caustics for inner and outer billiards with a plane closed convex curve t we associate two area preserving twist maps the classical inner billiard int and the outer billiard in the exterior oft the invariant circles of these twist maps correspond to certain plane curves the inner and the outer caustics oft we investigate how the shape oft determines the possible location of caustics establish the existence of open regions which are free of caustics and estimate fro below the size of these regions in terms of the geometry oft

multiple testing of general contrasts powerful improvements are possible for multiple testing procedures when the hypotheses are logically related closed testing with exhaustive tests provides a unifying framework for developing such procedures but can be computationally difficult and can be nonmonotonic in p values royen introduced a truncated closed testing method for the case of all pairwise comparisons in the analysis of variance that is monotonic in p values shaffer developed a similar truncated procedure for more general comparisons but using bonferroni tests rather than exhaustive tests and westfall extended shaffer s method to allow exhaustive tests this article extends royen s method to general contrasts and proves that it is equivalent to the extended shaffer procedure for k contrasts the method generally requires evaluation of o 2k critical values that correspond to subset intersection hypotheses and is computationally infeasible for large k the set of intersections is represented using a tree structure and a

blind source separation in the time frequency domain based on multiple hypothesis testing this paper considers a time frequency t f based approach for blind separation of nonstationary signals in particular we propose a time frequency point selection algorithm based on multiple hypothesis testing which allows automatic selection of auto or cross source locations in the time frequency plane the selected t f points are then used via a joint diagonalization and off diagonalization algorithm to perform source separation the proposed algorithm is developed assuming deterministic signals with additive white complex gaussian noise a performance comparison of the proposed and existing approaches is provided

finite time transport in aperiodic flows abstract we study the transport of particles in a general two dimensional incompressible flow in the presence of a transient eddy i e a bounded set of closed streamlines with a finite time of existence using quantities obtained from eulerian observations we provide explicit conditions for the existence of a hyperbolic structure in the flow which induces mixing between the eddy and its environment our results can be used directly to study finite time transport in numerically or experimentally generated vector fields with general time dependence

an integrable deformation of an ellipse of small eccentricity is an ellipse the classical birkho conjecture claims that the boundary of a strictly convex integrable billiard table is necessarily an ellipse or a circle as a special case in this article we show that a version of this conjecture is true for tables bounded by small perturbations of ellipses of small eccentricity

a sharper bonferroni procedure for multiple tests of significance summary a simple procedure for multiple tests of significance based on individual p values is derived this simple procedure is sharper than holm s 1979 sequentially rejective procedure both procedures contrast the ordered p values with the same set of critical values holm s procedure rejects an hypothesis only if its p value and each of the smaller p values are less than their corresponding critical values the new procedure rejects all hypotheses with smaller or equal p values to that of any one found less than its critical value

break up of resonant invariant curves in billiards and dual billiards associated to perturbed circular tables two area preserving twist maps are associated to a smooth closed convex table the classical billiard map and the dual billiard map when the table is circular these maps are integrable and their phase spaces are foliated by invariant curves the invariant curves with rational rotation numbers are resonant and do not persist under generic perturbations of the circle we present a sufficient condition for the break up of these curves this condition is expressed directly in terms of the fourier coefficients of the perturbation it follows from a standard melnikov argument

distinguished material surfaces and coherent structures in three dimensional fluid flows we prove analytic criteria for the existence of finite time attracting and repelling material surfaces and lines in three dimensional unsteady flows the longest lived such structures define coherent structures in a lagrangian sense our existence criteria involve the invariants of the velocity gradient tensor along fluid trajectories an alternative approach to coherent structures is shown to lead to their characterization as local maximizers of the largest finite time lyapunov exponent field computed directly from particle paths both approaches provide effective tools for extracting distinguished lagrangian structures from three dimensional velocity data we illustrate the results on steady and unsteady abc type flows

