growth rates for the linearized motion of fluid interfaces away from equilibrium we consider the motion of a two dimensional interface separating an inviscid incompressible irrotational fluid influenced by gravity from a region of zero density we show that under certain conditions the equations of motion linearized about a presumed time dependent solution are wellposed that is linear disturbances have a bounded rate of growth if surface tension is neglected the linear equations are well posed provided the underlying exact motion satisfies a condition on the acceleration of the interface relative to gravity similar to the criterion formulated by g i taylor if surface tension is included the linear equations are well posed without qualifications whether the fluid is above or below the interface an interesting qualitative structure is found for the linear equations a lagrangian approach is used like that of numerical work such as 3 except that the interface is assumed horizontal at infinity certain integral equations which occur involving double layer potentials are shown to be solvable in the present case 1993 john wiley sons inc

the l2 boundedness of pseudodifferential operators on donne une nouvelle demonstration du theoreme de calderon vaillancourt on obtient la l 2 continuite de a x d si le symbole a x satisfait a certaines conditions

splash singularity for water waves we exhibit smooth initial data for the two dimensional 2d water wave equation for which we prove that smoothness of the interface breaks down in finite time moreover we show a stability result together with numerical evidence that there exist solutions of the 2d water wave equation that start from a graph turn over and collapse in a splash singularity self intersecting curve in one point in finite time

a new approach to analyticity of dirichlet neumann operators this paper outlines the theoretical background of a new approach towards an accurate and well conditioned perturbative calculation of dirichlet neumann operators dnos on domains that are perturbations of simple geometries previous work on the analyticity of dnos has produced formulae that as we have found are very ill conditioned we show how a simple change of variables can lead to recursions that satisfy analyticity estimates without relying on subtle cancellation properties at the heart of previous formulae

breakdown of smoothness for the muskat problem in this paper we show that there exists analytic initial data in the stable regime for the muskat problem such that the solution turns to the unstable regime and later breaks down that is no longer belongs to c4

efficient computation of trees with minimal atom bond connectivity index the atom bond connectivity abc index is one of the recently most investigated degree based molecular structure descriptors that have applications in chemistry for a graph g the abc index is defined as u v e g d u d v 2 d u d v where d u is the degree of vertex u in g and e g is the set of edges of g despite many attempts in the last few years it is still an open problem to characterize trees with minimal abc index in this paper we present an efficient approach of computing trees with minimal abc index by considering the degree sequences of trees and some known properties of the trees with minimal abc index the obtained results disprove some existing conjectures and suggest new ones to be set

global solvability of a free boundary three dimensional incompressible viscoelastic fluid system with surface tension motivated by beale commun pure appl math 34 359 392 1981 arch ration mech anal 84 307 352 1983 1984 we investigate the global well posedness of a free boundary problem of a three dimensional incompressible viscoelastic fluid system in an infinite strip and with surface tension on the upper free boundary provided that the initial data is sufficiently close to the equilibrium state

well posedness in smooth function spaces for moving boundary 1 d compressible euler equations in physical vacuum we prove well posedness for the 3 d compressible euler equations with moving physical vacuum boundary with an equation of state given by the so called gamma gas law for gamma 1 the physical vacuum singularity requires the sound speed c to go to zero as the square root of the distance to the moving boundary and thus creates a degenerate and characteristic hyperbolic free boundary system wherein the density vanishes on the free boundary the uniform kreiss lopatinskii condition is violated and manifest derivative loss ensues nevertheless we are able to establish the existence of unique solutions to this system on a short time interval which are smooth in sobolev spaces all the way to the moving boundary and our estimates have no derivative loss with respect to initial data our proof is founded on an approximation of the euler equations by a degenerate parabolic regularization obtained from a specific choice of a degenerate artificial viscosity term chosen to preserve as much of the geometric structure of the euler equations as possible we first construct solutions to this degenerate parabolic regularization using a new higher order hardy type inequality we then establish estimates for solutions to this degenerate parabolic system which are independent of the artificial viscosity parameter solutions to the compressible euler equations are found in the limit as the artificial viscosity tends to zero our regular solutions can be viewed as degenerate viscosity solutions out methodology can be applied to many other systems of degenerate and characteristic hyperbolic systems of conservation laws

contour dynamics of incompressible 3 d fluids in a porous medium with different densities we consider the problem of the evolution of the interface given by two incompressible fluids through a porous medium which is known as the muskat problem and in two dimensions it is mathematically analogous to the two phase hele shaw cell we focus on a fluid interface given by a jump of densities being the equation of the evolution obtained using darcy s law we prove local well posedness when the smaller density is above stable case and in the unstable case we show ill posedness

