on the measure of totally real algebraic integers ii for a certain measure s2 a defined for totally real algebraic integers a 0 1 we find the four smallest values of q a the methods used involve linear programming and the results are verified using interval arithmetic

lower bounds for the heights of solutions of linear equations let k be a number eld of degree d we de ne as usual the eld height hk of a number k by hk v m k max 1 dv v here the v adic valuations v are normalized such that on q they coincide with either the standard absolute value or with the p adic absolute value p having p p p 1 moreover dv denotes the local degree the absolute height of then is de ned by h hk lehmer s conjecture says the following let be a nonzero algebraic number of degree d then if is not a root of unity we have

on the asymptotic integrability of a higher order evolution equation describing internal waves in a deep fluid a higher order nonlocal evolution equation describing internal waves in a deep fluid is shown to be asymptotically integrable only if the coefficients of the higher order terms satisfy certain constraints in this case the nonlocal equation can be transformed to the integrable benjamin ono equation the asymptotic integrability of the reductions of the higher order evolution equation to a complex burgers equation to an envelope wave equation and to a finite dimensional dynamical system is also considered

on the spectrum of the zhang zagier height from recent work of zhang and of zagier we know that their height n is bounded away from 1 for every algebraic number a different from 0 1 1 2 3 2 the study of the related spectrum is especially interesting for it is linked to lehmer s problem and to a conjecture of bogomolov after recalling some definitions we show an improvement of the so called zhang zagier inequality to achieve this we need some algebraic numbers of small height so in the third section we describe an algorithm able to find them and we give an algebraic number with height 1 2875274z discovered in this way this search up to degree 64 suggests that the spectrum of n may have a limit point less than 1 292 we prove this fact in the fourth part

dark soliton generation for the intermediate nonlinear schrodinger equation the generation of dark solitons due to a small change of the initial data is studied within the framework of the intermediate nonlinear schrodinger inls equation in particular we analyze the spectral problem associated with the inls equation when the potential consists of a small perturbation imposed on a constant background we derive a criterion for the perturbation to generate a pair of new discrete eigenvalues as well as their explicit expressions in terms of the perturbation in addition we demonstrate that the eigenvalues appear without a threshold on the magnitude of the perturbation we also consider both the shallow and deep water limits of various results obtained for the inls spectral problem in the former case the limiting procedure can be performed smoothly whereas in the latter case the spectral equation exhibits a new feature of bound states showing that the eigenvalue is exponentially small compared with the magnitude of the perturbation

nonlinear evolution equations associated with a riemann hilbert scattering problem in an earlier paper nonlinear evolution equations associated with a riemann hilbert scattering problem which reduces in an appropriate limit to the zakharov shabat akns scattering problem were considered here we discuss certain necessary constraints associated with the scattering problem and their impact upon the associated evolution equations moreover the direct linearization of the nonlinear evolution equations and an algorithm to construct ann soliton solution are given

on the initial value problem for the modified benjamin ono equation the initial value problem for the modified benjamin ono equation is solved by exploiting the fokas ablowitz inverse scattering transform for a certain complex extension of the benjamin ono equation the asymptotics of the jost functions for small positive values of the spectral parameter are sharpened and lead to a previously overlooked nongeneric case

sums of heights of algebraic numbers for at x f x tg x we consider the set at 0 h t q the polynomials f x g x are in z x with only mild restrictions and h is the weil height of we show that this set is dense in d for some effectively computable limit point d

periodic ilw equation with discrete laplacian we study an integro differential equation which generalizes the periodic intermediate long wave ilw equation the kernel of the singular integral involved is an elliptic function written as a second order difference of the weierstrass function using sato s formulation we show the integrability and construct some special solutions an elliptic solution is also obtained we present a conjecture based on a poisson structure that gives an alternative description of this integrable hierarchy we note that this poisson algebra in turn is related to a quantum algebra related to the family of macdonald difference operators

le probleme de lehmer en dimension superieure we study a higher dimensional lehmer problem or alternatively the lehmer problem for a power of the multiplicative group more precisely if 1 n are multiplicatively independent algebraic numbers we provide a lower bound for the product of the heights of the i s in terms of the degree d of the number field generated by the i s this enables us to study the successive minima for the height function in a given number field our bound is a generalisation of an earlier result of dobrowolski and is best possible up to a power of log d this in particular shows that the lehmer problem is true for number fields having a small galois group