exponentially small separatrix splittings and almost invisible homoclinic bifurcations in some billiard tables abstract we present a numerical study of some billiard tables depending on a perturbative parameter 0 and a hyperbolicity parameter h 0 these tables are ellipses for 0 and circumferences in the limit h 0 elliptic billiard tables are integrable and have four separatrices which break up when 0 we conjecture based on numerical experiments that as h 0 the area of the main lobes of the resulting turnstile which can be interpreted as the difference of the lengths of the symmetric primary homoclinic billiard trajectories behaves like an exponential term e 2 h times an asymptotic series j 0 j h 2 j such that 0 0 this series is gevrey 1 of type 1 2 2 so that its borel transform is convergent on a disk of radius 2 2 in the limit 0 the series j 0 j 0 h 2 j is an analytic function which can be explicitly computed with a discrete melnikov method the asymptotic series j 0 j h 2 j associated to the second exponential term e 2 2 h has the same properties finally we have detected some almost invisible homoclinic bifurcations that take place in an exponentially small region of the parameter space our computations have been performed in multiple precision arithmetic namely with several thousands decimal digits and rely strongly on the expansion of the local invariant curves up to very high orders namely with several hundreds taylor coefficients our programs have been written using the pari system

total positivity properties of absolute value multinormal variables with applications to confidence interval estimates and related probabilistic inequalities total positivity properties of multivariate densities are useful in deducing positive dependence of random vector components and related probability inequalities in this paper we determine necessary and sufficient conditions for total positivity of absolute value multinormal variables the results are applied to obtain positive dependence and associated inequalities for the multinormal and related distributions e g the multivariate t and wishart distributions inequalities of this type yield bounds for multivariate confidence set probabilities

the break up of a heteroclinic connection in a volume preserving mapping abstract we consider a family of three dimensional volume preserving maps depending on a small parameter e as e 0 these maps asymptote to flows which attain a heteroclinic connection we show that for small e the heteroclinic connection breaks up and that the splitting between its components scales with e like e exp e we estimate using the singularities of the e 0 heteroclinic orbit in the complex plane we then estimate using linearization about orbits in the complex plane while these estimates are not proven they are well supported by our numerical calculations the work described here is a special case of the theory derived by amick et al which applies to q dimensional volume preserving mappings

existence of invariant tori in three dimensional maps with degeneracy abstract we prove a kam type result for the persistence of two dimensional invariant tori in perturbations of integrable action angle angle maps with degeneracy satisfying the intersection property such degenerate action angle angle maps arise upon generic perturbation of three dimensional volume preserving vector fields which are invariant under volume preserving action of s 1 when there is no motion in the group action direction for the unperturbed map this situation is analogous to degeneracy in hamiltonian systems the degenerate nature of the map and the unequal number of action and angle variables make the persistence proof non standard the persistence of the invariant tori as predicted by our result has implications for the existence of barriers to transport in three dimensional incompressible fluid flows simulation results indicating existence of two dimensional tori in a perturbation of swirling hill s spherical vortex flow are presented

modified sequentially rejective multiple test procedures abstract suppose that n hypotheses h 1 h 2 h n with associated test statistics t 1 t 2 t n are to be tested by a procedure with experimentwise significance level the probability of rejecting one or more true hypotheses smaller than or equal to some specified value a commonly used procedure satisfying this condition is the bonferroni b procedure which consists of rejecting h i for any i iff the associated test statistic t i is significant at the level n holm 1979 introduced a modified bonferroni procedure with greater power than the b procedure under holm s sequentially rejective bonferroni srb procedure if any hypothesis is rejected at the level n the denominator of for the next test is n 1 and the criterion continues to be modified in a stagewise manner with the denominator of reduced by 1 each time a hypothesis is rejected so that tests can be conducted at successively higher significance levels holm proved that the experimentwise significance level

an improved bonferroni procedure for multiple tests of significance summary a modification of the bonferroni procedure for testing multiple hypotheses is presented the method based on the ordered p values of the individual tests is less conservative than the classical bonferroni procedure but is still simple to apply a simulation study shows that the probability of a type i error of the procedure does not exceed the nominal significance level a for a variety of multivariate normal and multivariate gamma test statistics for independent tests the procedure has type i error probability equal to a the method appears particularly advantageous over the classical bonferroni procedure when several highly correlated test statistics are involved

