the isometry group of outer space abstract we prove analogues of royden s theorem for the lipschitz metrics of outer space namely that isom cv n out f n

forced lattice vibrations part ii this is the first part of a two part series on forced lattice vibrations in which a semi infinite lattice of one dimensional particles xn n 1 is driven from one end by a particle x0 this particle undergoes a given periodically perturbed uniform motion x0 t at h yt where a and are constants and h has period 2 for a wide variety of restoring forces f i e f 0 numerical calculations indicate the existence of a sequence of thresholds 1 1 a h f 2 2 a h f k k a h f k 0 as k if k k 1 a k phase wave that is well described by the wave form emerges and travels through the lattice the goal of this series is to describe the emergence and calculate some properties of these wave forms in part i the authors first consider the case where f x ex i e toda forces but h is arbitrary and show how to compute a basic diagnostic see j formula 1 26 for the system in terms of the solution of an associated scalar riemann hilbert problem once a certain finite set of numbers is known in another direction the authors consider the case where f x is restoring but arbitrary and h is small here the authors prove a general result asserting that if there exists a sufficiently ample family of traveling wave solutions of the doubly infinite lattice then it is possible to construct time periodic k phase wave solutions with asymptotics in n of type iii for the driven system i in part ii the authors prove that sufficiently ample families of traveling wave solutions of the system iv exist in the cases 1 and 1 2 for general restoring forces f in the case with toda forces f x ex the authors prove that sufficiently ample families of traveling wave solutions

on hamiltonian perturbations of hyperbolic systems of conservation laws ii universality of critical behaviour hamiltonian perturbations of the simplest hyperbolic equation u t a u u x 0 are studied we argue that the behaviour of solutions to the perturbed equation near the point of gradient catastrophe of the unperturbed one should be essentially independent on the choice of generic perturbation neither on the choice of generic solution moreover this behaviour is described by a special solution to an integrable fourth order ode

why nonconservative schemes converge to wrong solutions error analysis this paper attempts to give a qualitative and quailtitative description of the numerical error introduced by using finite difference schemes in nonconservative form for scalar conservation laws we show that these schemes converge strongly in li norm to the solution of an inhomogeneous conservation law containing a borel measure source term moreover we analyze the properties of this borel measure and derive a sharp estimate for the l1 error between the limit function given by the scheme and the correct solution in general the measure source term is of the order of the entropy dissipation measure associated with the scheme in certain cases the error can be small for short times which makes it difficult to detect numerically but generically such an error will grow in time and this would lead to a large error for large time calculations finally we show that a local correction of any high order accurate scheme in nonconservative form is sufficient to ensure its convergence to the correct solution

the semiclassical limit of the defocusing nls hierarchy we establish the semiclassical limit of the one dimensional defocusing cubic nonlinear schrodinger nls equation complete integrability is exploited to obtain a global characterization of the weak limits of the entire nls hierarchy of conserved densities as the field evolves from reflectionless initial data under all the associated commuting flows consequently this also establishes the zero dispersion limit of the modified korteweg de vries equation that resides in that hierarchy we have adapted and clarified the strategy introduced by lax and levermore to study the zero dispersion limit of the korteweg de vries equation expanding it to treat entire integrable hierarchies and strengthening the limits obtained a crucial role is played by the convexity of the underlying log determinant with respect to the times associated with the commuting flows c 1999 john wiley sons inc

parageometric outer automorphisms of free groups we study those fully irreducible outer automorphisms o of a finite rank free group f r which are parageometric meaning that the attracting fixed point of o in the boundary of outer space is a geometric r tree with respect to the action of f r but o itself is not a geometric outer automorphism in that it is not represented by a homeomorphism of a surface our main result shows that the expansion factor of o is strictly larger than the expansion factor of o 1 as corollaries proved independently by guirardel the inverse of a parageometric outer automorphism is neither geometric nor parageometric and a fully irreducible outer automorphism o is geometric if and only if its attracting and repelling fixed points in the boundary of outer space are geometric r trees

contracting orbits in outer space we show that strongly contracting geodesics in outer space project to parameterized quasigeodesics in the free factor complex this result provides a converse to a theorem of bestvina feighn and is used to give conditions for when a subgroup of mathrm out mathbb f has a quasi isometric orbit map into the free factor complex it also allows one to construct many new examples of strongly contracting geodesics in outer space

