boundary layer analysis in the semiclassical limit of a quantum drift diffusion model abstract we study a singularly perturbed elliptic second order system in one space variable as it appears in a stationary quantum drift diffusion model of a semiconductor we prove the existence of solutions and their uniqueness as minimizers of a certain functional and determine rigorously the principal part of an asymptotic expansion of a boundary layer of those solutions we prove analytical estimates of the remainder terms of this asymptotic expansion and confirm by means of numerical simulations that these remainder estimates are sharp

elliptic operators with unbounded drift coefficients and neumann boundary condition abstract we study the realization an of the operator a 1 2 du d in l 2 with neumann boundary condition where is a possibly unbounded convex open set in r n u is a convex unbounded function du x is the element with minimal norm in the subdifferential of u at x and dx c exp 2u x dx is a probability measure infinitesimally invariant for a we show that an is a dissipative self adjoint operator in l 2 log sobolev and poincare inequalities allow then to study smoothing properties and asymptotic behavior of the semigroup generated by an

cahn hilliard and thin film equations with nonlinear mobility as gradient flows in weighted wasserstein metrics abstract in this paper we establish a novel approach to proving existence of non negative weak solutions for degenerate parabolic equations of fourth order like the cahn hilliard and certain thin film equations the considered evolution equations are in the form of a gradient flow for a perturbed dirichlet energy with respect to a wasserstein like transport metric and weak solutions are obtained as curves of maximal slope our main assumption is that the mobility of the particles is a concave function of their spatial density a qualitative difference of our approach to previous ones is that essential properties of the solution non negativity conservation of the total mass and dissipation of the energy are automatically guaranteed by the construction from minimizing movements in the energy landscape

ten methods to bound multiple roots of polynomials given a univariate polynomial p with a k fold multiple root or a k fold root cluster near some z we discuss nine different methods to compute a disc near z which either contains exactly or contains at least k roots of p many of the presented methods are known some are new we are especially interested in the behaviour of methods when implemented in a rigorous way that is when taking into account all possible effects of rounding errors in other words every result shall be mathematically correct we display extensive test sets comparing the methods under different circumstances based on the results we present a tenth a hybrid method combining five of the previous methods which for given z i detects the number k of roots near z and ii computes an including disc with in most cases a radius of the order of the numerical sensitivity of the root cluster therefore the resulting discs are numerically nearly optimal

dependence of eigenvalues on the boundary conditions of sturm liouville problems with one singular endpoint abstract this paper is concerned with continuous and differentiable dependence of isolated eigenvalues on the boundary conditions of self adjoint sturm liouville problems with one singular endpoint locally continuous dependence of eigenvalues on the boundary conditions is proved especially in the limit point case the continuous dependence of isolated eigenvalues is shown by the relationships between weyl titchmarsh m function and the spectrum of the singular problems then continuous eigenvalue branches through each isolated eigenvalue over the space of boundary conditions are formed it is rigorously shown that the eigenfunctions for the eigenvalues along a continuous simple eigenvalue branch can be continuously chosen with uniform bound in l w 2 norm its proof is different from that in the regular sturm liouville problems the derivative formulas of the continuous simple eigenvalue branch with respect to all the parameters in the boundary conditions are given and thus its monotonicity with respect to some parameters is derived

homology computation by reduction of chain complexes abstract a new algorithm for computing the homology module of a finitely generated chain complex is given it is based on local one step reductions of the size of the initial chain complex and it has a clear geometrical interpretation the complexity of the algorithm is discussed in special cases

shooting methods and topological transversality we show that shooting methods for homoclinic or heteroclinic orbits in dynamical systems may automatically guarantee the topological transversality of the stable and unstable manifolds the interest of such results is twofold first these orbits persist under perturbations which destroy the structure allowing the shooting method and second topological transversality is often sufficient when some kind of transversality is required to obtain chaotic dynamics we shall focus on heteroclinic solutions in the extended fisher kolmogorov equation

the geometry of optimal transportation introduction 113 1 summary of main results 120 2 background on optimal measures 126 part i strictly convex costs 133 3 existence and uniqueness of optimal maps 133 4 characterization of the optimal map 137 part ii costs which are strictly concave as a function of d i s t a n c e 141 5 the role of optimal maps 141 6 uniqueness of optimal solutions 144 part iii appendices 148 a legendre transforms and conjugate costs 148 b examples of c concave potentials 152 c regularity of c concave potentials 154 references 159