expander graphs and their applications a major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists since expander graphs the protagonists of our story come up in numerous and often surprising contexts in both fields but perhaps we should start with a few words about graphs in general they are of course one of the prime objects of study in discrete mathematics however graphs are among the most ubiquitous models of both natural and human made structures in the natural and social sciences they model relations among species societies companies etc in computer science they represent networks of communication data organization computational devices as well as the flow of computation and more in mathematics cayley graphs are useful in group theory graphs carry a natural metric and are therefore useful in geometry and though they are just one dimensional complexes they are useful in certain parts of topology e g knot theory in statistical physics graphs can represent local connections between interacting parts of a system as well as the dynamics of a physical process on such systems the study of these models calls then for the comprehension of the significant structural properties of the relevant graphs but are there nontrivial structural properties which are universally important expansion of a graph requires that it is simultaneously sparse and highly connected expander graphs were first defined by bassalygo and pinsker and their existence first proved by pinsker in the early 70s the property of being an expander seems significant in many of these mathematical computational and physical contexts it is not surprising that expanders are useful in the design and analysis of communication networks what is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness in mathematics we will encounter e g their role in the study of metric embeddings and in particular in work around the baum connes conjecture expansion is closely related to the convergence rates of markov chains and so they play a key role in the study of monte carlo algorithms in statistical mechanics and in a host of practical computational applications the list of such interesting and fruitful connections goes on and on with so many applications we will not even

on the regularity of the flow map for the gravity capillary equations abstract we prove via explicitly constructed initial data that solutions to the gravity capillary wave system in r 3 representing a 2 d air water interface immediately fail to be c 3 with respect to the initial data if the initial data h 0 0 h s 1 2 h s for s 3 similar results hold in r 2 domains with a 1 d interface furthermore we discuss the related threshold for the pure gravity water wave system

expansion of random graphs new proofs new results we present a new approach to showing that random graphs are nearly optimal expanders this approach is based on recent deep results in combinatorial group theory it applies to both regular and irregular random graphs let gamma be a random d regular graph on n vertices and let lambda be the largest absolute value of a non trivial eigenvalue of its adjacency matrix it was conjectured by alon combinatorica 6 2 83 96 1986 that a random d regular graph is almost ramanujan in the following sense for every varepsilon 0 lambda 0 a a s lambda rho varepsilon where rho is the spectral radius of the universal cover of omega when omega is regular we get a bound of rho 0 84 and for an arbitrary omega we prove a nearly optimal upper bound of sqrt 3 rho this is a substantial improvement upon all known results by friedman linial puder lubetzky sudakov vu and addario berry griffiths

finite time singularities for the free boundary incompressible euler equations in this paper we prove the existence of smooth initial data for the 2d free boundary incompressible euler equations also known for some particular scenarios as the water wave problem for which the smoothness of the interface breaks down in nite time into a splash singularity or a splat singularity

turning waves and breakdown for incompressible flows we consider the evolution of an interface generated between two immiscible incompressible and irrotational fluids specifically we study the muskat and water wave problems we show that starting with a family of initial data given by f0 the interface reaches a regime in finite time in which is no longer a graph therefore there exists a time t where the solution of the free boundary problem parameterized as f t blows up f l t in particular for the muskat problem this result allows us to reach an unstable regime for which the rayleigh taylor condition changes sign and the solution breaks down

sharp estimates for pseudo differential operators with symbols of limited smoothness and commutators we consider here pseudo differential operators whose symbol x is not infinitely smooth with respect to x decomposing such symbols into four sometimes five components and using tools of paradifferential calculus we derive sharp estimates on the action of such pseudo differential operators on sobolev spaces and give explicit expressions for their operator norm in terms of the symbol x we also study commutator estimates involving such operators and generalize or improve the so called kato ponce and calderon coifman meyer estimates in various ways

well posedness and stability results for a quasilinear periodic muskat problem abstract we study the muskat problem describing the spatially periodic motion of two fluids with equal viscosities under the effect of gravity in a vertical unbounded two dimensional geometry we first prove that the classical formulation of the problem is equivalent to a nonlocal and nonlinear evolution equation expressed in terms of singular integrals and having only the interface between the fluids as unknown secondly we show that this evolution equation has a quasilinear structure which is at a formal level not obvious and we also disclose the parabolic character of the equation exploiting these aspects we establish the local well posedness of the problem for arbitrary initial data in h s s with s 3 2 2 determine a new criterion for the global existence of solutions and uncover a parabolic smoothing property besides we prove that the zero steady state solution is exponentially stable