walks on groups counting reducible matrices polynomials and surface and free group automorphisms we prove sharp limit theorems on random walks on graphs with values in finite groups we then apply these re sults together with some elementary algebraic geometry number theory and representation theory to finite quotients of lattices in semisimple lie groups specifically sl n z and sp 2n z to show that a random element in one of these lattices has irre ducible characteristic polynomials overz the term random can be defined in at least two ways in terms of height and also in terms of word length in terms of a generating set we show the result using both definitions we use the above results to show that a random in terms of word length element of the mapping class group of a surface is pseudo anosov and that a a random free group automorphism is irreducible with irreducible powers or strongly irreducible 1

new proofs of some results of nielsen given a compact oriented surface m with negative euler characteristic and a homeomorphism z m2 m2 what is the best representative cp m2 m2 in the isotopy class of s thurston s answered this question by defining pseudo anosov homeomorphisms a generalization of anosov diffeomorphisms on t2 which may be characterized as follows the homeomorphism rp is pseudo anosov iff there is a pair of transverse measured geodesic laminations i and au which intersect each closed nonperipheral geodesic in h4 which are topologically preserved by cp and which have their respective transverse measures multiplied under the action of cp by l n and d for some l 1 thurston showed that cp m m2 can be pieced together from homeomorphisms of subsurfaces which are either periodic or pseudoanosov more precisely thurston proved theorem 0 1 the preprint s provides an introduction to this theorem describing many of the useful properties of pseudo anosov diffeomorphisms the proof outlined in s is contained in 111

binary and ternary oscillations in a cubic numerical scheme abstract we study a central difference semi discretization of the cubic scalar conservation law both spatial period 2 binary and period 3 ternary oscillations are stationary solutions of this scheme and we find where each type is linearly stable we observe numerically the formation of ternary oscillations to the left of riemann shock initial data with ur 0 while binary oscillations form to the right of riemann rarefaction data having ul 0 we derive modulation equations for both wave patterns using them to show that binary oscillations generated by the scheme are numerical artifacts while computing an explicit solution for the ternary modulation equations for positive initial data the ternary modulation equations remain hyperbolic while the solutions enter an elliptic region for data of both signs conditions under which solutions of the ternary modulation equations can be inverted to yield period 3 oscillations are also discussed

existence and modulation of traveling waves in particles chains we consider an infinite particle chain whose dynamics are governed by the following system of differential equations where qn t is the displacement of the nth particle at time t along the chain axis and denotes differentiation with respect to time we assume that all particles have unit mass and that the interaction potential v between adjacent particles is a convex c function for this system we prove the existence of c time periodic traveling wave solutions of the form qn t q wt kn where q is a periodic function q z q z 1 the period is normalized to equal 1 and k are respectively the frequency and the wave number is the mean particle spacing and can be chosen to be an arbitrary parameter we present two proofs one based on a variational principle and the other on topological methods in particular degree theory for small amplitude waves based on perturbation techniques we describe the form of the traveling waves and we derive the weakly nonlinear dispersion relation for the fully nonlinear case when the amplitude of the waves is high we use numerical methods to compute the traveling wave solution and the non linear dispersion relation we finally apply whitham s method of averaged lagrangian to derive the modulation equations for the wave parameters k and 1999 john wiley sons inc

the analysis of categorical data from complex sample surveys chi squared tests for goodness of fit and independence in two way tables abstract the effect of stratification and clustering on the asymptotic distributions of standard pearson chi squared test statistics for goodness of fit simple hypothesis and independence in a two way contingency table denoted as x 2 and xi 2 respectively is investigated it is shown that both x 2 and xi 2 are asymptotically distributed as weighted sums of independent 1 2 random variables the weights are then related to the familiar design effects deffs used by survey samplers a simple correction to x 2 which requires only the knowledge of variance estimates or deffs for individual cells in the goodness of fit problem is proposed and empirical results on the performance of corrected x 2 provided empirical work on xi 2 indicated that the distortion of nominal significance level is substantially smaller with xi 2 than with x 2 some results under simple models for clustering are also given

