weak stability of ricci expanders with positive curvature operator we prove the l infty stability of expanding gradient ricci solitons with positive curvature operator and quadratic curvature decay at infinity

elimination of infrequent variables improves average case performance of satisfiability algorithms preprocessing a random instance i of cnf satisfiability in order to remove infrequent variables those which appear once or twice in an instance from i is considered the model used to generate random instances is the popular random clause size model with parameters n the number of clauses r the number of boolean variables from which clauses are composed and p the probability that a variable appears in a clause as a positive or negative literal it is shown that exhaustive search over such preprocessed instances runs in polynomial average time over a significantly larger parameter space than has been shown for any other algorithm under the random clause size model when n r epsilon epsilon epsilon 0 and pr epsilon geqslant frac 2 3

elliptic equations strongly degenerate at a point abstract in this paper we will mainly propose some problems for a class of degenerate elliptic equations either linear or nonlinear we will study some special cases of these problems and reveal some phenomena which may not have been noticed previously our problems originated from the self similar solutions of the heat flow of harmonic maps we will prove that the self similar solutions or the so called quasi harmonic spheres are discontinuous at infinity for the equivariant case in other words the equivariant quasi harmonic spheres are not continuous images of topological spheres

cnf satisfiability test by counting and polynomial average time the average case performance of an algorithm for cnf sat recently introduced by the author is discussed it is shown that the algorithm takes polynomial average time for a class of cnf equations satisfying the condition that for a constant c p 2 v geqq ln t c where v is the number of variables t is the number of clauses and p is the probability that a given literal appears in a clause it was known that backtracking plus the pure literal rule a common way of solving cnf sat takes polynomial average time if p geqq varepsilon any small constant or p leqq c ln v v 3 2 but no algorithms were known to take polynomial average time for all t in the range c ln v v 3 2 0 the new algorithm runs in polynomial average time for p alpha ln v v 1 2 so the unfavorable region is reduced to c ln v v 3 2 p alpha ln v v 1 2

regularity of solutions to axisymmetric navier stokes equations with a slightly supercritical condition abstract consider an axisymmetric suitable weak solution of 3d incompressible navier stokes equations with nontrivial swirl v v r e r v e v z e z let z denote the axis of symmetry and r be the distance to the z axis if the solution satisfies a slightly supercritical assumption that is v c ln ln r r for 0 0 028 when r is small then we prove that v is regular this extends the results in 6 16 18 where regularities under critical assumptions such as v c r were proven as a useful tool in the proof of our main result an upper bound estimate to the fundamental solution of the parabolic equation with a critical drift term will be given in the last part of this paper

probe order backtracking the algorithm for constraint satisfaction problems probe order backtracking has an average running time much faster than any previously analyzed algorithm under conditions where solutions are common the algorithm uses a probing assignment a preselected test assignment to unset variables to help guide the search for a solution if the problem is not satisfied when the unset variables are temporarily set to the probing assignment the algorithm selects one of the relations which is not satisfied by the probing assignment and selects an unset variable which affects the value of that relation it then does a backtracking splitting step where it generates subproblems by setting the selected variable each possible way each subproblem is simplified and then solved recursively for random problems with v variables t clauses and probability p that a literal appears in a clause the average time for probe order backtracking is no more than v n when p geq ln t v plus lower order terms the best previous result was p geq sqrt ln t v when the algorithm is combined with an algorithm of franco that makes selective use of resolution the average time for solving random problems is no more than v n for all values of p when t leq o n 1 3 v ln v 2 3 the best previous result was t leq o n 1 3 v ln v 1 6 probe order backtracking also runs in polynomial average time when p leq 1 v compared with the best previous result of p leq 1 2v with probe order backtracking the range of p that leads to more than polynomial time is much smaller than that for previously analyzed algorithms

erratum to on the weak l p hodge decomposition and beurling ahlfors transforms on complete riemannian manifolds based on a new martingale representation formula we prove some quantitative upper bound estimates of the l p norm of some singular integral operators on complete riemannian manifolds this leads us to establish the weak l p hodge decomposition theorem and to prove the l p boundedness of the beurling ahlfors transforms on complete non compact riemannian manifolds with non negative weitzenbock curvature operator