asymmetrical three dimensional travelling gravity waves we consider periodic travelling gravity waves at the surface of an infinitely deep perfect fluid the pattern is non symmetric with respect to the propagation direction of the waves and we consider a general non resonant situation defining a couple of amplitudes varepsilon 1 varepsilon 2 along the basis of wave vectors which satisfy the dispersion relation we first give the formal asymptotic expansion of bifurcating solutions in powers of varepsilon 1 varepsilon 2 then introducing an additional equation for the unknown diffeomorphism of the torus associated with an irrational rotation number which allows us to transform the differential at the successive points of the newton iteration method into a differential equation with two constant main coefficients we are able to use a descent method leading to an invertible differential then by using an adapted nash moser theorem we prove the existence of solutions with the above asymptotic expansion for values of the couple varepsilon 1 2 varepsilon 2 2 in a subset of the first quadrant of the plane with asymptotic full measure at the origin

interface evolution the hele shaw and muskat problems we study the dynamics of the interface between two incompressible 2 d ows where the evolution equation is obtained from darcy s law the free boundary is given by the discontinuity among the densities and viscosities of the uids this physical scenario is known as the two dimensional muskat problem or the two phase hele shaw ow we prove local existence in sobolev spaces when initially the dierence of the gradients of the pressure in the normal direction has the proper sign an assumption which is also known as the rayleigh taylor condition

global solutions for the gravity water waves equation in dimension 3 we show existence of global solutions for the gravity water waves equation in dimension 3 in the case of small data the proof combines energy estimates which yield control of l 2 related norms with dispersive estimates which give decay in l 8 to obtain these dispersive estimates we use an analysis in fourier space the study of space and time resonances is then the crucial point

a maximum principle for the muskat problem for fluids with different densities we study the fluid interface problem given by two incompressible fluids with different densities evolving by darcy s law this scenario is known as the muskat problem for fluids with the same viscosities being in two dimensions mathematically analogous to the two phase hele shaw cell we prove in the stable case the denser fluid is below a maximum principle for the l norm of the free boundary

approximate counting uniform generation and rapidly mixing markov chains the paper studies effective approximate solutions to combinatorial counting and uniform generation problems using a technique based on the simulation of ergodic markov chains it is shown that for self reducible structures almost uniform generation is possible in polynomial time provided only that randomised approximate counting to within some arbitrary polynomial factor is possible in polynomial time it follows that for self reducible structures polynomial time randomised algorithms for counting to within factors of the form 1 n are available either for all fl e r or for no fi e r a substantial part of the paper is devoted to investigating the rate of convergence of finite ergodic markov chains and a simple but powerful characterisation of rapid convergence for a broad class of chains based on a structural property of the underlying graph is established finally the general techniques of the paper are used to derive an almost uniform generation procedure for labelled graphs with a given degree sequence which is valid over a much wider range of degrees than previous methods this in turn leads to randomised approximate counting algorithms for these graphs with very good

global existence singular solutions and ill posedness for the muskat problem the muskat or muskat leibenzon problem describes the evolution of the interface between two immiscible fluids in a porous medium or hele shaw cell under applied pressure gradients or fluid injection extraction in contras t to the hele shaw problem the one phase version of the muskat problem there are few nontrivial exact solutions or analytic results for the muskat problem for the stable forward muskat problem in which the higher viscosity fluid ex pands into the lower viscosity fluid we show global in time existence for initial data that is a small perturbation of a flat interface the initial data in this result ma y contain weak e g curvature singularities for the unstable backwa rd problem in which the higher viscosity fluid contracts we construct singular solution s that start off with smooth initial data but develop a point of infinite curvature at fi nite time c 2004 wiley periodicals inc

entropy waves the zig zag graph product and new constant degree expanders the main contribution of this work is a new type of graph product which we call the zig zag product taking a product of a large graph with a small graph the resulting graph inherits roughly its size from the large one its degree from the small one and its expansion properties from both iteration yields simple explicit constructions of constant degree expanders of arbitrary size starting from one constant size expander crucial to our intuition and simple analysis of the properties of this graph product is the view of expanders as functions which act as entropy wave propagators they transform probability distributions in which entropy is concentrated in one area to distributions where that concentration is dissipated in these terms the graph product affords the constructive interference of two such waves

on the cauchy problem for gravity water waves we are interested in the system of gravity water waves equations without surface tension our purpose is to study the optimal regularity thresholds for the initial conditions in terms of sobolev embeddings the initial surfaces we consider turn out to be only of c 3 2 epsilon class for some epsilon 0 and consequently have unbounded curvature while the initial velocities are only lipschitz we reduce the system using a paradifferential approach

the free boundary euler equations with large surface tension abstract we study the free boundary euler equations with surface tension in three spatial dimensions showing that the equations are well posed if the coefficient of surface tension is positive then we prove that under natural assumptions the solutions of the free boundary motion converge to solutions of the euler equations in a domain with fixed boundary when the coefficient of surface tension tends to infinity

large time existence for 1d green naghdi equations we consider here the 1d green naghdi equations that are commonly used in coastal oceanography to describe the propagation of large amplitude surface waves we show that the solution of the green naghdi equations can be constructed by a standard picard iterative scheme so that there is no loss of regularity of the solution with respect to the initial condition

