the equations of motion of curved pipes conveying fluid abstract the non linear differential equations of motion of a fluid conveying pipe the no flow configuration of which is in the shape of a one dimensional line arbitrarily curved and twisted in space are derived by making use of the newtonian approach the extensible dynamics of the beam is based on the timoshenko beam theory some significant differences with the equations available in the open literature are discussed the stressed by flow configuration is found to be the only equilibrium state adequate for the study of the linear stability of such pipe fluid systems

discrete integrable systems and deformations of associative algebras interrelations between discrete deformations of the structure constants for associative algebras and discrete integrable systems are reviewed theory of deformations for associative algebras is presented closed left ideal generated by the elements representing the multiplication table plays a central role in this theory deformations of the structure constants are generated by the deformation driving algebra and governed by the central system of equations it is demonstrated that many discrete equations such as discrete boussinesq equation discrete wdvv equation discrete schwarzian kp and bkp equations discrete hirota miwa equations for kp and bkp hierarchies are particular realizations of the central system an interaction between the theories of discrete integrable systems and discrete deformations of associative algebras is reciprocal and fruitful an interpretation of the menelaus relation discrete schwarzian kp equation discrete hirota miwa equation for kp hierarchy consistency around the cube as the associativity conditions and the concept of gauge equivalence for instance between the menelaus and kp configurations are particular examples

lagrangian multiforms and multidimensional consistency we show that well chosen lagrangians for a class of two dimensional integrable lattice equations obey a closure relation when embedded in a higher dimensional lattice on the basis of this property we formulate a lagrangian description for such systems in terms of lagrangian multiforms we discuss the connection of this formalism with the notion of multidimensional consistency and the role of the lattice from the point of view of the relevant variational principle

a hybrid proximal extragradient self concordant primal barrier method for monotone variational inequalities this paper presents a hybrid proximal extragradient hpe self concordant primal barrier method for solving a monotone variational inequality over a closed convex set endowed with a self concordant barrier and with an underlying map that has lipschitz continuous derivative in contrast to the iteration of a previous method developed by the first and third authors that has to compute an approximate solution of a linearized variational inequality the one of the present method solves a simpler newton system of linear equations the method performs two types of iterations namely those that follow ever changing interior paths and those that correspond to large step hpe iterations due to its first order nature the present method is shown to have a better iteration complexity than its zeroth order counterparts such as korpelevich s algorithm and tseng s modified forward backward splitting method although its work per iteration is larger than the one for the latter methods

linear integral equations and nonlinear difference difference equations abstract in this paper we present a systematic method to obtain various integrable nonlinear difference difference equations and the associated linear integral equations from which their solutions can be inferred it is argued that these difference difference equations can be regarded as arising from bianchi identities expressing the commutativity of backlund transformations applying an appropriate continuum limit we first obtain integrable nonlinear differential difference equations together with the associated linear integral equations and after a second continuum limit we can obtain the corresponding integrable nonlinear partial differential equations and their linear integral equations as special cases we treat the difference difference versions and the differential difference versions of the korteweg de vries equation the modified korteweg de vries equation the nonlinear schrodinger equation the isotropic classical heisenberg spin chain and the complex and real sine gordon equation

superposition principles associated with the moutard transformation an integrable discretization of a 2 1 dimensional sine gordon system superposition principles both linear and nonlinear associated with the moutard transformation are found on suitable reinterpretation these constitute an integrable discrete nonlinear system and its associated linear system further it is shown that in a particular form this system is an integrable discretization of a 2 1 dimensional sine gordon system solutions of the discrete nonlinear system are constructed by means of a discrete analogue of the moutard transformation included in these solutions are discrete analogues of the kink solutions of the continuous system

a new family of four dimensional symplectic and integrable mappings abstract we investigate the generalisations of the quispel roberts and thompson qrt family of mappings in the plane leaving a rational quadratic expression invariant to the case of four variables we assume invariance of the rational expression under a cyclic permutation of variables and we impose a symplectic structure with poisson brackets of the weyl type all mappings satisfying these conditions are shown to be integrable either as four dimensional mappings with two explicit integrals which are in involution with respect to the symplectic structure and which can also be inferred from the periodic reductions of the double discrete versions of the modified korteweg devries mkdv and sine gordon sg equations or by reduction to two dimensional mappings with one integral of the symmetric qrt family

