discrete spline methods for solving two point fractional bagley torvik equation a new discrete spline method is developed to solve the two point fractional bagley torvik equation the method is based on discrete spline function and a nonstandard grunwald letnikov difference nsgd and the weighted and shifted grunwald letnikov difference wsgd operators to approximate the fractional derivative bounds for grunwald letnikov weights are considered convergence analysis is discussed and a class of second order and third order methods are obtained illustrative examples are presented to validate the practical usefulness of the methods

the feller property on riemannian manifolds abstract the asymptotic behavior of the heat kernel of a riemannian manifold gives rise to the classical concepts of parabolicity stochastic completeness or conservative property and feller property or c 0 diffusion property both parabolicity and stochastic completeness have been the subject of a systematic study which led to discovering not only sharp geometric conditions for their validity but also an incredible rich family of tools techniques and equivalent concepts ranging from maximum principles at infinity function theoretic tests khas minskii criterion comparison techniques etc the present paper aims to move a number of steps forward in the development of a similar apparatus for the feller property

representations p adiques et equations differentielles in this paper we associate to every p adic representation v a p adic differential equation d rig v that is to say a module with a connection over the robba ring we do this via the theory of fontaine s k modules this construction enables us to relate the theory of k modules to p adic hodge theory we explain how to construct dcris v and dst v from d rig v which allows us to recognize semi stable or crystalline representations the connection is then unipotent or trivial on d rig v 1 t in general the connection has an infinite number of regular singularities but v is de rham if and only if those are apparent singularities a structure theorem for modules over the robba ring allows us to get rid of all singularities at once and to obtain a classical differential equation with a frobenius structure using this we construct a functor from the category of de rham representations to that of classical p adic differential equations with frobenius structure a recent theorem of y andre gives a complete description of the structure of the latter object this allows us to prove fontaine s p adic monodromy conjecture every de rham representation is potentially semi stable as an application we can extend to the case of arbitrary perfect residue fields some results of hyodo h1g h1st of perrin riou the semi stability of ordinary representations of colmez absolutely crystalline representations are of finite height and of bloch and kato if the weights of v are 2 then bloch kato s exponential expv is an isomorphism

sharp estimates for capacities and applications to symmetric diffusions we give sharp estimates for the capacity of compact setf x wherex is the state space of a strongly regular dirichlet form of diffusion type these estimates are in terms of the reference measurem and the caratheodory metric onx for instance cap 0 f leqq left mathop smallint limits 0 infty frac dr v r right 1 leqq 2 left mathop smallint limits 0 infty frac r dr v r right 1 wherev r m 0 x f r from these estimates one easily obtains lower estimates for the green function as well as sharp criteria for polarity and for recurrence the latter we apply to the question of hitting the nodal set 0 for the process associated with the operator delta 2 tfrac nabla varphi varphi nabla and to recurrence questions for divergence form operators a ond n witha a ij which degenerate at d

symmetric green s functions on complete manifolds on demontre d une facon constructive que l operateur de laplace admet une fonction de green symetrique g x y sur une variete de riemann complete

the inverse mean curvature flow and the riemannian penrose inequality let m be an asymptotically flat 3 manifold of nonnegative scalar curvature the riemannian penrose inequality states that the area of an outermost minimal surface n in m is bounded by the adm mass m according to the formula n 16 m 2 we develop a theory of weak solutions of the inverse mean curvature flow and employ it to prove this inequality for each connected component of n using geroch s monotonicity formula for the adm mass our method also proves positivity of bartnik s gravitational capacity by computing a positive lower bound for the mass purely in terms of local geometry

generalized liouville property for schrodinger operator on riemannian manifolds in this paper we prove that the dimension of the space of positive bounded respectively mathcal l harmonic functions on a complete riemannian manifold with mathcal l regular ends is equal to the number of ends mathcal l nonparabolic ends respectively this result is a solution of an open problem of grigor yan related to the liouville property for the schrodinger operator mathcal l we also prove that if a given complete riemannian manifold is roughly isometric to a complete riemannian manifold satisfying the volume doubling condition the poincare inequality and the finite covering condition on each end then the dimension of the space of positive bounded respectively solutions for the schrodinger operator with a potential satisfying a certain decay rate on the manifold is equal to the number of ends mathcal l nonparabolic ends respectively this is a partial answer of the question suggested by li related to the regularity of ends of a complete riemannian manifold especially our results directly generalize various earlier results of yau of li and tam of grigor yan and of present authors but with different techniques that the peculiarity of the schrodinger operator demands