well posedness in smooth function spaces for the moving boundary three dimensional compressible euler equations in physical vacuum we prove well posedness for the three dimensional compressible euler equations with moving physical vacuum boundary with an equation of state given by p c for 1 the physical vacuum singularity requires the sound speed c to go to zero as the square root of the distance to the moving boundary and thus creates a degenerate and characteristic hyperbolic free boundary system wherein the density vanishes on the free boundary the uniform kreiss lopatinskii condition is violated and manifest derivative loss ensues nevertheless we are able to establish the existence of unique solutions to this system on a short time interval which are smooth in sobolev spaces all the way to the moving boundary and our estimates have no derivative loss with respect to initial data our proof is founded on an approximation of the euler equations by a degenerate parabolic regularization obtained from a specific choice of a degenerate artificial viscosity term chosen to preserve as much of the geometric structure of the euler equations as possible we first construct solutions to this degenerate parabolic regularization using a higher order version of hardy s inequality we then establish estimates for solutions to this degenerate parabolic system which are independent of the artificial viscosity parameter solutions to the compressible euler equations are found in the limit as the artificial viscosity tends to zero our regular solutions can be viewed as degenerate viscosity solutions our methodology can be applied to many other systems of degenerate and characteristic hyperbolic systems of conservation laws

cutoff phenomena for random walks on random regular graphs the cutoff phenomenon describes a sharp transition in the convergence of a family of ergodic finite markov chains to equilibrium many natural families of chains are believed to exhibit cutoff and yet establishing this fact is often extremely challenging an important such family of chains is the random walk on g n d a random d regular graph on n vertices it is well known that almost every such graph for d 3 is an expander and even essentially ramanujan implying a mixing time of o logn according to a conjecture of peres the simple random walk on g n d for such d should then exhibit cutoff with high probability whp as a special case of this durrett conjectured that the mixing time of the lazy random walk on a random 3 regular graph is whp 6 o 1 log2n in this work we confirm the above conjectures and establish cutoff in total variation its location and its optimal window both for simple and for non backtracking random walks on g n d namely for any fixed d 3 the simple random walk on g n d whp has cutoff at d d 2 logd 1n with window order logn surprisingly the non backtracking random walk on g n d whp has cutoff already at logd 1n with constant window order we further extend these results to g n d for any d no 1 that grows with n beyond which the mixing time is o 1 where we establish concentration of the mixing time on one of two consecutive integers

polynomial growth estimates for multilinear singular integral operators on demontre un theoreme pour des mesures de carleson et on l applique a l operateur de kato en dimension 1 on demontre un critere de limitation general on donne une application aux commutateurs de coolderon on etudie les d commutateurs

on structural properties of abc minimal chemical trees abstract the atom bond connectivity abc index of a connected graph g is defined as a b c g u v e g d u d v 2 d u d v where d w is the degree of vertex w in g and e g is the set of edges of g given greedy trees perturbed by different degree sequences we investigate the behaviors on their abc index in this paper moreover we also give the minimum abc index and its structural properties of the chemical trees with n vertices and p pendant vertices for n 3 p 2

some free boundary problems recast as nonlocal parabolic equations abstract in this work we demonstrate that a class of some one and two phase free boundary problems can be recast as nonlocal parabolic equations on a submanifold the canonical examples would be one phase hele shaw flow as well as its two phase analog we also treat nonlinear versions of both one and two phase problems in the special class of free boundaries that are graphs over r d we give a precise characterization that shows their motion is equivalent to that of a solution of a nonlocal fractional nonlinear parabolic equation for functions on r d our main observation is that the free boundary condition defines a nonlocal operator having what we call the global comparison property a consequence of the connection with nonlocal parabolic equations is that for free boundary problems arising from translation invariant elliptic operators in the positive and negative phases one obtains in a uniform treatment for all of the problems one and two phase a propagation of modulus of continuity for viscosity solutions of the free boundary flow

on abc eigenvalues and abc energy abstract let g be a graph with n vertices and let d i be the degree of its i th vertex the abc matrix of g is the square matrix of order n whose i j entry is equal to d i d j 2 d i d j if the i th vertex and the j th vertex of g are adjacent and 0 otherwise this matrix related closely to the atom bond connectivity abbreviated as abc index was recently introduced by estrada as a matrix representation of the probability of visiting a nearest neighbor edge from one side or the other of a given edge in a graph which in the context of molecular graphs can be related to the polarizing capacity of the bond considered the abc eigenvalues of g are the eigenvalues of its abc matrix and the abc energy of g is the sum of the absolute values of its abc eigenvalues in this paper some new results for the abc eigenvalues and the abc energy of a graph are presented

