new singular solutions of the nonlinear schrodinger equation we present numerical simulations of a new type of singular solutions of the critical nonlinear schrodinger equation nls that collapse with a quasi self similar ring profile at a square root blowup rate we find and analyze the equation of the ring profile we observe that the self similar ring profile is an attractor for a large class of radially symmetric initial conditions but is unstable under symmetry breaking perturbations the equation for the ring profile admits also multi ring solutions that give rise to collapsing self similar multi ring solutions but these solutions are unstable even in the radially symmetric case and eventually collapse with a single ring profile collapsing ring solutions are also observed in the supercritical nls

moving mesh methods for problems with blow up in this paper we consider the numerical solution of pdes with blow up for which scaling invariance plays a natural role in describing the underlying solution structures it is a challenging numerical problem to capture the qualitative behaviour in the blow up region and the use of nonuniform meshes is essential we consider moving mesh methods for which the mesh is determined using so called moving mesh partial differential equations mmpdes specifically the underlying pde and the mmpde are solved for the blow up solution and the computational mesh simultaneously motivated by the desire for the mmpde to preserve the scaling invariance of the underlying problem we study the effect of different choices of mmpdes and monitor functions it is shown that for suitable ones the mmpde solution evolves towards a moving mesh which close to the blow up point automatically places the mesh points in such a manner that the ignition kernel which is well known to be a natural coordinate in describing the behaviour of blow up approaches a constant as t to t the blow up time several numerical examples are given to verify the theory for these mmpde methods and to illustrate their efficacy

on the numerical solution of one dimensional pdes using adaptive methods based on equidistribution numerical experiments are described that illustrate some important features of the performance of moving mesh methods for solving one dimensional partial differential equations pdes the particular method considered here is an adaptive finite difference method based on the equidistribution of a monitor function and it is one of the moving mesh methods proposed by w huang y ren and r d russell 1994 siam j numer anal 31 709 we show how the accuracy of the computations is strongly dependent on the choice of monitor function and we present a monitor function that yields an optimal rate of convergence motivated by efficiency considerations for problems in two or more space dimensions we demonstrate a robust and efficient algorithm in which the mesh equations are uncoupled from the physical pde the accuracy and efficiency of the various formulations of the algorithm are considered and a novel automatic time step control mechanism is integrated into the scheme

existence and stability of a solution blowing up on a sphere for an l 2 supercritical nonlinear schrodinger equation we consider the quintic two dimensional focusing nonlinear schrodinger equation iut u u 4u which is l2 supercritical even though the existence of finite time blow up solutions in the energy space h1 is known very little is understood concerning the singularity formation numerics suggest the existence of a stable blow up dynamic corresponding to a self similar blowup at one point in space we prove the existence of a different type of dynamic and exhibit an open set among the h1 radial distributions of initial data for which the corresponding solution blows up in finite time on a sphere this is the first result of an explicit description of a blow up dynamic in the l2 supercritical setting

reversals and palindromes in continued fractions several results on continued fractions expansions are on indirect consequences of the mirror formula we survey occurrences of this formula for sturmian real numbers for simultaneous diophantine approximation and for formal power series

mesures de transcendance et aspects quantitatifs de la methode de thue siegel roth schmidt a proof of the transcendence of a real number based on the thue siegel roth schmidt method involves generally a sequence n n 1 of algebraic numbers of bounded degree or a sequence xn n 1 of integer r tuples in the present paper we show how such a proof can produce a transcendence measure for if one is able to quantify the growth of the heights of the algebraic numbers n or of the points xn our method rests on the quantitative schmidt subspace theorem we further give several applications including to certain normal numbers and to the extremal numbers introduced by roy

recurrence in infinite partial words the recurrence function r w n of an infinite word w was introduced by morse and hedlund in relation to symbolic dynamics it is the size of the smallest window such that wherever its position on w all length n subwords of w will appear at least once inside that window the recurrence quotient w of w defined as lim sup r w n n is useful for studying the growth rate of r w n it is known that if w is periodic then w 1 while if w is not then w 3 a long standing conjecture from rauzy states that the latter can be improved to w 5 5 2 3 618 this bound being true for each sturmian word and being reached by the fibonacci word in this paper we study in particular the spectrum of values taken by the recurrence quotients of infinite partial words which are sequences that may have some undefined positions in this case we determine exactly the spectrum of values which turns out to be 1 every real number greater than or equal to 2 and more precisely if an infinite partial word w is ultimately factor periodic then w 1 while if w is not then w 2 and we give constructions of infinite partial words achieving each value