function theory random paths and covering spaces pour des varietes de riemann connexes m on etudie la relation entre la theorie de fonction harmonique sur m les proprietes statistiques des chemins aleatoires sur m et la structure geometrique globale de m

analytic and geometric background of recurrence and non explosion of the brownian motion on riemannian manifolds we provide an overview of such properties of the brownian motion on complete non compact riemannian manifolds as recurrence and non explosion it is shown that both properties have various analytic characterizations in terms of the heat kernel green function liouville properties etc on the other hand we consider a number of geometric conditions such as the volume growth isoperimetric inequalities curvature bounds etc which are related to recurrence and non explosion

on the growth of supersolutions of nonlinear pde s on exterior domains abstract we obtain a comparison principle on annuli with catenoid like functions for supersolutions of non linear elliptic pdes over exterior domains in a non positively curved manifold with a pole this result is applied to get an upper estimate on the growth of such supersolutions and in particular of exterior graphs of non negative mean curvature

wavelet operational matrix method for solving fractional differential equations with variable coefficients abstract in this paper another operational matrix method based on haar wavelet is proposed to solve the fractional differential equations with variable coefficients the haar wavelet operational matrix of fractional order integration is derived without using the block pulse functions considered in li and zhao 2010 1 the operational matrix of fractional order integration is utilized to reduce the initial equations to a system of algebraic equations some examples are included to demonstrate the validity and applicability of the method moreover compared with the known technique the methodology is shown to be much more efficient and accurate

hardy littlewood theory for semigroups soit t t t 0 un semi groupe symetrique sousmarkovien et soit n 2 alors on a les proprietes equivalentes a dim t t n b il existe c 2 0 s t t t f c 2 t n 2 f 1 f l 1 x t 0 en fait a b meme dans le cas n 2

instability of the liouville property for quasi isometric riemannian manifolds and reversible markov chains on montre par un exemple que la propriete de liouville n est pas un invariant quasi isometrique des varietes de riemann et des chaines de markov reversibles

drift diffusion equations on domains in rd essential self adjointness and stochastic completeness abstract we consider the problem of quantum and stochastic confinement for drift diffusion equations on domains r d we obtain various sufficient conditions on the behavior of the coefficients near the boundary of which ensure the essential self adjointness or stochastic completeness of the symmetric form of the drift diffusion operator 1 d the proofs are based on the method developed in 31 for quantum confinement on bounded domains in r d in particular for stochastic confinement we combine the liouville property with agmon type exponential estimates for weak solutions

a two dimensional haar wavelets method for solving systems of pdes in this paper we modify the idea of haar wavelets method to obtain semi analytical solutions for the systems of three dimensional nonlinear partial differential equations theoretical considerations are discussed to demonstrate the efficiency of the method a test problem is presented the approximate solutions of the system of three dimensional nonlinear partial differential equations are compared with the exact solutions as well as existing numerical solutions found in the literature the numerical solutions which are obtained using the suggested method show that numerical solutions are in a very good coincidence with the exact solutions

new exact solutions of nonlinear fractional acoustic wave equations in ultrasound in this paper new exact solutions of fractional nonlinear acoustic wave equations have been devised the travelling periodic wave solutions of fractional burgers hopf equation and khokhlov zabolotskaya kuznetsov kzk equation have obtained by first integral method nonlinear ultrasound modelling is found to have an increasing number of applications in both medical and industrial areas where due to high pressure amplitudes the effects of nonlinear propagation are no longer negligible taking nonlinear effects into account the ultrasound beam analysis makes more accurate in these applications the burgers hopf equation is one of the extensively studied models in mathematical physics in addition the kzk parabolic nonlinear wave equation is one of the most widely employed nonlinear models for propagation of 3d diffraction sound beams in dissipative media in the present analysis these nonlinear equations have solved by first integral method as a result new exact analytical solutions have been obtained first time ever for these fractional order acoustic wave equations the obtained results are presented graphically to demonstrate the efficiency of this proposed method

gaussian upper bounds for the heat kernels of some second order operators on riemannian manifolds abstract we describe a method of obtaining gaussian upper bounds on heat kernels which unifies and improves recent results for hypoelliptic operators in divergence form and for laplace beltrami operators on complete riemannian manifolds