liouville theorems for quasi harmonic functions abstract let n be a compact riemannian manifold a self similar solution for the heat flow is a harmonic map from r n e x 2 2 n 2 d s 0 2 to n n 3 which was also called a quasi harmonic sphere cf lin and wang 1999 1 here d s 0 2 is the euclidean metric in r n it arises from the blow up analysis of the heat flow at a singular point when n r and without the energy constraint we call this a quasi harmonic function in this paper we prove that there is neither a nonconstant positive quasi harmonic function nor a nonconstant l p r n e x 2 2 n 2 d s 0 2 p n n 2 quasi harmonic function however for all 1 p n n 2 there exists a nonconstant quasi harmonic function in l p r n e x 2 2 n 2 d s 0 2

on a navier stokes allen cahn model with inertial effects abstract a mathematical model describing the flow of two phase fluids in a bounded container is considered under the assumption that the phase transition is influenced by inertial effects the model couples a variant of the navier stokes system for the velocity u with an allen cahn type equation for the order parameter relaxed in time in order to introduce inertia at the microscopic level the resulting model is characterized by second order material derivatives which constitute the main difficulty in the mathematical analysis in order to obtain a tractable problem a viscous relaxation term is then included in the phase equation the mathematical results consist in existence of weak solutions in 3d and under additional assumptions existence and uniqueness of strong solutions in 2d a partial characterization of the long time behavior of solutions is also given and in particular some issues related to dissipation of energy are discussed

average time analysis of clause order backtracking backtracking aigorithms solve problems by selecting a variable and assigning each possible value to the variable the resulting subproblems are simplified and solved recursively simple backtracking selects variables in a fixed order clause order backtracking selects variables from the first nontrivial clause that has not yet been satisfied formulas are given for the average time used by clause order backtracking when solving random cnf satisfiability problems where the problem sets have v variables t clauses and a probability p of a literal being in a clause the average time for clause order backtracking is always less than that for simple backtracking it leads to polynomial time under many conditions where simple backtracking uses exponential average time cases where clause order backtracking uses average time less than v n in the limit of v going to infinity include p sqrt ln t ln v 2 v the second resul

an analysis of a phase field model of a free boundary a mathematical analysis of a new approach to solidification problems is presented a free boundary arising from a phase transition is assumed to have finite thickness the physics leads to a system of nonlinear parabolic differential equations existence and regularity of solutions are proved invariant regions of the solution space lead to physical interpretations of the interface a rigorous asymptotic analysis leads to the gibbs thompson condition which relates the temperature at the interface to the surface tension and curvature

singularity and regularity of solutions to a certain class of degenerate elliptic equations abstract in this note we consider the solution of the degenerate elliptic system 1 div f x u 0 x b 1 0 u b 1 c b 1 where b 1 denotes the unit ball in r n and f is smooth and increasing on 0 1 with f 0 0 f r 0 r 0 it is known that there exists a unique solution u c b 1 0 l b 1 to this elliptic system here we will study the property of u at the origin at first we give the necessary and sufficient condition such that u can be extended to a continuous function in b furthermore we also give the necessary and sufficient condition such that u is a holder continuous function in b 1

three circles theorems for harmonic functions we proved two three circles theorems for harmonic functions on manifolds in integral sense as one application on manifold with nonnegative ricci curvature whose tangent cone at infinity is the unique metric cone with unique conic measure we showed the existence of nonconstant harmonic functions with polynomial growth this existence result recovered and generalized the former result of ding and led to a complete answer of ni s conjecture furthermore in similar context combining the techniques of estimating the frequency of harmonic functions with polynomial growth which were developed by colding and minicozzi we confirmed their conjecture about the uniform bound of frequency

on nonnegatively curved hypersurfaces in hyperbolic space author s bonini v ma s qing j abstract 2018 springer verlag gmbh germany part of springer nature in this paper we prove a conjecture of alexander and currier that states except for covering maps of equidistant surfaces in hyperbolic 3 space a complete nonnegatively curved immersed hypersurface in hyperbolic space is necessarily properly embedded