passive scalars and three dimensional liouvillian maps abstract global aspects of the motion of passive scalars in time dependent incompressible fluid flows are well described by volume preserving liouvillian three dimensional maps in this paper the possible invariant structures in liouvillian maps and the two most interesting nearly integrible cases are investigated in addition the fundamental role of invariant lines in organizing the dynamics of this type of system is exposed bifurcations involving the destruction of some invariant lines and tubes and the creation of new ones are described in detail

an adaptive confirmatory trial with interim treatment selection practical experiences and unbalanced randomization this article has been motivated by an ongoing international adaptive confirmatory trial at the interim analysis of this two stage trial none one or two active treatment regimens are selected for further study in the second stage a combination test approach is used in this practical setting with an extension of the theory to unbalanced randomization we show that a combination test with suitable weights can still preserve the overall type i error rate provided that the test statistic is chosen appropriately and the unpooled z test for proportions is not used the accuracy of stagewise p values is also discussed in a more general framework monte carlo simulations confirm the validity of the approach retained and evaluate the necessary sample size additional issues addressed during the design of the trial are also examined such as multiplicity due to testing hypotheses on key secondary endpoints a non inferiority comparison to an active treatment and covariate adjusted analyses for various types of outcome

a stagewise rejective multiple test procedure based on a modified bonferroni test summary simes 1986 has proposed a modified bonferroni procedure for the test of an overall hypothesis which is the combination of n individual hypotheses in contrast to the classical bonferroni procedure it is not obvious how statements about individual hypotheses are to be made for this procedure in the present paper a multiple test procedure allowing statements on individual hypotheses is proposed it is based on the principle of closed test procedures marcus peritz gabriel 1976 and controls the multiple level a

barriers to transport and mixing in volume preserving maps with nonzero flux abstract we identify some geometric structures secondary tori that restrict transport and prevent mixing in perturbations of integrable volume preserving systems with nonzero net flux unlike the customary kam tori secondary tori cannot be continued to the tori present in the integrable system but are generated by resonances and have a contractible direction we also note that secondary tori persist under the addition of a net flux which destroys all customary kam tori we introduce a remarkably simple algorithm to analyze the behavior of volume preserving maps and to obtain quantitative properties of the secondary tori we then implement the algorithm and after running it present assertions regarding the distribution of the escape times of the unbounded orbits the abundance of secondary tori the size of the resonant regions and the robustness of the tori under the addition of a mean flux

non integrability and continuation of fixed points of 2 n dimensional perturbed twist maps abstract in this paper a simple criterion to prove non integrability of symplectic perturbed twist mappings in 2 n dimensions is developed for sufficiently small perturbations in addition an upper bound for the number of isolating integrals the system can possess is provided a criterion for the analytic continuation of isolated periodic orbits in case of a small nonzero perturbation of twist maps is found the evaluation of their linear stability character by obtaining a simplified expression of the eigenvalues of the jacobian matrix concludes the theoretical part the theory is finally applied to a twist map perturbed by the suris potential and also a four dimensional map with physical interest

some probability inequalities for ordered rm mtp sb 2 random variables a proof of the simes conjecture some new probability inequalities involving the ordered components of an mtp 2 random vector are derived which provide an analytical proof of an important conjecture in the field of multiple hypothesis testing this conjecture has been mostly validated so far using simulation

geometry of the ergodic quotient reveals coherent structures in flows abstract dynamical systems that exhibit diverse behaviors can rarely be completely understood using a single approach however by identifying coherent structures in their state spaces i e regions of uniform and simpler behavior we could hope to study each of the structures separately and then form the understanding of the system as a whole the method we present in this paper uses trajectory averages of scalar functions on the state space to a identify invariant sets in the state space and b to form coherent structures by aggregating invariant sets that are similar across multiple spatial scales first we construct the ergodic quotient the object obtained by mapping trajectories to the space of the trajectory averages of a function basis on the state space second we endow the ergodic quotient with a metric structure that successfully captures how similar the invariant sets are in the state space finally we parametrize the ergodic quotient using intrinsic diffusion modes on it by segmenting the ergodic quotient based on the diffusion modes we extract coherent features in the state space of the dynamical system the algorithm is validated by analyzing the arnold beltrami childress flow which was the test bed for alternative approaches the ulam s approximation of the transfer operator and the computation of lagrangian coherent structures furthermore we explain how the method extends the poincare map analysis for periodic flows as a demonstration we apply the method to a periodically driven three dimensional hill s vortex flow discovering unknown coherent structures in its state space finally we discuss differences between the ergodic quotient and alternatives propose a generalization to analysis of quasi periodic structures and lay out future research directions

