inseparable local uniformization abstract it is known since the works of zariski in the early 40s that desingularization of varieties along valuations called local uniformization of valuations can be considered as the local part of the desingularization problem it is still an open problem if local uniformization exists in positive characteristic and dimension larger than three in this paper we prove that zariski local uniformization of algebraic varieties is always possible after a purely inseparable extension of the field of rational functions and therefore any valuation can be uniformized by a purely inseparable alteration

when is negativity not a problem for the ultradiscrete limit the ultradiscrete limit has provided a link between integrable difference equations and cellular automata displaying soliton like solutions in particular this procedure generally turns strictly positive solutions of algebraic difference equations with positive coefficients into corresponding solutions to equations involving the max operator although it certainly is the case that dropping these positivity conditions creates potential difficulties it is still possible for solutions to persist under the ultradiscrete limit even in their absence to recognize when this will occur one must consider whether a certain expression involving a measure of the rates of convergence of different terms in the difference equation and their coefficients is equal to zero applications discussed include the solution of elementary ordinary difference equations a discretization of the hirota bilinear difference equation and the identification of integrals of motion for ultradiscrete equations

fundamental cycle of a periodic box ball system we investigate a box ball system with periodic boundary conditions since the box ball system is a deterministic dynamical system that takes only a finite number of states it will exhibit periodic motion we determine its fundamental cycle for a given initial state

combinatorial representation of invariants of a soliton cellular automaton abstract the structure of the soliton cellular automaton is studied by means of combinatorial techniques it is shown that the shape of the young tableaux gives an infinite number of time invariants of the automaton the employed combinatorial materials are the dyck language stack representable sequences and the robinson schensted algorithm

supersymmetric extension of the integrable box ball system we introduce the supersymmetric extension of the box ball system by use of the isomorphism of the crystal base for the super lie algebra we define the time evolution operator and give the evolution equation explicitly we also construct the soliton solutions

ramification of valuations abstract in this paper we develop a very explicit theory of ramification of general valuations in algebraic function fields in characteristic zero and arbitrary dimension we obtain the strongest possible generalization of the classical ramification theory of local dedekind domains we further develop a ramification theory of algebraic functions fields of dimension two in positive characteristic we prove that local monomialization and simultaneous resolution hold under very mild assumptions and give pathological examples

linearizable cellular automata the initial value problem for a class of reversible elementary cellular automata with periodic boundaries is reduced to an initial boundary value problem for a class of linear systems on a finite commutative ring moreover a family of such linearizable cellular automata is given

the box ball system and the n soliton solution of the ultradiscrete kdv equation any state of the box ball system bbs together with its time evolution is described by the n soliton solution with appropriate choice of n of the ultradiscrete kdv equation it is shown that simultaneous elimination of all 10 walls in a state of the bbs corresponds exactly to reducing the parameters that determine the size of a soliton by one this observation leads to an expression for the solution to the initial value problem ivp for the bbs expressions for the solution to the ivp for the ultradiscrete toda molecule equation and the periodic bbs are also presented

darboux and binary darboux transformations for the nonautonomous discrete kp equation it is shown how darboux and binary darboux transformations for a nonautonomous discrete kp equation can be obtained from fermion analysis this equation is obtained by considering a generalized miwa transformation it is also shown to be linked to the discrete kp equation by a special gauge transformation the darboux and binary darboux transformations are used to discuss general classes of solutions in the form of casorati and gramm type determinants n soliton solutions are discussed as well

on a periodic soliton cellular automaton we propose a box and ball system with a periodic boundary condition periodic box and ball system pbbs the time evolution rule of the pbbs is represented as a boolean recurrence formula an inverse ultradiscretization of which is shown to be equivalent to the algorithm of the calculus for the 2nth root the relations to the pbbs of the combinatorial r matrix of u q an 1 are also discussed

correlation functions for a periodic box ball system we investigate correlation functions in a periodic box ball system for the two point functions of short distance we give explicit formulae obtained by combinatorial methods we give expressions for general n point functions in terms of ultradiscrete theta functions

constructing solutions to the ultradiscrete painleve equations we investigate the nature of particular solutions to the ultradiscrete painleve equations we start by analysing the autonomous limit and show that the equations possess an explicit invariant which leads naturally to the ultradiscrete analogue of elliptic functions for the ultradiscrete painleve equations ii and iii we present special solutions reminiscent of the casorati determinant ones which exist in the continuous and discrete cases finally we analyse the discrete painleve equation i and show how it contains both the continuous and the ultradiscrete ones as particular limits

