lifespan theorem for constrained surface diffusion flows we consider closed immersed hypersurfaces in mathbb r 3 and mathbb r 4 evolving by a class of constrained surface diffusion flows our result similar to earlier results for the willmore flow gives both a positive lower bound on the time for which a smooth solution exists and a small upper bound on a power of the total curvature during this time by phrasing the theorem in terms of the concentration of curvature in the initial surface our result holds for very general initial data and has applications to further development in asymptotic analysis for these flows

optimal rigidity estimates for nearly umbilical surfaces let r 3 be a smooth compact connected surface without boundary and denote by a its second fundamental form we prove the existence of a universal constant c such that

conservation laws for conformally invariant variational problems we succeed in writing 2 dimensional conformally invariant non linear elliptic pde harmonic map equation prescribed mean curvature equations etc in divergence form these divergence free quantities generalize to target manifolds without symmetries the well known conservation laws for weakly harmonic maps into homogeneous spaces from this form we can recover without the use of moving frame all the classical regularity results known for 2 dimensional conformally invariant non linear elliptic pde see hel it enables us also to establish new results in particular we solve a conjecture by e heinz asserting that the solutions to the prescribed bounded mean curvature equation in arbitrary manifolds are continuous and we solve a conjecture by s hildebrandt hil1 claiming that critical points of continuously differentiable elliptic conformally invariant lagrangian in two dimensions are continuous

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

receiver operating characteristic analysis under tree orderings of disease classes receiver operating characteristic roc curve and its summary statistics e g the area under curve auc are commonly used to evaluate the diagnostic accuracy for disease processes with binary classification the roc curve has been extended to roc surface for scenarios with three ordinal classes or to hyper surface for scenarios with more than three classes for classifier under tree or umbrella ordering in which the marker measurement for one class is lower or higher than those for the other classes the commonly adopted diagnostic measures are the naive auc nauc based on a pooled class of all the unordered classes and the umbrella volume uv based on the concept of volume under surface however both nauc and uv have some limitations for example nauc depends on the sampling weights for all the classes in population and uv has only been introduced for three class settings in this article we initiate the idea of a new roc framework for tree or umbrella ordering denoted as troc and propose the area under troc curve denoted as tauc as an appropriate diagnostic measure the proposed troc and tauc share many nice features with the traditional roc and auc both parametric and nonparametric approaches are explored to construct the confidence interval estimation of tauc the performances of these methods are compared in simulation studies under a variety settings at the end the proposed methods are applied to a published microarray data set

euler lagrange equation and regularity for flat minimizers of the willmore functional let input amssym s subset bbb r 2 be a bounded domain with boundary of class c and let gij ij denote the flat metric on input amssym bbb r 2 let u be a minimizer of the willmore functional within a subclass defined by prescribing boundary conditions on parts of s of all w2 2 isometric immersions of the riemannian manifold s g into input amssym bbb r 3 in this article we derive the euler lagrange equation and study the regularity properties for such u our main regularity result is that minimizers u are c3 away from a certain singular set and c away from a larger singular set 0 we obtain a geometric characterization of these singular sets and we derive the scaling of u and its derivatives near 0 our main motivation to study this problem comes from nonlinear elasticity on isometric immersions the willmore functional agrees with kirchhoff s energy functional for thin elastic plates 2010 wiley periodicals inc

