on the k tuple domination of generalized de brujin and kautz digraphs a vertex u in a digraph g v a is said to dominate itself and vertices v such that u v a for a positive integer k a k tuple dominating set of g is a subset d of vertices such that every vertex in g is dominated by at least k vertices in d the k tuple domination number of g is the minimum cardinality of a k tuple dominating set of g this paper deals with the k tuple domination problem on generalized de bruijn and kautz digraphs we establish bounds on the k tuple domination number for the generalized de bruijn and kautz digraphs and we obtain some conditions for the k tuple domination number attaining the bounds

iterative bregman projections for regularized transportation problems this article details a general numerical framework to approximate so lutions to linear programs related to optimal transport the general idea is to introduce an entropic regularization of the initial linear program this regularized problem corresponds to a kullback leibler bregman di vergence projection of a vector representing some initial joint distribu tion on the polytope of constraints we show that for many problems related to optimal transport the set of linear constraints can be split in an intersection of a few simple constraints for which the projections can be computed in closed form this allows us to make use of iterative bregman projections when there are only equality constraints or more generally bregman dykstra iterations when inequality constraints are in volved we illustrate the usefulness of this approach to several variational problems related to optimal transport barycenters for the optimal trans port metric tomographic reconstruction multi marginal optimal trans port and in particular its application to brenier s relaxed solutions of in compressible euler equations partial un balanced optimal transport and optimal transport with capacity constraints

generalized solutions for inextensible string equations we study the system of equations of motion for inextensible strings this system can be recast into a discontinuous system of conservation laws as well as into the total variation wave equation we prove existence of generalized young measure solutions with non negative tension after transforming the problem into a system of conservation laws and approximating it with a regularized system for which we obtain uniform estimates of the energy and the tension we also discuss sufficient conditions for non negativity of the tension for strong solutions

positive semidefinite zero forcing numbers of two classes of graphs abstract the positive semidefinite zero forcing number is a variant of the zero forcing number which is an important parameter in the study of minimum rank maximum nullity problems in this paper we consider two classes of graphs matching chain graphs and claw free graphs we first introduce the propagation decomposition of graphs then we use this decomposition to prove a lower bound for the positive semidefinite zero forcing number of a graph we apply this lower bound to find the positive semidefinite zero forcing number of matching chain graphs we show that the positive semidefinite zero forcing number and the zero forcing number agree for matching chain graphs as a consequence we prove the conjecture about the positive semidefinite zero forcing number of the cartesian product of two paths and partially prove the conjecture about the positive semidefinite zero forcing number of the cartesian product of a cycle and a path for claw free graphs we establish a relationship between the positive semidefinite zero forcing number and the zero forcing number we also prove that it is np complete to find the positive semidefinite zero forcing number of line graphs

generalized incompressible flows multi marginal transport and sinkhorn algorithm starting from brenier s relaxed formulation of the incompress ible euler equation in terms of geodesics in the group of measure preserving diffeomorphisms we propose a numerical method based on sinkhorn s algorithm for the entropic regularization of optimal transport we also make a detailed comparison of this entropic regular ization with the so called bredinger entropic interpolation problem see 1 numerical results in dimension one and two illustrate the feasibility of the method

dynamic theory of suspensions with brownian effects we consider a suspension of particles in a fluid settling under the influence of gravity and dispersing by brownian motion a mathematical description is provided by the stokes equations and a fokker planck equation for the one particle phase space density this is a nonlinear system that depends on a number of parametric functions of the spatial concentration of the particles these functions are known only empirically or for dilute suspensions we analyze the system its stability its asymptotic behavior under different scalings and its validity from more microscopic description we summarize our conclusions at the end

least action principle and the incompressible euler equations with variable density in this article we study a variational formulation proposed by v i arnold and by y brenier for the incompressible euler equations with variable density we consider the problem of minimizing an action functional the integral in time of the kinetic energy among fluid motions considered as trajectories in the group of volume preserving diffeomorphisms beginning at the identity and ending at some fixed diffeomorphism at a given time we show that a relaxed version of this variational problem always has a solution and we derive an euler lagrange system for the relaxed minimization problem which we call the relaxed euler equations finally we prove consistency between the relaxed euler equations and the classical euler system showing that weak solutions of the relaxed euler equations with the appropriate geometric structure give rise to classical euler solutions and that classical solutions of the euler system induce weak solutions of the relaxed euler equations the first consistency result is new even in the constant density case the remainder of our analysis is an extension of the work of y brenier 1999 to the variable density case