factorization reduction and embedding in integrable cellular automata factorized dynamics in soliton cellular automata with quantum group symmetry is identified with a motion of particles and anti particles exhibiting pair creation and annihilation an embedding scheme is presented showing that the d 1 n automaton contains as certain subsectors the box ball systems and all the other automata associated with the crystal bases of non exceptional affine lie algebras the results extend the earlier ones to higher representations by a certain reduction and to a wider class of boundary conditions

the two dimensional periodic box ball system and its fundamental cycle we study a two dimensional box ball system which is an ultradiscrete analogue of the discrete kp equation we construct an algorithm to calculate the fundamental cycle which is an important conserved quantity of the two dimensional box ball system with periodic boundary condition by using the tropical curve theory

solving the ultradiscrete kdv equation we show that a generalized cellular automaton exhibiting solitonic interactions can be explicitly solved by means of techniques first introduced in the context of the scattering problem for the kdv equation we apply this method to calculate the phase shifts caused by interactions between the solitonic and non solitonic parts into which arbitrary initial states separate in time

on the convergence of a new trust region algorithm in this paper we present a new trust region algorithm for general nonlinear constrained optimization problems the algorithm is based on the l infty exact penalty function under very mild conditions global convergence results for the algorithm are given local convergence properties are also studied it is shown that the penalty parameter generated by the algorithm will be eventually not less than the l 1 norm of the lagrange multipliers at the accumulation point it is proved that the method is equivalent to the sequential quadratic programming method for all large k hence superlinearly convergent results of the sqp method can be applied numerical results are also reported

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

box and ball system as a realization of ultradiscrete nonautonomous kp equation cellular automata which are realized by dynamics of several kinds of balls in an infinite array of boxes are investigated they show soliton patterns even in the case when each box has arbitrary capacity the analytical expression for the soliton patterns are obtained using ultradiscretization of the nonautonomous discrete kp equation

crystal graphs for representations of the q analogue of classical lie algebras the explicit form of the crystal graphs for the finite dimensional representations of the q analogue of the universal enveloping algebras of type a b c and d is given in terms of semi standard tableaux and its analogues

stable rigged configurations for quantum affine algebras of nonexceptional types for an affine algebra of nonexceptional type in the large rank we show the fermionic formula depends only on the attachment of the node 0 of the dynkin diagram to the rest and the fermionic formula of not type a can be expressed as a sum of that of type a with littlewood richardson coefficients combining this result with kirillov et al 2002 13 and lecouvey et al 2011 18 we settle the x m conjecture under the large rank hypothesis

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

the relative approximation degree in valued function fields we continue the work of kaplansky on immediate valued field extensions and determine special properties of elements in such extensions in particular we are interested in the question when an immediate valued function field of transcendence degree 1 is henselian rational i e generated modulo henselization by one element if so then ramification can be eliminated in this valued function field the results presented in this paper are crucial for the first author s proof of henselian rationality over tame fields which in turn is used in his work on local uniformization

noncommutative schur polynomials and the crystal limit of the u_ q widehat mathfrak sl 2 vertex model starting from the verma module of we consider the evaluation module for affine and discuss its crystal limit q 0 there exists an associated integrable statistical mechanics model on a square lattice defined in terms of vertex configurations its transfer matrix is the generating function for noncommutative complete symmetric polynomials in the generators of the affine plactic algebra an extension of the finite plactic algebra first discussed by lascoux and schutzenberger the corresponding noncommutative elementary symmetric polynomials were recently shown to be generated by the transfer matrix of the so called phase model discussed by bogoliubov izergin and kitanine here we establish that both generating functions satisfy baxter s tq equation in the crystal limit by tying them to special solutions of the yang baxter equation the tq equation amounts to the well known jacobi trudi formula leading naturally to the definition of noncommutative schur polynomials the latter can be employed to define a ring which has applications in conformal field theory and enumerative geometry it is isomorphic to the fusion ring of the wess zumino novikov witten model whose structure constants are the dimensions of spaces of generalized functions over the riemann sphere with three punctures

parameter optimization using the l exact penalty function and strictly convex quadratic programming problems abstract this paper presents an algorithm for the numerical solution of constrained parameter optimization problems the solution strategy is based on a sequential quadratic programming sqp technique that uses the l exact penalty function unlike similar sqp algorithms the method proposed here solves only strictly convex quadratic programs to obtain the search directions the global convergence properties of the algorithm are enhanced by the use of a nonmonotone line search and second order corrections to avoid the maratos effect the paper also presents an ansi c implementation of the algorithm the effectiveness of the proposed method is demonstrated by solving numerous parameter optimization and optimal control problems that have appeared in the literature