iteration complexity of block decomposition algorithms and the alternating direction method of multipliers in this paper we consider the monotone inclusion problem consisting of the sum of a continuous monotone map and a point to set maximal monotone operator with a separable two block structure and introduce a framework of block decomposition prox type algorithms for solving it which allows for each one of the single block proximal subproblems to be solved in an approximate sense moreover by showing that any method in this framework is also a special instance of the hybrid proximal extragradient hpe method introduced by solodov and svaiter we derive corresponding convergence rate results we also describe some instances of the framework based on specific and inexpensive schemes for solving the single block proximal subproblems finally we consider some applications of our methodology to establish for the first time i the iteration complexity of an algorithm for finding a zero of the sum of two arbitrary maximal monotone operators and as a consequence the ergodic iteration complexity of the douglas

a modified forward backward splitting method for maximal monotone mappings we consider the forward backward splitting method for finding a zero of the sum of two maximal monotone mappings this method is known to converge when the inverse of the forward mapping is strongly monotone we propose a modification to this method in the spirit of the extragradient method for monotone variational inequalities under which the method converges assuming only the forward mapping is lipschitz continuous on some closed convex subset of its domain the modification entails an additional forward step and a projection step at each iteration applications of the modified method to decomposition in convex programming and monotone variational inequalities are discussed

discrete time lagrangian mechanics on lie groups with an application to the lagrange top we develop the theory of discrete time lagrangian mechanics on lie groups originated in the work of veselov and moser and the theory of lagrangian reduction in the discrete time setting the results thus obtained are applied to the investigation of an integrable time discretization of a famous integrable system of classical mechanics the lagrange top we recall the derivation of the euler poinsot equations of motion both in the frame moving with the body and in the rest frame the latter ones being less widely known we find a discrete time lagrange function turning into the known continuous time lagrangian in the continuous limit and elaborate both descriptions of the resulting discrete time system namely in the body frame and in the rest frame this system naturally inherits poisson properties of the continuous time system the integrals of motion being deformed the discrete time lax representations are also found kirchhoff s kinetic analogy between elastic curves and motions of the lagrange top is also generalised to the discrete context

integrability conditions for nonautonomous quad graph equations this paper presents a systematic investigation of the integrability conditions for nonautonomous quad graph maps using the lax pair approach the ultra local singularity confinement criterion and direct construction of conservation laws we show that the integrability conditions derived from each of the methods are the one and the same suggesting that there exists a deep connection between these techniques for partial difference equations

integrable maps from galois differential algebras borel transforms and number sequences abstract a new class of integrable maps obtained as lattice versions of polynomial dynamical systems is introduced these systems are obtained by means of a discretization procedure that preserves several analytic and algebraic properties of a given differential equation in particular symmetries and integrability see tempesta 2010 40 our approach is based on the properties of a suitable galois differential algebra that we shall call a rota algebra a formulation of the procedure in terms of category theory is proposed in order to render the lattice dynamics confined a borel regularization is also adopted as a byproduct of the theory a connection between number sequences and integrability is discussed

new variants of bundle methods in this paper we describe a number of new variants of bundle methods for nonsmooth unconstrained and constrained convex optimization convex concave games and variational inequalities we outline the ideas underlying these methods and present rate of convergence estimates

reduction groups and related integrable difference systems of nonlinear schrodinger type we extend the reduction group method to the lax darboux schemes associated with nonlinear schrodinger type equations we consider all possible finite reduction groups and construct corresponding lax operators darboux transformations hierarchies of integrable differential difference equations integrable partial difference systems and associated scalar partial difference equations

on an elliptic extension of the kadomtsev petviashvili equation a generalization of the lattice potential kadomtsev petviashvili lpkp equation is presented using the method of direct linearization based on an elliptic cauchy kernel this yields a 3 1 dimensional lattice system with one of the lattice shifts singled out the integrability of the lattice system is considered presenting a lax representation and soliton solutions an associated continuous system is also derived yielding a 3 1 dimensional generalization of the potential kp equation associated with an elliptic curve