embedded area constrained willmore tori of small area in riemannian three manifolds ii morse theory the synthesis of cationic rhodium and iridium complexes of a bis imidazol 2 thione functionalised calix 4 arene ligand and their surprising capacity for potassium binding is described in both cases uptake of the alkali metal into the calix 4 arene cavity occurs despite adverse electrostatic interactions associated with close proximity to the transition metal fragment rh k 3 715 1 a ir k 3 690 1 a the formation and constituent bonding of these unusual heterobimetallic adducts has been interrogated through extensive solution and solid state characterisation examination of the host guest chemistry of the ligand and its upper rim unfunctionalised calix 4 arene analogue and computationally using dft based energy decomposition analysis eda this is the second part of a series of two papers where we construct embedded willmore tori with small area constraint in riemannian three manifolds in both papers the construction relies on a lyapunov schmidt reduction the difficulty being the moebius degeneration of the tori in the first paper the construction was performed via minimization here by morse theory to this aim we establish new geometric expansions of the derivative of the willmore functional on small clifford tori in geodesic normal coordinates which degenerate to small geodesic spheres with a small handle under the action of the moebius group by using these sharp asymptotics we give sufficient conditions in terms of the ambient curvature tensors and morse inequalities for having existence multiplicity of embedded tori which are stationary for the willmore functional under the constraint of prescribed sufficiently small area

an energy gap phenomenon for the whitney sphere in this paper we study lagrangian surfaces satisfying nabla t 0 where t 2 nabla check a lrcorner omega and check a is the lagrangian trace free second fundamental form we obtain a gap lemma for such a lagrangian surface

comparing the areas under two or more correlated receiver operating characteristic curves a nonparametric approach methods of evaluating and comparing the performance of diagnostic tests are of increasing importance as new tests are developed and marketed when a test is based on an observed variable that lies on a continuous or graded scale an assessment of the overall value of the test can be made through the use of a receiver operating characteristic roc curve the curve is constructed by varying the cutpoint used to determine which values of the observed variable will be considered abnormal and then plotting the resulting sensitivities against the corresponding false positive rates when two or more empirical curves are constructed based on tests performed on the same individuals statistical analysis on differences between curves must take into account the correlated nature of the data this paper presents a nonparametric approach to the analysis of areas under correlated roc curves by using the theory on generalized u statistics to generate an estimated covariance matrix

embedded area constrained willmore tori of small area in riemannian three manifolds i minimization we construct embedded willmore tori with small area constraint in riemannian three manifolds under some curvature condition used to prevent mobius degeneration the construction relies on a lyapunov schmidt reduction to this aim we establish new geometric expansions of exponentiated small symmetric clifford tori and analyze the sharp asymptotic behaviour of degenerating tori under the action of the mobius group in this first work we prove two existence results by minimizing or maximizing a suitable reduced functional in particular we obtain embedded area constrained willmore tori or equivalently toroidal critical points of the hawking mass under area constraint in compact 3 manifolds with constant scalar curvature and in the double schwarzschild space in a forthcoming paper new existence theorems will be achieved via morse theory

variational principles for immersed surfaces with l 2 bounded second fundamental form in this work we present new fundamental tools for studying the variations of the willmore functional of immersed surfaces into r m this approach gives for instance a new proof of the existence of a willmore minimizing embedding of an arbitrary closed surface in arbitrary codimension we explain how the same approach can solve constraint minimization problems for the willmore functional we show in particular that for a given closed surface and a given conformal class for this surface there is an immersion in r m away possibly from isolated branched points which minimizes the willmore energy among all possible lipschitz immersions in r m having an l 2 bounded second fundamental form and realizing this conformal class this branched immersion is either a smooth conformal willmore branched immersion or an isothermic branched immersion we show that branched points do not exist whenever the minimal energy in the conformal class is less than 8 and that these immersions extend to smooth conformal willmore embeddings or global isothermic embeddings of the surface in that case finally as a by product of our analysis we establish that inside a compact subspace of the moduli space the following holds weak limit of palais smale willmore sequences are conformal willmore that weak limits of palais smale sequences of conformal willmore are either conformal willmore or global isothermic and finally we observe also that weakly converging palais smale sequences of global isothermic immersions are global isothermic the analysis developped along the paper in particular these last results opens the door to the possibility of constructing new critical saddle points of the willmore functional without or with constraints using min max methods

the gradient flow of the l 2 curvature energy near the round sphere abstract we investigate the low energy behavior of the gradient flow of the l 2 norm of the riemannian curvature on four manifolds we show long time existence and exponential convergence to a metric of constant sectional curvature when the initial metric has positive yamabe constant and small initial energy