no splash theorems for fluid interfaces the article 1 by gancedo and strain in pnas studies how singularities may develop in the initially smooth interfaces separating two or more incompressible fluids the fluids and interfaces are assumed to evolve by either of the two standard systems of equations from fluid mechanics namely the surface quasi geostrophic sqg sharp front equation 2 or the muskat equation 3 gancedo and strain prove that initially smooth fluid interfaces evolving by either of those two equations cannot form a splash singularity 4

on structural properties of trees with minimal atom bond connectivity index iv solving a conjecture about the pendent paths of length three abstract the atom bond connectivity abc index is one of the most investigated degree based molecular structure descriptors with a variety of chemical applications it is known that among all connected graphs the trees minimize the abc index however a full characterization of trees with a minimal abc index is still an open problem by now one of the proved properties is that a tree with a minimal abc index may have at most one pendent path of length 3 with the conjecture that it cannot be a case if the order of a tree is larger than 1178 here we provide an affirmative answer of a strengthened version of that conjecture showing that a tree with minimal abc index cannot contain a pendent path of length 3 if its order is larger than 415

spectra of lifted ramanujan graphs abstract a random n lift of a base graph g is its cover graph h on the vertices n v g where for each edge uv in g there is an independent uniform bijection and h has all edges of the form i u i v a main motivation for studying lifts is understanding ramanujan graphs and namely whether typical covers of such a graph are also ramanujan let g be a graph with largest eigenvalue 1 and let be the spectral radius of its universal cover friedman 2003 12 proved that every new eigenvalue of a random lift of g is o 1 2 1 1 2 with high probability and conjectured a bound of o 1 which would be tight by results of lubotzky and greenberg 1995 15 linial and puder 2010 17 improved friedman s bound to o 2 3 1 1 3 for d regular graphs where 1 d and 2 d 1 this translates to a bound of o d 2 3 compared to the conjectured 2 d 1 here we analyze the spectrum of a random n lift of a d regular graph whose nontrivial eigenvalues are all at most in absolute value we show that with high probability the absolute value of every nontrivial eigenvalue of the lift is o log this result is tight up to a logarithmic factor and for d 2 3 e it substantially improves the above upper bounds of friedman and of linial and puder in particular it implies that a typical n lift of a ramanujan graph is nearly ramanujan

finite time analyticity for the two and three dimensional kelvin helmholtz instability the well posed property for the finite time vortex sheet problem with analytic initial data was first conjectured by birkhoff in two dimensions and is shown here to hold both in two and three dimensions incompressible inviscid and irrotational flow with a velocity jump across an interface is assumed in two dimensions global existence of a weak solution to the euler equation with such initial conditions is established in three dimensions a lagrangian representation of the vortex sheet analogous to the birkhoff equation in two dimensions is presented

rayleigh taylor breakdown for the muskat problem with applications to water waves the muskat problem models the evolution of the interface between two different fluids in porous media the rayleigh taylor condition is natural to reach linear stability of the muskat problem we show that the rayleigh taylor condition may hold initially but break down in finite time as a consequence of the method used we prove the existence of water waves turning

well posedness and analyticity of solutions to a water wave problem with viscosity abstract the water wave problem models the free surface evolution of an ideal fluid under the influence of gravity and surface tension the governing equations are a central model in the study of open ocean wave propagation but they possess a surprisingly difficult and subtle well posedness theory in this paper we establish the existence and uniqueness of spatially periodic solutions to the water wave equations augmented with physically inspired viscosity suggested in the recent work of dias et al 2008 16 as we show this viscosity which can be arbitrarily weak not only delivers an enormously simplified well posedness theory for the governing equations but also justifies a greatly stabilized numerical scheme for use in studying solutions of the water wave problem

difference equations isoperimetric inequality and transience of certain random walks the difference laplacian on a square lattice in rn has been stud ied by many authors in this paper an analogous difference operator is studied for an arbitrary graph it is shown that many properties of the laplacian in the continuous setting e g the maximum principle the harnack inequality and cheeger s bound for the lowest eigenvalue hold for this difference oper ator the difference laplacian governs the random walk on a graph just as the laplace operator governs the brownian motion as an application of the theory of the difference laplacian it is shown that the random walk on a class of graphs is transient the random walks we consider are defined as follows let k be a connected graph i e a one dimensional simplicial complex for a vertex x e k let m x denote the number of edges emanating from x the probability that a particle moves from x to another vertex y e k is l m x if x and y are connected by an edge and it is zero otherwise as observed by courant friedrichs and lewy cfl for the case of a square lattice in the plane this random walk is intimately related to the difference analog of the laplacian