moving mesh methods based on moving mesh partial differential equations several versions of a moving mesh method are developed based on a mesh spatial smoothing technique and on the moving mesh pdes derived in a previous paper these versions are quite simple and easy to program they are applied to three bench mark one dimensional problems which show different solution behaviour the numerical results clearly demonstrate that the present methods are capable of accurately tracking rapid spatial and temporal transitions

periodicity repetitions and orbits of an automatic sequence we revisit a technique of s lehr on automata and use it to prove old and new results in a simple way we give a very simple proof of the 1986 theorem of honkala that it is decidable whether a given k automatic sequence is ultimately periodic we prove that it is decidable whether a given k automatic sequence is overlap free or squarefree or cubefree etc we prove that the lexicographically least sequence in the orbit closure of a k automatic sequence is k automatic and use this last result to show that several related quantities such as the critical exponent irrationality measure and recurrence quotient for sturmian words with slope a have automatic continued fraction expansions if a does

singular standing ring solutions of nonlinear partial differential equations abstract we present a general framework for constructing singular solutions of nonlinear evolution equations that become singular on a d dimensional sphere where d 1 the asymptotic profile and blowup rate of these solutions are the same as those of solutions of the corresponding one dimensional equation that become singular at a point we provide a detailed numerical investigation of these new singular solutions for the following equations the nonlinear schrodinger equation i t t x 2 0 with 2 the biharmonic nonlinear schrodinger equation i t t x 2 2 0 with 4 the nonlinear heat equation t t x 2 0 with 0 and the nonlinear biharmonic heat equation t t x 2 2 0 with 0

spectrum of the exponents of best rational approximation using the new theory of w m schmidt and l summerer called parametric geometry of numbers we show that the going up and going down transference inequalities of w m schmidt and m laurent describe the full spectrum of the n exponents of best rational approximation to points in mathbb r n 1

computer simulation of wave collapses in the nonlinear schro dinger equation abstract the self tuning numerical scheme for simulation of collapse phenomena in the framework of the nonlinear schrodinger equation is developed the scheme allows to reach extremely high field amplitudes 2 0 2 10 15 without lack of validity numerical results are compared with analytical predictions

on the least number of palindromes contained in an infinite word we investigate the least number of palindromic factors in an infinite word we first consider general alphabets and give answers to this problem for periodic and non periodic words closed or not under reversal of factors we then investigate the same problem when the alphabet has size two

abelian square rich words abstract an abelian square is the concatenation of two words that are anagrams of one another a word of length n can contain at most n 2 distinct factors and there exist words of length n containing n 2 distinct abelian square factors that is distinct factors that are abelian squares this motivates us to study infinite words such that the number of distinct abelian square factors of length n grows quadratically with n more precisely we say that an infinite word w is abelian square rich if for every n every factor of w of length n contains on average a number of distinct abelian square factors that is quadratic in n and uniformly abelian square rich if every factor of w contains a number of distinct abelian square factors that is proportional to the square of its length of course if a word is uniformly abelian square rich then it is abelian square rich but we show that the converse is not true in general we prove that the thue morse word is uniformly abelian square rich and that the function counting the number of distinct abelian square factors of length 2 n of the thue morse word is 2 regular as for sturmian words we prove that a sturmian word s of angle is uniformly abelian square rich if and only if the irrational has bounded partial quotients that is if and only if s has bounded exponent

nonparametric density estimation the l 1 view differentiation of integrals consistency lower bounds for rates of convergence rates of convergence in l1 the automatic kernel estimate l1 and pointwise convergence estimates related to the kernel estimate and the histogram estimate simulation inequalities and random variate generation the transformed kernel estimate applications in discrimination operations on density estimates estimators based on orthogonal series index

limit values of the recurrence quotient of sturmian sequences abstract the recurrence quotient is a real number associated to any symbolic sequence in the case of sturmian sequences morse and hedlund proved that the smallest possible value is 5 5 2 we study here the structure of the set of values taken by the recurrence quotient for a sturmian sequence

approximation to real numbers by cubic algebraic integers i in 1969 h davenport and w m schmidt studied the problem of approximation to a real number xi by algebraic integers of degree at most 3 they did so using geometry of numbers by resorting to the dual problem of finding simultaneous approximations to xi and xi 2 by rational numbers with the same denominator in this paper we show that their measure of approximation for the dual problem is optimal and that it is realized for a countable set of real numbers xi we give several properties of these numbers including measures of approximation by rational numbers by quadratic real numbers and by algebraic integers of degree at most 3

