fully implicit methodology for the dynamics of biomembranes and capillary interfaces by combining the level set and newton methods abstract this framework is concerned with the numerical modeling of the dynamics of individual biomembranes and capillary interfaces in a surrounding newtonian fluid a level set approach helps to follow the interface motion our method features the use of high order fully implicit time integration schemes that enable to overcome stability issues related to the explicit discretization of the highly non linear bending force or capillary force at each time step the tangent systems are derived and the resulting nonlinear problems are solved by a newton raphson method based on the signed distance assumption several inexact newton strategies are employed to solve the capillary and vesicle problems and guarantee the second order convergence behavior we address in detail the main features of the proposed method and we report several experiments in the two dimensional case with the aim of illustrating its accuracy and efficiency comparative investigations with respect to the fully explicit scheme depict the stabilizing effect of the new method which allows to use significantly larger time step sizes

embedded surfaces of arbitrary genus minimizing the willmore energy under isoperimetric constraint the isoperimetric ratio of an embedded surface in mathbb r 3 is defined as the ratio of the area of the surface to power three to the squared enclosed volume the aim of the present work is to study the minimization of the willmore energy under fixed isoperimetric ratio when the underlying abstract surface has fixed genus g geqq 0 the corresponding problem in the case of spherical surfaces that is g 0 was recently solved by schygulla see schygulla arch ration mech anal 203 901 941 2012 with different methods

the λ coalescent speed of coming down from infinity consider a coalescent that comes down from infinity meani ng that it starts from a configuration containing infinitely many blocks at time 0 yet it has a finite number nt of blocks at any positive time t 0 we exhibit a deterministic function v 0 0 such that nt v t 1 almost surely and in l p for any p 1 as t 0 our approach relies on a novel martingale technique

on the convergence and representation of random fuzzy number integrals in this paper a new kind of convergence w s convergence is introduced for certain type of fuzzy numbers taking values in separable reflexive banach spaces the convergence theorem of random fuzzy number integrals in the w s sense is given and a condition under which a fuzzy number function can be represented by random fuzzy number integrals is obtained

on the topology and index of minimal surfaces we show that for an immersed two sided minimal surface in r 3 there is a lower bound on the index depending on the genus and number of ends using this we show the nonexistence of an embedded minimal surface in r 3 of index 2 as conjectured by choe cho90 moreover we show that the index of an immersed two sided minimal surface with embedded ends is bounded from above and below by a linear function of the total curvature of the surface

strings harmonic maps and hyperbolic systems abstract we review a number of cases where strings and hyperbolic systems arise in the study of harmonic maps of pseudo riemannian manifolds

rigidity and non rigidity results for conformal immersions abstract in this paper we show a quantitative rigidity result for the minimizer of the willmore functional among all projective planes in r n with n 4 we also construct an explicit counterexample to a corresponding rigidity result in codimension one by showing that an enneper surface might split off during a blow up process for conformal immersions of spheres with large enough willmore energies we construct explicit counterexamples to a quantitative rigidity result and this complements the recently obtained rigidity results in 14

stochastic flows associated to coalescent processes we study a class of stochastic flows connected to the coalescent processes that have been studied recently by mohle pitman sagitov and schweinsberg in connection with asymptotic models for the genealogy of populations with a large fixed size we define a bridge to be a right continuous process b r r 0 1 with nondecreasing paths and exchangeable increments such that b 0 0 and b 1 1 we show that flows of bridges are in one to one correspondence with the so called exchangeable coalescents this yields an infinite dimensional version of the classical kingman representation for exchangeable partitions of we then propose a poissonian construction of a general class of flows of bridges and identify the associated coalescents we also discuss an important auxiliary measure valued process which is closely related to the genealogical structure coded by the coalescent and can be viewed as a generalized fleming viot process

the controlled convergence theorems for the strong henstock integrals of fuzzy number valued functions in this paper we discuss the properties of the strong henstock integrals of the fuzzy number valued functions using the notion of generalized derivatives and prove the controlled convergence theorem for such integrals also we present the concept of equi integrability of a sequence for fuzzy number valued functions under this concept we prove another controlled convergence theorem and establish the relationships between the above two controlled convergence theorems