strongly damped wave equations with critical nonlinearities abstract this note deals with the strongly damped nonlinear wave equation u t t u t u f u t g u h with dirichlet boundary conditions where both the nonlinearities f and g exhibit a critical growth while h is a time independent forcing term the existence of an exponential attractor of optimal regularity is proven as a corollary a regular global attractor of finite fractal dimension is obtained

blow up and global existence for heat flows of harmonic maps in this paper it is proved that the solution to the evolution problem for harmonic maps blows up in finite time if the initial map belongs to some nontrivial homotopy class and the initial energy is sufficiently small

boundary behavior of harmonic functions on manifolds in this paper we mainly set up a kind of representation theorem of harmonic functions on manifolds with ricci curvature bounded below and study non tangential limits of harmonic functions

a phase field model for the mixture of two incompressible fluids and its approximation by a fourier spectral method abstract a phase field model for the mixture of two incompressible fluids is presented in this paper the model is based on an energetic variational formulation it consists of a navier stokes system linear momentum equation coupled with a cahn hilliard equation phase field equation through an extra stress term and the transport term the extra stress represents the phase induced capillary effect for the mixture due to the surface tension a fourier spectral method for the numerical approximation of this system is proposed and analyzed numerical results illustrating the robustness and versatility of the model are presented

results related to threshold phenomena research in satisfiability lower bounds we present a history of results related to the threhold phenomena which arise in the study of random conjunctive normal form cnf formulas in a companion paper d achlioptas theoret comput sci this volume the major ideas used to achieve many of the lower bounds results on the location of the threshold are described in an informal intuitive manner

global existence of weak solutions to a nonlocal cahn hilliard navier stokes system abstract a well known diffuse interface model consists of the navier stokes equations nonlinearly coupled with a convective cahn hilliard type equation this system describes the evolution of an incompressible isothermal mixture of binary fluids and it has been investigated by many authors here we consider a variant of this model where the standard cahn hilliard equation is replaced by its nonlocal version more precisely the gradient term in the free energy functional is replaced by a spatial convolution operator acting on the order parameter while the potential f may have any polynomial growth therefore the coupling with the navier stokes equations is difficult to handle even in two spatial dimensions because of the lack of regularity of we establish the global existence of a weak solution in the two dimensional case we also prove that such a solution satisfies the energy identity and a dissipative estimate provided that f fulfills a suitable coercivity condition

on a diffuse interface model for two phase flows of viscous incompressible fluids with matched densities we study a diffuse interface model for the flow of two viscous incompressible newtonian fluids of the same density in a bounded domain the fluids are assumed to be macroscopically immiscible but a partial mixing in a small interfacial region is assumed in the model moreover diffusion of both components is taken into account this leads to a coupled navier stokes cahn hilliard system which is capable of describing the evolution of droplet formation and collision during the flow we prove the existence of weak solutions of the non stationary system in two and three space dimensions for a class of physical relevant and singular free energy densities which ensures in contrast to the usual case of a smooth free energy density that the concentration stays in the physical reasonable interval furthermore we find that unique strong solutions exist in two dimensions globally in time and in three dimensions locally in time moreover we show that for any weak solution the concentration is uniformly continuous in space and time because of this regularity we are able to show that any weak solution becomes regular for large times and converges as t to a solution of the stationary system these results are based on a regularity theory for the cahn hilliard equation with convection and singular potentials in spaces of fractional time regularity as well as on maximal regularity of a stokes system with variable viscosity and forces in l2 0 hs s in 0 frac12 which are new themselves