theory of singular vortex solutions of the nonlinear schrodinger equation we present a systematic study of singular vortex solutions of the critical and supercritical two dimensional nonlinear schrodinger equation in particular we study the critical power for collapse and the asymptotic blowup profile of singular vortices

a simple adaptive technique for nonlinear wave problems abstract a method for the numerical integration of the nonlinear schrodinger equation is derived which uses variable time steps and a moving spatial grid the benefits of adaptation are clearly demonstrated in the numerical experiments reported the simple technique employed to move the nodes can be applied with little coding effort to general one dimensional systems of pdes

an adaptive method of lines solution of the korteweg de vries equation abstract following a method of lines formulation the korteweg de vries equation is solved using a static spatial remeshing algorithm based on the equidistribution principle which allows the number of nodes to be significantly reduced as compared to a fixed grid solution several finite difference schemes including direct and stagewise procedures are compared and the results of a large number of computational experiments are presented which demonstrate that the selection of a spatial approximation scheme for the third order derivative term is the primary determinant of solution accuracy

singular ring solutions of critical and supercritical nonlinear schrodinger equations abstract we present new singular solutions of the nonlinear schrodinger equation nls i t t r r r d 1 r r 2 0 1 d 2 d 2 these solutions collapse with a quasi self similar ring profile q i e q where q 1 l 1 t q r r m t l exp i 0 t d s l 2 s i l t 4 l r 2 1 r r m t 2 l t is the ring width that vanishes at the singularity r m t r 0 l t is the ring radius and 2 d 1 the blowup rate of these solutions is 1 1 for 2 d 2 and 1 d 0 1 and a square root with a loglog correction the loglog law when 2 and 1 d 0 therefore the nls has solutions that collapse with any blowup rate p for 1 2 p 1 this study extends the results of g fibich n gavish x wang new singular solutions of the nonlinear schrodinger equation physica d 211 2005 193 220 for 1 and d 2 and of p raphael existence and stability of a solution blowing up on a sphere for a l 2 super critical non linear schrodinger equation duke math j 134 2 2006 199 258 for 2 and d 2 to all 2 d 2 and 1 d

strong approximation of quantile processes by iterated kiefer processes the notion of a kth iterated kiefer process k t k for k n and v t r is introduced we show that the uniform quantile process n t may be approximated on 0 1 by n 1 2 k n t k at an optimal uniform almost sure rate of o n 1 2 1 2k 1 0 1 for each k n our arguments are based in part on a new functional limit law of independent interest for the increments of the empirical process applications include an extended version of the uniform bahadur kiefer representation together with strong limit theorems for nonparametric functional estimators

diophantine approximation on planar curves and the distribution of rational points let c be a nondegenerate planar curve and for a real positive decreasing function let c denote the set of simultaneously approximable points lying on c we show that c is of khintchine type for divergence i e if a certain sum diverges then the one dimensional lebesgue measure on c of c is full we also obtain the hausdorff measure analogue of the divergent khintchine type result in the case that c is a rational quadric the convergence counterparts of the divergent results are also obtained furthermore for functions with lower order in a critical range we determine a general exact formula for the hausdorff dimension of c these results constitute the first precise and general results in the theory of simultaneous diophantine approximation on manifolds

new self similar solutions of the nonlinear schrodinger equation with moving mesh computations we study the blow up self similar solutions of the radially symmetric nonlinear schrodinger equation nls given by iut urr d 1rur u u 2 with dimension d 2 these solutions become infinite in a finite time t by a series of careful numerical computations partly supported by analytic results we demonstrate that there is a countably infinite set of blow up self similar solutions which satisfy a second order complex ordinary differential equation with an integral constraint these solutions are characterised by the number of oscillations in their amplitude when d is close to 2 the solutions are computed as functions of d and their behaviour in the critical limit as d 2 is investigated the stability of these solutions is then studied by solving the nls by using an adaptive numerical method this method uses moving mesh partial differential equations and exploits the scaling invariance properties of the underlying equation we demonstrate that the single humped self similar solution is globally stable whereas the multi humped solutions all appear to be unstable