on the fuzzy initial value problem abstract the initial value problem x t f t x t x 0 x 0 with fuzzy initial value and with deterministic or fuzzy function f is considered two different approaches viz the extension principle and the use of extremal solutions of deterministic initial value problems are applied generalizations to fuzzy integral equations and fuzzy functional differential equations are indicated

beta coalescents and continuous stable random trees coalescents with multiple collisions also known as a coalescents were introduced by pitman and sagitov in 1999 these processes describe the evolution of particles that undergo stochastic coagulation in such a way that several blocks can merge at the same time to form a single block in the case that the measure a is the beta 2 a a distribution they are also known to describe the genealogies of large populations where a single individual can produce a large number of offspring here we use a recent result of birkner et al to prove that beta coalescents can be embedded in continuous stable random trees about which much is known due to the recent progress of duquesne and le gall our proof is based on a construction of the donnelly kurtz lookdown process using continuous random trees which is of independent interest this produces a number of results concerning the small time behavior of beta coalescents most notably we recover an almost sure limit theorem of the present authors for the number of blocks at small times and give the multifractal spectrum corresponding to the emergence of blocks with atypical size also we are able to find exact asymptotics for sampling formulae corresponding to the site frequency spectrum and the allele frequency spectrum associated with mutations in the context of population genetics

on the willmore energy of shells under infinitesimal deformations in this paper we apply tensor calculus and differential geometry to consider shell structure using tensor calculus we examine the stationarity of the willmore energy under infinitesimal deformations of the surfaces in 3 we obtain the class of the surfaces which does not change its willmore energy under infinitesimal deformations in particular a special kind of deformation is considered infinitesimal bending which preserves the arc length the change of the willmore energy under such deformations is determined also we give a new proof of a well known theorem that reads that the total mean curvature of a surface is stationary under an infinitesimal bending applying tensor calculus

controlled convergence theorems for infinite dimension henstock integrals of fuzzy valued functions based on weak equi integrability abstract in this paper we give the concept of weak henstock equi integrability for a sequence of fuzzy number valued functions under this notion we investigate a new version of the henstock s lemma of fuzzy number valued functions thus it is possible to discuss the controlled convergence theorems of fuzzy henstock integral in sense of vitali covering moreover we prove that a uniform version of sklyarenko s and lusin s integrability condition of fuzzy henstock integrals together with pointwise convergence of a sequence of integrable functions is sufficient for a convergence theorem of fuzzy henstock integrals as the applications of the controlled convergence theorem we discuss the existence theorems of generalized solution for a class of discontinuous fuzzy differential equations

on donaldson s flow of surfaces in a hyperkahler four manifold we prove some basic properties of donaldson s flow of surfaces in a hyperkahler 4 manifold when the initial submanifold is symplectic with respect to one kahler form and lagrangian with respect to another we show that certain kinds of singularities cannot form and we prove a convergence result under a condition related to one considered by m t wang for the mean curvature flow

on ruelle s probability cascades and an abstract cavity method we construct in this work a markov process which describes a clustering mechanism through which equivalence classes on are progressively lumped together this clustering process gives a new description of ruelle s continuous probability cascades it also enables to introduce an abstract cavity method which mimicks certain features of the cavity method developed by physicists in the context of the sherrington kirkpatrick model

caratheodory solutions of fuzzy differential equations in this paper the existence of solutions for fuzzy differential equations with right hand side satisfying a caratheodory condition on a class of fuzzy sets is established

measurablility concepts for fuzzy mappings abstract in a series of recent works alternative measurability concepts for fuzzy mappings were introduced and studied in this paper we show that some of these concepts are equivalent or strongly inter related

on fuzzy boundary value problems the statement that a two point boundary value problem of fuzzy differential equation is equivalent to a fuzzy integral equation was pointed out by lakshmikantham et al and o regan et al recently bede gave a counterexample to show that this statement does not hold and he also argued that in many cases two point boundary value problems have no solutions under a new structure and certain conditions we show that a two point boundary value problem is equivalent to a fuzzy integral equation we also prove the existence of solutions to the two point boundary value problem in some sense this is an amendment to results of lakshmikantham et al and o regan et al and it is an answer to one of bede s problems