single step searching in weighted block graphs in this paper three types of problems for single step searching weighted graphs are investigated the summation minimization s type for short bottleneck minimization b type for short and hybrid h type for short weighted single step graph searching problems all three types are shown to be np hard but polynomial solvable for block graphs the s type problem is proved to be linearly equivalent to the optimum weight 2 independent set problem then we solve the s type problem on a block graph g in linear time by solving the optimum weight 2 independent set problem on g to solve the b type problem the first phase computes the bottleneck cost and the second phase constructs the searching plan by applying the s type algorithm using the bottleneck cost derived in the first phase finally an o e log v time algorithm for solving the h type problem on weighted block graphs is proposed

computable elastic distances between shapes we dene distances between geometric curves by the square root of the minimal energy required to transform one curve into the other the energy is formally dened from a left invariant riemannian distance on an innite dimensional group acting on the curves which can be explicitly computed the obtained distance boils down to a variational problem for which an optimal matching between the curves has to be computed an analysis of the distance when the curves are polygonal leads to a numerical procedure for the solution of the variational problem which can eciently be implemented as illustrated by experiments 1 1 generalities the problem of matching two objects together is very impor tant in computer vision and shape recognition in many cases recognition is based on shapes outlines with the help of some suitably designed distance a general principle is to associate with any pair o1 o 2 of objects to be compared a measure of discrepancy d o1 o 2 the recognition of an observed object o may be done by nding from a dictionary of templates the previously recorded object otemp for which d o otemp is minimal clearly the denition of the distance is the crucial step of the method and much research has been done in this direction we shall not try here to provide a review of the huge literature existing on the subject see for example 17 but rather focus on methods related to deformable templates with which we are directly concerned instead of basing recognition on a nite collection of points of interest primitives taken from the outline of an object corners inflexion points etc which is a popular way of handling the problem our purpose is to base the comparison on the whole outline considered as a plane curve the distance we shall dene incorporates some deformation energy between the curves the approach as we will see turns out to be intrinsic and robust to usual euclidean transformations the method is related to the wide literature on snakes 14 7 21 etc in the way that our distance corresponds to some continuous process of deformation of one curve into another it is also related to papers on elastic matching such as 8 however we provide an elastic matching algorithm which is based on a true distance between intrinsic properties of the shapes taking into account possible invariance to scaling or euclidean transformations in the case they are required from this point of view our results are indebted to the seminal work of grenander on group theory applied to pattern recognition cf 10 and 11 in particular see also 1 2 12 another source of inspiration may come from mathematical physics since we are going to look to the path process of lowest energy which deforms one object into

generalized solutions and hydrostatic approximation of the euler equations abstract solutions to the euler equations on a 3d domain d 3 typically the unit cube or the periodic unit cube can be formally obtained by minimizing the action of an incompressible fluid moving inside d 3 between two given configurations when these two configurations are very close to each other classical solutions do exist as shown by ebin and marsden however shnirelman found a class of data essentially 2d in the sense that they trivially depend on the vertical coordinate for which there cannot be any classical minimizer for such data generalized solutions can be shown to exist as a substitute for classical solutions these generalized solutions have unusual features that look highly unphysical in particular different fluid parcels can cross at the same point and at the same time but the pressure field which does not depend on the vertical coordinate is well and uniquely defined in the present paper we show that these generalized solutions are actually quite conventional in the sense they obey up to a suitable change of variable a well known variant widely used for geophysical flows of the 3d euler equations for which the vertical acceleration is neglected according to the so called hydrostatic approximation

on the h s theory of hydrostatic euler equations in this paper we study the two dimensional hydrostatic euler equations in a periodic channel we prove the local existence and uniqueness of hs solutions under the local rayleigh condition this extends brenier s nonlinearity 12 3 495 512 1999 existence result by removing an artificial condition and proving uniqueness in addition we prove weak strong uniqueness mathematical justification of the formal derivation and stability of the hydrostatic euler equations these results are based on weighted hs a priori estimates which come from a new type of nonlinear cancellation between velocity and vorticity

the reconstruction problem for the euler poisson system in cosmology the motion of a continuum of matter subject to gravitational interaction is classically described by the euler poisson system prescribing the density of matter at initial and final times we are able to obtain weak solutions for this equation by minimizing the action of the lagrangian which is a convex functional through this variational formulation the reconstruction problem becomes very similar to an optimal transportation problem then we see that such minimizing solutions are consistent with smooth solutions of the euler poisson system and enjoy some special regularity properties