on weighted hochberg procedures summary we consider different ways of constructing weighted hochberg type step up multiple test procedures including closed procedures based on weighted simes tests and their conservative step up short cuts and step up counterparts of two weighted holm procedures it is shown that the step up counterparts have some serious pitfalls such as lack of familywise error rate control and lack of monotonicity in rejection decisions in terms of p values therefore an exact closed procedure appears to be the best alternative its only drawback being lack of simple stepwise structure a conservative step up short cut to the closed procedure may be used instead but with accompanying loss of power simulations are used to study the familywise error rate and power properties of the competing procedures for independent and correlated p values although many of the results of this paper are negative they are useful in highlighting the need for caution when procedures with similar pitfalls may be used

invariant geometric properties of a class of 3d chaotic flows this article extends the analysis developed by giona and adrover m giona a adrover phys rev lett 81 1998 3864 for 2d area preserving diffeomorphisms to 3d volume preservingc 1 diffeomorphisms of the 3d torus topologically conjugate to a linear map the article analyzes the invariant geometric properties of vector dynamics and surface element evolution in 3d systems and provides an analytic expression for the probability measure describing pointwise statistical properties of the unstable foliations in the hyperbolic case the convergence properties of this measure are addressed starting from the dynamics of surface elements the application of the methods developed to physically realizable 3d chaotic flows such as abc flow is discussed in detail 2000 elsevier science b v all rights reserved

mixing in the stokes flow in a cylindrical container kinematic features of three dimensional mixing by advection of passive particles in time periodic flows are the primary subject of this study a classification of periodic points providing important information about the mixing properties of a flow is presented and the dynamics of the poincare map in the vicinity of periodic points is analysed for all identified types three examples of stokes flow in a finite cylindrical cavity with discontinuous periodic motion of its end walls are used to illustrate the determination of both periodic lines and isolated periodic points in the flow domain the stable and unstable manifolds of points on the periodic lines create two surfaces in the flow a numerical technique based on tracking of a material surface is presented to study the manifold surfaces and their intersections it is illustrated with numerical examples that flows with periodic lines possess only quasi two dimensional mechanisms of chaotic advection

familywise robustness criteria for multiple comparison procedures a criticism of multiple comparison procedures is that the family of inferences over which an error rate is controlled is often arbitrarily selected yet the conclusion may depend heavily on the choice of the family such ambiguity is most likely in large exploratory studies requiring numerous simultaneous inferences in ambiguous situations it is desirable that results of multiple comparison procedures depend little on the chosen family to assess this we propose several familywise robustness criteria to evaluate such procedures and we find some of their properties theoretically and by simulation procedures that control the false discovery rate seem to be familywise robust

greene s residue criterion for the breakup of invariant tori of volume preserving maps abstract invariant tori play a fundamental role in the dynamics of symplectic and volume preserving maps codimension one tori are particularly important as they form barriers to transport such tori foliate the phase space of integrable volume preserving maps with one action and d angles for the area preserving case greene s residue criterion is often used to predict the destruction of tori from the properties of nearby periodic orbits even though kam theory applies to the three dimensional case the robustness of tori in such systems is still poorly understood we study a three dimensional reversible volume preserving analogue of chirikov s standard map with one action and two angles we investigate the preservation and destruction of tori under perturbation by computing the residue of nearby periodic orbits we find tori with diophantine rotation vectors in the spiral mean cubic algebraic field the residue is used to generate the critical function of the map and find a candidate for the most robust torus

frictional granular mechanics a variational approach the mechanical properties of a cohesionless granular material are evaluated from grain scale simulations intergranular interactions including friction and sliding are modeled by a set of contact rules based on the theories of hertz mindlin and deresiewicz a computer generated three dimensional irregular pack of spherical grains is loaded by incremental displacement of its boundaries deformation is described by a sequence of static equilibrium configurations of the pack a variational approach is employed to find the equilibrium configurations by minimizing the total work against the intergranular loads effective elastic moduli are evaluated from the intergranular forces and the deformation of the pack good agreement between the computed and measured moduli achieved with no adjustment of material parameters establishes the physical soundness of the proposed model

