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

on some results in fuzzy metric spaces abstract we define a hausdorff topology on a fuzzy metric space introduced by kramosil and michalek kybernetica 11 1975 326 334 and prove some known results of metric spaces including baire s theorem for fuzzy metric spaces

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

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

regularity criterion for a bona colin lannes system abstract we prove a regularity criterion of the strong solutions for a bona colin lannes system in besov space as a byproduct we show the existence of globally smooth solutions for a symmetric bona colin lannes system

integral points on elliptic curves and 3 torsion in class groups we give new bounds for the number of integral points on elliptic curves the method may be said to interpolate between approaches via diophantine techniques bp hbr and methods based on quasi orthogonality in the mordell weil lattice sil6 gs he we apply our results to break previous bounds on the number of elliptic curves of given conductor and the size of the 3 torsion part of the class group of a quadratic field the same ideas can be used to count rational points on curves of higher genus

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 for the motion of an incompressible liquid with free surface boundary we study the motion of an incompressible perfect liquid body in vacuum this can be thought of as a model for the motion of the ocean or a star the free surface moves with the velocity of the liquid and the pressure vanishes on the free surface this leads to a free boundary problem for euler s equations where the regularity of the boundary enters to highest order we prove local existence in sobolev spaces assuming a physical condition related to the fact that the pressure of a fluid has to be positive

weak topologies for the closed subsets of a metrizable space the purpose of this article is to propose a unified theory for topologies on the closed subsets of a metrizable space it can be shown that all of the standard hyperspace topologies including the hausdorff metric topology the vietoris topology the attouch wets topology the fell topology the locally finite topology and the topology of mosco convergence arise as weak topologies generated by families of geometric functionals defined on closed sets a key ingredient is the simple yet beautiful interplay between topologies determined by families of gap functionals and those determined by families of hausdorff excess functionals

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

some properites of fuzzy metric spaces abstract we prove that the topology induced by any complete fuzzy metric space in the sense of george and veeramani is completely metrizable we also show that every separable fuzzy metric space admits a precompact fuzzy metric and that a fuzzy metric space is compact if and only if it is precompact and complete finally we generalize among other classical results the niemytzki tychonoff theorem to fuzzy metric spaces

the hausdorff fuzzy quasi metric removing the condition of symmetry in the notion of a fuzzy pseudo metric in kramosil and michalek s sense one has the notion of a fuzzy quasi pseudo metric then for each fuzzy quasi pseudo metric on a set x we construct a fuzzy quasi pseudo metric on the collection of all nonempty subsets of x called the hausdorff fuzzy quasi pseudo metric we investigate several properties of this structure and present several illustrative examples as well as an application to the domain of words the notion of hausdorff fuzzy quasi pseudo metric when quasi pseudo metric fuzziness is considered in the sense of george and veeramani is also discussed

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

a note on the quantitative measure of image enhancement through fuzziness the index of fuzziness and entropy of an image reflect a kind of quantitative measure of its enhancement quality their values are found to decrease with enhancement of an image when different sets of s type membership functions with appropriate crossover points were considered for extracting the fuzzy property plane from the spatial domain of the image

class numbers of quadratic fields and shimura s correspondence it is well known see fueter 8 and aigner 1 that fermat s equation x y 3 1 has no solutions in a quadratic field k q d for d 1 mod 3 provided the class number of k is not divisible by 3 this subject was further investigated by frey 6 who gave estimates for the 3 rank of the class group of k in terms of 3 descent on a curve y 2 x d in this paper we consider curves ed dy 2 4x 27 which are quadratic twists of the fermat s curve x y 3 1 we give a precise formula for the rank of the selmer group corresponding to the complex multiplication 3 ed e 3d in terms of the 3 rank of the class group of q d resp q 3d this result may be considered as a quantitative version of 1 and 8 we discuss also birch and swinnerton dyer s conjecture for curves ed according to a theorem of waldspurger 22 23 natural rational factor of l ed q 1 may be expressed in terms of coefficients of certain modular forms of weight 3 2 we identify these forms explicitely and verify mod 3 part of birch and swinnertondyer s conjecture for ed in most cases for d of density 5 6