on exact analytical solutions for the few particle schrodinger equation iv the asymptotic form and normalisability of the wavefunction for pt iii see ibid vol 16 nol 18 p 4237 53 1983 formal series solutions for the schrodinger equations of few particle systems contain infinitely many associated with the normalisability i e square integrability of the wavefunction the energy is one of these parameters a method for determining the parameters by examining the asymptotic behaviour of the series has been developed no integration matrix inversions or trial and error procedures are involved the method is directly applicable to the 1s states of two electron atoms if the wavefunction is expressed as a multipole expansion in spherical polar coordinates the radial functions have asymptotic properties which give rise to relations between the parameters values assigned to the parameters not specified by these relations one per multipole ensure that the wavefunction is normalisable

planar formulae and their uses we define the set of planar boolean formulae and then show that the set of true quantified planar formulae is polynomial space complete and that the set of satisfiable planar formulae is np complete using these results we are able to provide simple and nearly uniform proofs of np completeness for planar node cover planar hamiltonian circuit and line geometric connected dominating set and of polynomial space completeness for planar generalized geography the np completeness of planar node cover and planar hamiltonian circuit and line were first proved elsewhere m r garey and d s johnson the rectilinear steiner tree is np complete siam j appl math 32 1977 pp 826 834 and m r garey d s johnson and r e tarjan the planar hamilton circuit problem is np complete siam j comp 5 1976 pp 704 714

axisymmetric flow of ideal fluid moving in a narrow domain a study of the axisymmetric hydrostatic euler equations abstract in this article we will introduce a new model to describe the leading order behavior of an ideal and axisymmetric fluid moving in a very narrow domain after providing a formal derivation of the model we will prove the well posedness and provide a rigorous mathematical justification for the formal derivation under a new sign condition finally a blowup result regarding this model will be discussed as well

ideal hydrodynamics on lie groups abstract in this paper we consider the reduction to a lie group of geodesic flows of left or right invariant metrics for a fixed value of momentum we describe the reduced vector field on the lie group as well as its right or left invariant symmetry fields the reduced vector field has an important hydrodynamic interpretation it is a stationary flow of an ideal fluid with a constant pressure we use the explicit expressions for the reduced vector field and its symmetry fields to define secondary hydrodynamics i e we study the reduction for the euler equations of an ideal fluid that describe the geodesics of a right invariant metric on a lie group s diff m of the volume preserving diffeomorphisms of a riemannian manifold m to the group s diff m for a typical coadjoint orbit we find all symmetry fields of a reduced flow and as a corollary we get a simple proof for nonexistence of new invariants of coadjoint orbits which are the integrals of local densities over the flow domain

oscillations and concentrations in weak solutions of the incompressible fluid equations the authors introduce a new concept of measure valued solution for the 3 d incompressible euler equations in order to incorporate the complex phenomena present in limits of approximate solutions of these equations one application of the concepts developed here is the following important result a sequence of leray hopf weak solutions of the navier stokes equations converges in the high reynolds number limit to a measure valued solution of 3 d euler defined for all positive times the authors present several explicit examples of solution sequences for 3 d incompressible euler with uniformly bounded local kinetic energy which exhibit complex phenomena involving both persistence of oscillations and development of concentrations an extensions of the concept of young measure is developed to incorporate these complex phenomena in the measure valued solutions constructed here

generalized compressible flows and solutions of the h div geodesic problem we study the geodesic problem on the group of diffeomorphism of a domain m rd equipped with the h div metric the geodesic equations coincide with the camassa holm equation when d 1 and represent one of its possible multi dimensional generalizations when d 1 we propose a relaxation a la brenier of this problem in which solutions are represented as probability measures on the space of continuous paths on the cone over m we use this relaxation to prove that smooth h div geodesics are globally length minimizing for short times we also prove that there exists a unique pressure field associated to solutions of our relaxation finally we propose a numerical scheme to construct generalized solutions on the cone and present some numerical results illustrating the relation between the generalized camassa holm and incompressible euler solutions

a priori estimates for fluid interface problems we consider the regularity of an interface between two incompressible and inviscid fluid flows in the presence of surface tension we obtain local in time estimates on the interface in h 3 2 k 1 and the velocity fields in h 3 2 k these estimates are obtained using geometric considerations which show that the kelvin helmholtz instabilities are a consequence of a curvature calculation 2007 wiley periodicals inc

well posedness of the epdiff equation with a pseudo differential inertia operator abstract in this article we study the class of right invariant fractional order sobolev type metrics on groups of diffeomorphisms of a compact manifold m our main result concerns well posedness properties for the corresponding euler arnold equations also called the epdiff equations which are of importance in mathematical physics and in the field of shape analysis and template registration depending on the order of the metric we will prove both local and global well posedness results for these equations as a result of our analysis we will also obtain new commutator estimates for elliptic pseudo differential operators