new fluid velocity expression in an extensible semi circular pipe conveying fluid a new expression for fluid velocity is presented for a fluid conveying semi circular pipe with an extensible centreline the proposed fluid velocity expression is obtained from the material derivative while the previous velocity expressions are from love s kinematical relations in order to show that the new fluid velocity expression is more reasonable than the previous velocity expressions the equations of in plane motion derived with the new expression are compared to the equations derived with the previous expressions furthermore the equilibrium positions natural frequencies and mode shapes obtained with both the new and previous expressions are analysed and discussed

prox method with rate of convergence o 1 t for variational inequalities with lipschitz continuous monotone operators and smooth convex concave saddle point problems we propose a prox type method with efficiency estimate o epsilon 1 for approximating saddle points of convex concave c 1 1 functions and solutions of variational inequalities with monotone lipschitz continuous operators application examples include matrix games eigenvalue minimization and computing the lovasz capacity number of a graph and these are illustrated by numerical experiments with large scale matrix games and lovasz capacity problems

backward forward algorithms for structured monotone inclusions in hilbert spaces abstract in this paper we study the backward forward algorithm as a splitting method to solve structured monotone inclusions and convex minimization problems in hilbert spaces it has a natural link with the forward backward algorithm and has the same computational complexity since it involves the same basic blocks but organized differently surprisingly enough this kind of iteration arises when studying the time discretization of the regularized newton method for maximally monotone operators first we show that these two methods enjoy remarkable involutive relations which go far beyond the evident inversion of the order in which the forward and backward steps are applied next we establish several convergence properties for both methods some of which were unknown even for the forward backward algorithm this brings further insight into this well known scheme finally we specialize our results to structured convex minimization problems the gradient projection algorithms and give a numerical illustration of theoretical interest

complexity of variants of tseng s modified f b splitting and korpelevich s methods for hemivariational inequalities with applications to saddle point and convex optimization problems in this paper we consider both a variant of tseng s modified forward backward splitting method and an extension of korpelevich s method for solving hemivariational inequalities with lipschitz continuous operators by showing that these methods are special cases of the hybrid proximal extragradient method introduced by solodov and svaiter we derive iteration complexity bounds for them to obtain different types of approximate solutions in the context of saddle point problems we also derive complexity bounds for these methods to obtain another type of an approximate solution namely that of an approximate saddle point finally we illustrate the usefulness of the above results by applying them to a large class of linearly constrained convex programming problems including for example cone programming and problems whose objective functions converge to infinity as the boundaries of their effective domains are approached

generalized self concordant functions a recipe for newton type methods we study the smooth structure of convex functions by generalizing a powerful concept so called self concordance introduced by nesterov and nemirovskii in the early 1990s to a broader class of convex functions which we call generalized self concordant functions this notion allows us to develop a unified framework for designing newton type methods to solve convex optimization problems the proposed theory provides a mathematical tool to analyze both local and global convergence of newton type methods without imposing unverifiable assumptions as long as the underlying functionals fall into our class of generalized self concordant functions first we introduce the class of generalized self concordant functions which covers the class of standard self concordant functions as a special case next we establish several properties and key estimates of this function class which can be used to design numerical methods then we apply this theory to develop several newton type methods for solving a class of smooth convex optimization problems involving generalized self concordant functions we provide an explicit step size for a damped step newton type scheme which can guarantee a global convergence without performing any globalization strategy we also prove a local quadratic convergence of this method and its full step variant without requiring the lipschitz continuity of the objective hessian mapping then we extend our result to develop proximal newton type methods for a class of composite convex minimization problems involving generalized self concordant functions we also achieve both global and local convergence without additional assumptions finally we verify our theoretical results via several numerical examples and compare them with existing methods

iteration complexity of a newton proximal extragradient method for monotone variational inequalities and inclusion problems in a recent paper by monteiro and svaiter a hybrid proximal extragradient hpe framework was used to study the iteration complexity of a first order or in the context of optimization second order method for solving monotone nonlinear equations the purpose of this paper is to extend this analysis to study a prox type first order method for monotone smooth variational inequalities and inclusion problems consisting of the sum of a smooth monotone map and a maximal monotone point to set operator each iteration of the method computes an approximate solution of a proximal subproblem obtained by linearizing the smooth part of the operator in the corresponding proximal equation for the original problem which is then used to perform an extragradient step as prescribed by the hpe framework both pointwise and ergodic iteration complexity results are derived for the aforementioned first order method using corresponding results obtained here for a subfamily of the hpe framework