well posedness for the linearized motion of a compressible liquid with free surface boundary we study the motion of a compressible perfect liquid body in vacuum this can be through of as a model for the motion of the ocean or a star the free surface moves with the velocity of the liquid and the pressure vanishes on the free surface this leads to a free boundary problem for euler s equations where the regularity of the boundary enters to highest order we prove linearized stability in sobolev space assuming a physical condition related to the fact that the pressure of a fluid has to be positive

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

on critical cases of sobolev s inequalities abstract we present a new form of the trudinger type inequality which shows an explicit dependence moreover we give an alternative proof of the brezis gallouet wainger inequality

the zero surface tension limit of two dimensional water waves we consider two dimensional water waves of infinite depth periodic in the horizontal direction it has been proven by wu in the slightly different nonperiodic setting that solutions to this initial value problem exist in the absence of surface tension recently ambrose has proven that solutions exist when surface tension is taken into account in this paper we provide a shorter more elementary proof of existence of solutions to the water wave initial value problem both with and without surface tension our proof requires estimating the growth of geometric quantities using a renormalized arc length parametrization of the free surface and using physical quantities related to the tangential velocity of the free surface using this formulation we find that as surface tension goes to 0 the water wave without surface tension is the limit of the water wave with surface tension far from being a simple adaptation of previous works our method requires a very original choice of variables these variables turn out to be physical and well adapted to both cases c

the critical sobolev inequalities in besov spaces and regularity criterion to some semi linear evolution equations we show the critical sobolev inequalities in the besov spaces with the logarithmic form such as brezis gallouet wainger and beale kato majda as an application of those inequalities the regularity problem under the critical condition to the navier stokes equations the euler equations in mathbb r n and the gradient flow to the harmonic map to the sphere are discussed namely the serrin ohyama type regularity criteria are improved in the terms of the besov spaces

how to do a p descent on an elliptic curve in this paper we describe an algorithm that reduces the computation of the full p selmer group of an elliptic curve e over a number field to standard number field computations such as determining the p torsion of the s class group and a basis of the s units modulo pth powers for a suitable set s of primes in particular we give a result reducing this set s of bad primes to a very small set which in many cases only contains the primes above p as of today this provides a feasible algorithm for performing a full 3 descent on an elliptic curve over q but the range of our algorithm will certainly be enlarged by future improvements in computational algebraic number theory when the galois module structure of e p is favorable simplifications are possible and p descents for larger p are accessible even today to demonstrate how the method works several worked examples are included

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

canonical periods and congruence formulae the purpose of this article is to show how congruences between the fourier coefficients of hecke eigenforms give rise to corresponding congruences between the algebraic parts of the critical values of the associated l functions this study was initiated by b mazur in his fundamental work on the eisenstein ideal see maz77 and maz79 where it was made clear that congruences for analytic l values were closely related to the integral structure of certain hecke rings and cohomology groups the results of maz79 also showed that congruences were useful in the study of nonvanishing of l functions this idea was then further developed by stevens ste82 and rubin wiles rw82 the work of rubin and wiles in particular used congruences to study the behavior of elliptic curves in towers of cyclotomic fields a key ingredient here was a theorem of washington which states roughly that almost l values in certain families are nonzero modulo p this theme has recently been taken up again in the work of ono skinner osa osb james jam and kohnen koh97 while the earlier history was primarily concerned with cyclotomic twists the current emphasis is on families of twists by quadratic characters here one wants quantitative estimates for the number of quadratic twists of a given modular form which have nonvanishing l function at s 1 we continue this trend in the present work by using our general results to obtain a strong nonvanishing theorem for the quadratic twists of modular elliptic curves with rational points of order three this generalizes a beautiful example due to kevin james and provides new evidence for a conjecture of goldfeld gol79 it should however be pointed out that even the study of quadratic twists may be traced back to mazur the reader is urged to look at pages 212 213 of maz79 and especially at the footnote at the bottom of page 213 the theorems of davenport heilbronn dh71 and washington was78 which are crucial in this paper are both mentioned in mazur s article we want to begin by discussing the congruences that lie at the heart of this article thus let f anq n be an elliptic modular cuspform of level m and weight k 2 assume that f is a simultaneous eigenform for all the hecke operators and that a1 f 1 the l function associated to f is defined by the dirichlet series l s f ann s which converges for the real part of s sufficiently large and has analytic continuation to s c a fundamental theorem of shimura shi76 states that l s f enjoys the following algebraicity property