the automorphism group of a free group is not linear a linear group has ascending chain condition on centralizers malcev s p 511 finitely generated linear groups are residually finite malcev s p 453 a solvable linear group is nilpotent by abelian byfinite tits 5 pp 14551461 a linear group either contains a free group of rank two or is solvable by locally finite lubotzky 3 has characterized finitely generated linear groups over a field of characteristic zero by purely group theoretic conditions these conditions appear difficult to check although they have been used to show linearity for certain groups however we do not use his theorem the main result of this paper theorem 5 is that for n 2 3 the automorphism group of a free group of rank n is not a linear group the proof uses the representation theory of algebraic groups to show that a kind of diophantine equation between the irreducible representations of

the small dispersion limit of the korteweg de vries equation ii in part i the scattering transform method is used to study the weak limit of solutions to the initial value problem for the korteweg devries kdv equation as the dispersion tends to zero in that limit the associated schrodinger operator becomes semiclassical so the exact scattering data is replaced by its corresponding wkb expressions only nonpositive initial data are considered in that case the limiting reflection coefficient vanishes the explicit solution of kay and moses for the reflectionless inverse transform is then analyzed and the weak limit valid for all time is characterized by a quadratic minimum problem with constraints this minimum problem is reduced to a riemann hilbert problem in function theory in parts ii and iii we use function theoretical methods to solve the riemann hilbert problem in terms of solutions to an auxiliary initial value problem the weak limit satisfies the kdv equation with the dispersive term dropped until its derivatives become infinite up to that time the weak limit is a strong l2 limit at later times the weak limit is locally described by whitham s averaged equations or more generally by the equations found by flaschka et al using multiphase averaging for large times behavior of the weak limit is studied directly from the minimum problem

large oscillations arising in a dispersive numerical scheme abstract we study the oscillatory behavior that arises in solutions of a dispersive numerical scheme for the hopf equation whenever the classical solution of the equation develops a singularity modulation equations are derived that describe period two oscillations so long as the solution of those equations takes values for which the equations are hyperbolic however those equations have an elliptic region that may be entered by its solutions in a finite time after which the corresponding period two oscillations are seen to break down this kind of phenomenon has not been observed for integrable schemes the generation and propagation of period two oscillations are asymptotically analyzed and a matching formula is found for the transition between oscillatory and nonoscillatory regions modulation equations are also presented for period three oscillations numerical experiments are carried out that illustrate our analysis

automorphisms of free groups have finitely generated fixed point sets an automorphism of a finitely generated free group f extends to a homeomorphism of the end completion e of f the set of fixed points of this homeomorphism is finitely generated in a certain sense in particular this implies that the subgroup of elements fixed by the automorphism is finitely generated in the usual sense the emphasis on e instead of f in this paper is analogous to thurston s study of surface groups where measured laminations are studied instead of simple closed curves the techniques in this paper arose from a study of the behaviour of an automorphism under iteration that is the dynamics of the automorphism see 11 i thank bill thurston for explaining some of his ideas to me for analysing automorphisms of free groups in particular the result on bounded cancellation below is due to thurston and matt grayson peter scott asked whether the fixed subgroup is always finitely generated in 3 and gersten first answered this question in 2 using quite different methods

some metric properties of automorphisms of groups abstract study of the dynamics of automorphisms of a group is usually focused on their growth and or finite orbits including fixed points in this paper we introduce properties of a different kind using somewhat informal language we call them metric properties two principal characteristics of this kind are called here the curl and the flux there seems to be very little correlation between these and the growth of an automorphism which means they are likely to be an essentially new tool for studying automorphisms we also observe that our definitions of the curl and flux are sufficiently general to be applied to mappings of arbitrary metric spaces

automorphisms of free groups have asymptotically periodic dynamics we show that every automorphism alpha of a free group f k of finite rank k has it asymptotically periodic dynamics on f k and its boundary partial f k there exists a positive power alpha q such that every element of the compactum f k cup partial f k converges to a fixed point under iteration of alpha q