existence of solutions for a class of fourth order cross diffusion systems of gradient flow type abstract this article is concerned with the existence and the long time behavior of weak solutions to certain coupled systems of fourth order degenerate parabolic equations of gradient flow type the underlying metric is a wasserstein like transportation distance for vector valued functions with nonlinear mobilities in each component under the hypothesis of flat convexity of the driving free energy functional weak solutions are constructed by means of the variational minimizing movement scheme for metric gradient flows the essential regularity estimates are derived by variational methods

from a large deviations principle to the wasserstein gradient flow a new micro macro passage we study the connection between a system of many independent brownian particles on one hand and the deterministic diffusion equation on the other for a fixed time step h 0 a large deviations rate functional j h characterizes the behaviour of the particle system at t h in terms of the initial distribution at t 0 for the diffusion equation a single step in the time discretized entropy wasserstein gradient flow is characterized by the minimization of a functional k h we establish a new connection between these systems by proving that j h and k h are equal up to second order in h as h 0 this result gives a microscopic explanation of the origin of the entropy wasserstein gradient flow formulation of the diffusion equation simultaneously the limit passage presented here gives a physically natural description of the underlying particle system by describing it as an entropic gradient flow

entropy stable and entropy dissipative approximations of a fourth order quantum diffusion equation structure preserving numerical schemes for a nonlinear parabolic fourth order equation modeling the electron transport in quantum semiconductors with periodic boundary conditions are analyzed first a two step backward differentiation formula bdf semi discretization in time is investigated the scheme preserves the nonnegativity of the solution is entropy stable and dissipates a modified entropy functional the existence of a weak semi discrete solution and in a particular case its temporal second order convergence to the continuous solution is proved the proofs employ an algebraic relation which implies the g stability of the two step bdf second an implicit euler and q q step bdf discrete variational derivative method are considered this scheme which exploits the variational structure of the equation dissipates the discrete fisher information or energy numerical experiments show that the discrete relative entropies and fisher information decay even exponentially fast to zero

on the existence of an eigenvalue below the essential spectrum this paper is concerned with a computer assisted proof of eigenvalues below the essential spectrum of the sturm liouville problem on the half line it uses methods of functional analysis and interval analysis to derive algorithms that may be used to prove the existence of such eigenvalues

fractal boundary for the existence of invariant circles for area preserving maps observations and renormalisation explanation abstract breakup of a golden invariant circle is studied in an extended version of the standard map with two parameters the critical line in parameter space turns out to have a cantor set of cusps which can be organized in a self similar tree a theoretical explanation is conjectured in terms of a horseshoe for a renormalisation operator the results of some other researchers on similar systems are shown to fit this explanation as well

existence and positivity of solutions of a fourth order nonlinear pde describing interface fluctuations we study the partial differential equation which arose originally as a scaling limit in the study of interface fluctuations in a certain spin system in that application x lies in r but here we study primarily the periodic case x e s we establish existence uniqueness and regularity of solutions locally in time for positive initial data in h s and prove the existence of several families of lyapunov functions for the evolution from the latter we establish a sharp connection between existence globally in time and positivity preservation if o t is a maximal half open interval of existence for a positive solution of the equation with t 1 16r2 then t 03 and lim m w r exists and is constant we discuss also some explicit solutions and propose a generalization to higher dimensions 0 1994 john wiley

large time behavior of solution to the three dimensional quantum bipolar drift diffusion model from semiconductors abstract in this study we consider the three dimensional quantum bipolar drift diffusion model arising from the semiconductor device simulation which consists of the coupled nonlinear fourth order parabolic equation and poisson equation based on the results of the self similar stability for the one dimensional quantum bipolar drift diffusion equation we show the stability of planar self similar wave for the three dimensional quantum bipolar drift diffusion model using the energy methods we present the global existence of smooth solutions for the initial value problem of the three dimensional quantum bipolar drift diffusion equation when the initial data are close to the planar self similar wave we also show that in large time the solution of the three dimensional quantum bipolar drift diffusion equations tends to the planar self similar wave at an algebraic time decay rate

on the geometry of transport in phase space i transport in k degree of freedom hamiltonian systems 2 k abstract over the past several years the effect of geometrical structures such as kam tori resonance zones and cantori on phase space transport in two degree of freedom hamiltonian systems has been studied in three degree of freedom systems and higher these structures no longer form complete or partial barriers to transport in the sense that they or their associated stable and unstable manifolds are no longer codimension one in the level set of the hamiltonian in this paper we show that in the 2 k 1 dimensional level sets of the hamiltonian of a class of k degree of freedom hamiltonian systems the 2 k 2 dimensional stable and unstable manifolds of normally hyperbolic invariant 2 k 3 dimensional spheres form partial barriers to transport under certain conditions we show that lobes and turnstiles can be formed from such geometrical structures in a way that is exactly analogous to the situation for two degree of freedom hamiltonian systems considering a foliation of the level set of the hamiltonian into disjoint regions separated by such partial barriers we can then give exact formulas for the transport of volumes of phase space throughout the different regions based on the dynamics of these generalized lobes and turnstiles on the other hand despite the fact that these partial barriers are codimension one for k degree of freedom systems k 3 they may not intersect in such a way so as to divide the phase space