the tree length of an evolving coalescent a well established model for the genealogy of a large population in equilibrium is kingman s coalescent for the population together with its genealogy evolving in time this gives rise to a time stationary tree valued process we study the sum of the branch lengths briefly denoted as tree length and prove that the suitably compensated sequence of tree length processes converges as the population size tends to infinity to a limit process with cadlag paths infinite infinitesimal variance and a gumbel distribution as its equilibrium

complete embedded minimal surfaces of finite total curvature we will survey what is known about minimal surfaces s r 3 which are complete embedded and have finite total curvature int s k da infty the only classically known examples of such surfaces were the plane and the catenoid the discovery by costa 14 15 early in the last decade of a new example that proved to be embedded sparked a great deal of research in this area many new examples have been found even families of them as will be described below the central question has been transformed from whether or not there are any examples except surfaces of rotation to one of understanding the structure of the space of examples

smaller population size at the mrca time for stationary branching processes we consider an elementary model of random size varying population governed by a stationary continuous state branching process we compute the distributions of various variables related to the most recent common ancestor mrca the time to the mrca the size of the current population and the size of the population just before the mrca in particular we observe a natural mild bottleneck effect as the size of the population just before the mrca is stochastically smaller than the size of the current population we also compute the number of individuals involved in the last coalescent event of the genealogical tree that is the number of individuals at the time of the mrca who have descendants in the current population by studying more precisely the genealogical structure of the population we get asymptotics for the number of ancestors just before the current time we give explicit computations in the case of the quadratic branching mechanism in this case the size of the population at the mrca is in mean 2 3 of the size of the current population we also provide in this case the fluctuations for the renormalized number of ancestors

isothermic submanifolds of symmetric r spaces we extend the classical theory of isothermic surfaces in conformal 3 space due to bour christoffel darboux bianchi and others to the more general context of submanifolds of symmetric r spaces with essentially no loss of integrable structure

differentials of fuzzy functions abstract in this paper the radstrom embedding theorem proc amer math soc 3 1952 165 is generalized and is used to define the concept of the differential of a fuzzy function

a new conformal invariant and its applications to the willmore conjecture and the first eigenvalue of compact surfaces let m be a compact riemannian manifold with a fixed conformal structure then we introduce the concept of conformal volume of m in the following manner for each branched conformal immersion q9 of m into the unit sphere s n we consider the set of all branched conformal immersions obtained by composi t ion of qo with the conformal automorphisms of s we let vc n qg be the max imum volume of these branched immersions the conformal volume of m is defined to be the infimum of v n q0 where qo ranges over all branched conformal immersions of m into the unit sphere s in this paper we study the case when m is a compact surface and we call the conformal volume of m to be the conformal area of m we demonst ra te that this conformal invariant is non trivial in fact we prove that if there exists a minimal immersion of m into s where coordinate functions are first eigenfunctions then the conformal area of m is given by the area of m with respect to the induced metric this enables us to compute the conformal area for several surfaces for example the conformal area of r p z is 6n and the conformal area of the square torus is 27c 2 we believe that the computa t ion of the conformal area for general surfaces will be very impor tant in studying the geometry of compact surfaces we demonstrate this claim by applying the concept of conformal area to two different branches of surface theory the first application is to study the total curvature of a compact surface in r this problem has a long history fenchel and fary 9 proved that for a closed curve o in r slk 27c where k is its curvature then milnor 12

solutions with time dependent singular sets for the heat equation with absorption abstract we consider the heat equation with a superlinear absorption term t u u u p in r n and study the existence of nonnegative solutions with an m dimensional time dependent singular set where n m 3 we prove that if 1 p n m n m 2 then there are two types of singular solutions moreover we show the uniqueness of the solutions and specify the exact behavior of the solutions near the singular set

a willmore functional for compact surfaces of complex projective plane where h denotes the mean curvature vector of the immersion k the sectional curvature of m restricted to and da the canonical measure of the induced metric this functional has been extensively studied when m is the euclidean space r or the sphere s or the hyperbolic space rh because w is invariant under conformal transformations of ambient space the type of problems which have been studied for this ambient space r are of a different nature on the one hand are the papers written with the object of obtaining lower bounds for the functional w given that w has 4 as an absolute minimum and is only reached by the round spheres of r willmore proposed to study this functional on tori and conjectured that 2 is the minimum value for these surfaces and it is attained only by the clifford torus this problem which still has not been resolved can be considered as the starting point of a series of important papers in which minimization