integrable vs nonintegrable geodesic soliton behavior abstract we study confined solutions of certain evolutionary partial differential equations pde in 1 1 space time the pde we study are lie poisson hamiltonian systems for quadratic hamiltonians defined on the dual of the lie algebra of vector fields on the real line these systems are also euler poincare equations for geodesic motion on the diffeomorphism group in the sense of the arnold program for ideal fluids but where the kinetic energy metric is different from the l2 norm of the velocity these pde possess a finite dimensional invariant manifold of particle like measure valued solutions we call pulsons we solve the particle dynamics of the two pulson interaction analytically as a canonical hamiltonian system for geodesic motion with two degrees of freedom and a conserved momentum the result of this two pulson interaction for rear end collisions is elastic scattering with a phase shift as occurs with solitons the results for head on antisymmetric collisions of pulsons tend to be singularity formation numerical simulations of these pde show that their evolution by geodesic dynamics for confined or compact initial conditions in various nonintegrable cases possesses the same type of multi soliton behavior elastic collisions asymptotic sorting by pulse height as the corresponding integrable cases do we conjecture this behavior occurs because the integrable two pulson interactions dominate the dynamics on the invariant pulson manifold and this dynamics dominates the pde initial value problem for most choices of confined pulses and initial conditions of finite extent

quantitative linearization results for the monge ampere equation this paper is about quantitative linearization results for the monge ampere equation with rough data we develop a large scale regularity theory and prove that if a measure is close to the lebesgue measure in wasserstein distance at all scales then the displacement of the macroscopic optimal coupling is quantitatively close at all scales to the gradient of the solution of the corresponding poisson equation the main ingredient we use is a harmonic approximation result for the optimal transport plan between arbitrary measures this is used in a campanato iteration which transfers the information through the scales

non differentiable embedding of lagrangian systems and partial differential equations abstract we develop the non differentiable embedding theory of differential operators and lagrangian systems using a new operator on non differentiable functions we then construct the corresponding calculus of variations and we derive the associated non differentiable euler lagrange equation and apply this formalism to the study of pdes first we extend the characteristics method to the non differentiable case we prove that non differentiable characteristics for the navier stokes equation correspond to extremals of an explicit non differentiable lagrangian system second we prove that the solutions of the schrodinger equation are non differentiable extremals of the newton s lagrangian

the bottleneck independent domination on the classes of bipartite graphs and block graphs let g v e w denote a graph with vertex set v and edge set e and each vertex v is associated with a cost w v for any set v v the bottleneck cost of v is defined as max w x x v this paper considers the bottleneck independent dominating set problem the bids problem which determines an independent dominating set of g such that its bottleneck cost is minimized this paper studies the problem on the classes of bipartite graphs and block graphs this paper first proves that the problem is np hard on chordal bipartite graphs second a linear time algorithm on convex bipartite graphs is proposed next a linear time algorithm on trees is designed then we generalize the algorithmic result to block graphs all algorithms are designed by the dynamic programming strategy

structure preserving discretization of incompressible fluids the geometric nature of euler fluids has been clearly identified and extensively studied over the years culminating with lagrangian and hamiltonian descriptions of fluid dynamics where the configuration space is defined as the volume preserving diffeomorphisms and kelvin s circulation theorem is viewed as a consequence of noether s theorem associated with the particle relabeling symmetry of fluid mechanics however computational approaches to fluid mechanics have been largely derived from a numerical analytic point of view and are rarely designed with structure preservation in mind and often suffer from spurious numerical artifacts such as energy and circulation drift in contrast this paper geometrically derives discrete equations of motion for fluid dynamics from first principles in a purely eulerian form our approach approximates the group of volume preserving diffeomorphisms using a finite dimensional lie group and associated discrete euler equations are derived from a variational principle with non holonomic constraints the resulting discrete equations of motion yield a structure preserving time integrator with good long term energy behavior and for which an exact discrete kelvin s circulation theorem holds

stochastic least action principle for the incompressible navier stokes equation abstract we formulate a stochastic least action principle for solutions of the incompressible navier stokes equation which formally reduces to hamilton s principle for the incompressible euler solutions in the case of zero viscosity we use this principle to give a new derivation of a stochastic kelvin theorem for the navier stokes equation recently established by constantin and iyer which shows that this stochastic conservation law arises from particle relabelling symmetry of the action we discuss issues of irreversibility energy dissipation and the inviscid limit of navier stokes solutions in the framework of the stochastic variational principle in particular we discuss the connection of the stochastic kelvin theorem with our previous martingale hypothesis for fluid circulations in turbulent solutions of the incompressible euler equations