involutivity of integrals of sine gordon modified kdv and potential kdv maps closed form expressions in terms of multi sums of products have been given in tran et al 2009 j phys a math theor 42 225201 and van der kamp et al 2007 j phys a math theor 39 12789 98 of integrals of sine gordon modified korteweg de vries and potential korteweg de vries maps obtained as so called p 1 travelling wave reductions of the corresponding partial difference equations we prove the involutivity of these integrals with respect to recently found symplectic structures for those maps the proof is based on explicit formulae for the poisson brackets between multi sums of products

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

linear integral equations and multicomponent nonlinear integrable systems i abstract a systematic method for obtaining multicomponent generalizations of integrable nonlinear partial differential equations pde s is developed the method starts from a general type of linear integral equations containing integrations over an arbitrary contour with an arbitrary measure in the complex plane of k the spectral parameter special k dependent factors in the integrand are shown to induce an extra coupling between solutions of these integral equations with a different source term in this way the direct linearization is obtained of multicomponent generalizations of various nonlinear pde s such as the nonlinear schrodinger equation the derivative nonlinear schrodinger equations the isotropic heisenberg spin chain equation the complex sine gordon equation and the massive thirring model equations in the present paper i we present the general framework to derive finite matrix pde s and we also discuss backlund transformations and multicomponent lattice versions in a subsequent paper ii we treat a variety of examples of multicomponent pde s and discuss miura transformations and gauge equivalences

integrable mappings derived from soliton equations abstract we derive a hierarchy of ibtegrable mappings integrable ordinary difference equations corresponding to solutions of the initial value problem of an integrable partial difference equation with periodic initial data for each n n this hierarchy contains at least one integrable mapping r n r n the integrals of these mappings are constructed using the lax pair of the underlying partial difference equation our approach is illustrated for the integrable partial difference analogues of the sine gordon and the modified korteweg de vries equations

cubic regularization of newton method and its global performance in this paper we provide theoretical analysis for a cubic regularization of newton method as applied to unconstrained minimization problem for this scheme we prove general local convergence results however the main contribution of the paper is related to global worst case complexity bounds for different problem classes including some nonconvex cases it is shown that the search direction can be computed by standard linear algebra technique

on some linear integral equations generating solutions of nonlinear partial differential equations abstract two types of linear inhomogeneous integral equations which yield solutions of a broad class of nonlinear evolution equations are investigated one type is characterized by a two fold integration with an arbitrary measure and contour over a complex variable k and thier complex conjugates whereas the other one has a two fold integration over one and the same contour the inhomogeneous term which may contain an arbitrary function of k makes it possible to define a matrix structure on the solutions of the integral equations the elements of these matrices are shown to obey a system of partial differential equations the special form of which depends on the choice of the dispersion relation occurring in the integral equations for special elements of the matrices closed partial differential equations can be derived such as e g the nonlinear schrodinger equation and the real and complex modified korteweg de vries and sine gordon equations the relations between the matrix elements are shown to lead to miura transformations between the various partial differential equations

lattice equations hierarchies and hamiltonian structures ii kp type of hierarchies on 2d lattices abstract we investigate the continuum limit properties of an integrable lattice version of the kadomtsev petviashvili kp equation by applying continuum limits with vertex operators involving an infinite number of continous variables to the lattice kp we obtain hierarchies of integrable equations for fields depending on one continous variable at the sites of a two dimensional lattice the direct linearization of the hierarchies is obtained applying the same limit to the free wave function in the integral equation for lattice kp on the basis of the relation between the generator of the hierarchy and the monodromy factor for the scattering problem on a line which is not involved in the continuum limit we establish the hamiltonian structure of the equations of the hierarchy