on otsuki tori and their willmore energy abstract otsuki tori in the three dimensional unit sphere are studied from calculations of the willmore energy the lower and upper bound estimates for the otsuki tori are established invariance of the otsuki tori under antipodal maps is also considered and in such cases better lower bound estimates of their willmore energies are obtained

constant mean curvature tori as stationary solutions to the davey stewartson equation a well known result of da rios and levi civita says that a closed planar curve is elastic if and only if it is stationary under the localized induction or smoke ring equation where stationary means that the evolution under the localized induction equation is by rigid motions we prove an analogous result for surfaces an immersion of a torus into the conformal 3 sphere has constant mean curvature with respect to a space form subgeometry if and only if it is stationary under the davey stewartson flow

holomorphic differentials and laguerre deformation of surfaces a laguerre geometric local characterization is given of l minimal surfaces and laguerre deformations t transforms of l minimal isothermic surfaces in terms of the holomorphicity of a quartic and a quadratic differential this is used to prove that via their l gauss maps the t transforms of l minimal isothermic surfaces have constant mean curvature h r in some translate of hyperbolic 3 space mathbb h 3 r 2 subset mathbb r 4 1 de sitter 3 space mathbb s 3 1 r 2 subset mathbb r 4 1 or have mean curvature h 0 in some translate of a time oriented lightcone in mathbb r 4 1 as an application we show that various instances of the lawson isometric correspondence can be viewed as special cases of the t transformation of l isothermic surfaces with holomorphic quartic differential

global divergence of spatial coalescents we study several fundamental properties of a class of stochastic processes called spatial coalescents in these models a number of particles perform independent random walks on some underlying graph g in addition particles on the same vertex merge randomly according to a given coalescing mechanism a remarkable property of mean field coalescent processes is that they may come down from infinity meaning that starting with an infinite number of particles only a finite number remains after any positive amount of time almost surely we show here however that in the spatial setting on any infinite and bounded degree graph the total number of particles will always remain infinite at all times almost surely moreover if g mathbb z d and the coalescing mechanism is kingman s coalescent then starting with n particles at the origin the total number of particles remaining is of order log n d at any fixed positive time where log is the inverse tower function at sufficiently large times the total number of particles is of order log n d 2 when d 2 we provide parallel results in the recurrent case d 2 the spatial beta coalescents behave similarly where log log n is replacing log n

a variational problem for surfaces in laguerre geometry we consider the variational problem defined by the functional h k k da on immersed surfaces in euclidean space using the invariance of the functional under the group of laguerre transformations we study the extremal surfaces by the method of moving frames

existence of immersed spheres minimizing curvature functionals in compact 3 manifolds we study curvature functionals for immersed 2 spheres in a compact three dimensional riemannian manifold m under the assumption that the sectional curvature k m is strictly positive we prove the existence of a smooth immersion f mathbb s 2 rightarrow m minimizing the l 2 integral of the second fundamental form assuming instead that k m le 2 and that there is some point overline x in m with scalar curvature r m overline x 6 we obtain a smooth minimizer f mathbb s 2 rightarrow m for the functional int frac 1 4 h 2 1 where h is the mean curvature

analysis aspects of willmore surfaces a new formulation for the euler lagrange equation of the willmore functional for immersed surfaces in m is given as a nonlinear elliptic equation in divergence form with non linearities comprising only jacobians letting vec h be the mean curvature vector of the surface our new formulation reads mathcal l vec h 0 where mathcal l is a well defined locally invertible self adjoint elliptic operator several consequences are studied in particular the long standing open problem asking for a meaning to the willmore euler lagrange equation for immersions having only l 2 bounded second fundamental form is now solved the regularity of weak willmore immersions with l 2 bounded second fundamental form is also established its proof relies on the discovery of conservation laws which are preserved under weak convergence a weak compactness result for willmore surfaces with energy less than 8 the li yau condition ensuring the surface is embedded is proved via a point removability result established for wilmore surfaces in m thereby extending to arbitrary codimension the main result in ks3 finally from this point removability result the strong compactness of willmore tori below the energy level 8 is proved both in dimension 3 this had already been settled in ks3 and in dimension 4