on budaev and bogy s approach to diffraction by the 2d traction free elastic wedge several semianalytical approaches are now available for describing diffraction of a plane wave by the 2d two dimensional traction free isotropic elastic wedge in this paper we follow budaev and bogy who reformulated the original diffraction problem as a singular integral one this comprises two algebraic and two singular integral equations each integral equation involves two unknowns a function and a constant we discuss the underlying integral operators and develop a new semianalytical scheme for solving the integral equations we investigate the properties of the solution obtained and argue that it is the solution of the original diffraction problem we describe a comprehensive code verification and validation program

a frame energy for immersed tori and applications to regular homotopy classes the paper is devoted to study the dirichelet energy of moving frames on 2 dimensional tori immersed in the euclidean 3 m dimensional space this functional called frame energy is naturally linked to the willmore energy of the immersion and on the conformal structure of the abstract underlying surface as first result a willmore conjecture type lower bound is established namely for every torus immersed in r m m 3 and any moving frame on it the frame energy is at least 2 2 and equalty holds if and only if m 4 the immersion is the standard clifford torus up to rotations and dilations and the frame is the flat one smootheness of the critical points of the frame energy is proved after the discovery of hidden conservation laws and as application the minimization of the frame energy in regular homotopy classes of immersed tori in r 3 is performed

the geometric triharmonic heat flow of immersed surfaces near spheres abstract we consider closed immersed surfaces in r 3 evolving by the geometric triharmonic heat flow using local energy estimates we prove interior estimates and a positive absolute lower bound on the lifespan of solutions depending solely on the local concentration of curvature of the initial immersion in l 2 we further use an e regularity type result to prove a gap lemma for stationary solutions using a monotonicity argument we then prove that a blowup of the flow approaching a singular time is asymptotic to a non umbilic embedded stationary surface this allows us to conclude that any solution with initial l 2 norm of the tracefree curvature tensor smaller than an absolute positive constant converges exponentially fast to a round sphere with radius equal to 3 v 0 4 3 where v 0 denotes the signed enclosed volume of the initial data

exact inference for growth curves with intraclass correlation structure summary we consider repeated observations taken over time for each of several subjects for example one might consider the growth curve of a cohort of babies over time we assume a simple linear growth curve model exact results based on sufficient statistics exact tests of the null hypothesis that a coefficient is zero or exact confidence intervals for coefficients are not available to make inference on regression coefficients when an intraclass correlation structure is assumed this paper will demonstrate that such exact inference is possible using generalized inference

notes on the overlap measure as an alternative to the youden index how are they related the receiver operating characteristic roc curve is frequently used to evaluate and compare diagnostic tests as one of the roc summary indices the youden index measures the effectiveness of a diagnostic marker and enables the selection of an optimal threshold value cut off point for the marker recently the overlap coefficient which captures the similarity between 2 distributions directly has been considered as an alternative index for determining the diagnostic performance of markers in this case a larger overlap indicates worse diagnostic accuracy and vice versa this paper provides a graphical demonstration and mathematical derivation of the relationship between the youden index and the overlap coefficient and states their advantages over the most popular diagnostic measure the area under the roc curve furthermore we outline the differences between the youden index and overlap coefficient and identify situations in which the overlap coefficient outperforms the youden index numerical examples and real data analysis are provided

estimation of the area under the roc curve the area under the receiver operating characteristic curve is frequently used as a measure for the effectiveness of diagnostic markers in this paper we discuss and compare estimation procedures for this area these are based on i the mann whitney statistic ii kernel smoothing iii normal assumptions iv empirical transformations to normality these are compared in terms of bias and root mean square error in a large variety of situations by means of an extensive simulation study overall we find that transforming to normality usually is to be preferred except for bimodal cases where kernel methods can be effective