fast verification of solutions of matrix equations in this paper we are concerned with a matrix equation ax b where a is an n times n real matrix and x and b are n vectors assume that an approximate solution tilde x is given together with an approximate lu decomposition we will present fast algorithms for proving nonsingularity of a and for calculating rigorous error bounds for a 1 b tilde x infty the emphasis is on rigour of the bounds the purpose of this paper is to propose different algorithms the fastest with frac 2 3 n 3 flops computational cost for the verification step the same as for the lu decomposition the presented algorithms exclusively use library routines for lu decomposition and for all other matrix and vector operations

free boundaries in optimal transport and monge ampere obstacle problems given compactly supported 0 f g 2 l 1 r n the problem of trans porting a fraction m min kfkl1 kgkl1 of the mass of f onto g as cheaply as possible is considered where cost per unit mass transported is given by a cost function c typically quadratic c x y x y 2 2 this question is shown to be equivalent to a double obstacle problem for the monge ampere equation for which sucient conditions are given to guarantee uniqueness of the solution such as f vanishing on sptg in the quadratic case the part of f to be transported increases monotonically with m and if sptf and sptg are separated by a hyperplane h then this part will separated from the balance of f by a semiconcave lipschitz graph over the hyperplane if f f and g g are bounded away from zero and infinity on separated strictly convex domains r n for the quadratic cost this graph is shown to be a c 1 loc hypersurface in whose normal coincides with the direction transported the optimal map between f and g is shown to be holder continuous up to this free bound ary and to those parts of the fixed boundary which map to locally convex parts of the path connected target region

existence and stability for fokker planck equations with log concave reference measure we study markov processes associated with stochastic differential equations whose non linearities are gradients of convex functionals we prove a general result of existence of such markov processes and a priori estimates on the transition probabilities the main result is the following stability property if the associated invariant measures converge weakly then the markov processes converge in law the proofs are based on the interpretation of a fokker planck equation as the steepest descent flow of the relative entropy functional in the space of probability measures endowed with the wasserstein distance

a shooting approach to the lorenz equations we announce and outline a proof of the existence of a homoclinic orbit of the lorenz equations in addition we develop a shooting technique and two key conditions which lead to the existence of a one to one correspondence between a set of solutions and the set of all infinite sequences of 1 s and 3 s

periodic orbits symbolic dynamics and topological entropy for the restricted 3 body problem this paper concerns the restricted 3 body problem by applying topological methods we give a computer assisted proof of the existence of some classes of periodic orbits the existence of symbolic dynamics and we give a rigorous lower estimate for the topological entropy

exact numerical studies of hamiltonian maps iterating without roundoff error abstract for many important hamiltonian maps e g the standard map it is possible to construct related mappings that i carry a lattice into itself ii approach the original map as the lattice spacing is decreased iii can be iterated exactly using integer arithmetic and iv are hamiltonian themselves we compare these lattice maps to maps that use floating point arithmetic to evaluate the original map we discuss the problems associated with roundoff error and we argue that lattice maps are superior to floating point maps for the study of the long term behaviour of hamiltonian dynamical systems

automatic differentiation for fourier series and the radii polynomial approach abstract in this work we develop a computer assisted technique for proving existence of periodic solutions of nonlinear differential equations with non polynomial nonlinearities we exploit ideas from the theory of automatic differentiation in order to formulate an augmented polynomial system we compute a numerical fourier expansion of the periodic orbit for the augmented system and prove the existence of a true solution nearby using an a posteriori validation scheme the radii polynomial approach the problems considered here are given in terms of locally analytic vector fields i e the field is analytic in a neighborhood of the periodic orbit hence the computer assisted proofs are formulated in a banach space of sequences satisfying a geometric decay condition in order to illustrate the use and utility of these ideas we implement a number of computer assisted existence proofs for periodic orbits of the planar circular restricted three body problem pcrtbp

eigenvalues of limit point sturm liouville problems abstract the dependence of the eigenvalues of self adjoint sturm liouville problems on the boundary conditions when each endpoint is regular or in the limit circle case is now due to some surprisingly recent results well understood here we study this dependence for singular problems with one endpoint in the limit point case

