elliptic c n and bcn ruijsenaars schneider models which are elliptic generalization of systems given in previous paper by the present authors chen j math phys 41 8132 2000 the lax pairs for these models are constructed by hamiltonian reduction technology we show that the spectral curves can be parametrized by the involutive integrals of motion for these models taking nonrelativistic limit and scaling limit we verify that they lead to the systems corresponding to calogero moser and toda types c 2001 american institute of physics

matrices differential operators and polynomials it is noted that the matrix z of order n defined in terms of the n arbitrary numbers xj by the formula zjk jk j l 1 xj xl 1 1 jk xj xk 1 may be considered in an appropriate framework to correspond to the differential operator d dx there follow prescriptions to construct explicit matrices of arbitrary order n in terms of n or more arbitrary numbers or of the n roots of given polynomials matrices whose eigenvalues and eigenvectors are given fully or in part by very simple formulas novel representations of the classical polynomials hermite laguerre lagrange gegenbauer jacobi are also obtained such as the formula for hermite polynomials hn x 2n det xi h where i is the unit matrix of order n and the matrix h of order n is defined by hjk 2n 1 2 jk n 1 exp 2i j 1 2 exp 2i j 1 jk exp 2i j 2i sin j k 1 exp i j k with j j n arbitrary

integrable discretizations of the spin ruijsenaars schneider models integrable discretizations are introduced for the rational and hyperbolic spin ruijsenaars schneider models these discrete dynamical systems are demonstrated to belong to the same integrable hierarchies as their continuous time counterparts explicit solutions are obtained for arbitrary flows of the hierarchies including the discrete time ones

asymptotic properties for a class of partially identified models we propose inference procedures for partially identified population features for which the population identification region can be written as a transformation of the aumann expectation of a properly defined set valued random variable svrv an svrv is a mapping that associates a set rather than a real number with each element of the sample space examples of population features in this class include interval identified scalar parameters best linear predictors with interval outcome data and parameters of semiparametric binary models with interval regressor data we extend the analogy principle to svrvs and show that the sample analog estimator of the population identification region is given by a transformation of a minkowski average of svrvs using the results of the mathematics literature on svrvs we show that this estimator converges in probability to the population identification region with respect to the hausdorff distance we then show that the hausdorff distance and the directed hausdorff distance between the population identification region and the estimator when properly normalized by n converge in distribution to functions of a gaussian process whose covariance kernel depends on parameters of the population identification region we provide consistent bootstrap procedures to approximate these limiting distributions using similar arguments as those applied for vector valued random variables we develop a methodology to test assumptions about the true identification region and its subsets we show that these results can be used to construct a confidence collection and a directed confidence collection those are respectively collection of sets that when specified as a null hypothesis for the true value a subset of values of the population identification region cannot be rejected by our tests

quasi lagrangian systems of newton equations systems of newton equations of the form q 12a 1 q k with an integral of motion quadratic in velocities are studied these equations generalize the potential case when a i the identity matrix and they admit a curious quasi lagrangian formulation which differs from the standard lagrange equations by the plus sign between terms a theory of such quasi lagrangian newton qln systems having two functionally independent integrals of motion is developed with focus on two dimensional systems such systems admit a bi hamiltonian formulation and are proved to be completely integrable by embedding into five dimensional integrable systems they are characterized by a linear second order partial differential equation pde which we call the fundamental equation fundamental equations are classified through linear pencils of matrices associated with qln systems the theory is illustrated by two classes of systems separable potential systems and driven systems new separation variables for driven systems are foun

the schrodinger langevin equation special solutions and nonexistence of solitary waves two types of solutions of the schrodinger langevin equation are investigated it is proved that a special type of gaussian solutions exist globally in time for the harmonic oscillator hamiltonian furthermore it is shown that the schrodinger langevin equation can have no solitary wave type solutions in the damped free particle case which lie in l2