nonlinear stability of the current vortex sheet to the incompressible mhd equations in this paper we solve a long standing open problem nonlinear stability of the current vortex sheet in the ideal incompressible magnetohydrodynamics under the syrovatskij stability condition this result gives the first rigorous confirmation of the stabilizing effect of the magnetic field on kelvin helmholtz instability 2017 wiley periodicals inc

long wave approximations for water waves in this paper we obtain new nonlinear systems describing the interaction of long water waves in both two and three dimensions these systems are symmetric and conservative rigorous convergence results are provided showing that solutions of the complete free surface euler equations tend to associated solutions of these systems as the amplitude becomes small and the wavelength large using this result as a tool a rigorous justification of all the two dimensional approximate systems recently put forward and analysed by bona chen and saut is obtained in the two dimensional context our methods also allows a significant improvement of the convergence estimate obtained by schneider and wayne in their justification of the decoupled korteweg de vries approximation of the two dimensional euler equations it also follows from our theory that coupled models provide a better description than the decoupled ones over short time scales results are obtained both on an unbounded domain for solutions that evanesce at infinity as well as for solutions that are spatially periodic

computer search for trees with minimal abc index the abc index is a degree based molecular structure descriptor that found chemical applications finding the connected graph s of a given order whose abc index is minimal is a hitherto unsolved problem but it is known that these must be trees in this paper by combining mathematical arguments and computer based modeling we establish the basic structural features of the minimum abc trees

on the edge version atom bond connectivity and geometric arithmetic indices of certain graph operations let dl g e be the degree of an edge e in line graph l g of a graph g the edge versions of atom bond connectivity abce and geometric arithmetic gae indices of g are defined as efe l g dl g e dl g f 2dl g e dl g f and gae g efe l g 2dl g e dl g f dl g e dl g f in this paper we study the abce and gae indices for joint graphs and certain graph operations

on structural properties of trees with minimal atom bond connectivity index iii the atom bond connectivity abc index is a degree based graph topological index that found chemical applications including those of predicting the stability of alkanes and the strain energy of cycloalkanes several structural properties of the trees with minimal abc index were proved recently however the complete characterization of the minimal abc trees is still an open problem it is known that minimal abc trees can have at most one pendent path of length 3 it is also known that the minimal abc trees that have a pendent path of length 3 do not contain so called bk branches with k 4 and do not contain more than two b2 branches here we improve the latter result by showing that minimal abc trees of order larger than 168 and with a pendent path of length 3 do not contain b2 branches moreover we show that trees with minimal abc index with a pendent path of length 3 do not contain b1 branches

a note on interface dynamics for convection in porous media we studied the fluid interface problem through porous media given by two incompressible 2 d fluids of different densities this problem is mathematically analogous to the dynamics interface for convection in porous media where the free boundary evolves between fluids with different temperatures we found a new formula for the evolution equation of the free boundary parameterized as a function in the periodic case in this formula there is not a principal value in the non local integral operator involved in the equation giving a simpler system using this formulation we perform numerical simulations in the stable case denser fluid below which shows a strong regularity effect in the periodic interface

on the finite time splash and splat singularities for the 3 d free surface euler equations we prove that the 3 d free surface incompressible euler equations with regular initial geometries and velocity fields have solutions which can form a finite time splash or splat singularity first introduced in castro et al splash singularity for water waves http arxiv org abs 1106 2120v2 2011 wherein the evolving 2 d hypersurface the moving boundary of the fluid domain self intersects at a point or on surface such singularities can occur when the crest of a breaking wave falls unto its trough or in the study of drop impact upon liquid surfaces our approach is founded upon the lagrangian description of the free boundary problem combined with a novel approximation scheme of a finite collection of local coordinate charts as such we are able to analyze a rather general set of geometries for the evolving 2 d free surface of the fluid we do not assume the fluid is irrotational and as such our method can be used for a number of other fluid interface problems including compressible flows plasmas as well as the inclusion of surface tension effects

interface evolution water waves in 2 d abstract we study the free boundary evolution between two irrotational incompressible and inviscid fluids in 2 d without surface tension we prove local existence in sobolev spaces when initially the difference of the gradients of the pressure in the normal direction has the proper sign an assumption which is also known as the rayleigh taylor condition the well posedness of the full water wave problem was first obtained by wu 1997 20 the methods introduced in this paper allow us to consider multiple cases with or without gravity but also a closed boundary or a periodic boundary with the fluids placed above and below it it is assumed that the initial interface does not touch itself being a part of the evolution problem to check that such property prevails for a short time as well as it does the rayleigh taylor condition depending conveniently upon the initial data the addition of the pressure equality to the contour dynamic equations is obtained as a mathematical consequence and not as a physical assumption from the mere fact that we are dealing with weak solutions of euler s equation in the whole space