foliations the ergodic theorem and brownian motion regarding the leaves of a compact foliated riemannian manifold as the orbits of a dynamical system we will prove a birkhoff ergodic theorem asserting that the spatial average of a function equals its time average along random paths in the leaves there are several unexpected developments in the foundations first there always are measures on any compact foliated riemannian manifold m f with respect to which an adequate brownian motion along the leaves ergodic theorem holds second relative to these measures which are called harmonic there is a liouville type theorem for any bounded bore1 function on the manifold which is harmonic on each leaf namely for almost all leaves relative to any one of these harmonic measures this leaf harmonic function is constant on each leaf third the harmonic measures have a very reasonable local characterization in any foliation coordinate system they are transversal sums of h l dx l where h l is a positive harmonic function on the leaf l and dx l is the riemannian measure on l these statements comprise theorem 1 which is stated more precisely later in the introduction the first fact is surprising because up to now the only measures attached to foliations were the holonomy invariant transverse measures of plante 141 and ruelle sullivan 151 and some foliations do not admit these the second fact is unexpected because there are foliations where all the leaves are hyperbolic planes and each hyperbolic plane being conformal to the disk admits 285 0022 1236183 3 00

lower bounds for nodal sets of eigenfunctions we prove lower bounds for the hausdorff measure of nodal sets of eigenfunctions 0 introduction let m be a smooth closed riemannian manifold and the laplace operator a function u is said to be an eigenfunction with eigenvalue if u u 0 1 with our convention on the sign of the eigenvalues are non negative and go to infinity one of the most fundamental questions about eigenfunctions is to understand the sets where they vanish these sets are called nodal sets nodal sets are n 1 dimensional manifolds away from n 2 dimensional singular sets where the gradient also vanishes so it is natural to estimate the n 1 dimensional hausdorff measure hn 1 the main result of this short note is the following lower bound theorem 1 given a closed n dimensional riemannian manifold m there exists c so that hn 1 u 0 c 3 n 4 0 2 in particular theorem 1 gives a uniform lower bound in dimension n 3 in y s t yau conjectured the lower bound c in all dimensions this was proven for surfaces by bruning b and yau independently and for real analytic metrics by donnelly and fefferman in df but remains open in the smooth case the authors were partially supported by nsf grants dms 0606629 dms 0906233 and nsf frg grants dms 0854774 and dms 0853501 778 t h colding w p minicozzi ii the donnelly fefferman argument leads to exponentially decaying lower bounds in the smooth case see also hl in m mangoubi considered eigenfunctions on a ball in a manifold and proved lower bounds for the volume of the subset of the ball where the function is positive and the subset where it is negative combined with the isoperimetric inequality one can get the lower bound c 3 n 2 1 2n for the measure of the nodal set on the entire manifold recently sogge and zelditch sz proved the lower bound c 7 3n 8 their argument is completely different and is based in part on a beautiful new integral formula relating the l1 norm of u on the nodal set and the l1 norm of u on m finally note that some papers use 2 for the eigenvalue i e u 2 u with that convention our bound is c 3 n 2 1 finding good balls throughout m will be a fixed closed manifold with laplace operator and u will be an eigenfunction with eigenvalue and m u 2 1 we will always assume that 1 since our interest is in what happens when goes to infinity the first step is to fix a scale r depending on lemma 1 there exists a 0 so that u has a zero in every ball of radius a 3 this lemma is standard but we will include a proof since it is so short proof if u does not vanish on b r 3 then barta s theorem implies that the lowest dirichlet eigenvalue on this ball is at least let be a function that is identically one on b r 6 and that cuts off linearly to zero on the annulus b r 3 b r 6 it follows that 2 2 36r 2 vol b r 3 vol b r 6 c r2 1 1 where c comes from the bishop gromov volume comparison see p 275 of g and depends only on n the ricci curvature of m and an upper bound for r since this is impossible for r2 c 1 the lemma follows from now on we set r a 1 2 with a given by lemma 1 next we use a standard covering argument to decompose the manifold m into small balls of radius r if b is a ball in m then we write 2b for the ball with the same center as b and twice the radius lemma 2 there exists a collection bi of balls of radius r in m so that m i bi and each point of m is contained in at most cm cm m of the double balls 2bi s proof choose a maximal disjoint collection of balls b r 2 pi it follows immediately from maximality that the double balls bi br pi cover m suppose that p m is contained in balls b2r p1 b2r pk in particular the disjoint balls b r 2 p1 b 2 pk are all contained in b3r p so that k i 1 vol b r 2 pi vol b3r p 1 2