dispersive approximations in fluid dynamics we have compared the results of a von neumann richtmyer calculation with the weak limit of calculations performed by von neumann s original method although there are discernible differences they are surprisingly small we analyze von newmann s algorithm for solving the initial value problem for the lagrangian equations of compressible flow

wavelet methods to eliminate resonances in the galerkin truncated burgers and euler equations it is well known that solutions to the fourier galerkin truncation of the inviscid burgers equation and other hyperbolic conservation laws do not converge to the physically relevant entropy solution after the formation of the first shock this loss of convergence was recently studied in detail ray et al phys rev e 84 016301 2011 and traced back to the appearance of a spatially localized resonance phenomenon perturbing the solution in this work we propose a way to remove this resonance by filtering a wavelet representation of the galerkin truncated equations a method previously developed with a complex valued wavelet frame is applied and expanded to embrace the use of real valued orthogonal wavelet bases which we show to yield satisfactory results only under the condition of adding a safety zone in wavelet space we also apply the complex valued wavelet based method to the two dimensional euler equation problem showing that it is able to filter the resonances in this case as well

operator splitting methods for generalized korteweg de vries equations we apply the method of operator splitting on the generalized korteweg de vries kdv equation ut f u x uxxx 0 by solving the nonlinear conservation law ut f u x 0 and the linear dispersive equation ut uxxx 0 sequentially we prove that if the approximation obtained by operator splitting converges then the limit function is a weak solution of the generalized kdv equation convergence properties are analyzed numerically by studying the effect of combining different numerical methods for each of the simplified problems

stable representatives for symmetric automorphisms of groups and the general form of the scott conjecture let g be a group acting on a tree x such that all edge stabilizers are finite we extend bestvina handel s theory of train tracks for automorphisms of free groups to automorphisms of g which permute vertex stabilizers using this extension we show that there is an upper bound depending only on g for the complexity of the graph of groups decomposition of the fixed subgroups of such automorphisms of g

an analysis of density dependent viability selection abstract generalized linear modeling techniques and resistant fitting methods were used to analyze the results of two experiments to test for density dependent viability selection on chromosomal variants in drosophila pseudoobscura fruit flies in the framework of the generalized linear model a series of nested hypotheses were fitted by the maximum likelihood method the hypotheses describe how the viabilities of two genotypes relative to a third varied with density and other factors an additional analysis of the absolute viabilities of the three genotypes was also performed by the maximum likelihood method estimates were computed by the em algorithm resistant methods of fitting were then performed for the best fitting relative viability models the analyses confirm that the relative viability of one genotype increased linearly with density with slope and intercept viability differences between genotypes can thus be explained by variation in density

on a nonlinear hyperbolic variational equation ii the zero viscosity and dispersion limits we consider viscosity and dispersion regularizations of the nonlinear hyperbolic partial differential equation u t uux x 1 2u x 2 with the simplest initial data such that u x blows up in finite time we prove that the zero viscosity limit selects a unique global weak solution of the partial differential equation without viscosity we also present numerical experiments which indicate that the zero dispersion limit selects a different global weak solution of the same initial value problem

resonance phenomenon for the galerkin truncated burgers and euler equations it is shown that the solutions of inviscid hydrodynamical equations with suppression of all spatial fourier modes having wave numbers in excess of a threshold k mathrm g exhibit unexpected features the study is carried out for both the one dimensional burgers equation and the two dimensional incompressible euler equation for large k mathrm g and smooth initial conditions the first symptom of truncation a localized short wavelength oscillation which we call a tyger is caused by a resonant interaction between fluid particle motion and truncation waves generated by small scale features shocks layers with strong vorticity gradients etc these tygers appear when complex space singularities come within one galerkin wavelength ensuremath lambda mathrm g 2 ensuremath pi k mathrm g from the real domain and typically arise far away from preexisting small scale structures at locations whose velocities match that of such structures tygers are weak and strongly localized at first in the burgers case at the time of appearance of the first shock their amplitudes and widths are proportional to k mathrm g ensuremath 2 3 and k mathrm g ensuremath 1 3 respectively but grow and eventually invade the whole flow they are thus the first manifestations of the thermalization predicted by t d lee q j appl math 10 69 1952 the sudden dissipative anomaly the presence of a finite dissipation in the limit of vanishing viscosity after a finite time t ensuremath star which is well known for the burgers equation and sometimes conjectured for the three dimensional euler equation has as counterpart in the truncated case the ability of tygers to store a finite amount of energy in the limit k mathrm g ensuremath rightarrow ensuremath infty this leads to reynolds stresses acting on scales larger than the galerkin wavelength and thus prevents the flow from converging to the inviscid limit solution there are indications that it may eventually be possible to purge the tygers and thereby to recover the correct inviscid limit behavior