on schmidt and summerer parametric geometry of numbers recently w m schmidt and l summerer introduced a new theory that allowed them to recover the main known inequalities relating the usual exponents of diophantine approximation to a point in r n and to discover new ones they rst note that these exponents can be computed in terms of the successive minima of a parametric family of convex bodies attached to the given point then they prove that the n tuple of these successive minima can in turn be approximated up to bounded dierence by a function from a certain class in this paper we show that the same is true within a smaller and simpler class of functions which we call rigid systems we also show that conversely given a rigid system there exists a point in r n whose associated family of convex bodies has successive minima that approximate that rigid system up to bounded dierence as a consequence the problem of describing the joint spectrum of a family of exponents of diophantine approximation is reduced to combinatorial analysis

a grid redistribution method for singular problems many physical phenomena develop singular or nearly singular behavior in localized regions e g boundary layers or blowup solutions using uniform grids for such problems becomes computationally prohibitive as the solution approaches singularity ren and wang developed a semi static adaptive grid method w ren x p wang an iterative grid redistribution method for singular problems in multiple dimensions j comput phys 159 2000 246 273 for the solution of these problems known as the iterative grid redistribution igr method in this study we develop a theoretical basis for semi static adaptive grid method for singular problems based on this theory we obtain the key result of this study a methodology for designing robust weight functionals which ensures grid resolution in the singular region as well as control of the maximal grid spacing in the outer region using this methodology we introduce a semi static adaptive grid method which does not involve an iterative procedure for grid redistribution as in the igr method we demonstrate the efficacy of this method with numerical examples of solutions which localize by more than nine orders of magnitude

an iterative grid redistribution method for singular problems in multiple dimensions we introduce an iterative grid redistribution method based on the variational approach the iterative procedure enables us to gain more precise control of the grid distribution near the regions of large solution variations the method is particularly effective for solving partial differential equations with singular solutions e g blowup solutions our method requires little prior information of the singular solutions and can handle multiple singularities the method is successfully applied to the nonlinear schrodinger equation and the keller segal equations where solutions with multiple blowup points can be solved up to times very close to the blowup time

a natural bem with non uniform grids on the elliptic boundary and its coupling method in this paper for solving poisson equation in 2 d unbounded domains a natural bem with non uniform grids on the elliptic boundary is investigated the coupling of nbem fem with non uniform grids on the elliptic artificial boundary is discussed and the corresponding convergence theorems of the nbem and the coupling method are proved the moving mesh method based on the arc length equidistribution principle is introduced into the nbem and the coupling method respectively to solve exterior problems numerical examples confirm the convergence and demonstrate the advantage in accuracy and efficiency for the proposed methods

an efficient dynamically adaptive mesh for potentially singular solutions abstract we develop an efficient dynamically adaptive mesh generator for time dependent problems in two or more dimensions the mesh generator is motivated by the variational approach and is based on solving a new set of nonlinear elliptic pdes for the mesh map when coupled to a physical problem the mesh map evolves with the underlying solution and maintains high adaptivity as the solution develops complicated structures and even singular behavior the overall mesh strategy is simple to implement avoids interpolation and can be easily incorporated into a broad range of applications the efficacy of the mesh is first demonstrated by two examples of blowing up solutions to the 2 d semilinear heat equation these examples show that the mesh can follow with high adaptivity a finite time singularity process the focus of applications presented here is however the baroclinic generation of vorticity in a strongly layered 2 d boussinesq fluid a challenging problem the moving mesh follows effectively the flow resolving both its global features and the almost singular shear layers developed dynamically the numerical results show the fast collapse to small scales and an exponential vorticity growth

exponents of diophantine approximation in dimension 2 for numbers of sturmian type we generalize the construction of roy s fibonacci type numbers to the case of a sturmian recurrence and we determine the classical exponents of approximation omega 2 xi widehat omega 2 xi lambda 2 xi widehat lambda 2 xi associated with these real numbers this also extends similar results established by bugeaud and laurent in the case of sturmian continued fractions more generally we provide an almost complete description of the combined graph of parametric successive minima functions defined by schmidt and summerer in dimension two for such sturmian type numbers as a side result we obtain new information on the joint spectra of the above exponents as well as a new family of numbers for which it is possible to construct the sequence of the best rational approximations

diophantine approximation on planar curves the convergence theory the convergence theory for the set of simultaneously approximable points lying on a planar curve is established our results complement the divergence theory developed in 1 and thereby completes the general metric theory for planar curves

singular solutions of the biharmonic nonlinear schrodinger equation we consider singular solutions of the l 2 critical biharmonic nonlinear schrodinger equation we prove that the blowup rate is bounded by a quartic root the solution approaches a quasi self similar profile and a finite amount of l 2 norm which is no less than the critical power concentrates into the singularity we also prove the existence of a ground state solution we use asymptotic analysis to show that the blowup rate of peak type singular solutions is slightly faster than that of a quartic root and the self similar profile is given by the ground state standing wave these findings are verified numerically up to focusing levels of 10 8 using an adaptive grid method we also use the spectral renormalization method to compute the ground state of the standing wave equation and the critical power for collapse in one two and three dimensions