componentwise pseudospectrum of a matrix abstract given a nonnegative weight matrix e defining componentwise structured perturbations e e we introduce a componentwise pseudospectrum of a square matrix a as contour sets of the function c a i 1 e where m is the spectral radius of a square matrix m and discuss how to compute it

example of a physical system with a hyperbolic attractor of the smale williams type a simple and transparent example of a nonautonomous flow system with a hyperbolic strange attractor is suggested the system is constructed on the basis of two coupled van der pol oscillators the characteristic frequencies differ twice and the parameters controlling generation in both oscillators undergo a slow periodic counterphase variation in time in terms of stroboscopic poincare sections the respective 4d mapping has a hyperbolic strange attractor of the smale williams type qualitative reasoning and quantitative data of numerical computations are presented and discussed e g lyapunov exponents and their parameter dependencies a special test for hyperbolicity based on analysis of distributions of angles between stable and unstable subspaces of a chaotic trajectory is performed

a second order equation for schrodinger bridges with applications to the hot gas experiment and entropic transportation cost the schrodinger problem is obtained by replacing the mean square distance with the relative entropy in the monge kantorovich problem it was first addressed by schrodinger as the problem of describing the most likely evolution of a large number of brownian particles conditioned to reach an unexpected configuration its optimal value the entropic transportation cost and its optimal solution the schrodinger bridge stand as the natural probabilistic counterparts to the transportation cost and displacement interpolation moreover they provide a natural way of lifting from the point to the measure setting the concept of brownian bridge in this article we prove that the schrodinger bridge solves a second order equation in the riemannian structure of optimal transport roughly speaking the equation says that its acceleration is the gradient of the fisher information using this result we obtain a fine quantitative description of the dynamics and a new functional inequality for the entropic transportation cost that generalize talagrand s transportation inequality finally we study the convexity of the fisher information along schrodigner bridges under the hypothesis that the associated reciprocal characteristic is convex the techniques developed in this article are also well suited to study the feynman kac penalisations of brownian motion

long time behavior for a nonlinear fourth order parabolic equation we study the asymptotic behavior of solutions of the initial boundary value problem with periodic boundary conditions for a fourth order nonlinear degenerate diffusion equation with a logarithmic nonlinearity for strictly positive and suitably small initial data we show that a positive solution exponentially approaches its mean as time tends to infinity these results are derived by analyzing the equation verified by the logarithm of the solution

the range of a structural projection abstract let a be a jbw triple a linear subspace j of a is called an inner ideal in a provided that the subspace j a j is contained in j a subtriple b in a is said to be complemented if a b ker b where ker b a a b a b 0 a complemented subtriple in a is a weak closed inner ideal a linear projection on a is said to be structural if for all elements a b and c in a formula the range of a structural projection is a complemented subtriple and conversely a complemented subtriple is the range of a unique structural projection we analyze the structure of the weak closed inner ideal generated by two arbitrary tripotents in a jbw triple in terms of the simultaneous peirce spaces of three suitably chosen pairwise compatible tripotents this result is then used to show that every weak closed inner ideal j in a jbw triple a is a complemented subtriple in a and therefore the range of a unique structural projection on a as an application structural projections on w algebras are considered

on the effects of the bohm potential on a macroscopic system of self interacting particles abstract we consider a nonstationary macroscopic system of self interacting particles with an additional potential the so called bohm potential we study the existence of nonnegative global solutions to the system of equations and allude to the differences to results obtained for classical models the problem is considered on a bounded domain up to three spatial dimensions subject to initial and neumann boundary conditions for the particle density and the dirichlet boundary condition for the self interacting potential moreover the initial datum is only assumed to be nonnegative and to satisfy a weak integrability condition

grothendieck s inequalities for real and complex jbw triples we prove that if m 4 1 2 3 and e 0 if v and w are complex jbw triples with preduals v andw respectively and if u is a separately weak continuous bilinear form on v w then there exist norm one functionals 1 2 v and 1 2 w satisfying u x y m u x 2 2 e 2 x 2 1 1 2 y 2 2 e 2 y 2 1 1 2 for all x y v w here for a norm one functional on a complex jb triple v stands for the prehilertian seminorm on v associated to in bf1 we arrive in this grothendieck s inequality through results of c h chu b iochum and g loupias cil and a corrected version of the little grothendieck s inequality for complex jb triples due to t barton and y friedman bf1 we also obtain extensions of these results to the setting of real jb triples 2000 mathematics subject classification 17c65 46k70 46l05 46l10 and 46l70