mesures et equidistribution sur les espaces de berkovich the proof by ullmo and zhang of bogomolov s conjecture about points of small height in abelian varieties made a crucial use of an equidistribution property for small points in the associated complex abelian variety we study the analogous equidistribution property at p adic places our results can be conveniently stated within the framework of the analytic spaces defined by berkovich the first one is valid in any dimension but is restricted to algebraic metrics the second one is valid for curves but allows for more general metrics in particular to the normalized heights with respect to dynamical systems

positive line bundles on arithmetic varieties p 5 u integers is defined to count the volume of the lattice i l n of integral sections in the space j7 lon of real sections with supremum norm we want to prove that the leading term of this hilbert function as n f oo is given in terms of the height of x in as 1 this is known as a theorem of gillet and soule gs2 if x has a regular generic fiber beside this known result our proof uses hironaka s theorem on resolutions of singularities and minkowski s theorem on successive minima by hironaka s theorem we may construct

quasi likelihood and its application a general approach to optimal parameter estimation the general framework an alternative approach e sufficiency asymptotic confidence zones of minimum size asymptotic quasi likelihood combining estimating functions projected quasi likelihood bypassing the likelihood hypothesis testing infinite dimensional problems miscellaneous applications consistency and asymptotic normality for estimating functions complements and strategies for application

singularity confinement analysis of integrodifferential equations of benjamin ono type we analyse the integrability of various types of intregrodifferential equations belonging to the benjamin ono family the method used is a combination of two well known integrability detectors the painleve method for continuous systems and the singularity confinement for discrete systems we confirm the results of hietarinta based on the study of multisoliton solutions in particular we show that the third order extension to benjamin ono that he proposed does pass the test and is thus an excellent candidate for integrability

eigenfunctions and eigenvalues for a scalar riemann hilbert problem associated to inverse scattering a complete set of eigenfunctions is introduced within the riemann hilbert formalism for spectral problems associated to some solvable nonlinear evolution equations in particular we consider the time independent and time dependent schrodinger problems which are related to the kdv and kpi equations possessing solitons and lumps respectively non standard scalar products orthogonality and completeness relations are derived for these problems the complete set of eigenfunctions is used for perturbation theory and bifurcation analysis of eigenvalues supported by the potentials under perturbations we classify two different types of bifurcations of new eigenvalues and analyze their characteristic features one type corresponds to thresholdless generation of solitons in the kdv equation while the other predicts a threshold for generation of lumps in the kpi equation

multidimensional nonlinear evolution equations and inverse scattering abstract in this paper we will review some recent work done in the field of integrable nonlinear evolution equations and inverse scattering we will concentrate on the basic underlying areas and refer interested readers to suitable references for complete details specifically background material can be found in various texts on this subject e g 1 by ablowitz and segur more recent references will be given as necessary the outline of the paper is as follows 1 1 introductory remarks 2 2 a discussion of two separate but related issues namely a solving certain nonlinear evolutions equations in infinite space and b inverse scattering these are important problems having many physical applications moreover they are related to each other by what we refer to as the inverse scattering transform ist 3 3 at the end of the paper we will make some remarks on the possibility of solving nonlinear evolution equations in high dimensions i e equations with more than two spatial and one time variable by using the ist method as we now understand it

an application of the inverse scattering transform to the modified intermediate long wave equation the modified intermediate long wave milw equation is a 1 1 dimensional nonlinear singular integro differential equation that possesses soliton solutions in an appropriate limit the milw equation reduces to the well known modified korteweg de vries equation in this paper we solve the initial value problem for the milw equation through a suitable implementation of the inverse scattering transform and use of the miura type transformation that maps solutions of the milw equation into solutions of a complexified version of the standard intermediate long wave ilw equation the initial value used for the milw equation is assumed to be real valued sufficiently smooth and decaying to zero as the absolute value of the spatial variable approaches large values an interesting feature of the procedure we develop is that soliton solutions for the ilw and milw equations can be derived by appropriate specializations of a master set of equations