remarks on the graovac ghorbani index of bipartite graphs the atom bond connectivity abc index is a well known degree based molecular structure descriptor with a variety of chemical applications in 2010 graovac and ghorbani introduced a distance based analog of this index the graovac ghorbani gg index which yielded promising results when compared to analogous descriptors in this paper we investigate the structure of graphs that maximize and minimize the gg index specifically we show that amongst all bipartite graphs the minimum gg index is attained by a complete bipartite graph while the maximum gg index is attained by a path or a cycle like graph the structure of the resulting graph depends on the number of vertices through the course of the research we also derive an asymptotic estimate of the gg index of paths in order to obtain our results we introduce a normalized version of the gg index and call it the normalized graovac ghorbani ngg index finally we discuss some related open questions as a potential extension of our work

validity of the korteweg de vries approximation for the two dimensional water wave problem in the arc length formulation we consider the two dimensional water wave problem in an infinitely long canal of finite depth both with and without surface tension it has been proven by several authors that long wavelength solutions to this problem can be approximated over a physically relevant timespan by solutions of the korteweg de vries equation or for certain values of the surface tension by solutions of the kawahara equation these proofs are formulated either in lagrangian or in eulerian coordinates in this paper we provide a new proof which is simpler more elementary and shorter moreover the rigorous justification of the kdv approximation can be given for the cases with and without surface tension together by one proof in our proof we parametrize the free surface by arc length and use some geometrically and physically motivated variables with good regularity properties this formulation of the water wave problem has already been of great usefulness for ambrose and masmoudi to simplify the proof of the local well posedness of the water wave problem in sobolev spaces 2011 wiley periodicals inc

nonstrictly hyperbolic nonlinear systems on etudie le caractere bien pose du probleme aux valeurs initiales localement en temps pour une classe de systemes hyperboliques non lineaires

almost global wellposedness of the 2 d full water wave problem we consider the problem of global in time existence and uniqueness of solutions of the 2 d infinite depth full water wave equation it is known that this equation has a solution for a time period 0 t e for initial data of the form e where t depends only on in this paper we show that for such data there exists a unique solution for a time period 0 e t e this is achieved by better understandings of the nature of the nonlinearity of the full water wave equation

optimal construction of edge disjoint paths in random graphs given a graph g v e with n vertices m edges and a family of kappa pairs of vertices in v we are interested in finding for each pair ai bi a path connecting ai to bi such that the set of kappa paths so found is edge disjoint for arbitrary graphs the problem is cal np complete although it is in cal p if kappa is fixed we present a polynomial time randomized algorithm for finding the optimal number of edge disjoint paths up to constant factors in the random graph gn m for all edge densities above the connectivity threshold the graph is chosen first then an adversary chooses the pairs of endpoints our results give the first tight bounds for the edge disjoint paths problem for any nontrivial class of graphs

on extremality of abc spectral radius of a tree abstract the abc matrix of a graph g recently introduced by estrada is the square matrix of order g whose i j entry is equal to d i d j 2 d i d j if the i th vertex and the j th vertex of g are adjacent and 0 otherwise where d i is the degree of the i th vertex of g the abc spectral radius of g is the largest eigenvalue of the abc matrix of g which is denoted by g in this paper we prove that for any tree t of order n 3 2 cos n 1 t n 2 with equality in the left resp right inequality if and only if t is isomorphic to the path p n resp the star k 1 n 1

functional limit theorems for random regular graphs consider d uniformly random permutation matrices on n labels consider the sum of these matrices along with their transposes the total can be interpreted as the adjacency matrix of a random regular graph of degree 2d on n vertices we consider limit theorems for various combinatorial and analytical properties of this graph or the matrix as n grows to infinity either when d is kept fixed or grows slowly with n in a suitable weak convergence framework we prove that the finite but growing in length sequences of the number of short cycles and of cyclically non backtracking walks converge to distributional limits we estimate the total variation distance from the limit using stein s method as an application of these results we derive limits of linear functionals of the eigenvalues of the adjacency matrix a key step in this latter derivation is an extension of the kahn szemeredi argument for estimating the second largest eigenvalue for all values of d and n

well posedness in sobolev spaces of the full water wave problem in 2 d we consider the motion of the interface of 2 d irrotational incompressible inviscid water wave with air above water and surface tension zero we show that the interface is always not accelerating into the water region normal to itself as rapidly as the normal acceleration of gravity as long as the interface is nonself intersect we therefore obtain the existence and uniqueness of solutions of the full water wave problem locally in time for any initial interface which is nonself intersect including the case that the interface is of multiple heights

well posedness of the water waves equations 1 1 presentation of the problem the water waves problem for an ideal liquid consists of describing the motion of the free surface and the evolution of the velocity field of a layer of perfect incompressible irrotational fluid under the influence of gravity in this paper we restrict our attention to the case when the surface is a graph parameterized by a function x where t denotes the time variable and x xi xd rd the horizontal spatial variables the method developed here works equally well for any integer d 1 but the only physically relevant cases are of course d 1 and d 2 the layer of fluid is also delimited from below by a not necessarily flat bottom parameterized by a time independent function b x we denote by q t the fluid domain at time t the incompressibility of the fluids is expressed by 1 1 div v 0 in qu t 0