existence of global weak solutions to one component vlasov poisson and fokker planck poisson systems in one space dimension with measures as initial data we consider cauchy problems for the 1 d one component vlasov poisson and fokker planck poisson equations with the initial electron density being in the natural space of arbitrary non negative finite measures in particular the initial density can be a dirac measure concentrated on a curve which we refer to as electron sheet initial data these problems resemble both structurally and functional analytically cauchy problems for the 2 d euler and navier stokes equations in vorticity formulation with vortex sheet initial data here we need to define weak solutions more specifically than usual since the product of a finite measure with a function of bounded variation is involved we give a natural definition of the product establish its weak stability and existence of weak solutions follows our concept of weak solutions through the newly defined product is justified since solutions to the fokker planck poisson equation the analogue of navier stokes equation are shown to converge to weak solutions of the vlasov poisson equation as the fokker planck term vanishes the main difficulty is the aforementioned weak stability which we establish through a careful analysis of the explicit structure of these equations this is needed because the problem studied here is beyond the range of applicability of the velocity averaging compactness methods of diperna lions 1994 john wiley sons inc

power domination throttling abstract a power dominating set of a graph g v e is a set s v that colors every vertex of g according to the following rules in the first timestep every vertex in n s becomes colored in each subsequent timestep every vertex which is the only non colored neighbor of some colored vertex becomes colored the power domination throttling number of g is the minimum sum of the size of a power dominating set s and the number of timesteps it takes s to color the graph in this paper we determine the complexity of power domination throttling and give some tools for computing and bounding the power domination throttling number some of our results apply to very general variants of throttling and to other aspects of power domination

coordinate systems and analytic expansions for three body atomic wavefunctions ii closed form wavefunction to second order in r for pt i see ibid vol 20 no 8 p 2043 75 1987 several coordinate systems for solving the few electron schrodinger equation are presented formal solutions corresponding to each coordinate system are given in terms of the fock expansion and their interrelationships and general structure are examined attention is focused on the solutions obtained using spherical polar coordinates for a coulomb potential of arbitrary symmetry the wavefunction is obtained up to second order in the hyperradius r r21 r sup 22 12 and the special case of 1s states is then reduced to a closed form using classical techniques the insight gained from this reduction suggests methods for solving the wavefunction to all orders the results hint at the existence of closed form wavefunctions for few body systems

on rainbow domination numbers of graphs given a graph g and a set of t colors assume that we assign an arbitrary subset of these colors to each vertex of g if we require that each vertex to which an empty set is assigned has in its neighborhood all t colors then this assignment is called a t rainbow dominating function of the graph g the corresponding invariant c r t g which is the minimum sum of numbers of assigned colors over all vertices of g is called the t rainbow domination number of g in this paper bounds for the t rainbow domination number of an arbitrary graph for an arbitrary positive integer t are given the 3 rainbow domination numbers of several classes of graphs such as paths cycles and the generalized petersen graphs p n k are investigated the 3 rainbow domination number of p n 1 is determined and the upper bounds for p n 2 and p n 3 are provided by computer search we obtain that the upper bounds for p n 2 match their exact values for n 17

on the regularity of the pressure field of relaxed solutions to euler equations with variable density abstract in this paper we improve the regularity in time of the gradient of the pressure field in the solution of relaxed version of variational formulation proposed by v i arnold and by y brenier for the incompressible euler equations with variable density we obtain that the pressure field is not only a measure but a function in l l o c 2 0 t b v l o c d as an extension of the work of ambrosio and figalli 2008 in 1 to the variable density case

analysis of a stochastic implicit interface model for an immersed elastic surface in a fluctuating fluid we present some mathematical analyses of a recently proposed stochastic implicit interface model for an elastic surface immersed in an incompressible viscous fluid subject to fluctuation forces we derive suitable a priori estimates and establish the well posedness of pathwise solutions and provide uniform control on the solutions in probability

on the error of a priori sampling zero forcing sets and propagation time abstract zero forcing is an iterative process on a graph used to bound the maximum nullity the process begins with select vertices as colored and the remaining vertices can become colored under a specific color change rule the goal is to find a minimum set of vertices such that after iteratively applying the rule all of the vertices become colored i e a minimum zero forcing set of particular interest is the propagation time of a chosen set which is the number of steps the rule must be applied in order to color all the vertices of a graph we give a purely linear algebraic view of zero forcing find a set of vertices s such that for any weighted adjacency matrix a whenever ax 0 the entirety of x can be recovered using only x s the entries corresponding to s the key here is that s must be chosen before a in this light we are able to give a linear algebraic interpretation of the propagation time any error in x s effects the error of x exponentially in the propagation time this error can be quantitatively measured using newly defined zero forcing related parameters the error polynomial vector and the variance polynomial vector in this sense the quality of two zero forcing sets can objectively be compared even if the sets are the same size and their propagation times are the same examples and constructions are given