accelerating the cubic regularization of newton s method on convex problems in this paper we propose an accelerated version of the cubic regularization of newton s method nesterov and polyak in math program 108 1 177 205 2006 the original version used for minimizing a convex function with lipschitz continuous hessian guarantees a global rate of convergence of order o big 1 over k 2 big where k is the iteration counter our modified version converges for the same problem class with order o big 1 over k 3 big keeping the complexity of each iteration unchanged we study the complexity of both schemes on different classes of convex problems in particular we argue that for the second order schemes the class of non degenerate problems is different from the standard class

integrable dynamics of a discrete curve and the ablowitz ladik hierarchy it is shown that the following elementary geometric properties of the motion of a discrete i e piecewise linear curve select the integrable dynamics of the ablowitz ladik hierarchy of evolution equations i the set of points describing the discrete curve lie in the sphere s3 ii the distance between any two subsequent points does not vary in time iii the equations of the dynamics do not depend explicitly on the radius of the sphere these results generalize to a discrete context the previous work on continuous curves a doliwa and p m santini phys lett a 185 373 1994 it is shown that the following elementary geometric properties of the motion of a discrete i e piecewise linear curve select the integrable dynamics of the ablowitz ladik hierarchy of evolution equations i the set of points describing the discrete curve lie in the sphere s3 ii the distance between any two subsequent points does not vary in time iii the equations of the dynamics do not depend explicitly on the radius of the sphere these results generalize to a discrete context the previous work on continuous curves a doliwa and p m santini phys lett a 185 373 1994

integrable discretizations of the euler top discretizations of the euler top sharing the integrals of motion with the continuous time system are studied those of them which are also poisson with respect to the invariant poisson bracket of the euler top are characterized for all these poisson discretizations a solution in terms of elliptic functions is found allowing a direct comparison with the continuous time case we demonstrate that the veselov moser discretization also belongs to our family and apply our methods to this particular example

duality for discrete integrable systems a new class of discrete dynamical systems is introduced via a duality relation for discrete dynamical systems with a number of explicitly known integrals the dual equation can be defined via the difference of an arbitrary linear combination of integrals and its upshifted version we give an example of an integrable mapping with two parameters and four integrals leading to a four dimensional dual mapping with four parameters and two integrals we also consider a more general class of higher dimensional mappings arising via a travelling wave reduction from the integrable mkdv partial difference equation by differencing the trace of the monodromy matrix we obtain a class of novel dual mappings which is shown to be integrable as level set dependent versions of the original ones

in plane and out of plane motions of an extensible semi circular pipe conveying fluid abstract in plane and out of plane motions of a semi circular pipe conveying fluid are analyzed in this paper assuming that the centerline of the semi circular pipe is extensible nonlinear equations of in plane and out of plane motions are derived according to the extended hamilton principle the lagrange nonlinear strain theory and the euler bernoulli beam theory are used to derive the equations the derived equations of motion are discretized by applying the galerkin method linearized equations around the equilibrium position are obtained from the discretized equations and then the dynamic characteristics of the pipe are investigated in addition some modelling issues which are related to the nonlinearity of the circumferential strain and stress are discussed this study finds that a semi circular pipe conveying fluid does not lose stability even at a high fluid velocity although a model using the lagrange nonlinear strain and the corresponding nonlinear stress yields the most accurate computational results of the natural frequencies a model using the lagrange strain and a linearized stress is recommended to compute the natural frequencies efficiently while still maintaining accuracy

k integrals and k lie symmetries in discrete dynamical systems abstract we generalize the concept of symplectic maps to that of k symplectic maps maps whose kth iterates are symplectic similarly k symmetries and k integrals are symmetries resp integrals of the kth iterate of the map it is shown that k symmetries and k integrals are related by the k symplectic structure as in the k 1 continuous case noether s theorem examples are given of k integrals and their related k symmetries for k 1 4

a unifying geometric solution framework and complexity analysis for variational inequalities in this paper we propose a concept of polynomiality for variational inequality problems and show how to find a near optimal solution of variational inequality problems in a polynomial number of iterations to establish this result we build upon insights from several algorithms for linear and nonlinear programs the ellipsoid algorithm the method of centers of gravity the method of inscribed ellipsoids and vaidya s algorithm to develop a unifying geometric framework for solving variational inequality problems the analysis rests upon the assumption of strong f monotonicity which is weaker than strict and strong monotonicity since linear programs satisfy this assumption the general framework applies to linear programs