on the initial value problem for a class of nonlinear integral evolution equations including the sine hilbert equation a method for solving a class of nonlinear singular integral evolution equations for decaying initial values on the line is presented the underlying scattering problem is a matrix riemann hilbert problem scattering analysis shows that the spectrum is purely discrete an application is to the so called sine hilbert equation h t c sin where c is a constant and h denotes the hilbert transform

the scattering transform for the benjamin ono equation in the small dispersion limit abstract using exact formulae for the scattering data of the benjamin ono equation valid for general rational potentials recently obtained in miller and wetzel 17 we rigorously analyze the scattering data in the small dispersion limit in particular we deduce precise asymptotic formulae for the reflection coefficient the location of the eigenvalues and their density and the asymptotic dependence of the phase constant associated with each eigenvalue on the eigenvalue itself our results give direct confirmation of conjectures in the literature that have been partly justified by means of inverse scattering and they also provide new details not previously reported in the literature

well posedness for the nonlocal nonlinear schrodinger equation abstract we establish local well posedness for small initial data in the usual sobolev spaces h s r s 1 and global well posedness in h 1 r for the cauchy problem associated to the nonlocal nonlinear schrodinger equation t u i x 2 u u x u 2 i u t h x u 2 i u 2 u where u u x t x t r t h is a singular integral operator 0 0 and 0 are real constants our method of proof is based on the smoothing effects produced by the linear schrodinger equation

a spectral transform for the intermediate nonlinear schrodinger equation a new spectral transform system is introduced to solve the initial value problem for the intermediate nonlinear schrodinger inls equation describing envelope waves in a deep stratified fluid the spectral system is a combination of the zakharov shabat linear system and a local riemann hilbert problem in a strip of the complex plane the inverse scattering transform technique is developed and the backlund darboux transformation soliton solutions and an infinite number of conservation laws are constructed it is shown that all these properties of the inls equation are closely related to those of the intermediate long wave equation

local well posedness for the nonlocal derivative nonlinear schrodinger equation in besov spaces abstract in this paper we study the cauchy problem associated with the one dimensional integro differential nonlocal derivative nonlinear schrodinger equation in the besov space b 2 1 2 1 r the local well posedness for small initial data in b 2 1 2 1 r is established our method of proof combines the contraction principle applied to the associated integral equation together with interpolations of some smoothing effects kato s smoothing effects strichartz estimate and estimates for the maximal function for phase localized functions associated to the linear dispersive part of the equation and a fractional vector valued leibniz s rule derived by molinet and ribaud in 2004

direct and inverse scattering problems of the nonlinear intermediate long wave equation the inverse scattering transformation method associated with a nonlinear singular integrodifferential equation is discussed the equation describes long internal gravity waves in a stratified fluid of finite depth and reduces to the korteweg de vries equation as shallow water limit and the benjamin ono equation as deep water limit both limits of the method and novel aspects of the theory are also discussed

backlund transformation conservation laws and inverse scattering transform of a model integrodifferential equation for water waves the backlund transformation bt an infinite number of conservation laws and the inverse scattering transform ist of a model integrodifferential equation for water waves in fluids of finite depth y matsuno j math phys 29 49 1989 are constructed by employing the bilinear transformation method the model equation is also shown to pass the painleve test these facts prove the complete integrability of the equation both the deep and shallow water limits of various results thus obtained are then investigated in detail in addition a new method to evaluate conserved quantities for pure n soliton is developed by utilizing actively the time part of the bt it is found that the structure of conservation laws exhibits peculiar characteristics in comparison with those of usual water wave equations such as the benjamin ono and the korteweg de vries equations the most important problem left open in this paper is to solve various ist equations

orbits of algebraic numbers with low heights we prove that the two smallest values of h h 1 1 h 1 1 are 0 and 0 4218 for any algebraic integer

on the limit from the intermediate long wave equation to the benjamin ono equation the intermediate long wave equation is a physically important singular integrodifferential equation containing a parameter referred to here as for it reduces to the benjamin ono equation it has been recently shown that the inverse scattering transform schemes of the above equations have certain significant differences here it is shown that for the inverse scattering transform scheme of the intermediate long wave equation reduces to that of the benjamin ono equation

two new points in the spectrum of the absolute mahler measure of totally positive algebraic integers for totally positive algebraic integers 0 1 of degree d we consider the set l of values of m 1 d where m is the mahler measure of c j smyth has found the four smallest values of l and conjectured that the fifth point is 2 cos 2 60 2 we prove that this is so and moreover we give the sixth point of l