inversible max plus algebras and integrable systems we present an extended version of max plus algebra which includes the inverse operator of max this algebra enables us to ultradiscretize the system including subtractions and obtain new ultradiscrete equations the known ultradiscrete equations can also be recovered by this construction

the algebra and model theory of tame valued fields a henselian valued field k is called a tame field if its algebraic closure k is a tame extension that is the ramification field of the normal extension k k is algebraically closed every algebraically maximal kaplansky field is a tame field but not conversely we develop the algebraic theory of tame fields and then prove ax kochen ershov principles for tame fields this leads to model completeness and completeness results relative to value group and residue field as the maximal immediate extensions of tame fields will in general not be unique the proofs have to use much deeper valuation theoretical results than those for other classes of valued fields which have already been shown to satisfy ax kochen ershov principles the results of this paper have been applied to gain insight in the zariski space of places of an algebraic function field and in the model theory of large fields

box ball system associated with antisymmetric tensor crystals a new box ball system associated with an antisymmetric tensor crystal of the quantum affine algebra of type a is considered this includes the so called coloured box ball system with capacity 1 as the simplest case an infinite number of conserved quantities is constructed and the scattering rule of two solitons is given explicitly

analysis of a particle antiparticle description of a soliton cellular automaton we present a derivation of a formula that gives dynamics of an integrable cellular automation associated with crystal bases this automaton is related to type d affine lie algebra and contains usual box ball systems as a special case the dynamics is described by means of such objects as carriers particles and antiparticles we derive it from an analysis of a recently obtained formula of the combinatorial r an intertwiner between tensor products of crystals that was found in a study of geometric crystals

large deviations and one sided scaling limit of randomized multicolor box ball system the basic kappa color box ball bbs system is an integrable cellular automaton on one dimensional lattice whose local states take 0 1 ldots kappa with 0 regarded as an empty box the time evolution is defined by a combinatorial rule of quantum group theoretical origin and the complete set of conserved quantities is given by a kappa tuple of young diagrams in the randomized bbs a probability distribution on 0 1 ldots kappa to independently fill the consecutive n sites in the initial state induces a highly nontrivial probability measure on the kappa tuple of those invariant young diagrams in a recent work kuniba et al nucl phys b 937 240 271 2018 their large n equilibrium shape has been determined in terms of schur polynomials by a markov chain method and also by a very different approach of thermodynamic bethe ansatz tba in this paper we establish a large deviations principle for the row lengths of the invariant young diagrams as a corollary they are shown to converge almost surely to the equilibrium shape at an exponential rate we also refine the tba analysis and obtain the exact scaling form of the vacancy the row length and the column multiplicity which exhibit nontrivial factorization in a one parameter specialization

elimination of ramification i the generalized stability theorem we prove a general version of the stability theorem if k is a valued field such that the ramification theoretical defect is trivial for all of its finite extensions and if f k is a finitely generated transcendental extension of valued fields for which equality holds in the abhyankar inequality then the defect is also trivial for all finite extensions of f this theorem is applied to eliminate ramification in such valued function fields it has applications to local uniformization and to the model theory of valued fields in positive characteristic

conserved quantities and generalized solutions of the ultradiscrete kdv equation we construct generalized solutions to the ultradiscrete kdv equation including the so called negative solition solutions the method is based on the ultradiscretization of soliton solutions to the discrete kdv equation with gauge transformation the conserved quantities of the ultradiscrete kdv equation are shown to be constructed in a similar way to those for the box ball system

a numerically stable dual method for solving strictly convex quadratic programs an efficient and numerically stable dual algorithm for positive definite quadratic programming is described which takes advantage of the fact that the unconstrained minimum of the objective function can be used as a starting point its implementation utilizes the cholesky and qr factorizations and procedures for updating them the performance of the dual algorithm is compared against that of primal algorithms when used to solve randomly generated test problems and quadratic programs generated in the course of solving nonlinear programming problems by a successive quadratic programming code the principal motivation for the development of the algorithm these computational results indicate that the dual algorithm is superior to primal algorithms when a primal feasible point is not readily available the algorithm is also compared theoretically to the modified simplex type dual methods of lemke and van de panne and whinston and it is illustrated by a numerical example

conserved quantities of generalized periodic box ball systems constructed from the ndkp equation we investigate periodic box ball systems pbbss with several kinds of balls and box capacity greater than or equal to one conserved quantities of the pbbss are constructed from those of the nonautonomous discrete kp ndkp equation using the lax representation of the ndkp equation