kiefer s theorem via the hungarian construction a proof of kiefer s theorem estimating the difference between the empirical and quantile processes is given by appealing to the hungarian construction

some observations on a static spatial remeshing method based on equidistribution principles this paper proposes a method of lines solution procedure with time and space adaptation for one dimensional systems of partial differential equations whose solutions display steep moving fronts the spatial remeshing algorithm which is a variation of the method published by sanz serna and christie and an extension suggested by revilla is a static remeshing method based on equidistribution principles the selection of the several algorithm components i e grid placement criterion spatial discretization scheme time integrator adaptation frequency and parameter tuning are investigated and illustrated with several test examples i e the cubic schrodinger equation a model of a single step reaction with diffusion and a model of flame propagation

a three dimensional adaptive method based on the iterative grid redistribution in this paper we develop a three dimensional adaptive method based on iterative grid redistribution technique introduced in j comput phys 159 2000 246 the key step for the successful implementation is a fast algorithm for grid generation which is composed of solving the linear grid equation systems and an efficient method for inverting a map by computing iso surface intersections to carry out the three dimensional calculations the whole procedure is also parallelized on a pc cluster the improved and extended three dimensional adaptive method is applied to solve pdes with singular solutions that have three dimensional structures numerical experiments have demonstrated the method s effectiveness

a robust moving mesh finite volume method applied to 1d hyperbolic conservation laws from magnetohydrodynamics in this paper we describe a one dimensional adaptive moving mesh method and its application to hyperbolic conservation laws from magnetohydrodynamics mhd the method is robust because it employs automatic control of mesh adaptation when a new model is considered without manually set parameters adaptive meshes are a common tool for increasing the accuracy and reducing computational costs when solving time dependent partial differential equations pdes mesh points are moved towards locations where they are needed the most to obtain a time dependent adaptive mesh monitor functions are used to automatically monitor the importance of the various parts of the domain by assigning a weight value to each location based on the equidistribution principle all mesh points are distributed according to their assigned weights we use a sophisticated monitor function that tracks both small local phenomena as well as large shocks in the same solution the combination of the moving mesh method and a high resolution finite volume solver for hyperbolic pdes yields a serious gain in accuracy at relatively no extra costs the results of several numerical experiments including comparisons with h refinement are presented which cover many intriguing aspects typifying nonlinear magneto fluid dynamics with higher accuracy than often seen in similar publications

markoff lagrange spectrum and extremal numbers let gamma frac 1 2 left 1 sqrt 5 right denote the golden ratio h davenport and w m schmidt showed in 1969 that for each non quadratic irrational real number there exists a constant c 0 with the property that for arbitrarily large values of x the inequalities left x 0 right leqslant x left x 0 xi x 1 right leqslant c x 1 left gamma right text and left x 0 xi 2 x 2 right leqslant c x 1 left gamma right admit no non zero solution left x 0 x 1 x 2 right in mathbb z 3 their result is best possible in the sense that conversely there are countably many non quadratic irrational real numbers such that for a larger value of c the same inequalities admit a non zero integer solution for each x 1 such extremal numbers are transcendental and their set is stable under the action of text g text l 2 left mathbb z right on mathbb r backslash mathbb q by linear fractional transformations in this paper it is shown that there exist extremal numbers for which the lagrange constant liminfq q q is frac 1 3 the largest possible value for a non quadratic number and that there is a natural bijection between the text g text l 2 left mathbb z right equivalence classes of such numbers and the non trivial solutions of markoff s equation

an adaptive coupling method for exterior anisotropic elliptic problems in this paper we propose an adaptive coupling method for solving anisotropic elliptic pdes in unbounded domains firstly the existence and the uniqueness of the solution for the coupling method are proven and the a priori error estimates in h1 norm and l2 norm that depend on the size of the fem mesh the location of the elliptic artificial boundary and the truncation of the infinite series in the artificial boundary integral condition are derived secondly the a posteriori error estimates and the a posteriori error indicator of the coupling method are obtained finally the adaptive coupling method refines the mesh distribution by the arc length equidistribution principle and the a posteriori error indicator successively numerical examples confirm the advantage in accuracy and efficiency for the proposed method