cubulating hyperbolic free by cyclic groups the irreducible case let v be a nite graph and let v v be an irreducible train track map whose mapping torus has word hyperbolic fundamental group g then g acts freely and cocompactly on a cat 0 cube complex hence if f is a nite rank free group and f f an irreducible monomorphism so that g f is word hyperbolic then g acts freely and cocompactly on a cat 0 cube complex this holds in particular if is an irreducible automorphism with g f o z word hyperbolic

mapping class groups and outer automorphism groups of free groups are c simple abstract we prove that the reduced c algebras of centerless mapping class groups and outer automorphism groups of free groups are simple as are the irreducible pure subgroups of mapping class groups and the analogous subgroups of outer automorphism groups of free groups

binary modulated oscillations in a semi discrete version of burgers equation abstract in this paper we study both analytically and numerically the binary modulated oscillations which arise in an integrable semi discrete approximation to inviscid burgers explicit solutions of the binary modulation equations derived in goodman and lax 1988 are found for several choices of initial data observations based on numerical experiments are presented as to which initial data lead to solutions containing purely binary oscillations comparisons are made to the integrable theory of the kac van moerbeke and toda lattices non integrable schemes for which the binary oscillations also remain hyperbolic are presented and explicit solutions of their binary modulation equations are found

topology of finite graphs this paper derives from a course in group theory which i gave at berkeley in 1982 i wanted to prove the standard theorems on free groups and discovered that after a few preliminaries the notion of locally injective map or immersion of graphs was very useful this enables one to see in an effective easy algorithmic way just what happens with finitely generated free groups one can understand in this way 1 howson s theorem that if a and b are finitely generated subgroups of a free group then a b is finitely generated and 2 m hall s theorem that finitely generated subgroups of free groups are closed in the profinite topology during this course s m gersten came up with a simple proof of h neumann s inequality on the ranks in howson s theorem one of the ideas in gersten s proof was to use core graphs graphs with no trees hanging on subsequently i found that some consequences of a paper of greenberg s could be proved using core graphs and covering translations of immersions the most striking such result is that if a and b are finitely generated subgroups of a free group and if a c b is of finite index in both a and b then a b is of finite index in a v b the subgroup generated by a ub

fixed points of automorphisms of free groups abstract the set of fixed points of an automorphism of a finitely generated free group is a finitely generated group settling a conjecture of g p scott s

detecting automorphic orbits in free groups abstract we present an effective algorithm for detecting automorphic orbits in free groups as well as a number of algorithmic improvements of train tracks for free group automorphisms

the zero dispersion limit a deterministic analogue of turbulence weak limits of oscillatory sequences of dispersive compressible flows show a remarkable resemblance to the ensemble averages of classical turbulence theory how much relevance such ideas have for incompressible flows is not clear

dispersive shock waves and modulation theory there is growing physical and mathematical interest in the hydrodynamics of dissipationless dispersive media since g b whitham s seminal publication fifty years ago that ushered in the mathematical study of dispersive hydrodynamics there has been a significant body of work in this area however there has been no comprehensive survey of the field of dispersive hydrodynamics utilizing whitham s averaging theory as the primary mathematical tool we review the rich mathematical developments over the past fifty years with an emphasis on physical applications the fundamental large scale coherent excitation in dispersive hydrodynamic systems is an expanding oscillatory dispersive shock wave or dsw both the macroscopic and microscopic properties of dsws are analyzed in detail within the context of the universal integrable and foundational models for uni directional korteweg de vries equation and bi directional nonlinear schrodinger equation dispersive hydrodynamics a dsw fitting procedure that does not rely upon integrable structure yet reveals important macroscopic dsw properties is described dsw theory is then applied to a number of physical applications superfluids nonlinear optics geophysics and fluid dynamics finally we survey some of the more recent developments including non classical dsws dsw interactions dsws in perturbed and inhomogeneous environments and two dimensional oblique dsws