pseudo differential energy estimates of singular perturbations in this paper we study singular limits of hyperbolic systems which exhibit large time oscillations and in particular pseudo differential energy estimates that enable us to obtain uniform existence time and a priori bounds on the solutions 1997 john wiley sons inc

evolution of microstructure in unstable porous media flow a relaxational approach we study the flow of two immiscible fluids of different density and mobility in a porous medium if the heavier phase lies above the lighter one the interface is observed to be unstable the two phases start to mix on a mesoscopic scale and the mixing zone grows in time an example of evolution of microstructure a simple set of assumptions on the physics of this two phase flow in a porous medium leads to a mathematically ill posed problem when used to establish a continuum free boundary problem we propose and motivate a relaxation of this nonconvex constraint of a phase distribution with a sharp interface on a macroscopic scale we prove that this approach leads to a mathematically well posed problem that predicts shape and evolution of the mixing profile as a function of the density difference and mobility quotient 1999 john wiley sons inc

power domination in block graphs the problem of monitoring an electric power system by placing as few measurement devices in the system as possible is closely related to the well known domination problem in graphs in 2002 haynes et al considered the graph theoretical representation of this problem as a variation of the domination problem they defined a set s to be a power dominating set of a graph if every vertex and every edge in the system is monitored by the set s following a set of rules for power system monitoring the power domination number p g of a graph g is the minimum cardinality of a power dominating set of g this problem was proved np complete even when restricted to bipartite graphs and chordal graphs in this paper we present a linear time algorithm for solving the power domination problem in block graphs as an application of the algorithm we establish a sharp upper bound for power domination number in block graphs and characterize the extremal graphs

minimal geodesics on groups of volume preserving maps and generalized solutions of the euler equations the three dimensional motion of an incompressible inviscid fluid is classically described by the euler equations but can also be seen following arnold 1 as a geodesic on a group of volume preserving maps local existence and uniqueness of minimal geodesics have been established by ebin and marsden 16 in the large for a large class of data the existence of minimal geodesics may fail as shown by shnirelman 26 for such data we show that the limits of approximate solutions are solutions of a suitable extension of the euler equations or equivalently are sharp measure valued solutions to the euler equations in the sense of diperna and majda 14 1999 john wiley sons inc

continuity equations and ode flows with non smooth velocity in this paper we review many aspects of the well posedness theory for the cauchy problem for the continuity and transport equations and for the ordinary differential equation ode in this framework we deal with velocity fields that are not smooth but enjoy suitable weak differentiability assumptions we first explore the connection between the partial differential equation pde and the ode in a very general non smooth setting then we address the renormalization property for the pde and prove that such a property holds for sobolev velocity fields and for bounded variation velocity fields finally we present an approach to the ode theory based on quantitative estimates

contractions in the 2 wasserstein length space and thermalization of granular media an algebraic decay rate is derived which bounds the time required for velocities to equilibrate in a spatially homogeneous flow through model representing the continuum limit of a gas of particles interacting through slightly inelastic collisions this rate is obtained by reformulating the dynamical problem as the gradient flow of a convex energy on an infinite dimensional manifold an abstract theory is developed for gradient flows in length spaces which shows how degenerate convexity or even non convexity if uniformly controlled will quantify contractivity limit expansivity of the flow

the least action principle and the related concept of generalized flows for incompressible perfect fluids the link between the euler equations of perfect incompressible flows and the least action principle has been known for a long time i solu tions can be considered as geodesic curves along the manifold of volume pre serving mappings here the shortest path problem is investigated given two different volume preserving mappings at two different times find for the inter mediate times an incompressible flow map that minimizes the kinetic energy or more generally the action in its classical formulation this problem has been solved 7 only when the two different mappings are sufficiently close in some very strong sense in this paper a new framework is introduced where gen eralized flows are defined in the spirit of l c young as probability measures on the set of all possible trajectories in the physical space then the minimiza tion problem is generalized as the continuous linear programming problem that is much easier to handle the existence problem is completely solved in this content downloaded from 157 55 39 159 on sun 18 sep 2016 05 02 20 utc all use subject to http about jstor org terms the least action principle 255 the case of the d dimensional torus it is also shown that under natural re strictions a classical solution to the euler equations is the unique optimal flow in the generalized framework finally a link is established with the concept of measure valued solutions to the euler equations 6 and an example is provided where the unique generalized solution can be explicitly computed and turns out to be genuinely probabilistic inria bp 105 78153 le chesnay cedex france this content downloaded from 157 55 39 159 on sun 18 sep 2016 05 02 20 utc all use subject to http about jstor org terms