removability of point singularities of willmore surfaces we investigate point singularities of willmore surfaces which for example appear as blowups of the willmore flow near singularities and prove that closed willmore surfaces with one unit density point singularity are smooth in codimension one as applications we get in codimension one that the willmore flow of spheres with energy less than 8 exists for all time and converges to a round sphere and further that the set of willmore tori with energy less than

willmore surfaces with nonremovable singularities and number of critical levels in this paper we show that for every prescribed genus p in mathbb n and n in mathbb n with n ge 3 there exists a closed and orientable willmore surface in mathbb r n of genus p with at least two nonremovable point singularities furthermore we prove an energy gap theorem for smooth closed and orientable willmore surfaces in mathbb r n of prescribed genus p in mathbb n with n in mathbb n and n ge 3 moreover we show that the willmore functional as a map on the set of smooth closed and orientable surfaces in mathbb r n of genus p in mathbb n with n in 3 4 has only a finite number of critical levels strictly below a douglas condition on the willmore energy in order to exclude topological splitting

on the structure of solutions for fuzzy initial value problem in this paper by means of banach fixed point theorem we establish the existence and uniqueness of solutions for set differential equations the continuous dependence of solutions on initial values and the structural stability of solutions on the basis of these results we define level sets of solutions to fuzzy differential equations by the solutions of set differential equations and then we obtain the corresponding results for the large solutions of the fuzzy initial value problem fivp finally discuss the relationship between small solutions and large solutions of fivp

min max theory and the willmore conjecture in 1965 t j willmore conjectured that the integral of the square of the mean curvature of a torus immersed in r 3 is at least 2 2 we prove this conjecture using the min max theory of minimal surfaces

a phase transition in excursions from infinity of the fast fragmentation coalescence process an important property of kingman s coalescent is that starting from a state with an infinite number of blocks over any positive time horizon it transitions into an almost surely finite number of blocks this is known as coming down from infinity moreover of the many different exchangeable stochastic coalescent models kingman s coalescent is the fastest to come down from infinity in this article we study what happens when we counteract this fastest coalescent with the action of an extreme form of fragmentation we augment kingman s coalescent where any two blocks merge at rate with a fragmentation mechanism where each block fragments at constant rate into its constituent elements we prove that there exists a phase transition at between regimes where the resulting fast fragmentation coalescence process is able to come down from infinity or not in the case that we develop an excursion theory for the fast fragmentation coalescence process out of which a number of interesting quantities can be computed explicitly

willmore blowups are never compact we prove a ojasiewicz simon gradient inequality for the willmore functional near willmore surfaces and we apply this inequality to exclude compact blowups for the willmore flow

fuzzy differential equations abstract this paper deals with fuzzy set valued mappings of a real variable whose values are normal convex upper semicontinuous and compactly supported fuzzy sets in rn we study differentiability and integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy differential equation

sequences of willmore surfaces in this paper we develop the theory of willmore sequences for willmore surfaces in the 4 sphere we show that under appropriate conditions this sequence has to terminate in this case the willmore surface either is the twistor projection of a holomorphic curve into mathbb c mathbb p 3 or the inversion of a minimal surface with planar ends in mathbb r 4 these results give a unified explanation of previous work on the characterization of willmore spheres and willmore tori with non trivial normal bundles by various authors

a gap theorem for willmore tori and an application to the willmore flow abstract in 1965 willmore conjectured that the integral of the square of the mean curvature of a torus immersed in r 3 is at least 2 2 and attains this minimal value if and only if the torus is a mobius transform of the clifford torus this was recently proved by marques and neves 2012 in this paper we show for tori that there is a gap to the next critical point of the willmore energy and we discuss an application to the willmore flow we also prove an energy gap from the clifford torus to surfaces of higher genus

random fuzzy number integrals in banach spaces abstract in this paper a new kind of random fuzzy number integral is introduced by the pettis aumann integral of the set valued functions in banach spaces the existence of the random fuzzy integral is also shown levi s convergence theorem of the integral is given a special mention

energy quantization for willmore surfaces and applications we prove a bubble neck decomposition together with an energy quantization result for sequences of willmore surfaces into r m with uniformly bounded energy and nondegenerating conformal type we deduce the strong compactness of willmore closed surfaces of a given genus modulo