quad equations and auto backlund transformations of nls type systems treating an integrable quad equation along with its two generalized symmetries as a compatible system allows one to construct an auto backlund transformation for solutions of the related nls type system a fixed periodic reduction of the quad equation yields a quasi periodic reduction of its generalized symmetries that turns them into differential constraints compatible with the nls type system

linearizing integral transform for the multicomponent lattice kp abstract we present a generalization of the direct linearization method for the kadomtsev petviashvili kp equation and its lattice analogues such as the two dimensional toda equation the generalization consists of a riemann hilbert type of integral transform which relates solutions of the associated spectral problem to any given solution of the spectral problem by dimensional reduction we obtain the integral transform for the korteweg de vries kdv equation and its lattice analogues the integral transform is well suited for the investigation of the finite matrix generalizations of the above mentioned equations such finite matrix generalizations may be of use in connection with partial difference equations for spin correlation functions in the two dimensional ising model

the staircase method integrals for periodic reductions of integrable lattice equations we show in full generality that the staircase method papageorgiou et al 1990 phys lett a 147 106 14 quispel et al 1991 physica a 173 243 66 provides integrals for mappings and correspondences obtained as traveling wave reductions of systems of integrable partial difference equations we apply the staircase method to a variety of equations including the korteweg de vries equation the five point bruschi calogero droghei equation the quotient difference qd algorithm and the boussinesq system we show that in all these cases if the staircase method provides r integrals for an n dimensional mapping with then one can introduce q 2r variables which reduce the dimension of the mapping from n to q these dimension reducing variables are obtained as joint invariants of k symmetries of the mappings our results support the idea that often the staircase method provides sufficiently many integrals for the periodic reductions of integrable lattice equations to be completely integrable we also study reductions on other quad graphs than the regular lattice and we prove linear growth of the multi valuedness of iterates of high dimensional correspondences obtained as reductions of the qd algorithm

continuous symmetric reductions of the adler bobenko suris equations continuously symmetric solutions of the adler bobenko suris class of discrete integrable equations are presented initially defined by their invariance under the action of both of the extended three point generalized symmetries admitted by the corresponding equations these solutions are shown to be determined by an integrable system of partial differential equations the connection of this system to the nijhoff hone joshi generating partial differential equations is established and an auto backlund transformation and a lax pair for it are constructed applied to the h1 and q1 0 members of the adler bobenko suris family the method of continuously symmetric reductions yields explicit solutions determined by the painleve trancendents

new non linear modelling for vibration analysis of a straight pipe conveying fluid abstract a new non linear model of a straight pipe conveying fluid is presented for vibration analysis when the pipe is fixed at both ends using the euler bernoulli beam theory and the non linear lagrange strain theory from the extended hamilton s principle the coupled non linear equations of motion for the longitudinal and transverse displacements are derived these equations of motion are discretized by using the galerkin method after the discretized equations are linearized in the neighbourhood of the equilibrium position the natural frequencies are computed from the linearized equations on the other hand the time histories for the displacements are also obtained by applying the generalized time integration method to the non linear discretized equations the validity of the new modelling is provided by comparing results from the proposed non linear equations with those from the equations proposed by paidoussis

discretization of nonlinear evolution equations over associative function algebras abstract a general approach is proposed for discretizing nonlinear dynamical systems and field theories on suitable functional spaces defined over a regular lattice of points in such a way that both their symmetry and integrability properties are preserved a class of discrete kdv equations is introduced also new hierarchies of discrete evolution equations of gelfand dickey type are defined

soliton solutions for abs lattice equations i cauchy matrix approach in recent years there have been new insights into the integrability of quadrilateral lattice equations i e partial difference equations which are the natural discrete analogues of integrable partial differential equations in 1 1 dimensions in the scalar i e single field case there now exist classification results by adler bobenko and suris abs leading to some new examples in addition to the lattice equations of kdv type that were known since the late 1970s and early 1980s in this paper we review the construction of soliton solutions for the kdv type lattice equations and use those results to construct n soliton solutions for all lattice equations in the abs list except for the elliptic case of q4 which is left to a separate treatment