dissipative anomalies in singular euler flows abstract we discuss onsager s conjecture that non vanishing energy dissipation in high reynolds number turbulence is associated to singular distributional solutions of the incompressible euler equations we carefully explain the physical and mathematical meaning of the conjecture and also review relevant theoretical experimental and numerical work emphasizing some of the dramatic successes of onsager s point of view finally we present several new ideas and results on lagrangian dynamics of circulations and vortex lines that we believe will be important for future progress

geodesics in the space of measure preserving maps and plans we study brenier s variational models for incompressible euler equations these models give rise to a relaxation of the arnold distance in the space of measure preserving maps and more generally measure preserving plans we analyze the properties of the relaxed distance we show a close link between the lagrangian and the eulerian model and we derive necessary and sufficient optimality conditions for minimizers these conditions take into account a modified lagrangian induced by the pressure field moreover adapting some ideas of shnirelman we show that even for non deterministic final conditions generalized flows can be approximated in energy by flows associated to measure preserving maps

spontaneous stochasticity and anomalous dissipation for burgers equation we develop a lagrangian approach to conservation law anomalies in weak solutions of inviscid burgers equation motivated by previous work on the kraichnan model of turbulent scalar advection we show that the entropy solutions of burgers possess markov stochastic processes of generalized lagrangian trajectories backward in time for which the burgers velocity is a backward martingale this property is shown to guarantee dissipativity of conservation law anomalies for general convex functions of the velocity the backward stochastic burgers flows with these properties are not unique however we construct infinitely many such stochastic flows both by a geometric construction and by the zero noise limit of the constantin iyer stochastic representation of viscous burgers solutions the latter proof yields the spontaneous stochasticity of lagrangian trajectories backward in time for burgers at unit prandtl number it is conjectured that existence of a backward stochastic flow with the velocity as martingale is an admissibility condition which selects the unique entropy solution for burgers we also study linear transport of passive densities and scalars by inviscid burgers flows we show that shock solutions of burgers exhibit spontaneous stochasticity backward in time for all finite prandtl numbers implying conservation law anomalies for linear transport we discuss the relation of our results for burgers with incompressible navier stokes turbulence especially lagrangian admissibility conditions for euler solutions and the relation between turbulent cascade directions and time asymmetry of lagrangian stochasticity

small noise limit and convexity for generalized incompressible flows schrodinger problems and optimal transport this paper is concerned with six variational problems and their mutual connections the quadratic monge kantorovich optimal transport the schrodinger problem brenier s relaxed model for incompressible fluids the so called brodinger problem recently introduced by arnaudon et al an entropic interpolation problem for incompressible viscid fluids arxiv preprint arxiv 1704 02126 2017 the multiphase brenier model and the multiphase brodinger problem all of them involve the minimization of a kinetic action and or a relative entropy of some path measures with respect to the reversible brownian motion as the viscosity parameter nu rightarrow 0 we establish gamma convergence relations between the corresponding problems and prove the convergence of the associated pressures arising from the incompressibility constraints we also present new results on the time convexity of the entropy for some of the dynamical interpolations along the way we extend previous results by lavenant calc var partial differ equ 56 6 170 2017 and benamou et al generalized incompressible flows multi marginal transport and sinkhorn algorithm arxiv preprint arxiv 1710 08234 2017

young measures generated by ideal incompressible fluid flows in their seminal paper diperna and majda commun math phys 108 4 667 689 1987 introduced the notion of a measure valued solution for the incompressible euler equations in order to capture complex phenomena present in limits of approximate solutions such as persistence of oscillation and development of concentrations furthermore they gave several explicit examples exhibiting such phenomena in this paper we show that any measure valued solution can be generated by a sequence of exact weak solutions in particular this gives rise to a very large arguably too large set of weak solutions of the incompressible euler equations