2d universality of period doubling bifurcations in 3d conservative reversible mappings abstract infinite sequences of period doubling bifurcations are known to occur generically i e with codimension 1 not only in dissipative 1d systems but also in 2d conservative systems described by area preserving mappings in this paper we study a 3d volume preserving reversible mapping and show that it does possess period 2 m m 1 2 orbits with stability intervals whose length decreases rapidly with increasing m varying one parameter of the system we find that these orbits always bifurcate out of one another with the usual stability exchange and universal properties of period doubling sequences of 2d conservative maps this raises the interesting question whether these 3d reversible maps possess an analytic integral which would render them essentially 2 dimensional

a general multistage procedure for k out of n gatekeeping we generalize a multistage procedure for parallel gatekeeping to what we refer to as k out of n gatekeeping in which at least k out of n hypotheses 1 k n in a gatekeeper family must be rejected in order to test the hypotheses in the following family this gatekeeping restriction arises in certain types of clinical trials for example in rheumatoid arthritis trials it is required that efficacy be shown on at least three of the four primary endpoints we provide a unified theory of multistage procedures for arbitrary k with k 1 corresponding to parallel gatekeeping and k n to serial gatekeeping the theory provides an insight into the construction of truncated separable multistage procedures using the closure method explicit formulae for calculating the adjusted p values are given the proposed procedure is simpler to apply for this particular problem using a stepwise algorithm than the mixture procedure and the graphical procedure with memory using entangled graphs

passive scalars three dimensional volume preserving maps and chaos the dynamics of a medium sized particle passive scalar suspended in a general time periodic incompressible fluid flow can be described by three dimensional volume preserving maps in this paper these maps are studied in limiting cases in which some of the variables change very little in each iteration and others change quite a lot the former are called slow variables or actions the latter fast variables or angles the maps are classified by their number of actions for maps with only one action we find strong evidence for the existence of invariant surfaces that survive the nonlinear perturbation in a kam like way on the other hand for the two action case the motion is confined to invariant lines that break for arbitrary small size of the nonlinearity instead we find that adiabatic invariant surfaces emerge and typically intersect the resonance sheet of the fast motion at these intersections surfaces are locally broken and transitions from one to another can occur we call this process which is analogous to arnold diffusion singularity induced diffusion it is characteristic of two action maps in one action maps this diffusion is blocked by kam like surfaces

footprints of lagrangian flow structures in eulerian concentration distributions in periodic mixing flows abstract transport of passive tracers may be described through the spatio temporal evolution of eulerian concentration distributions or via the geometrical composition of the lagrangian flow structure the present study seeks to deepen insight into the connection between the eulerian and lagrangian perspectives by investigating the role of lagrangian coherent structures lcss in the eulerian concentration distributions in time periodic and spatially periodic mixing flows representation of the eulerian transport by the mapping method describing realistic transport problems by distribution matrices admits a generic analysis based on matrix and graph theory this reveals that lcss and the transport barriers that separate them leave a distinct footprint in the eigenmode spectrum of the distribution matrix and by proxy of the underlying eulerian transport operator transport barriers impart a block diagonal structure upon the mapping matrix where each block matrix a corresponds with a given lcs its kind is reflected in the spectrum of a higher order periodicity yields a distinct permutation within a the composition of the distribution matrix versus the lagrangian flow structure thus predicted is demonstrated by way of examples these findings increase fundamental understanding of transport phenomena and have great practical potential for e g flow and mixing control

fractal dimensions of fast dynamo magnetic fields abstract recent studies of the fast kinematic dynamo find that without diffusion the magnetic flux tends to concentrate on a fractal set and generally exhibits an arbitrarily fine scaled alternation in the orientation of the magnetic field the presence of small magnetic diffusivity in conjunction with this arbitrarily fine scaled alternation in the magnetic field orientation radically changes the small scale spatial structure of the field motivated by this consideration we study the multifractal dimension spectra associated with fast dynamo magnetic fields both with and without diffusion these spectra are shown to be different in the two cases in addition using a partition function technique we obtain results giving the dimension spectra in terms of the distribution of finite time lyapunov exponents numerical experiments illustrating and testing these results are presented