well posedness of the free surface incompressible euler equations with or without surface tension we provide a new method for treating free boundary problems in perfect fluids and prove local in time well posedness in sobolev spaces for the free surface incompressible 3d euler equations with or without surface tension for arbitrary initial data and without any irrotationality assumption on the fluid this is a free boundary problem for the motion of an incompressible perfect liquid in vacuum wherein the motion of the fluid interacts with the motion of the free surface at highest order

well posedness of the linearized problem for mhd contact discontinuities abstract we study the free boundary problem for contact discontinuities in ideal compressible magnetohydrodynamics mhd they are characteristic discontinuities with no flow across the discontinuity for which the pressure the magnetic field and the velocity are continuous whereas the density and the entropy may have a jump under the rayleigh taylor sign condition p n 0 on the jump of the normal derivative of the pressure satisfied at each point of the unperturbed contact discontinuity we prove the well posedness in sobolev spaces of the linearized problem for 2d planar mhd flows

justification of the nonlinear schrodinger equation for the evolution of gravity driven 2d surface water waves in a canal of finite depth in 1968 v e zakharov derived the nonlinear schrodinger equation for the two dimensional water wave problem in the absence of surface tension that is for the evolution of gravity driven surface water waves in order to describe slow temporal and spatial modulations of a spatially and temporarily oscillating wave packet in this paper we give a rigorous proof that the wave packets in the two dimensional water wave problem in a canal of finite depth can be approximated over a physically relevant timespan by solutions of the nonlinear schrodinger equation

well posedness in sobolev spaces of the full water wave problem in 3 d we consider the motion of the interface separating an inviscid incompressible irrotational fluid from a region of zero density in three dimensional space we assume that the fluid region is below the vacuum the fluid is under the influence of gravity and the surface tension is zero assume that the density of mass of the fluid is one the gravitational field is 0 0 1 the free interface is t at time t 0 and the fluid occupies the region t the motion of the fluid is described by vt v v 0 0 1 p on t t 0 euler s equation 1 1 div v 0 on t t 0 incompressible 1 2 curl v 0 on t t 0 irrotational 1 3 where v v1 v2 v3 is the fluid velocity p is the fluid pressure since we neglect the surface tension the pressure is zero on the interface so on the interface p 0 on t 1 4 1 v is tangent to the free surface t t 1 5 we want to find solutions of system 1 1 1 5 taking prescribed initial data such that for every fixed t 0 t approaches the xy plane at infinity and v x y z t 0 vt x y z t 0 as x y z since the fluid is assumed irrotational incompressible we can reduce the study of the entire motion to the motion of the free surface the above model is a 3 d water wave model it is generally known that when surface tension is neglected the motion of the interface between an inviscid fluid and vacuum under the influence of gravity can be subject to taylor instability 8 22 in a previous work 24 we studied the 2 dimensional water wave model we showed that for a 2 d water wave the sign condition relating to taylor instability always holds for nonself intersecting interface that is the motion of the interface is not subject to taylor instability we showed further that the 2 d full nonlinear water wave problem is uniquely solvable in sobolev spaces locally in time for any initially nonself intersecting interface earlier works on the well posedness in sobolev spaces of the 2 d water wave problem include nalimov 18 yosihara 25 and walter craig 6 where the main results concern the well posedness of the

the long wave limit for the water wave problem i the case of zero surface tension the korteweg de vries equation boussinesq equation and many other equations can be formally derived as approximate equations for the two dimensional water wave problem in the limit of long waves here we consider the classical problem concerning the validity of these equations for the water wave problem in an infinitely long canal without surface tension we prove that the solutions of the water wave problem in the long wave limit split up into two wave packets one moving to the right and one to the left where each of these wave packets evolves independently as a solution of a korteweg de vries equation our result allows us to describe the nonlinear interaction of solitary waves 2000 john wiley sons inc

global classical solution of muskat free boundary problem abstract in this paper the muskat problem which describes a two phase flow of two fluids for example oil and water in porous media is discussed the problem involves in seeking two time dependent harmonic functions u1 x y t and u2 x y t in oil and water regions respectively and the interface between oil and water i e the free boundary y x t such that on the free boundary u 1 u 2 v n k 1 u 1 n k 2 u 2 n where n the unit normal vector on the free boundary toward oil region vn is the normal velocity of the free boundary k1 and k2 are positive constants satisfying k1 k2 we prove the existence of classical solution globally in time under some reasonable assumptions the argument developed in this paper can be used in any multidimensional case