soliton solutions for q3 we construct n soliton solutions to the equation called q3 in the recent adler bobenko surisclassification anessentialingredientintheconstruction is the relationship of q3 0 to the equation proposed by nijhoff quispel and capel in 1983 the nqc equation this latter equation has two extra parameters and depending on their sign choices we get a 4 to 1 relationship from nqc to q3 0 this leads to a four term background solution and then to a 1 soliton solution using a b acklund transformation using the 1ss as a guide allows us to get the n soliton solution in terms of the function of the hirota miwa equation

on the lagrangian formulation of multidimensionally consistent systems multidimensional consistency has emerged as a key integrability property for partial difference equations p es defined on the space time lattice it has led among other major insights to a classification of scalar affine linear quadrilateral p es possessing this property leading to the so called adler bobenko suris abs list recently a new variational principle has been proposed that describes the multidimensional consistency in terms of discrete lagrangian multi forms this description is based on a fundamental and highly non trivial property of lagrangians for those integrable lattice equations namely the fact that on the solutions of the corresponding p e the lagrange forms are closed i e they obey a closure relation here we extend those results to the continuous case it is known that associated with the integrable p es there exist systems of partial differential equations pdes in fact differential equations with regard to the parameters of the lattice as independent variables which equally possess the property of multidimensional consistency in this paper we establish a universal lagrange structure for affine linear quad lattices alongside a universal lagrange multi form structure for the corresponding continuous pdes and we show that the lagrange forms possess the closure property

dual extrapolation and its applications to solving variational inequalities and related problems in this paper we suggest new dual methods for solving variational inequalities with monotone operators we show that with an appropriate step size strategy our method is optimal both for lipschitz continuous operators o 1 over epsilon iterations and for the operators with bounded variations o 1 over epsilon 2 iterations our technique can be applied for solving non smooth convex minimization problems with known structure in this case the worst case complexity bound is o 1 over epsilon iterations

discrete painleve equations and their lax pairs as reductions of integrable lattice equations we describe a method to obtain lax pairs for periodic reductions of a rather general class of integrable non autonomous lattice equations the method is applied to obtain reductions of the non autonomous discrete korteweg de vries equation and non autonomous discrete schwarzian korteweg de vries equation which yield a discrete analogue of the fourth painleve equation a q analogue of the sixth painleve equation and the q painleve equation with a symmetry group of affine weyl type e 1 6

on linear convergence of iterative methods for the variational inequality problem abstract we present new linear convergence results for iterative methods for solving the variational inequality problem the methods include the extragradient method the proximal point method a matrix splitting method and a certain feasible descent method the proofs of the results are based on certain error bounds related to the algorithmic mappings moreover we show that all these error bounds hold if a projection type error bound holds

piecewise linear soliton equations and piecewise linear integrable maps we study piecewise linear integrable systems the associated piecewise linear solitons piecewise linear integrable maps and piecewise linear lax representation are discussed

on the complexity of the hybrid proximal extragradient method for the iterates and the ergodic mean in this paper we analyze the iteration complexity of the hybrid proximal extragradient hpe method for finding a zero of a maximal monotone operator recently proposed by solodov and svaiter one of the key points of our analysis is the use of new termination criteria based on the varepsilon enlargement of a maximal monotone operator the advantage of using these termination criteria is that their definition do not depend on the boundedness of the domain of the operator we then show that korpelevich s extragradient method for solving monotone variational inequalities falls in the framework of the hpe method as a consequence using the complexity analysis of the hpe method we obtain new complexity bounds for korpelevich s extragradient method which do not require the feasible set to be bounded as assumed in a recent paper by nemirovski another feature of our analysis is that the derived iteration complexity bounds are proportional to the distance of the initial point to the solution set the hpe framework is also used to obtain the first iteration complexity result for tseng s modified forward backward splitting method for finding a zero of the sum of a monotone lipschitz continuous map with an arbitrary maximal monotone operator whose resolvent is assumed to be easily computable also using the framework of the hpe method we study the complexity of a variant of a newton type extragradient algorithm proposed by solodov and svaiter for finding a zero of a smooth monotone function with lipschitz continuous jacobian