coordinate systems and analytic expansions for three body atomic wavefunctions i partial summation for the fock expansion in hyperspherical coordinates a survey of analytic techniques for solving the two electron atomic schrodinger equation is presented the hyperspherical formalism is introduced and specialised to the case of two electrons and zero total angular momentum following fock the schrodinger equation is then converted to an infinite set of coupled second order differential equations by proposing an expansion including logarithmic functions of the interparticle coordinates the equivalence of the techniques of pluvinage and hylleraas 1985 to the fock expansion is demonstrated and the method for solution is illustrated the extension to states of arbitrary angular momentum and excited states is indicated methods for simplifying the recurrence relation generated by the fock expansion are used to determine the highest power logarithmic terms to sixth order finally the wavefunction for s states is given to second order as a singly infinite sum of legendre polynomials

a variational problem arising from speech recognition by following the general approach of deformable templates the problem of recognizing a single word independently of the speaker is shown to lead to the computation of the minimum value of some particular functional more precisely this allows us to recover for each possible word in a prespecified set the best matching with the recorded signal by selecting the minimum value the recognition problem can be solved in this paper we are concerned with the detailed study of the variational problem associated with this sort of functional namely the existence of a minimum point and the features of such a minimum finally we discuss the convergence of a discretized finite dimensional approximation suggested by the engineering literature on this subject

on the complexity of the positive semidefinite zero forcing number abstract the positive semidefinite zero forcing number of a graph is a graph parameter that arises from a non traditional type of graph colouring and is related to a more conventional version of zero forcing we establish a relation between the zero forcing and the fast mixed searching which implies some np completeness results for the zero forcing problem relationships between positive semidefinite zero forcing sets and clique coverings are well understood for chordal graphs building upon constructions associated with optimal tree covers and forest covers we present a linear time algorithm for computing the positive semidefinite zero forcing number of chordal graphs we also prove that it is np complete to determine whether a graph has a positive semidefinite zero forcing set with an additional property

on complex singularities of the 2d euler equation at short times abstract we present a study of complex singularities of a two parameter family of solutions for the two dimensional euler equation with periodic boundary conditions and initial conditions 0 z 1 z 2 f p cos p z f q cos q z in the short time asymptotic regime as has been shown numerically in pauls et al w pauls t matsumoto u frisch j bec nature of complex singularities for the 2d euler equation physica d 219 2006 40 59 the type of the singularities depends on the angle between the modes p and q thus the fourier coefficients of the solutions decrease as g k c k e k with the exponent depending on here we show for the two particular cases of going to zero and to that the type of the singularities can be determined very accurately being characterised by 5 2 and 3 respectively in these two cases we are also able to determine the subdominant corrections furthermore we find that the geometry of the singularities in these two cases is completely different the singular manifold being located over different points in the real domain

ricci curvature for metric measure spaces via optimal transport we dene a notion of a measured length space x having nonnegative n ricci curvature for n 2 1 1 or having1 ricci curvature bounded below byk fork2 r the denitions are in terms of the displacement convexity of certain functions on the associated wasserstein metric space p2 x of probability measures we show that these properties are preserved under measured gromov hausdor limits we give geometric and analytic consequences this paper has dual goals one goal is to extend results about optimal transport from the setting of smooth riemannian manifolds to the setting of length spaces a second goal is to use optimal transport to give a notion for a measured length space to have ricci curvature bounded below we refer to 11 and 44 for background material on length spaces and optimal transport respectively further bibliographic notes on optimal transport are in appendix f in the present introduction we motivate the questions that we address and we state the main results to start on the geometric side there are various reasons to try to extend notions of curvature from smooth riemannian manifolds to more general spaces a fairly general setting is that of length spaces meaning metric spaces x d in which the distance between two points equals the inmum of the lengths of curves joining the points in the rest of this introduction we assume that x is a compact length space alexandrov gave a good notion of a length space having curvature bounded below by k with k a real number in terms of the geodesic triangles in x in the case of a riemannian manifold m with the induced length structure one recovers the riemannian notion of having sectional curvature bounded below by k length spaces with alexandrov curvature bounded below by k behave nicely with respect to the gromovhausdor topology on compact metric spaces modulo isometries they form

on exact analytical solutions for the few particle schrodinger equation ii the ground state of helium analytical methods for solving the schrodinger equation directly can be applied to few particle systems this is illustrated by deriving a solution to the first order perturbation equation for the ground state of helium this solution is the in form of a partial wave expansion in spherical polar coordinates with legendre polynomials as the angular functions the radial functions include polynomials and exponential integral functions arbitrary parameters in the formal solution associated with the square integrability of the wavefunction are identified their values are determined by a least squares method the same arbitrary parameters occur in formal solutions of the higher order perturbation equations it is evident that a similar treatment can be applied to these equations and to the eigenvalue problem