splitting of separatrices for standard and semistandard mappings abstract we investigate chirikov s standard mapping which is a model for the onset of nonintegrability in nonlinear hamiltonian systems the asymptotic formula when the deviation from an integrable system is small for the angle of intersection of separatrices at a homoclinic point is derived on the basis of approximation by the green percival semistandard mapping in the complex domain computed data on the splitting of separatrices for the semi standard map are presented from which we derive the pre exponential coefficient in the formula for splitting of separatrices in the standard mapping

strange eigenmodes and decay of variance in the mixing of diffusive tracers we prove the existence of asymptotic spatial patterns for diffusive tracers advected by unsteady velocity fields the asymptotic patterns arise from convergence to a time dependent inertial manifold in the underlying advection diffusion equation for time periodic velocity fields we find that the inertial manifold is spanned by a finite number of floquet solutions the strange eigenmodes observed first numerically by pierrehumbert these strange eigenmodes only admit a regular asymptotic expansion in the diffusivity if the velocity field is completely integrable

on homoclinic structure and numerically induced chaos for the nonlinear schro dinger equation it has recently been demonstrated that standard discretizations of the cubic nonlinear schrodinger nls equation may lead to spurious numerical behavior in particular the origins of numerically induced chaos and the loss of spatial symmetry are related to the homoclinic structure associated with the nls equation in this paper an analytic description of the homoclinic structure via soliton type solutions is provided and some consequences for numerical computations are demonstrated differences between an integrable discretization and standard discretizations are highlighted

truncated simes tests it is known that the one sided simes test controls the error rate if the underlying distribution is multivariate totally positive of order 2 mtp2 but not in general the two sided test also controls the error rate when the coordinate absolute value has an mtp2 distribution which holds more generally we prove mathematically that when the coordinate absolute value controls the error rate at level 2 then certain kinds of truncated simes tests also control the one sided error rate at level we also compare the closure of the truncated tests with the holms hochberg and hommel procedures in many scenarios when the test statistics are multivariate normal

regular and chaotic motion of fluid particles in a two dimensional fluid the authors study the chaotic behaviour exhibited by particles which move in a two dimensional fluid the connection of this lagrangian chaos with the velocity field behaviour is discussed both in the lorenz model and in truncated navier stokes equations they indicate a possible method for the onset of lagrangian chaos which seems to be rather generic lagrangian chaos appears when the eulerian equation passes from a steady solution to a periodic one via hopf bifurcation it is also shown that the transition to chaos for the velocity field eulerian chaos does not affect the particle motion properties in some typical cases

nonlinear elasticity and pressure dependent wave speeds in granular media following is an analysis of the small strain nonlinear elasticity of granular media near states of zero stress as it relates to the pressure dependent incremental linear elasticity and wave speeds the main object is elucidation of the p dependence of incremental elastic moduli on pressure p a dependence observed in numerous experiments but found to be at odds with the p scaling predicted by various micromechanical models based on hertzian contact after presenting a power law continuum model for small strain nonlinear elasticity the present work develops micromechanical models based on two alternative mechanisms for the anomalous pressure scaling namely 1 departures at the single contact level from the hertzian contact due to point like or conical asphericity 2 variation in the number density of hertzian contacts due to buckling of particle chains both mechanisms result in p pressure scaling at low pressure and both exhibit a high pressure transition to p scaling at a characteristic transition pressure p for assemblages of nearly equal spheres a non hertzian contact model for mechanism 1 and percolation type model for 2 yield estimates of p of the form p c 3 here c is a non dimensional coefficient depending only on granular contact geometry while 1 is a small parameter representing spherical imperfections and is an appropriate elastic modulus of the particles then with r representing particle radius and h a characteristic spherical tolerance or asperity height it is found that h r for mechanism 1 as opposed to h r for 2 limited data from the classic experiments of duffy mindlin on sphere assemblages tend to support mechanism 1 but more exhaustive experiments are called for in addition to the above analysis of reversible elastic effects a percolation model of inelastic shake down or consolidation is given it serves to describe how prolonged mechanical vibration leading to the replacement of point like or inactive contacts by stiffer hertz contacts may change the pressure scaling behaviour of particulate media the present analysis suggests that pressure dependence of elasticity may provide a useful means of characterizing the state of consolidation and stability of dense particulate media