lattice equations hierarchies and hamiltonian structures by now many integrable systems with soliton solutions are known in two dimensions for fields u n1 n2 defined at the sites nln2 of a two dimensional 2d lattice for time dependent fields u n t defined at the sites n of a one dimensional chain and for fields u x t depending on two continuous variables x and t the relation between an integrable partial differential equation and its integrable discrete versions can be treated in the framework of the direct linearization method 1 2 which is based on a linear integral equation with arbitrary measure and contour 3 4 in the treatment use is made of backlund transformations bt s which are generated by scalar multiplications of the free wave function and or measure in the integral equation 5 the integrable lattice equations are obtained in the form of bianchi identities expressing the commutativity of bt s the integrable equations with one or more continuous variables and their direct linearizations are obtained applying suitable continuum limits to the lattice equation and the integral equation at the same time

dynamic stability of a semi circular pipe conveying harmonically oscillating fluid the dynamic stability of a semi circular pipe conveying harmonically oscillating fluid is investigated in this study for analysis harmonically oscillating flow is regarded as a parametrically excited system with time varying coefficients considering the extensibility and nonlinearity of the semi circular pipe the equations of motion and the associated boundary conditions are obtained by using the extended hamilton principle applying floquet theory to the derived equations the stability of the pipe is analyzed for variations of the amplitude and oscillating frequency of the fluid velocity the effects of the mean value for the fluid velocity on the stability are also analyzed it is found that the semi circular pipe may become unstable when the fluctuation frequency approaches some multiples or sum of the out of plane natural frequencies furthermore it is observed that the instability regions increase with the mean value of the velocity the results obtained via the stability analysis are verified by time responses computed by using a direct time integration method

a systematic method for constructing time discretizations of integrable lattice systems local equations of motion we propose a new method for discretizing the time variable in integrable lattice systems while maintaining the locality of the equations of motion the method is based on the zero curvature lax pair representation and the lowest order conservation laws in contrast to the pioneering work of ablowitz and ladik our method allows the auxiliary dependent variables appearing in the stage of time discretization to be expressed locally in terms of the original dependent variables the time discretized lattice systems have the same set of conserved quantities and the same structures of the solutions as the continuous time lattice systems only the time evolution of the parameters in the solutions that correspond to the angle variables is discretized the effectiveness of our method is illustrated using examples such as the toda lattice the volterra lattice the modified volterra lattice the ablowitz ladik lattice an integrable semi discrete nonlinear schrodinger system and the lattice heisenberg ferromagnet model for the modified volterra lattice we also present its ultradiscrete analogue

free non linear vibration of a rotating thin ring with the in plane and out of plane motions free non linear vibration of a rotating thin ring with a constant speed is analyzed when the ring has both the in plane and out of plane motions the geometric non linearity of displacements is considered by adopting the lagrange strain theory for the circumferential strain instead of the infinitesimal strain theory by using hamilton s principle the coupled non linear partial differential equations are derived which describe the out of plane bending and torsional motions as well as the in plane bending and extensional motions during deriving the equations of motion we discuss how to model the circumferential stress and strain in order to consider the geometric non linearity four models are established three non linear models and one linear model for the four models the linearized equations of motion are obtained in the neighbourhood of the steady state equilibrium position based on the linearized equations of the four cases the natural frequencies are computed at various rotational speeds and then they are compared through the comparison this study recommends which model is appropriate to describe the non linear behaviour more precisely

fredholm determinants and the mkdv sinh gordon hierarchies for a particular class of integral operatorsk we show that the quantity phi log det i k log det i k satisfies both the integrated mkdv hierarchy and the sinh gordon hierarchy this proves a conjecture of zamolodchikov

a semi discrete modified kdv equation in this paper we present an integrable semi discretization of the modified korteweg devries mkdv equation we discretize the time variable of the mkdv equation and get an integrable differential difference system under a standard limit the differential difference system converges to the continuous mkdv equation by hirota s bilinear method we find some explicit solutions including solitons and breather solutions from the semi discrete system we design a numerical scheme to the mkdv equation and carry out several numerical experiments with the 3 soliton solution and breather solution