lower bounds on the arithmetic self intersection number of the relative dualizing sheaf on arithmetic surfaces we give an explicitly computable lower bound for the arithmetic self intersection number of the dualizing sheaf on a large class of arithmetic surfaces if some technical conditions are satisfied then this lower bound is positive in particular these technical conditions are always satisfied for minimal arithmetic surfaces with simple multiplicities and at least one reducible fiber but we have also used our techniques to obtain lower bounds for some arithmetic surfaces with non reduced fibers

calculus on arithmetic surfaces in a2 and a3 arakelov introduces an intersection calculus for arithmetic surfaces that is for stable models of curves over a number field in this paper we intend to show that his intersection product has a lot of useful properties more precisely we show that the following properties from the theory of algebraic surfaces have an analogue in our situation

stability of nonlinear periodic internal waves in a deep stratified fluid stability of the nonlinear periodic solutions of the benjamin ono equation with respect to one dimensional perturbations and taking into account transverse modulations is analytically studied the full set of eigenfrequencies and of small perturbations is found in the one dimensional case the dispersion equation for long wave oblique perturbations is obtained these perturbations are shown to be unstable under either sign of the dispersion of the linear dispersion relation the instability limits and the maximum growth rates are determined in the extreme cases of strongly and weakly nonlinear waves

the number of irreducible factors of a polynomial i let f x be a polynomial with coefficients in an algebraic number field k we estimate the number of irreducible cyclotomic factors of f in k x the number of irreducible noncyclotomic factors of f the number of nth roots of unity among the roots of f and the number of primitive nth roots of unity among the roots of f all of these quantities are counted with multiplicity and estimated by expressions which depend explicitly on k on the degree of f and height of f and when appropriate on n we show by constructing examples that some of our results are essentially sharp

generalized autoregressive conditional heteroskedasticity a natural generalization of the arch autoregressive conditional heteroskedastic process introduced in engle 1982 to allow for past conditional variances in the current conditional variance equation is proposed stationarity conditions and autocorrelation structure for this new class of parametric models are derived maximum likelihood estimation and testing are also considered finally an empirical example relating to the uncertainty of the inflation rate is presented

generalized intermediate long wave hierarchy in zero curvature representation with noncommutative spectral parameter the simplest generalization of the intermediate long wave hierarchy ilw is considered to show how to extend the zakharov shabat dressing method to nonlocal i e integro partial differential equations the purpose is to give a procedure of constructing the zero curvature representation of this class of equations this result obtains by combining the drinfeld sokolov formalism together with the introduction of an operator valued spectral parameter namely a spectral parameter that does not commute with the space variable x this extension provides a connection between the ilwk hierarchy and the saveliev vershik continuum graded lie algebras in the case of ilw2 the fairlie zachos sinh algebra was found

on the absolute mahler measure of polynomials having all zeros in a sector ii kklet be an algebraic integer of degree d not 0 or a root of unity all of whose conjugates i are confined to a sector arg z in the paper on the absolute mahler measure of polynomials having all zeros in a sector g rhin and c smyth compute the greatest lower bound c of the absolute mahler measure d i 1 max 1 i 1 d of for belonging to nine subintervals of 0 2 3 in this paper we improve the result to thirteen subintervals of 0 and extend some existing subintervals

a class of integro differential equations constrained from the kp hierarchy in this paper we give a class of integro differential evolution equations which generalizes the ilwk hierarchy we show the generalized lax pairs the bilinear representations and the rational and n soliton solutions for these integro differential equations we also give the hamiltonian formalism for the simplest set of equations among these integro differential equations which is the coupling of the ilw equation and the time dependent schrodinger equations

the arithmetic hodge index theorem for adelic line bundles in this paper we prove index theorems for integrable metrized line bundles on projective varieties over complete fields and number fields respectively as applications we prove a non archimedean analogue of the calabi theorem and a rigidity theorem about the preperiodic points of algebraic dynamical systems

intersecting curves and algebraic subgroups conjectures and more results this paper solves in the affirmative up to dimension n 5 a question raised in an earlier paper by the authors the equivalence of the problem with a conjecture of shou wu zhang is proved in the appendix