a matrix li yau hamilton estimate for kahler ricci flow in this paper we prove a new matrix li yau hamilton lyh estimate for kahler ricci flow on manifolds with nonnegative bisectional curvature the form of this new lyh estimate is obtained by the interpolation consideration originated in ch by chow this new inequality is shown to be connected with perelman s entropy formula through a family of differential equalities in the rest of the paper we show several applications of this new estimate and its corresponding estimate for linear heat equation these include a sharp heat kernel comparison theorem generalizing the earlier result of li and tian lt a manifold version of stoll s theorem st on the characterization of algebraic divisors and a localized monotonicity formula for analytic subvarieties which sharpens the bishop volume comparison theorem motivated by the connection between the heat kernel estimate and the reduced volume monotonicity of perelman p we prove a sharp lower bound heat kernel estimate for the time dependent heat equation which is in a certain sense dual to perelman s monotonicity of the reduced volume as an application of this new monotonicity formula we show that the blow down limit of a certain type long time solution is a gradient expanding soliton we also illustrate the connection between the new lyh estimate and the hessian comparison theorem of fin on the forward reduced distance local monotonicity formulae on entropy and forward reduced volume are also derived

the pure literal rule and polynomial average time for a simple parameterized model of conjunctive normal form predicates we show that a simplified version of the davis putnam procedure can for many values of the parameters solve the satisfiability problem in polynomial average time let v be the number of variables t v the number of clauses in a predicate and p v the probability that a given literal appears in a clause p v is the same for all literals let varepsilon be any small positive constant and n any large positive integer then a version of the davis putnam procedure that uses only backtracking and the pure literal rule uses average time that is polynomial in the problem size when any of the following conditions are true for large v 1 t v leqq n ln v 2 t v geqq exp varepsilon v 3 p v geqq varepsilon or 4 p v leqq n ln v v 3 2 until recently the best previous bounds for cases 1 and 4 were t v leqq ln ln v ln3 and p v leqq exp v ln ln v these results show that t

on nonlinear wave equations with degenerate damping and source terms in this article we focus on the global well posedness of the differential equation u tt u u k j u t u p 1 u in x 0 t where j is a sub differential of a continuous convex function j under some conditions on j and the parameters in the equations we obtain several results on the existence of global solutions uniqueness nonexistence and propagation of regularity under nominal assumptions on the parameters we establish the existence of global generalized solutions with further restrictions on the parameters we prove the existence and uniqueness of a global weak solution in addition we obtain a result on the nonexistence of global weak solutions to the equation whenever the exponent p is greater than the critical value k m and the initial energy is negative we also address the issue of propagation of regularity specifically under some restriction on the parameters we prove that solutions that correspond to any regular initial data such that u 0 h 2 h 1 0 u 1 h 1 0 are indeed strong solutions

energy decay rates for the semilinear wave equation with nonlinear localized damping and source terms abstract in this paper we develop an intrinsic approach to derivation of energy decay rates for the semilinear wave equation with localized interior nonlinear monotone damping g u t and a source term f u the proposed approach allows to consider in an unified way much more general classes of hyperbolic problems than addressed before in the literature these generalizations refer to both geometric and topological aspects of the problem the method leads to optimal decay rates for solutions of semilinear hyperbolic equations driven by a source of critical exponent and subjected to a nonlinear damping localized in a small region adjacent to a portion of the boundary the distinct features of the model include i neumann boundary conditions are assumed and ii no growth conditions are imposed on the damping g s it is well known that neumann boundary does not satisfy lopatinski condition and therefore the analysis of propagation of energy in the absence of the damping on the neumann part of the boundary requires special geometric considerations in addition the sole conditions assumed on g s are monotonicity continuity and g 0 0 in particular no differentiability and no growth conditions are imposed on the damping both at the origin and at the infinity the asymptotic decay rates for the energy function are obtained from an intrinsic algorithm driven by solutions of simple ode several examples illustrate the theory by exhibiting various decay rates exponential algebraic rational logarithmic etc for the energy functional an important corollary of our energy decay theorem is a stability result which shows that under certain conditions when dissipation is sublinear at infinity the solution of the system remains uniformly bounded for all time in the norms above the finite energy level even in the presence of a nonlinear source term