on the dynamics and the fixed subgroup of a free group automorphism on considere des automorphismes f n f n ou f n f a 1 a n est le groupe libre de base a 1 a 2 a n n 2 on compose un champ vectoriel sur un graphe d on introduit des aretes repulsives attractives et des points fixes a l infini et on etudie la dynamique resultante

on the cubical geometry of higman s group we investigate the cocompact action of higman s group on a cat 0 square complex associated to its standard presentation we show that this action is in a sense intrinsic which allows for the use of geometric techniques to study the endomorphisms of the group and show striking similarities with mapping class groups of hyperbolic surfaces outer automorphism groups of free groups and linear groups over the integers we compute explicitly the automorphism group and outer automorphism group of higman s group and show that the group is both hop an and co hop an we actually prove a stronger rigidity result about the endomorphisms of higman s group every non trivial morphism from the group to itself is an automorphism we also study the geometry of the action and prove a surprising result although the cat 0 square complex acted upon contains uncountably many ats the higman group does not contain subgroups isomorphic to z finally we show that this action possesses features reminiscent of negative curvature which we use to prove a re ned version of the tits alternative for higman s group higman s group was constructed in 15 as the rst example of a nitely presented in nite group without non trivial nite quotients it is given by the following presentation h ai i z 4z aiai 1a i a 2 i 1 i z 4z if some partial results about higman s group are known the structure of the group and its geometry have remained mostly elusive the group is known to split as an amalgamated product over several non abelian free subgroups and such splittings can be used to derive various properties of the group for instance the group has nite asymptotic dimension by work of bell dranishnikov 1 a property that implies the novikov conjecture and the coarse baum connes conjecture for the group 36 37 such splittings have also been used to show that the higman group admits in a sense many quotients schupp developed small cancellation tools to show that the group is sq universal 30 that is every countable group embeds in a quotient of h minasyan osin used these splittings to show that the group is 2010 mathematics subject classi cation primary 20f65 secondary 20f28

resistant fits for some commonly used logistic models with medical application logistic regression type models are used in many applications some examples include the classical dose response experiment prospective and retrospective studies of disease incidence with and without matching and the analysis of ordinal data in most instances the model is fitted by the method of maximum likelihood which like least squares is sensitive to atypical observations an alternative to maximum likelihood is proposed and illustrated by examples

on dispersive difference schemes i etude du comportement oscillatoire et de la convergence faible des solutions des approximations aux differences dispersives de u t 1 2 u 2 x 0

non classical shocks and kinetic relations scalar conservation laws this paper analyzes the non classical shock waves which arise as limits of certain diffusive dispersive approximations to hyperbolic conservation laws such shocks occur for non convex fluxes and connect regions of different convexity they have negative entropy dissipation for a single convex entropy function but not all convex entropies and do not obey the classical oleinik entropy criterion we derive necessary conditions for the existence of non classical shock waves and construct them as limits of traveling wave solutions for several diffusive dispersive approximations

tits alternatives for graph products we discuss various types of tits alternative for subgroups of graph products of groups and prove that under some natural conditions a graph product of groups satisfies a given form of tits alternative if and only if each vertex group satisfies this alternative as a corollary we show that every finitely generated subgroup of a graph product of virtually solvable groups is either virtually solvable or large as another corollary we prove that every non abelian subgroup of a right angled artin group has an epimorphism onto the free group of rank 2 in the course of the paper we develop the theory of parabolic subgroups which allows to describe the structure of subgroups of graph products that contain no non abelian free subgroups we also obtain a number of results regarding the stability of some group properties under taking graph products

the expansion factors of an outer automorphism and its inverse a fully irreducible outer automorphism o of the free group f n of rank n has an expansion factor which often differs from the expansion factor of o 1 nevertheless we prove that the ratio between the logarithms of the expansion factors of o and o 1 is bounded above by a constant depending only on the rank n we also prove a more general theorem applying to an arbitrary outer automorphism of f n and its inverse and their two spectrums of expansion factors