rigorous verification of chaos in a molecular model the thiele wilson system a simple model of a linear triatomic molecule has been studied extensively in the past the system exhibits complex molecular dynamics including dissociation periodic trajectories and bifurcations in addition it has for a long time been suspected to be chaotic but this has never been proved with mathematical rigor in this paper we present numerical results that using interval methods rigorously verify the existence of transversal homoclinic points in a poincar e map of this system by a theorem of smale the existence of transversal homoclinic points in a map rigorously proves its mixing property i e the chaoticity of the system

computer assisted proof of chaos in the lorenz equations abstract in this paper we prove with computer assistance the existence of chaos in a suitable poincare map generated by the lorenz system of equations by chaos we mean the existence of symbolic dynamics with infinite number of periodic trajectories the proof combines abstract results based on the fixed point index and finite rigorous computer calculations discussion concerning numerical algorithms is also included

the banach lie algebra of multiplication operators on a jbw triple abstract the banach lie algebra l a of multiplication operators on the jb triple a is introduced and it is shown that the hermitian part l a h of l a is a unital gm space the base of the dual cone in the dual gl space l a h of which is affine isomorphic and weak homeomorphic to the state space of l a in the case in which a is a jbw triple it is shown that tripotents u and v in a are orthogonal if and only if the corresponding multiplication operators in the unital gm space l a h satisfy 0 d u u d v v id a and that u is a pre associate of v if and only if d u u d v v

chaotic trajectories in the standard map the concept of anti integrability a rigorous proof is given in the standard map associated with a frenkel kontorowa model for the existence of chaotic trajectories with unbounded momenta for large enough coupling constant k k0 these chaotic trajectories with finite entropy per site are coded by integer sequences mi such that the sequence bi mi 1 mi 1 2mi be bounded by some integer b the bound k0 in k depends on b and can be lowered for coding sequences mi fulfilling more restrictive conditions the obtained chaotic trajectories correspond to stationary configurations of the frenkel kontorowa model with a finite non zero photon gap called gap parameter in dimensionless units this property implies that the trajectory or the configuration ui can be uniquely continued as a uniformly continuous function of the model parameter k in some neighborhood of the initial configuration a non zero gap parameter implies that the lyapunov coefficient is strictly positive when it is defined in addition the existence of dilating and contracting manifolds is proven for these chaotic trajectories exotic trajectories such as ballistic trajectories are also proven to exist as a consequence of these theorems the concept of anti integrability emerges from these theorems in the anti integrable limit which can be only defined for a discrete time dynamical system the coordinates of the trajectory at time i do not depend on the coordinates at time i 1 thus at this singular limit the existence of chaotic trajectories is trivial and the dynamical system reduces to a bernoulli shift it is well known that the kam tori of symplectic dynamical originates by continuity from the invariant tori which exists in the integrible limit under certain conditions in a similar way it appears that the chaotic trajectories of dynamical systems originate by continuity from those which exists at the anti integrable limits also under certain conditions

self validating methods abstract we present some ideas on self validating sv methods this note is especially intended to give some background for the articles in this special volume of laa it is not intended to be a survey of the big variety of all possible aspects of sv methods but rather a summary of some basic concepts especially some common misconceptions are mentioned and explored

local discretization error bounds using interval boundary element method in this paper a method to account for the point wise discretization error in the solution for boundary element method is developed interval methods are used to enclose the boundary integral equation and a sharp parametric solver for the interval linear system of equations is presented the developed method does not assume any special properties besides the laplace equation being a linear elliptic partial differential equation whose green s function for an isotropic media is known numerical results are presented showing the guarantee of the bounds on the solution as well as the convergence of the discretization error copyright 2008 john wiley sons ltd

the wasserstein gradient flow of the fisher information and the quantum drift diffusion equation we prove the global existence of non negative variational solutions to the drift diffusion evolution equation partial t u div left u mathrm d left 2 frac delta sqrt u sqrt u f right right 0 under variational boundary condition despite the lack of a maximum principle for fourth order equations non negative solutions can be obtained as a limit of a variational approximation scheme by exploiting the particular structure of this equation which is the gradient flow of the perturbed fisher information functional fancyscript f f u frac 12 int left mathrm d log u right 2 u dx int fu dx with respect to the kantorovich rubinstein wasserstein distance between probability measures we also study long time behavior of the solutions proving their exponential decay to the equilibrium state g e v characterized by delta v frac12 left mathrm d v right 2 f quad int rm e v dx int u 0 dx when the potential v is uniformly convex in this case the functional fancyscript f f coincides with the relative fisher information fancyscript f f u frac12 fancyscript i u g int left mathrm d log u g right 2u dx