characteristic classes and hitchin systems general construction we consider topologically non trivial higgs g bundles over riemann surfaces g with marked points and the corresponding hitchin systems we show that if g is not simply connected then there exists a finite number of different sectors of the higgs bundles endowed with the hitchin hamiltonians they correspond to different characteristic classes of the underlying bundles defined as elements of h 2 sigma g mathcal z g mathcal z g is a center of g we define the conformal version cg of g an analog of gl n for sl n and relate the characteristic classes with degrees of cg bundles we describe explicitly bundles in the genus one g 1 case if 1 has one marked point and the bundles are trivial then the hitchin systems coincide with calogero moser cm systems for the nontrivial bundles we call the corresponding systems the modified calogero moser mcm systems their phase space has the same dimension as the phase space of the cm systems with spin variables but less number of particles and greater number of spin variables starting with these bundles we construct lax operators quadratic hamiltonians and define the phase spaces and the poisson structure using dynamical r matrices the latter are completion of the classification list of etingof varchenko corresponding to the trivial bundles to describe the systems we use a special basis in the lie algebras that generalizes the basis of t hooft matrices for sl n we find that the mcm systems contain the standard cm subsystems related to some unbroken subalgebras the configuration space of the cm particles is the moduli space of the stable holomorphic bundles with non trivial characteristic classes

distance based linear discriminant analysis for interval valued data interval valued observations arise in many real life situations as either the precise representation of the objective entity or the representation of incomplete knowledge thus given p features observed over a sample of objects belonging to one of two possible classes each observation can be perceived as a non empty closed and bounded hyperrectangle on r p the aim of the paper is to suggest a p dimensional classification method for random intervals when two or more classes are considered by the generalization of fisher s procedure for linear discriminant analysis the idea consists of finding a directional vector which maximizes the ratio of the dispersion between the classes and within the classes of the observed hyperrectangles a classification rule for new observations is also provided and some simulations are carried out to compare the behavior of the proposed classification procedure with respect to other methods known from the literature finally the suggested methodology are applied on a real life situation example

a solvable n body problem in the plane i we introduce and discuss an n body problem in the plane characterized by equations of motion of newtonian type rj k 1nfjk j 1 n with given forces fjk having the following characteristics fjk depends only on rj rk rj rk i e only one body and two body forces are present fjk behaves as a two dimensional vector under rotations in the plane i e the model is rotation invariant for j k fjk is linear in rj and rj for j k fjk rj rk 2fjk with fjk a homogeneous polynomial of third degree in rj rk rj rk hence fjk is homogeneous of degree one in rj rk rj rk fjk contains linearly 8 arbitrary coupling constants the n body problem is solvable for arbitrary n and for arbitrary values of the 8 coupling constants its solutions display a rich phenomenology if the 8 coupling constants are suitably restricted the model is translation invariant and or hamiltonian of course when it is hamiltonian it is integrable indeed in some ca

extended support vector interval regression networks for interval input output data in many applications it is natural to use interval data because of uncertainty existence in the measurements variability in defining terms such as the temperature during a given day description for extremely behavior such as the maximum wind speed in a given area etc in order to handle such interval data a novel approach called the interval support vector interval regression networks isvirns is proposed the isvirns is extended from our previous work the support vector interval regression networks svirns it is easy to find that svirns can handle interval output data but for input data they must be crisp in this study the hausdorff distance is employed as the distance measure of interval data and is incorporated into the kernel functions of svirns to determine the initial structure of isvirns because the proposed approach can provide a better initial structure for isvirns it can have a faster convergent speed the experimental results with real data sets show the validity of the proposed isvirns

liouville integrability of a class of integrable spin calogero moser systems and exponents of simple lie algebras in previous work we introduced a class of integrable spin calogero moser systems associated with the classical dynamical r matrices with spectral parameter as classified by etingof and varchenko for simple lie algebras here the main purpose is to establish the liouville integrability of these systems by a uniform method based on evaluating the primitive invariants of chevalley on the lax operators with spectral parameter as part of our analysis we will develop several results concerning the algebra of invariant polynomials on simple lie algebras and their expansions

dynamical r matrix for the elliptic ruijsenaars schneider system the classical r matrix structure for the generic elliptic ruijsenaars schneider model is presented it makes manifest the integrability of this model as well as of its discrete time version that was constructed in a recent paper

ermakov and non ermakov systems in quantum dissipative models via the hydrodynamical formulation of quantum mechanics a unified protocol to treat the quantum time dependent harmonic oscillator with friction is presented described by two different models an explicitly time dependent linear schrodinger equation caldirola kanai model and a logarithmic nonlinear schrodinger equation kostin model for the former model an ermakov system that makes it possible to obtain an invariant of ermakov lewis type is derived for the latter model a non ermakov system is derived instead and it is shown that neither an exact nor an approximate invariant of ermakov lewis type exists