existence of kam tori in degenerate hamiltonian systems abstract this article is devoted to the proof that invariant tori still exist in perturbed integrable hamiltonian systems with n degrees of freedom h p q n p p p q if the hypothesis of nondegeneracy is replaced by a much weaker one that the rank of the mapping n p can be any positive integer smaller than n provided the image of any open set under the mapping includes a cn 2 curved one dimensional submanifold

testing multivariate economic restrictions using quantiles the example of slutsky negative semidefiniteness abstract this paper is concerned with testing a core economic restriction negative semidefiniteness of the slutsky matrix we consider a system of nonseparable structural equations with infinite dimensional unobservables and employ quantile regression methods because they allow us to utilize the entire distribution of the data difficulties arise because the restriction involves several equations while the quantile is a univariate concept we establish that we may use quantiles of linear combinations of the dependent variable develop a new empirical process based test that applies kernel quantile estimators and investigate its finite and large sample behavior finally we apply all concepts to canadian microdata

hochberg s step up method cutting corners off holm s step down method holm s method and hochberg s method for multiple testing can be viewed as step down and step up versions of the bonferroni test we show that both are special cases of partition testing the difference is that while holm s method tests each partition hypothesis using the largest order statistic setting a critical value based on the bonferroni inequality hochberg s method tests each partition hypothesis using all the order statistics setting a series of critical values based on simes inequality geometrically hochberg s step up method cuts corners off the acceptance regions of holm s step down method by making assumptions on the joint distribution of the test statistics as can be expected partition testing making use of the joint distribution of the test statistics is more powerful than partition testing using probabilistic inequalities thus if the joint distribution of the test statistics is available through modelling for example we recommend partition step down testing setting exact critical values based on the joint distribution

an adaptive method for computing invariant manifolds in non autonomous three dimensional dynamical systems abstract we present a computational method for determining the geometry of a class of three dimensional invariant manifolds in non autonomous aperiodically time dependent dynamical systems the presented approach can be also applied to analyse the geometry of 3d invariant manifolds in three dimensional time dependent fluid flows the invariance property of such manifolds requires that at any fixed time they are given by surfaces in r 3 we focus on a class of manifolds whose instantaneous geometry is given by orientable surfaces embedded in r 3 the presented technique can be employed in particular to compute codimension one invariant stable and unstable manifolds of hyperbolic trajectories in 3d non autonomous dynamical systems which are crucial in the lagrangian transport analysis the same approach can also be used to determine evolution of an orientable material surface in a fluid flow these developments represent the first step towards a non trivial 3d extension of the so called lobe dynamics a geometric invariant manifold based framework which has been very successful in the analysis of lagrangian transport in unsteady two dimensional fluid flows in the developed algorithm the instantaneous geometry of an invariant manifold is represented by an adaptively evolving triangular mesh with piecewise c 2 interpolating functions the method employs an automatic mesh refinement which is coupled with adaptive vertex redistribution a variant of the advancing front technique is used for remeshing whenever necessary such an approach allows for computationally efficient determination of highly convoluted evolving geometry of codimension one invariant manifolds in unsteady three dimensional flows we show that the developed method is capable of providing detailed information on the evolving lagrangian flow structure in three dimensions over long periods of time which is crucial for a meaningful 3d transport analysis