random fibonacci sequences and the number 1 13198824 begin abstract for the familiar fibonacci sequence defined by f 1 f 2 1 and f n f n 1 f n 2 for n 2 f n increases exponentially with n at a rate given by the golden ratio 1 sqrt 5 2 1 61803398 ldots but for a simple modification with both additions and subtractions the it random fibonacci sequences defined by t 1 t 2 1 and for n 2 t n pm t n 1 pm t n 2 where each pm sign is independent and either or with probability 1 2 it is not even obvious if abs t n should increase with n our main result is that begin equation sqrt n abs t n rightarrow 1 13198824 ldots text as n rightarrow infty end equation with probability 1 finding the number 1 13198824 ldots involves the theory of random matrix products stern brocot division of the real line a fractal like measure a computer calculation and a rounding error analysis to validate the computer calculation end abstract

peirce inner ideals in jordan triples abstract a subspace j of an anisotropic jordan triple a is said to be an inner ideal if the subspace j a j is contained in j an inner ideal j in a is said to be complemented if a is equal to the sum of j and the kernel ker j of j defined to be the subspace of a consisting of elements a in a for which j a j is equal to 0 the annihilator j of an inner ideal j in a is the inner ideal consisting of elements a in a such that j a a is equal to 0 when both j and j are complemented a can be decomposed into the direct sum of j ker j ker j and j modulo six of the generalized peirce relations this decomposition is a grading of a of peirce type since an inner ideal in a jbw triple is complemented if and only if it is weak closed the result described above applies to all weak closed inner ideals j in a jbw triple a furthermore it can be shown that in this case all except five of the generalized peirce relations hold and an example is given of a weak closed inner ideal in a jbw triple for which all five fail to hold thereby showing that the result is the best possible it is also shown that the condition that a weak closed inner ideal in a jbw triple a leads to a grading of a which is of peirce type is equivalent to several other conditions all of a topological rather than algebraic nature these results are applied to w algebras spin triples and the bi cayley triple

on a computer assisted proof of the existence of eigenvalues below the essential spectrum of the sturm liouville problem abstract there is considerable interest in determining the existence of eigenvalues of the sturm liouville problem py qy wy where the independent variable x 0 and p q and w are real valued functions and is the spectral parameter in general an analytic attack on this problem is quite difficult and usually requires the use of the variational principal together with choice of suitable test functions we show how results from functional analysis together with interval analysis and interval arithmetic can be used not only to determine the existence of such eigenvalues but also to compute provably correct bounds on their values

high frequency limit of non autonomous gradient flows of functionals with time periodic forcing we study the high frequency limit of non autonomous gradient flows in metric spaces of energy functionals comprising an explicitly time dependent perturbation term which might oscillate in a rapid way on grounds of the existence results by ferreira and guevara 2015 on non autonomous gradient flows which we also extend to a broader range of energy functionals we prove that the associated solution curves converge to a solution of the time averaged evolution equation in the limit of infinite frequency under additional assumptions on the energy we obtain an explicit rate of convergence furthermore we specifically investigate nonlinear drift diffusion equations with time dependent drift which formally are gradient flows with respect to the l2 wasserstein distance we prove that a family of weak solutions obtained as a limit of the minimizing movements scheme exhibits the above mentioned behavior in the high frequency limit

chaos in the lorenz equators a computer assisted proof part ii details details of a new technique for obtaining rigorous results concerning the global dynamics of nonlinear systems is described the technique combines abstract existence results based on the conley index theory with rigorous computer assisted computations as an application of these methods it is proven that for some explicit parameter values the lorenz equations exhibit chaotic dynamics

principal minors part i a method for computing all the principal minors of a matrix an order o 2 n algorithm for computing all the principal minors of an arbitrary n n complex matrix is motivated and presented offering an improvement by a factor of n 3 over direct computation the algorithm uses recursive schur complementation and submatrix extraction storing the answer in a binary order an implementation of the algorithm in matlab is also given and practical considerations are discussed and treated accordingly

a strengthened central limit theorem for smooth densities abstract we consider the central limit theorem for the probability density function n x of the standardized sum of independent and identically distributed random variables with finite variance and regular probability density function by showing boundedness of different convex functionals along the sequence fn we prove convergence in various norms to the normal density

bohmian measures and their classical limit abstract we consider a class of phase space measures which naturally arise in the bohmian interpretation of quantum mechanics we study the classical limit of these so called bohmian measures in dependence on the scale of oscillations and concentrations of the sequence of wave functions under consideration the obtained results are consequently compared to those derived via semi classical wigner measures to this end we shall also give a connection to the theory of young measures and prove several new results on wigner measures themselves our analysis gives new insight on oscillation and concentration effects in the semi classical regime