the lax representation for an integrable class of relativistic dynamical systems we exhibit the lax pair for the class of relativistic dynamical systems recently introduced by ruijsenaars and schneider whose equations of motion read whereb is an arbitrary constant and the weierstrass elliptic function

associated integrable systems new class of integrable hamiltonian systems is obtained as systems directly defined over jacobi varieties this approach enables one to establish a link between associated discrete and continuous systems in particular regular procedure is found for the discretization of the integrable continuous systems hierarchies of the continuous systems associated with toda lattices relativistic toda lattices and volterra lattices are found and studied

the transition from regular to irregular motions explained as travel on riemann surfaces we introduce and discuss a simple hamiltonian dynamical system interpretable as a three body problem in the complex plane and providing the prototype of a mechanism explaining the transition from regular to irregular motions as travel on riemann surfaces the interest of this phenomenology illustrating the onset in a deterministic context of irregular motions is underlined by its generality suggesting its eventual relevance to understand natural phenomena and experimental investigations here only some of our main findings are reported without detailing their proofs a more complete presentation will be published elsewhere

partial identification using random set theory abstract this paper illustrates how the use of random set theory can benefit partial identification analysis we revisit the origins of manski s work in partial identification e g manski 1989 manski 1990 focusing our discussion on identification of probability distributions and conditional expectations in the presence of selectively observed data statistical independence and mean independence assumptions and shape restrictions we show that the use of the choquet capacity functional and the aumann expectation of a properly defined random set can simplify and extend previous results in the literature we pay special attention to explaining how the relevant random set needs to be constructed depending on the econometric framework at hand we also discuss limitations in the applicability of specific tools of random set theory to partial identification analysis

time discretization of the spin calogero moser model and the semi discrete matrix kp hierarchy we introduce the discrete time version of the spin calogero moser system the equations of motion follow from the dynamics of poles of rational solutions to the matrix kadomtsev petviashvili hierarchy with discrete time the dynamics of poles is derived using the auxiliary linear problem for the discrete flow

interval arithmetic based simple linear regression between interval data discussion and sensitivity analysis on the choice of the metric the prediction of a response random interval valued set from an explanatory one has been examined in previous developments these developments have considered an interval arithmetic based linear model between the random interval valued sets and a least squares regression analysis the least squares approach involves a generalized l 2 metric between interval data this metric weights squared distances between data location mid points centers and squared distances between data imprecision spread radius as a consequence estimators of the parameters in the linear model depend on the choice of the weights in the metric to investigate about a suitable choice of weighting in the generalized mid spread metric a theoretical conclusion is first obtained finally the impact of varying the weights is discussed by considering a monte carlo simulation study

the subalgebras of the rank two symplectic lie algebra abstract the semisimple subalgebras of the rank 2 symplectic lie algebra sp 4 c are well known and we recently classified its levi decomposable subalgebras in this article we classify the solvable subalgebras of sp 4 c up to inner automorphism this completes the classification of the subalgebras of sp 4 c more broadly speaking in completing the classification of the subalgebras of sp 4 c we have completed the classification of the subalgebras of the rank 2 semisimple lie algebras

the gervais neveu felder equation and the quantum calogero moser systems we quantize the spin calogero moser model in ther matrix formalism the quantumr matrix of the model is dynamical thisr matrix has already appeared in gervais neveu s quantization of toda field theory and in felder s quantization of the knizhnik zamolodchikov bernard equation

the dn ruijsenaars schneider model the lax pair of the ruijsenaars schneider model with interaction potential of trigonometric type based on dn lie algebra is presented we give a general form for the lax pair and prove partial results for small n liouville integrability of the corresponding system follows a series of involutive hamiltonians generated by the characteristic polynomial of the lax matrix the rational case appears as a natural degeneration and the non relativistic limit exactly leads to the well known calogero moser system associated with dn lie algebra