a general class of exponential inequalities for martingales and ratios in this paper we introduce a technique for obtaining exponential inequalities with particular emphasis placed on results involving ratios our main applications consist of approximations to the tail probability of the ratio of a martingale over its conditional variance or its quadratic variation for continuous martingales we provide examples that strictly extend several of the classical exponential inequalities for sums of independent random variables and martingales the spirit of this application is that when going from results for sums of independent random variables to martingales one should replace the variance by the conditional variance and the exponential of a function of the variance by the expectation of the exponential of the same function of the conditional variance the decoupling inequalities used to attain our goal are of independent interest they include a new exponential decoupling inequality with constraints and a sharp inequality for the probability of the intersection of a fixed number of dependent sets finally we also present an exponential inequality that does not require any integrability conditions involving the ratio of the sum of conditionally symmetric variables to its sum of squares

global geometry and topology of spacelike stationary surfaces in the 4 dimensional lorentz space abstract for spacelike stationary i e zero mean curvature surfaces in 4 dimensional lorentz space one can naturally introduce two gauss maps and a weierstrass type representation in this paper we investigate the global geometry of such surfaces systematically the total gaussian curvature is related with the surface topology as well as the indices of the so called good singular ends by a gauss bonnet type formula on the other hand as shown by a family of counterexamples to osserman s theorem finite total curvature no longer implies that gauss maps extend to the ends interesting examples include the deformations of the classical catenoid the helicoid the enneper surface and jorge meeks k noids each family of these generalizations includes embedded examples in the 4 dimensional lorentz space showing a sharp contrast with the 3 dimensional case

remarks on peano like theorems for fuzzy differential equations the validity of the classical peano theorem is noted for differential equations on the metric space en dp of normal fuzzy convex sets in rn where dp is the p th mean of the hausdorff distances between corresponding level sets this contrasts with differential equations on the space en d where d is the max metric which is not locally compact so more than just continuity is required to ensure the existence of solutions some additional properties differing from that given by kaleva are mentioned

fuzzy random variables abstract in this paper we define the concepts of fuzzy random variable and the expectation of a fuzzy random variable the new definition of expectation generalizes the integral of a set valued function we derive some properties of these new concepts by considering a suitable generalization of the hausdorff metric we derive the lebesgue dominated convergence type theorem

change of the willmore energy under infinitesimal bending of membranes change of the willmore energy as the special case of elastic bending energy under infinitesimal bending of the vesicle membrane is discussed a membrane is thought of as a smooth surface in r 3 because its thickness is much smaller than its lateral dimension variation of the willmore energy at the surface point under infinitesimal bending of that surface as well as the condition for the stationarity of the willmore energy are given some examples are visualized variation of the willmore energy of a compact path connected smooth surface in the euclidean 3 space is reduced to a line integral of a special vector field and it is found that the willmore energy of a boundary free surface is stationary under an infinitesimal bending also the willmore energy of a minimal surface is stationary under its infinitesimal bending

coagulation transport equations and the nested coalescents the nested kingman coalescent describes the dynamics of particles called genes contained in larger components called species where pairs of species coalesce at constant rate and pairs of genes coalesce at constant rate provided they lie within the same species we prove that starting from rn species the empirical distribution of species masses numbers of genes n at time t n converges as n rightarrow infty to a solution of the deterministic coagulation transport equation begin aligned partial t d partial x psi d a t left d star d d right end aligned where psi x cx 2 star denotes convolution and a t 1 t delta with delta 2 r the most interesting case when delta 0 corresponds to an infinite initial number of species this equation describes the evolution of the distribution of species of mass x where pairs of species can coalesce and each species mass evolves like dot x psi x we provide two natural probabilistic solutions of the latter ipde and address in detail the case when delta 0 the first solution is expressed in terms of a branching particle system where particles carry masses behaving as independent continuous state branching processes the second one is the law of the solution to the following mckean vlasov equation begin aligned dx t psi x t dt v t delta j t end aligned where j is an inhomogeneous poisson process with rate 1 t delta and v t t ge 0 is a sequence of independent random variables such that mathcal l v t mathcal l x t we show that there is a unique solution to this equation and we construct this solution with the help of a marked brownian coalescent point process when psi x x gamma we show the existence of a self similar solution for the pde which relates when gamma 2 to the speed of coming down from infinity of the nested kingman coalescent