blow up phenomena for the yamabe equation it is well known that the pde 1 has at least one positive solution for any choice of m g if n 6 and m g is not locally conformally flat this follows from results of t aubin 3 the remaining cases were solved by r schoen using the positive mass theorem 16 solutions to 1 are not usually unique as an example consider the product metric on 51 l x 5n 1 l if l is sufficiently small then the yamabe pde has a unique solution on the other hand there are many non minimizing solutions if l is large d pollack 14 has used gluing techniques to construct high energy solutions on more general background manifolds given any conformai class with positive yamabe constant and any positive integer iv there exists a new conformai class which is close to the original one in the c norm and contains at least n metrics of constant scalar curvature see 14 theorem 0 1 it is an interesting question whether the set of all solutions to the yamabe pde is compact in the c2 topology say a well known conjecture states that this should be true unless m p is conformally equivalent to the round sphere see 17 18 19 this conjecture has been verified in low dimensions and in the locally conformally flat case if m g is locally conformally flat compactness follows from work of r schoen 17 18 moreover schoen proposed a strategy for proving the conjecture in the non locally conformally flat case based on the pohozaev identity

diffraction by an elastic wedge with stress free boundary existence and uniqueness applying the spectral function techniques developed by croisille and lebeau we prove the existence of solutions to problems of plane and cylindrical waves diffraction by an elastic wedge with stress free boundary conditions we also formulate radiation conditions under which the uniqueness holds the latter implies absolute continuity of the spectrum of the lame operator in a wedge domain with stress free boundary

a theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity the energy functional of nonlinear plate theory is a curvature functional for surfaces rst proposed on physical grounds by g kirchhoff in 1850 we show that it arises as a 0 limit of three dimensional nonlinear elasticity theory as the thickness of a plate goes to zero a key ingredient in the proof is a sharp rigidity estimate for maps v v u r n u r n we show that the l 2 distance of rv from a single rotation matrix is bounded by a multiple of the l 2 distance from

isogeometric analysis of geometric partial differential equations abstract we consider the numerical approximation of geometric partial differential equations pdes defined on surfaces in the 3d space in particular we focus on the geometric pdes deriving from the minimization of an energy functional by l 2 gradient flow we analyze two energy functionals the area which leads to the mean curvature flow a nonlinear second order pde and the willmore energy leading to the willmore flow a nonlinear fourth order pde we consider surfaces represented by single patch tensor product nurbs and discretize the pdes by means of nurbs based isogeometric analysis in the framework of the galerkin method to approximate the high order geometric pdes we use high order continuous nurbs basis functions for the time discretization of the nonlinear geometric pdes we use backward differentiation formulas bdf with extrapolation of the geometric quantities involved in the weak formulation of the problem in this manner we solve a linear problem at each time step we report numerical results concerning the mean curvature and willmore flows on different geometries of interest and we show the accuracy and efficiency of the proposed approximation scheme

the wiener hopf solution of the isotropic penetrable wedge problem diffraction and total field the diffraction of an incident plane wave by an isotropic penetrable wedge is studied using generalized wiener hopf equations and the solution is obtained using analytical and numerical analytical approaches that reduce the wiener hopf factorization to fredholm integral equations of second kind mathematical aspects are described in a unified and consistent theory for angular region problems the formulation is presented in the general case of skew incidence and several numerical tests at normal incidence are reported to validate the new technique the solutions consider engineering applications in terms of gtd utd diffraction coefficients and total fields

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

a family of nonparametric statistics for comparing diagnostic markers with paired or unpaired data summary in this paper we study a broad class of nonparametric statistics for comparing two diagnostic markers one can compare the sensitivities of these diagnostic markers over restricted ranges of specificity by selecting an appropriate statistic from this class as special cases one can compare the entire area under the receiver operator curve hanley mcneil 1982 or one can compare the sensitivities at a fixed common specificity usually we would recommend a comparison based on an average of sensitivities over a restricted high level of specificities test procedures and confidence intervals are based on asymptotic normality these procedures are applicable for paired data in which both diagnostic markers are performed on each subject and for unpaired data the procedures may be used to compare two real functions of multiple diagnostic markers as well as to compare individual markers