on rational addition theorems abstract we investigate the functions which admit an addition theorem of the form f t s i 1 n y i t u i s j 1 m z j t v j s our approach is based on the reduction to a system of differential equations all functions are supposed to be continuously differentiable on some interval the concepts of joint linear dependence and joint quadratic dependence of two families of functions are introduced it s proved that there are only two possibilities in the case of jointly linearly independent ui and vj a the function f is a ratio of quasi polynomials b the families yi and zj are jointly quadratically dependent the second possibility is studied for m n 2 we apply our results to the solving of the functional equation f t s y 1 t y 2 s y 2 t y 1 s z 1 t z 2 s z 2 t z 1 s this equation and its special cases have important applications in mathematical physics and were studied by several authors with the assumption of analyticity of the solutions we show that the class of solutions remains the same if one looks for the differentiable solutions on an interval

on nonparametric maximum likelihood estimation with interval censoring and left truncation summary a graph theoretical approach is employed to describe the support set of the nonparametric maximum likelihood estimator for the cumulative distribution function given interval censored and left truncated data a necessary and sufficient condition for the existence of a nonparametric maximum likelihood estimator is then derived two previously analysed data sets are revisited

nonlinear harmonic oscillators the existence is noted of assemblies of an arbitrary number of complex oscillators or equivalently of an arbitrary even number of real oscillators characterized by newtonian equations of motion acceleration equal force with one body velocity dependent linear forces and many body velocity independent cubic forces all the nonsingular solutions of which are isochronous completely periodic with the same period as for the singular solutions as usual they emerge in the context of the initial value problem from a closed domain in phase space having lower dimensionality

integrable and solvable many body problems in the plane via complexification a simple prescription allows us to transform by appropriate complexification any one dimensional many body problem describing motions on a line with newtonian acceleration equals forces equations of motion featuring forces which depend analytically on the positions and possibly on the velocities of the particles and which are moreover scaling or translation invariant into a two dimensional many body problem describing motions in the plane with rotation invariant equations of motion if the original one dimensional model is hamiltonian the two dimensional model is also hamiltonian in fact bi hamiltonian in this manner starting from known integrable or solvable one dimensional many body problems integrable or solvable many body problems in the plane generally featuring much richer dynamics are obtained several examples are exhibited finally another more direct prescription is outlined to transform by complexification almost any one dimensional many body problem into a rota

inference on regressions with interval data on a regressor or outcome this paper examines inference on regressions when interval data are available on one variable the other variables being measured precisely let a population be characterized by a distribution p y x v v 0 v 1 where y r 1 x r k and the real variables v v 0 v 1 satisfy v 0 v v 1 let a random sample be drawn from p and the realizations of y x v 0 v 1 be observed but not those of v the problem of interest may be to infer e y x v or e v x this analysis maintains interval i monotonicity m and mean independence mi assumptions i p v 0 v v 1 1 m e y x v is monotone in v mi e y x v v 0 v 1 e y x v no restrictions are imposed on the distribution of the unobserved values of v within the observed intervals v 0 v 1 it is found that the immi assumptions alone imply simple nonparametric bounds on e y x v and e v x these assumptions invoked when y is binary and combined with a semiparametric binary regression model yield an identification region for the parameters that may be estimated consistently by a modified maximum score mms method the immi assumptions combined with a parametric model for e y x v or e v x yield an identification region that may be estimated consistently by a modified minimum distance mmd method monte carlo methods are used to characterize the finite sample performance of these estimators empirical case studies are performed using interval wealth data in the health and retirement study and interval income data in the current population survey

on the symplectic structure of instanton moduli spaces abstract we study the complex symplectic structure of the quiver varieties corresponding to the moduli spaces of su 2 instantons on both commutative and non commutative r 4 we identify global darboux coordinates and quadratic hamiltonians on classical phase spaces for which these quiver varieties are natural completions we also show that the group of non commutative symplectomorphisms of the corresponding path algebra acts transitively on the moduli spaces of non commutative instantons this paper should be viewed as a step towards extending known results for calogero moser spaces to the instanton moduli spaces

solvable nonrelativistic classical n body problems on the line i several solvable n body problems on the line are exhibited they are characterized by equations of motion of newtonian type mjxj fj j 1 n with the forces fj given as explicit functions of the particle coordinates xk and their velocities xk k 1 n the forces fj generally also depend on some free parameters so that in each case there is actually a class of solvable models in this paper the emphasis is on translation invariant models and on values of n sufficiently small few body problems to allow the display of the solution of the initial value problem in completely explicit form