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

large time existence for 3d water waves and asymptotics we rigorously justify in 3d the main asymptotic models used in coastal oceanography including shallow water equations boussinesq systems kadomtsev petviashvili kp approximation green naghdi equations serre approximation and full dispersion model we first introduce a variable nondimensionalized version of the water waves equations which vary from shallow to deep water and which involves four dimensionless parameters using a nonlocal energy adapted to the equations we can prove a well posedness theorem uniformly with respect to all the parameters its validity ranges therefore from shallow to deep water from small to large surface and bottom variations and from fully to weakly transverse waves

rough sets and boolean reasoning abstract in this article we discuss methods based on the combination of rough sets and boolean reasoning with applications in pattern recognition machine learning data mining and conflict analysis

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

rudiments of rough sets abstract worldwide there has been a rapid growth in interest in rough set theory and its applications in recent years evidence of this can be found in the increasing number of high quality articles on rough sets and related topics that have been published in a variety of international journals symposia workshops and international conferences in recent years in addition many international workshops and conferences have included special sessions on the theory and applications of rough sets in their programs rough set theory has led to many interesting applications and extensions it seems that the rough set approach is fundamentally important in artificial intelligence and cognitive sciences especially in research areas such as machine learning intelligent systems inductive reasoning pattern recognition mereology knowledge discovery decision analysis and expert systems in the article we present the basic concepts of rough set theory and point out some rough set based research directions and applications

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

remarks on the breakdown of smooth solutions for the 3 d euler equations the authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3 d euler equations in other words if a solution of the euler equations is initially smooth and loses its regularity at some later time then the maximum vorticity necessarily grows without bound as the critical time approaches equivalently if the vorticity remains bounded a smooth solution persists

nature inspired framework for measuring visual image resemblance a near rough set approach the problem considered in this paper is how to determine the degree of nearness between complex visual objects the proposed solution to this problem stems from a natural computing approach to solving the visual acuity problem in terms of a granular representation of visual information that is quantifiable as well as understandable for humans this is accomplished via a near rough set framework in the approximation of a pair of disjoint sets and measurement of distances between sets using various fuzzy pseudometrics pseudometrics in general and fuzzy pseudometrics in particular are useful in measuring the distance between pairs of objects such as sets such distances are indicators of the nearness of resemblance between visual objects these observations lead to a number of practical applications such as object recognition and object retrieval in digital image analysis one such application is reported in this article the contribution of this article is threefold introduction of a nature inspired framework for measurement of visual object resemblance four different incarnations of the standard fuzzy metric and application of fuzzy metrics in content based image retrieval experiments

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

semi metrics closure spaces and digital topology abstract we investigate certain generalized topological structures with the aim of showing that they can provide a suitable framework within which to compare various approaches to digital topology including tolerance geometry with ordinary topology within an appropriate category of these structures ordinary spaces arise as inverse limits of digital spaces in the first instance the structures can be taken to be a slight generalization of cech closure spaces the structures of this type which seem to be the most useful are those which can be realized as topological graphs see section 3 domain equations for the real unit interval provide our main detailed application section 4 another application area which seems promising is that of modal semantics