how to find the conserved quantities of nonlinear discrete equations a method of finding conserved quantities of nonlinear ordinary difference equations is briefly discussed the method is based on the hypothesis that the conserved quantities are expressed by a ratio of two polynomials of dependent variables we obtain the conserved quantities of the discrete equation of motion of an anharmonic oscillator and a nonlinear difference equation of third order

ultradiscretization of solvable one dimensional chaotic maps we consider the ultradiscretization of a solvable one dimensional chaotic map which arises from the duplication formula of the elliptic functions it is shown that the ultradiscrete limit of the map and its solution yield the tent map and its solution simultaneously a geometric interpretation of the dynamics of the tent map is given in terms of the tropical jacobian of a certain tropical curve generalization to the maps corresponding to the mth multiplication formula of the elliptic functions is also discussed

on reversibility of cellular automata with periodic boundary conditions reversibility of one dimensional cellular automata with periodic boundary conditions is discussed it is shown that there exist exactly 16 reversible elementary cellular automaton rules for infinitely many cell sizes by means of a correspondence between elementary cellular automaton and the de bruijn graph in addition a sufficient condition for reversibility of three valued and two neighbour cellular automaton is given

counting the number of distinct distances of elements in valued field extensions abstract the defect of valued field extensions is a major obstacle in open problems in resolution of singularities and in the model theory of valued fields whenever positive characteristic is involved we continue the detailed study of defect extensions through the tool of distances which measure how well an element in an immediate extension can be approximated by elements from the base field we show that in several situations the number of essentially distinct distances in fixed extensions or even just over a fixed base field is finite and we compute upper bounds we apply this to the special case of valued functions fields over perfect base fields in particular this provides important information used in forthcoming research on the ramification theory of two dimensional valued function fields

value groups residue fields and bad places of rational function fields we classify all possible extensions of a valuation from a ground field k to a rational function field in one or several variables over k we determine which value groups and residue fields can appear and we show how to construct extensions having these value groups and residue fields in particular we give several constructions of extensions whose corresponding value group and residue field extensions are not finitely generated in the case of a rational function field k x in one variable we consider the relative algebraic closure of k in the henselization of k x with respect to the given extension and we show that this can be any countably generated separable algebraic extension of k in the tame case we show how to determine this relative algebraic closure finally we apply our methods to power series fields and the p adics

box and ball system with a carrier and ultradiscrete modified kdv equation a new soliton cellular automaton is proposed it is defined by an array of an infinite number of boxes a finite number of balls and a carrier of balls moreover it reduces to a discrete equation obtained from the discrete modified korteweg de vries equation through a limit an algebraic expression of soliton solutions is also proposed

symmetry of integrable cellular automaton we study an integrable cellular automaton which is called the box ball system bbs the bbs can be derived directly from the integrable differential difference equation by either ultradiscretization or crystallization we clarify the integrable structure and the hidden symmetry of the bbs

the am 1 automata related to crystals of symmetric tensors a soliton cellular automaton associated with crystals of symmetric tensor representations of the quantum affine algebra uq am 1 is introduced it is a crystal theoretic formulation of the generalized box ball system in which capacities of boxes and carriers are arbitrary and inhomogeneous scattering matrices of two solitons coincide with the combinatorial r matrices of uq am 1 1 a piecewise linear evolution equation of the automaton is identified with an ultradiscrete limit of the nonautonomous discrete kadomtsev petviashivili equation a class of n soliton solutions is obtained through the ultradiscretization of soliton solutions of the latter

tropical krichever construction for the non periodic box and ball system a solution for an initial value problem of the box and ball system is constructed from a solution of the periodic box and ball system the construction is done through a specific limiting process based on the theory of tropical geometry this method gives a tropical analogue of the krichever construction which is an algebro geometric method to construct exact solutions to integrable systems for the non periodic system

a finite toda representation of the box ball system with box capacity a connection between the finite ultradiscrete toda lattice and the box ball system is extended to the case where each box has its own capacity and a carrier has a capacity parameter depending on time in order to consider this connection new carrier rules size limit for solitons and recovery of balls and a concept expansion map are introduced a particular solution to the extended system of a special case is also presented

solutions to the ultradiscrete kdv equation expressed as the maximum of a quadratic function we propose the functions defined by the maximum of a discrete quadratic form and satisfying the ultradiscrete kdv equation these functions include not only soliton solutions but also pseudo periodic solutions in the proof we employ some facts of discrete convex analysis