bifurcations of new eigenvalues for the benjamin ono equation a criterion for the emergence of new eigenvalues is found for the linear scattering problem associated with the benjamin ono bo equation this bifurcation occurs due to perturbations of nongeneric potentials which include the soliton solutions of the bo equation the asymptotic approximation of an exponentially small new eigenvalue is derived the method is based on the expansion of a localized function through a complete set of unperturbed eigenfunctions explicit expressions are obtained for the soliton potentials

a note on garch model identification financial returns are often modeled as autoregressive time series with innovations having conditional heteroscedastic variances especially with garch processes the conditional distribution in garch models is assumed to follow a parametric distribution typically this error distribution is selected without justification in this paper we have applied the results of thavaneswaran and ghahramani a thavaneswaran m ghahramani applications of combining estimating functions in proceedings of the international sri lankan conference visions of futuristic methodologies university of peradeniya and royal melbourne institute of technology rmit 2004 pp 515 532 on identification of garch models to a number of financial data sets

canonical heights transfinite diameters and polynomial dynamics let z be a polynomial of degree at least 2 with coefficients in a number field k iterating gives rise to a dynamical system and a corresponding canonical height function as defined by call and silverman we prove a simple product formula relating the transfinite diameters of the filled julia sets of over various completions of k and we apply this formula to give a generalization of bilu s equidistribution theorem for sequences of points whose canonical heights tend to zero

constraints of the kadomtsev petviashvili hierarchy for the kadomtsev petviashvili kp hierarchy constructed in terms of the famous sato theory a k constraint is proposed that leads the hierarchy to the nonlinear system involving a finite number of dynamical coordinates the eigenvalue problem of the k constrained system is naturally obtained from the linear system of the kp hierarchy which takes the form of kth order polynomial coupled with a first order one thus we are able to derive the correspondent lax pair recursion operator bi hamiltonian structures and conserved quantities the constraints for the bkp hierarchy are also sketched

new integrable nonlinear integrodifferential equations and related solvable finite dimensional dynamical systems a new integrable nonlinear integrodifferential equation nide is proposed this equation may be interpreted as a model equation for deep water waves the n periodic and n soliton solutions for the equation are constructed by means of the bilinear transformation method these solutions have the same structure as that for the benjamin ono equation which describes internal waves in stratified fluids of great depth furthermore it is shown that the motion of the positions of the poles of solutions is related to certain solvable finite dimensional dynamical systems described by first order nonlinear ordinary differential equations the discussion is also made on a more general nide that may be interpreted as a model equation describing nonlinear waves in fluids of finite depth

effective bounds of linear series on algebraic varieties and arithmetic varieties in this paper we prove effective upper bounds for effective sections of line bundles on projective varieties and hermitian line bundles on arithmetic varieties in terms of the volumes they are effective versions of the hilbert samuel formula and the arithmetic hilbert samuel formula the treatments are high dimensional generalizations of duke math j 162 2013 1723 1770 and relative noether inequality on fibered surfaces preprint 2013 similar results are obtained independently by huayi chen majorations explicites de fonctions de hilbert samuel geometrique et arithmetique preprint 2014 with less explicit error terms

multi scale expansions in the theory of systems integrable by the inverse scattering transform abstract it is shown that using multi scale expansions conventionally employed in the theory of nonlinear waves one can transform systems integrable by the ist method into other systems of this type

generalized internal long waves equations construction hamiltonian structure and conservation laws a general class of the ilw type equations is constructed we introduce a hamiltonian structure and construct an infinite number of conservation laws

on the zero dispersion limit of the benjamin ono cauchy problem for positive initial data we study the cauchy initial value problem for the benjamin ono equation in the zero dispersion limit and we establish the existence of this limit in a certain weak sense by developing an appropriate analogue of the method invented by lax and levermore to analyze the corresponding limit for the korteweg de vries equation 2010 wiley periodicals inc

nonlocal integrable partners to generalized mkdv and two dimensional toda lattice equation in the formalism of a dressing method with quantized spectral parameter two new hierarchies milw2 and a two dimensional nonlocal toda lattice are constructed the characteristic property of the first one is the connection with the ilw2 hierarchy by means ofgl 2 miura transformation on the other hand milw2 equations turn out to be symmetry equations for a two dimensional nonlocal toda lattice a new version of the dressing technique with quantized spectral parameter is proposed