eigenfunctions for quasi laplacian abstract to study the regularity of heat flow lin and wang 1999 introduced the quasi harmonic sphere which is a harmonic map from m r m e x 2 2 m 2 d s 0 2 to n with finite energy here d s 0 2 is euclidean metric in r m ding and yongqiang 2006 showed that if the target is a sphere any equivariant quasi harmonic spheres is discontinuous at infinity the metric g e x 2 2 m 2 d s 0 2 is quite singular at infinity and it is not complete in this paper we mainly study the eigenfunction of quasi laplacian g e x 2 2 m 2 g 0 g 0 h g 0 e x 2 2 m 2 h for h x 2 4 in particular we show that there are infinite number of eigenvalues of the quasi laplacian g of which the corresponding eigenfunctions are discontinuous at and any non constant eigenfunction of drifted laplacian h g 0 g 0 h g 0 is discontinuous at infinity

average time for the full pure literal rule the simplified pure literal algorithm solves satisfiability problems by choosing variables in a fixed order and then generating subproblems for various values of the chosen variable if some value satisfies every relation that depends on the chosen variable then only the subproblem for that preferred value is generated otherwise a subproblem is generated for every value of the variable the full pure literal algorithm chooses variables that have a preferred value before choosing those that do not a recurrence equation is found for the average time used by the full pure literal rule algorithm when solving random conjunctive normal form satisfiability problems the random problems are characterized by the number of variables v the number of clauses t and the probability that a literal is in a clause p an asymptotic lower bound analysis shows that running time is more than polynomial in v when t increases more rapidly than ln v 2 when p is set to maximize the running time a numerical study indicates that the results of the lower bound analysis are close to the true results thus the full pure literal rule is faster than the simple pure literal rule where the polynomial time boundary occurs at t lnv but slower than franco s infrequent variable algorithm where the time is polynomial for all p when t o v16

stabilization of the wave equation with localized nonlinear damping abstract we prove some decay estimates of the energy of the wave equation in a bounded domain the damping is nonlinear and is effective only in a neighborhood of a suitable subset of the boundary the method of proof is direct and is based on the multipliers technique on some integral inequalities due to haraux and komornik and on a judicious idea of conrad and rao

convergence properties of optimization algorithms for the sat problem the satisfiability sat problem is a basic problem in computing theory presently an active area of research on sat problem is to design efficient algorithms to find a solution for a satisfiable conjunctive normal form cnf formula a new formulation the universal sat problem model which transforms the sat problem on boolean space into an optimization problem on real space has been developed j gu 1988 1992 1994 many optimization techniques such as the steepest descent method newton s method and the coordinate descent method can be used to solve the universal sat problem we prove that when the initial solution is sufficiently close to the optimal solution the steepest descent method has a linear convergence ratio spl beta 1 newton s method has a convergence ratio of order two and the convergence ratio of the steepest descent method is approximately 1 spl beta m for the universal sat problem with m variables an algorithm based on the coordinate descent method for the universal sat problem is also presented experimental results show that this algorithm is more efficient than some previous ones in finding a solution for certain classes of the satisfiable cnf formulas

singularities of first kind in the harmonic map and yang mills heat flows by constructing examples which are explicit up to solving an ode we prove that singularities of first kind exist for the harmonic map and yang mills heat flows as a by product we also get a simplified proof of ratto s ding s theorem about the existence of harmonic hopf constructions

uniqueness of l 1 solutions for the laplace equation and the heat equation on riemannian manifolds on deduit une hypothese geometrique optimale sur une variete de riemann complete qui assure des proprietes d unicite pour des solutions l 1 de l equation de laplace et de la chaleur

the dirichlet problem at infinity for manifolds of negative curvature on considere l existence de fonctions harmoniques bornees sur des varietes simplement connexes n n de courbure negative