integrable time discretisation of the ruijsenaars schneider model an exactly integrable symplectic correspondence is derived which in a continuum limit leads to the equations of motion of the relativistic generalization of the calogero moser system that was introduced for the first time by ruijsenaars and schneider for the discrete time model the equations of motion take the form of bethe ansatz equations for the inhomogeneous spin 1 2 xyz heisenberg magnet we present a lax pair the sympletic structure and prove the involutivity of the invariants exact solutions are investigated in the rational and hyperbolic trigonometric limits of the system that is given in terms of elliptic functions these solutions are connected with discrete soliton equations the results obtained allow us to consider the bethe ansatz equations as ones giving an integrable symplectic correspondence mixing the parameters of the quantum integrable system and the parameters of the corresponding bethe wavefunction

a technique to identify solvable dynamical systems and another solvable extension of the goldfish many body problem we take advantage of the simple approach recently discussed which associates to solvable matrix equations solvable dynamical systems interpretable as interesting many body problems possibly involving auxiliary dependent variables in addition to those identifying the positions of the moving particles starting from a solvable matrix evolution equation we obtain the corresponding many body model and note that in one case the auxiliary variables can be altogether eliminated obtaining thereby an also hamiltonian extension of the goldfish model the solvability of this novel model and of its isochronous variant is exhibited a related as well solvable model is also introduced as well as its isochronous variant finally the small oscillations of the isochronous models around their equilibrium configurations are investigated and from their isochronicity certain diophantine relations are evinced

inclusion degree tests for the aumann expectation of a random interval an extension of the inclusion test of the expected value of a real random variable in an interval to the case of general random intervals is introduced the hypothesis of strict inclusion is relaxed by considering a measure of the degree of inclusion thus partial inclusions are also tested asymptotic and bootstrap techniques are established the performance of the bootstrap test is also analyzed by means of some simulations a case study regarding the blood pressure classification in adults is considered

novel solvable variants of the goldfish many body model a recent technique to identify solvable many body problems in two dimensional space yields via a new twist new many body problems of goldfish type some of these models are isochronous namely their generic solutions are completely periodic with a fixed period independent of the initial data the investigation of the behavior of some of these isochronous systems in the vicinity of their equilibrium configurations yields some amusing diophantine relations

a generalisation of the calogero moser system abstract a generalised calogero moser many body system in which the particles possess extra internal degrees of freedom is introduced and solved by the method of olshanetsky and perelomov further the hierarchy of flows commuting with this system is obtained from a hierarchy of linear systems possessing sl n symmetry by the usual method of reduction thus they are shown to be integrable

goldfishing a new solvable many body problem a recent technique allows one to identify and investigate solvable dynamical systems naturally interpretable as classical many body problems being characterized by equations of motion of newtonian type generally in two dimensional space in this paper we tersely review results previously obtained in this manner and present novel findings of this kind mainly solvable variants of the goldfish many body model including models that feature isochronous classes of completely periodic solutions different formulations of these models are presented the behavior of one of these isochronous dynamical systems in the neighborhood of its equilibrium configuration is investigated and in this manner some remarkable diophantine findings are obtained

the non dynamical r matrix structure for the elliptic calogero moser model in this paper we construct a new lax operator for the elliptic calogero moser model with general from the classical dynamical twisting in which the corresponding r matrix is purely numerical non dynamic the non dynamical r matrix structure of this lax operator is obtained which is the elliptic symmetric r matrix

energy level motion and a supersymmetric hamiltonian from the eigenvalue equation hlambda mod psi n lambda en lambda mod psi n lambda where hlambda identical to h0 lambda v one can derive an autonomous system of first order differential equations for the eigenvalues en lambda and the matrix elements vmn lambda psi m lambda mod v mod psi n lambda where lambda is the independent variable to solve the dynamical system the authors need the initial values en lambda 0 and mod psi n lambda 0 thus one finds the motion of the energy levels en lambda they derive the equations of motion for the extended case hlambda h0 lambda v1 lambda 2v2 and give an application to a supersymmetric hamiltonian