heat semigroup on a complete riemannian manifold let m be a complete riemannian manifold and p t x y the minimal heat kernel on m let p t be the associated semigroup we say that m is stochastically complete if int m p t x y dy 1 for all t 0 x in m we say that m has the c 0 diffusion property or the feller property if p tf vanishes at infinity for all t 0 whenever f is so let x 0 in m and let kappa r 2 geq inf ric x rho x x 0 leq r rho is the riemannian distance we prove that m is stochastically complete and has the c 0 diffusion property if int infty c kappa r 1 dr infty by studying the radial part of the riemannian brownian motion on m

computation of eigenvalues in proportionally damped viscoelastic structures based on the fixed point iteration abstract linear viscoelastic structures are characterized by dissipative forces that depend on the history of the velocity response via hereditary damping functions the free motion equation leads to a nonlinear eigenvalue problem characterized by a frequency dependent damping matrix in the present paper a novel and efficient numerical method for the computation of the eigenvalues of linear and proportional or lightly non proportional viscoelastic structures is developed the central idea is the construction of two complex valued functions of a complex variable whose fixed points are precisely the eigenvalues this important property allows the use of these functions in a fixed point iterative scheme with help of some results in fixed point theory necessary conditions for global and local convergence are provided it is demonstrated that the speed of convergence is linear and directly depends on the level of induced damping in addition under certain conditions the recursive method can also be used for the calculation of non viscous real eigenvalues in order to validate the mathematical results two numerical examples are analyzed one for single degree of freedom systems and another for multiple ones

superlinear elliptic inequalities on manifolds abstract let m be a complete non compact riemannian manifold and let be a radon measure on m we study the problem of existence or non existence of positive solutions to a semilinear elliptic inequality u u q in m where q 1 we obtain necessary and sufficient criteria for existence of positive solutions in terms of green function of in particular explicit necessary and sufficient conditions are given when m has nonnegative ricci curvature everywhere in m or more generally when green s function satisfies the 3g inequality

on the parabolic kernel of the schrodinger operator etude des equations paraboliques du type q x t t u x t 0 sur une variete riemannienne generale introduction estimations de gradients inegalites de harnack majorations et minorations des solutions fondamentales equation de la chaleur et noyau de green operateur de schrodinger

on the geometry of static spacetimes we review geometrical properties of a static spacetime m g including geodesic completeness causality standard splittings compact m closed geodesics and geodesic connectedness we pay special attention to the critical quadratic behaviour at infinity of the coefficients 1 g k k being k a timelike irrotational killing vector field which essentially control completeness causality and geodesic connectedness recent references are specially discussed

helicoidal minimal surfaces of prescribed genus for every genus g we prove that mathbf s 2 times mathbf r contains complete properly embedded genus g minimal surfaces whose two ends are asymptotic to helicoids of any prescribed pitch we also show that as the radius of the mathbf s 2 tends to infinity these examples converge smoothly to complete properly embedded minimal surfaces in mathbf r 3 that are helicoidal at infinity we prove that helicoidal surfaces in mathbf r 3 of every prescribed genus occur as such limits of examples in mathbf s 2 times mathbf r

a note on lattices in semi stable representations let p be a prime k a finite extension over mathbb q p and g rm gal overline k k we extend kisin s theory on varphi modules of finite e u height to give a new classification of g stable mathbb z 1 p lattices in semi stable representations

minimal graphs over riemannian surfaces and harmonic diffeomorphisms we construct a parabolic entire minimal graph s over a finite topology complete riemannian surface sigma of curvature 1 and infinite area thus of non parabolic conformal type the vertical projection of this graph yields a harmonic diffeomorphism from s onto sigma the proof uses the theory of divergence lines to construct minimal graphs we also generalize a theorem of r schoen let g 1 and g 2 be two complete metrics on a orientable surface s with compact boundary and suppose int s r 2 k g 2 d sigma g 2 le c ln 2 r for some c 0 and all r 0 if there is a harmonic diffeomorphism from s g 1 to s g 2 then s g 1 is parabolic

analytical solution of the bagley torvik equation by adomian decomposition method abstract the fractional derivative has been occurring in many physical problems such as frequency dependent damping behavior of materials motion of a large thin plate in a newtonian fluid creep and relaxation functions for viscoelastic materials the pi d controller for the control of dynamical systems etc phenomena in electromagnetics acoustics viscoelasticity electrochemistry and material science are also described by differential equations of fractional order the solution of the differential equation containing fractional derivative is much involved instead of application of the existing methods an attempt has been made in the present analysis to obtain the solution of bagley torvik equation r l bagley p j torvik on the appearance of the fractional derivative in the behavior of real materials asme j appl mech 51 1984 294 298 i podlubny fractional differential equations academic press san diego ca usa 1999 by the relatively new adomian decomposition method the results obtained by this method are then graphically represented and then compared with those available in the work of podlubny i podlubny fractional differential equations academic press san diego ca usa 1999 a good agreement of the results is observed