reflections on the mechanics of granular matter during recent years a rather basic conflict has emerged between departments of mechanics physics concerning the description of granular media experts from mechanics measure stress strain relations using the so called triaxial tests and then use these data to predict the behavior of a sample under given boundary conditions some physicists have a different view they have claimed that it is not possible to define a proper displacement field in a heap of sand and that the notion of strains is thus ambiguous in the present text we conclude that the crucial features are the following a a heap is usually formed from a flowing phase of sand b there is empirically a sharp interface between the flowing phase and the frozen heap below c we may define for each grain a displacement which is measured from the moment when it froze this displacement is due for instance to a compaction of the heap under its own weight we also present some aspects of the dynamics including dune motions and surface flows of grains bouchaud cates and coworkers have constructed a very compact description for thin flows we discuss the practical consequences of this picture emphasizing some possible extensions for thicker flows this suggests a number of possible experiments on avalanches

volume preserving schemes and numerical experiments abstract in this paper we discuss the conditions for euler midpoint rule to be volume preserving and present explicit volume preserving schemes some numerical experiments are done to test these schemes

birkhoff periodic orbits for small perturbations of completely integrable hamiltonian systems with convex hamiltonians pour des systemes a n degres de liberte on montre l existence d au moins n orbites periodiques distinctes avec un vecteur frequence rationnel admissible proche du tore correspondant du systeme non perturbe sous l hypothese que le hamiltonien du systeme non perturbe a des surfaces d energie convexes pour les variables action angle

time reversal symmetry in dynamical systems a survey abstract in this paper we survey the topic of time reversal symmetry in dynamical systems we begin with a brief discussion of the position of time reversal symmetry in physics after defining time reversal symmetry as it applies to dynamical systems we then introduce a major theme of our survey namely the relation of time reversible dynamical sytems to equivariant and hamiltonian dynamical systems we follow with a survey of the state of the art on the theory of reversible dynamical systems including results on symmetric periodic orbits local bifurcation theory homoclinic orbits and renormalization and scaling some areas of physics and mathematics in which reversible dynamical systems arise are discussed in an appendix we provide an extensive bibliography on the topic of time reversal symmetry in dynamical systems

break up of invariant surfaces in action angle angle maps and flows abstract an important class of three dimensional volume preserving maps and flows arises as a perturbation from integrable action action angle maps and flows the properties of this class of maps and flows are discussed while action angle angle volume preserving maps admit an analog of the kam theorem general results on nonexistence of two dimensional invariant manifolds of action action angle maps are proven here nonexistence of such two dimensional invariant manifolds means possibility of global transport and a mechanism for such transport the local mechanism of resonance induced dispersion phys rev lett 75 1995 3669 is studied perturbatively resonance induced dispersion is shown to arise from the existence of periodic orbits that survive perturbation at places where two dimensional invariant manifolds break down

comparison of hyperelastic models for granular materials three recently proposed hyperelastic models for granular materials are compared with experiment data though all three are formulated to give elastic moduli that are power law functions of the mean stress they have rather different dependencies on individual stresses and generally differ from well established experimental forms predicted static stress distributions are in qualitative agreement with experiments but do not differ greatly from isotropic linear elasticity and similarly fail to account for variability in experiment data that presumably occurs due to a preparation dependence of granular materials

shearless transport barriers in unsteady two dimensional flows and maps abstract we develop a variational principle that extends the notion of a shearless transport barrier from steady to general unsteady two dimensional flows and maps defined over a finite time interval this principle reveals that hyperbolic lagrangian coherent structures lcss and parabolic lcss or jet cores are the two main types of shearless barriers in unsteady flows based on the boundary conditions they satisfy parabolic barriers are found to be more observable and robust than hyperbolic barriers confirming widespread numerical observations both types of barriers are special null geodesics of an appropriate lorentzian metric derived from the cauchy green strain tensor using this fact we devise an algorithm for the automated computation of parabolic barriers we illustrate our detection method on steady and unsteady non twist maps and on the aperiodically forced bickley jet

nilpotent normal form for divergence free vector fields and volume preserving maps abstract we study the normal forms for incompressible flows and maps in the neighborhood of an equilibrium or fixed point with a triple eigenvalue we prove that when a divergence free vector field in r 3 has nilpotent linearization with maximal jordan block then to arbitrary degree coordinates can be chosen so that the nonlinear terms occur as a single function of two variables in the third component the analogue for volume preserving diffeomorphisms gives an optimal normal form in which the truncation of the normal form at any degree gives an exactly volume preserving map whose inverse is also polynomial with the same degree