nonlinear mobility continuity equations and generalized displacement convexity we consider the geometry of the space of borel measures endowed with a distance that is defined by generalizing the dynamical formulation of the wasserstein distance to concave nonlinear mobilities we investigate the energy landscape of internal potential and interaction energies for the internal energy we give an explicit sufficient condition for geodesic convexity which generalizes the condition of mccann we take an eulerian approach that does not require global information on the geodesics as by product we obtain existence stability and contraction results for the semigroup obtained by solving the homogeneous neumann boundary value problem for a nonlinear diffusion equation in a convex bounded domain for the potential energy and the interaction energy we present a non rigorous argument indicating that they are not displacement semiconvex

converse kam theory and practice we unify extend reinterpret and apply criteria of birkhoff 1 herman 9 mather 2 3 aubry et al 4 5 and newman and percival 6 for the nonexistence of invariant circles for area preserving twist maps the criteria enable one to establish regions of phase space through which no rotational invariant circles pass for families of maps the same can be done for regions of the combined space of phase points and parameters the criteria can be implemented rigorously on a computer and give a practical method of proving quite strong results as an example we present a computer program which proved that the standard map has no rotational invariant circles for any parameter value k 63 64

topological invariants mulitvalued maps and computer assisted proofs in dynamics abstract we introduce the concept of inheritable properties of multivalued maps and show how this concept when combined with interval arithmetic and representable multivalued approximations may be used in adapting topological methods to computer assisted proofs in dynamics as our first example we propose a simple geometric criterion for chaos a more sophisticated example is the conley index theory we show how the computation for the conley index for multivalued representable maps may be reduced to a finite computation this reduction was used in the computer assisted proof of chaos in the lorenz equations represented in 1

a family of fourth order q logarithmic equations we prove the existence of global in time weak nonnegative solutions to a family of nonlinear fourth order evolution equations parametrized by a real parameter q 0 1 which includes the well known thin film q 1 2 and the derrida lebowitz speer spohn dlss equation q 1 subject to periodic boundary conditions in one spatial dimension in contrast to the gradient flow approach in 25 our method relies on dissipation property of the corresponding entropy functionals tsallis entropies resulting in required a priori estimates and extends the existence result from 25 to a wider range of the family members namely to 0 q 1 2 generalized beckner type functional inequalities yield an exponential decay rate of relative entropies which in further implies the exponential stability in the l1 norm of the constant steady state finally we provide illustrative numerical examples supporting the analytical results

symbolic dynamics for the henon heiles hamiltonian on the critical level abstract we present a computer assisted proof of the existence of a rich symbolic dynamic structure for the henon heiles hamiltonian system at the critical energy e 1 6

initial time layer problem for quantum drift diffusion model we study the initial time layer problem to the quantum drift diffusion model the limit of vanishing debye length is established by entropy method and compactness argument as a byproduct the global weak solution is also obtained for this model

a convexity principle for interacting gases a new set of inequalities is introduced based on a novel but natural interpolation between borel probability measures on r d using these estimates in lieu of convexity or rearrangement inequalities the existence and uniqueness problems are solved for a family of attracting gas models in these models the gas interacts with itself through a force which increases with distance and is governed by an equation of state p p relating pressure to density p d 1 d is assumed non decreasing for a d dimensional gas by showing that the internal and potential energies for the system are convex functions of the interpolation parameter an energy minimizing stateunique up to translationis proven to exist the concavity established for t pd q as a function of t 0 1 generalizes the brunnminkowski inequality from sets to measures 1997 academic press

a planar topological horseshoe theory with applications to computer verifications of chaos in this paper we present an elementary theory on the existence and robustness of horseshoes under perturbations in terms of stability of crossing the framework is developed in the setting of 2d euclidean space but can be generalized to metric spaces as an application we give a rigorous verification of the existence of a horseshoe in the ikeda map

homology algorithm based on acyclic subspace we present a new reduction algorithm for the efficient computation of the homology of a cubical set the algorithm is based on constructing a possibly large acyclic subspace and then computing the relative homology instead of the plain homology we show that the construction of acyclic subspace may be performed in linear time this significantly reduces the amount of data that needs to be processed in the algebraic way and in practice it proves itself to be significantly more efficient than other available cubical homology algorithms