a probabilistic view on singularities the aim of this article is to promote the use of probabilistic methods in the study of problems in mathematical general relativity two new and simple singularity theorems whose features are different from the classical singularity theorems are proved using probabilistic methods under some energy conditions and without any causal or initial boundary assumption simple conditions on the energy flow imply probabilistic incompleteness

syntomic complexes and p adic nearby cycles we compute syntomic cohomology of semistable affinoids in terms of cohomology of varphi gamma modules which thanks to work of fontaine herr andreatta iovita and kedlaya liu is known to compute galois cohomology of these affinoids for a semistable scheme over a mixed characteristic local ring this implies a comparison isomorphism up to some universal constants between truncated sheaves of p adic nearby cycles and syntomic cohomology sheaves this generalizes the comparison results of kato kurihara and tsuji for small tate twists where no constants are necessary as well as the comparison result of tsuji that holds over the algebraic closure of the field as an application we combine this local comparison isomorphism with the theory of finite dimensional banach spaces and finiteness of etale cohomology of rigid analytic spaces proved by scholze to prove a semistable conjecture for formal schemes with semistable reduction

reverse khas minskii condition the aim of this paper is to present and discuss some equivalent characterizations of p parabolicity for complete riemannian manifolds in terms of existence of special exhaustion functions in particular khas minskii in ergodic properties of recurrent diffusion prossesses and stabilization of solution to the cauchy problem for parabolic equations theor prob appl 5 2 1960 proved that if there exists a 2 superharmonic function mathcal k defined outside a compact set on a complete riemannian manifold r such that lim x to infty mathcal k x infty then r is 2 parabolic and sario and nakai in classification theory of riemann surfaces springer berlin 1970 were able to improve this result by showing that r is 2 parabolic if and only if there exists an evans potential i e a 2 harmonic function e r setminus k to mathbb r with lim x to infty mathcal e x infty in this paper we will prove a reverse khas minskii condition valid for any p 1 and discuss the existence of evans potentials in the nonlinear case

the classical theory of minimal surfaces we present here a survey of recent spectacular successes in classical minimal surface theory we highlight this article with the theorem that the plane the helicoid the catenoid and the one parameter family frtgt2 0 1 of riemann minimal examples are the only complete properly embedded minimal planar domains in r 3 the proof of this result depends primarily on work of colding

fractional sturm liouville boundary value problems in unbounded domains recently zayernouri and karniadakis in 2013 78 investigated two classes of fractional sturm liouville eigenvalue problems on compact interval a b in more detail they were the first authors who not only obtained some explicit forms for the eigensolutions of these problems but also derived some useful spectral properties of the obtained eigensolutions until now to the best of our knowledge fractional sturm liouville eigenvalue problems on non compact interval such as 0 are not analyzed so our aim in this paper is to study these problems in detail to do so we study at first fractional sturm liouville operators fslos of the confluent hypergeometric differential equations of the first kind and then two special cases of fslos fslos 1 and fslos 2 are considered after this we obtain the analytical eigenfunctions for the cases and investigate the spectral properties of eigenfunctions and their corresponding eigenvalues also we derive two fractional types of the associated laguerre differential equations due to the non polynomial nature of the eigenfunctions obtained from the two fractional associated laguerre differential equations they are defined as generalized associated laguerre functions of the first and second kinds galfs 1 and galfs 2 furthermore we prove that these fractional sturm liouville operators are self adjoint and the obtained eigenvalues are all real the corresponding eigenfunctions are orthogonal with respect to the weight function associated to fslos 1 and fslos 2 and form two sets of non polynomial bases at the end two new quadrature rules and l 2 orthogonal projections with respect to and based on galfs 1 and galfs 2 are introduced the upper bounds of the truncation errors of these new orthogonal projections according to some prescribed norm are proved and then verified numerically with some test examples finally some fractional differential equations are provided and analyzed numerically

range of fluctuation of brownian motion on a complete riemannian manifold we investigate the escape rate of the brownian motion w x t on a complete noncompact riemannian manifold assuming that the manifold has at most polynomial volume growth and that its ricci curvature is bounded below we prove that dist w x t x ct log t for all large t with probability 1 on the other hand if the ricci curvature is nonnegative and the volume growth is at least polynomial of the order n 2 then formula math again for all large t with probability 1 where e 0