adaptive cellular integration of linearly implicit differential equations the use of cell to cell mapping methods for the global analysis of linearly implicit odes is addressed in this paper limitations of simple and adjoining cell mapping when applied to linearly implicit equations motivate the introduction of a finite time convergence model for cellular methods which unifies time and spatial discretization with numerical aspects in a comprehensive framework the resulting formal characterization of some properties of simple and adjoining cell mapping leads to a hybridization of both techniques which is shown to be well behaved for the analysis of linearly implicit odes

worst case bounds on the point wise discretization error in boundary element method for the elasticity problem in this work point wise discretization error is bounded via interval approach for the elasticity problem using interval boundary element formulation the formulation allows for computation of the worst case bounds on the boundary values for the elasticity problem from these bounds the worst case bounds on the true solution at any point in the domain of the system can be computed examples are presented to demonstrate the effectiveness of the treatment of local discretization error in elasticity problem via interval methods

long time asymptotics for strong solutions of the thin film equation in this paper we investigate the large time behavior of strong solutions to the one dimensional fourth order degenerate parabolic equation u t u u xxx x modeling the evolution of the interface of a spreading droplet for nonnegative initial values u 0 x h 1 both compactly supported or of finite second moment we prove explicit and universal algebraic decay in the l 1 norm of the strong solution u x t towards the unique among source type solutions strong source type solution of the equation with the same mass the method we use is based on the study of the time decay of the entropy introduced in 13 for the porous medium equation and uses analogies between the thin film equation and the porous medium equation

a geometric characterization of structural projections on a jbw triple a structural projection r on a jordan triple a is a linear projection such that for all elements a b and c in a r arbc rabrc the l orthogonal complement g of a subset g of a complex banach space e is the set of elements x in e such that for all elements y in g x y x y a contractive projection p on e is said to be neutral if the condition that px x implies that the elements px and x coincide and is said to be a gl projection if the l orthogonal complement pe of the range pe of p is contained in the kernel ker p of p it is shown that for a jbw triple a with predual a a linear projection r on a is structural if and only if it is the adjoint of a neutral gl projection on a thereby giving a purely geometric characterization of structural projections

detection of renewal system factors via the conley index let n be an isolating neighborhood for a map f if we can decompose n into the disjoint union of compact sets n 1 and n 2 then we can relate the dynamics on the maximal invariant set inv n to the shift on two symbols by noting which component of n each iterate of a point x inv n lies in we examine a method based on work by mischaikow szymczak et al for using the discrete conley index to detect explicit subshifts of the shift associated to n in essence we measure the difference between the conley index of inv n and the sum of the indices of inv n 1 and inv n 2

an extension of the derrida lebowitz speer spohn equation we show how the derivation of the derrida lebowitz speer spohn equation can be prolonged to obtain a new equation generalizing the models obtained in the paper by these authors we then investigate its properties from both an analytical and numerical perspective specifically a numerical method is presented to approximate solutions of the prolonged equation using this method we investigate the relationship between the solutions of the prolonged equation and the tracy widom goe distribution

polar factorization and monotone rearrangement of vector valued functions given a probability space x p and a bounded domain r in rd equipped with the lebesgue measure 1 i normalized so that 10 i i it is shown under additional technical assumptions on x and q that for every vector valued function u e lp x p rd there is a unique polar factorization u v s where is a convex function defined on r and s is a measure preserving mapping from x p into q i i provided that u is nondegenerate in the sense that p u e 0 for each lebesgue negligible subset e of rd through this result the concepts of polar factorization of real matrices helmholtz decomposition of vector fields and nondecreasing rearrangements of real valued functions are unified the monge amgre equation is involved in the polar factorization and the proof relies on the study of an appropriate monge kantorovich problem

a conley index study of the evolution of the lorenz strange set abstract in this paper we study the lorenz equations using the perspective of the conley index theory more specifically we examine the evolution of the strange set that these equations posses throughout the different values of the parameter we also analyze some natural morse decompositions of the global attractor of the system and the role of the strange set in these decompositions we calculate the corresponding morse equations and study their change along the successive bifurcations we see how the main features of the evolution of the lorenz system are explained by properties of the dynamics of the global attractor in addition we formulate and prove some theorems which are applicable in more general situations these theorems refer to poincare andronov hopf bifurcations of arbitrary codimension bifurcations with two homoclinic loops and a study of the role of the traveling repellers in the transformation of repeller attractor pairs into attractor repeller ones

rigorous chaos verification in discrete dynamical systems abstract in this paper we show how interval analysis can be used to calculate rigorously valid enclosures of transversal homoclinic points in discrete dynamical systems the existence of such points guarantees that a smale horseshoe is embedded in the mapping which therefore is chaotic we take special interest in the 2 dimensional case and describe a method to obtain a computer assisted proof of chaos for these systems numerical results are presented for the standard map