approximate solution of the bagley torvik equation by hybridizable discontinuous galerkin methods in this paper we introduce a hybridizable discontinuous galerkin method for numerically solving a boundary value problem associated with the bagley torvik equation that arises in the study of the motion of a plate immersed in a newtonian fluid one of the main features of these methods is that they are efficiently implementable since it is possible to eliminate all internal degrees of freedom and obtain a global linear system that only involves unknowns at the element interfaces we display the results of a series of numerical experiments to ascertain the performance of the method

on haar wavelet operational matrix of general order and its application for the numerical solution of fractional bagley torvik equation abstract in the present analysis the motion of an immersed plate in a newtonian fluid described by torvik and bagley s fractional differential equation 1 has been considered this bagley torvik equation has been solved by operational matrix of haar wavelet method the obtained result is compared with analytical solution suggested by podlubny 2 haar wavelet method is used because its computation is simple as it converts the problem into algebraic matrix equation

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

some bounds for ramification of p n torsion semi stable representations let p be an odd prime k a finite extension of q p g k gal kbar k its absolute galois group and e e k q p its absolute ramification index suppose that t is a p n torsion representation of g k that is isomorphic to a quotient of g k stable z p lattices in a semi stable representation with hodge tate weights in 0 r we prove that there exists a constant mu depending only on n e and r such that the upper numbering ramification group g k mu acts on t trivially

haar wavelet operational methods for the numerical solutions of fractional order nonlinear oscillatory van der pol system in this paper an efficient numerical method for the solution of nonlinear damped van der pol equation based on the haar wavelets approach is proposed the proposed scheme can be used for solving one of the widely studied and challenging equation in nonlinear dynamics like the damped van der pol equation we have applied a numerical procedure involving haar series for the highest order of derivative moreover the numerical result shows that the proposed mechanism is quite reasonable when compare to exact solution

sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative ricci curvature in this paper we consider complete noncompact riemannian manifolds m g with nonnegative ricci curvature and euclidean volume growth of dimension n ge 3 for every bounded open subset omega subset m with smooth boundary we prove that begin aligned int limits partial omega left frac mathrm h n 1 right n 1 mathrm d sigma ge mathrm avr g big mathbb s n 1 big end aligned where mathrm h is the mean curvature of partial omega and mathrm avr g is the asymptotic volume ratio of m g moreover the equality holds true if and only if m setminus omega g is isometric to a truncated cone over partial omega an optimal version of huisken s isoperimetric inequality for 3 manifolds is obtained using this result finally exploiting a natural extension of our techniques to the case of parabolic manifolds we also deduce an enhanced version of kasue s non existence result for closed minimal hypersurfaces in manifolds with nonnegative ricci curvature

on uniqueness class for a heat equation on graphs abstract we study the uniqueness class problem for a class of heat equations defined on weighted graphs as an application we recover the volume growth criterion for stochastic completeness of grigor yan huang and masamune in the case of weighted graphs

a probabilistic approach to martin boundaries for manifolds with ends we use martingale methods and coupling arguments to prove results of li and tam 1987 and donnelly 1986 characterizing positive and bounded harmonic functions respectively on certain manifolds with ends

dirichlet parabolicity and l1 liouville property under localized geometric conditions abstract we shed a new light on the l 1 liouville property for positive superharmonic functions by providing many evidences that its validity relies on geometric conditions localized on large enough portions of the space we also present examples in any dimension showing that the l 1 liouville property is strictly weaker than the stochastic completeness of the manifold the main tool in our investigations is represented by the potential theory of a manifold with boundary subject to dirichlet boundary conditions the paper incorporates under a unifying viewpoint some old and new aspects of the theory with a special emphasis on global maximum principles and on the role of the dirichlet green s kernel

surgery of the faber krahn inequality and applications to heat kernel bounds 2 cutting faber krahn inequalities 3 2 1 fk functions 3 2 2 cutting a manifold 6 2 3 non parabolic case 7 2 4 fk functions in balls 9 2 5 relative fk function 16

minimal and weakly trapped submanifolds in standard static spacetimes abstract in this article we obtain some rigidity results for spacelike submanifolds of arbitrary codimension in standard static spacetimes in particular under appropriate assumptions we prove that they must be contained in a spacelike slice which enables us to characterize minimal closed submanifolds of codimension two we also prove new rigidity results for complete non compact weakly trapped submanifolds of arbitrary codimension in these ambient spacetimes

haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations haar wavelet operational matrix has been widely applied in system analysis system identification optimal control and numerical solution of integral and differential equations in the present paper we derive the haar wavelet operational matrix of the fractional order integration and use it to solve the fractional order differential equations including the bagley torvik ricatti and composite fractional oscillation equations the results obtained are in good agreement with the existing ones in open literatures and it is shown that the technique introduced here is robust and easy to apply

kronecker operational matrices for fractional calculus and some applications the problems of systems identification analysis and optimal control have been recently studied using orthogonal functions the specific orthogonal functions used up to now are the walsh the block pulse the laguerre the legendre haar and many other functions in the present paper several operational matrices for integration and differentiation are studied we introduce the kronecker convolution product and expanded to the riemann liouville fractional integral of matrices for some applications it is often not necessary to compute exact solutions approximate solutions are sufficient because sometimes computational efforts rapidly increase with the size of matrix functions our method is extended to find the exact and approximate solutions of the general system matrix convolution differential equations the way exists which transform the coupled matrix differential equations into forms for which solutions may be readily computed finally several systems are solved by the new and other approaches and illustrative examples are also considered

dieudonne crystals and wach modules for p divisible groups let k be a perfect field of characteristic p 2 and k an extension of f fracw k contained in some f pr using crystalline dieudonne theory we provide a classification of p divisible groups over r ok t1 td in terms of finite height modules over s w k u t1 td when d 0 such a classification is a consequence of a special case of the theory of kisin ren in this setting our construction gives an independent proof of this result and moreover allows us to recover the dieudonne crystal of a p divisible group from the wach module associated to its tate module by berger breuil or by kisin ren

a liouville property for schrodinger operators 6 examples 27 6 1 thick and thin sets in r 27 6 2 big and small measures in r 29 6 3 a scattered measure in r which is big 31 6 4 counter example to theorem 1 2 32 6 5 counter example to theorem 5 2 34 6 6 harmonic functions with the third boundary condition 35

representations galoisiennes p adiques et phi tau modules let p be an odd prime number and k be a p adic field in this paper we develop an analogue of fontaine s theory of phi gamma modules replacing the p cyclotomic extension by the extension k infty obtained by adding to k a compatible system of p n th roots of a fixed uniformizer pi of k as a result we obtain a new classification of p adic representations of g k gal kbar k by some phi tau modules we then make a link between the theory of phi tau modules discussed above and the so called theory of phi n nabla modules developped by kisin as a corollary we answer a question of tong liu we prove that if k is a finite extension of q p every representation of g k of e u finite height is potentially semi stable

comparative experiment on the numerical solutions of hammerstein integral equation arising from chemical phenomenon in this paper efficient numerical techniques have been proposed to solve nonlinear hammerstein integral equation arising from chemical phenomenon the following integral equation forms the basis for the conductor like screening model for real solvent which appeared in chemical phenomenon our aim is to solve the hammerstein nonlinear integral equation s r t ln p s exp e i n t s r t d where r is the gas constant t is the temperature and the term e i n t denotes the interaction energy expression for the segments with screening charge density and respectively the molecular interaction in solvent is p s and the chemical potential of the surface segments is described by s which is to be determined this cosmo rs integral equation has been solved by bernstein collocation method haar wavelet method and sinc collocation method these methods reduce the integral equation to nonlinear system of algebraic equations and then this algebraic system has been solved numerically by newton s method comparison has been done for these methods illustrative examples have been discussed to demonstrate the validity and applicability of bernstein collocation method haar wavelet method and sinc collocation method

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

large time behavior of the heat equation on complete manifolds with non negative ricci curvature sur une variete de riemann complete m on considere le noyau de la chaleur h x y t qui resout l equation de la chaleur t f x t 0 sur mx 0 avec la donnee initiale f x 0 f x en posant f x t m h x y t f y dy

symmetric markov chains in ℤ d how fast can they move consider a reversible markov chain x n which takes values in a subset of d if the steps of the chain are uniformly bounded and the invariant measure satisfies a mild regularity condition varopoulos carne and kesten have obtained estimates on p x n x text 0 lambda n 1 2 which exhibit a gaussian tail in but blow up as n following kesten s approach we derive bounds which are uniform in n in some special cases our main result however is an example which shows that in general the estimates of varopoulos carne and kesten are essentially best possible