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

bubble tree of branched conformal immersions and applications to the willmore functional we develop a bubble tree construction and prove compactness results for w 2 2 branched conformal immersions of closed riemann surfaces with varying conformal structures whose limit may degenerate in bbb r n with uniformly bounded areas and willmore energies the compactness property is applied to construct willmore type surfaces in compact riemannian manifolds this includes a existence of a willmore 2 sphere in bbb s n with at least 2 nonremovable singular points b existence of minimizers of the willmore functional with prescribed area in a compact manifold n provided i the area is small when genus is 0 and ii the area is close to that of the area minimizing surface of schoen yau and sacks uhlenbeck in the homotopy class of an incompressible map from a surface of positive genus to n and pi 2 n is trivial c existence of smooth minimizers of the willmore functional if a douglas type condition is satisfied

roc analysis for markers with mass at zero the receiver operating characteristic roc curve and in particular the area under the curve auc is commonly used to examine the discriminatory ability of diagnostic markers certain markers while basically continuous and non negative have a positive probability mass spike at the value zero we discuss a flexible modelling approach to such data and contrast it with the standard non parametric approach we show how the modelling approach can be extended to take account of the effect of explanatory variables we motivate this problem and illustrate the modelling approach using data on the coronary calcium score measured by electron beam tomography which is a marker for atherosclerosis

generalized p values in significance testing of hypotheses in the presence of nuisance parameters abstract this article examines some problems of significance testing for one sided hypotheses of the form h 0 0 versus h 1 0 where is the parameter of interest in the usual setting let x be the observed data and let t x be a test statistic such that the family of distributions of t x is stochastically increasing in define cx as x t x t x 0 the p value is p x sup 0 pr x cx in the presence of a nuisance parameter there may not exist a nontrivial cx with a p value independent of we consider tests based on generalized extreme regions of the form cx x t x x t x x and conditions on t x x are given such that the p value p x sup 0 pr x cx is free of the nuisance parameter where t is stochastically increasing in we provide a solution to the problem of testing hypotheses about the differences in means of two independent exponential distributions a problem for which the fixed level testing approach

confidence intervals for p y1 y2 with normal outcomes in linear models recently there is an emerging interest in the inference of p y1 y2 where y1 and y2 stand for two independent continuous random variables so far most of the research in this field focuses on simply comparing two outcomes without adjusting for covariates this paper mainly presents a large sample approach based on a noncentral t distribution for the confidence interval estimation of p y1 y2 with normal outcomes in linear models furthermore the performance of the proposed large sample approach is compared with that of a generalized variable approach and a bootstrap approach simulation studies demonstrate that for small to medium sample sizes both the large sample approach and the generalized variable approach provide confidence intervals with satisfying coverage probabilities whereas the bootstrap approach can be slightly liberal for certain scenarios the proposed approaches are applied to three real life data sets copyright 2008 john wiley sons ltd

computational parametric willmore flow we propose a new algorithm for the computation of willmore flow this is the l 2 gradient flow for the willmore functional which is the classical bending energy of a surface willmore flow is described by a highly nonlinear system of pdes of fourth order for the parametrization of the surface the spatially discrete numerical scheme is stable and consistent the discretization relies on an adequate calculation of the first variation of the willmore functional together with a derivation of the second variation of the area functional which is well adapted to discretization techniques with finite elements the algorithm uses finite elements on surfaces we give numerical examples and tests for piecewise linear finite elements a convergence proof for the full algorithm remains an open question

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

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

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

flow by mean curvature of convex surfaces into spheres the motion of surfaces by their mean curvature has been studied by brakke 1 from the viewpoint of geometric measure theory other authors investigated the corresponding nonparametric problem 2 5 9 a reason for this interest is that evolutionary surfaces of prescribed mean curvature model the behavior of grain boundaries in annealing pure metal in this paper we take a more classical point of view consider a compact uniformly convex w dimensional surface m mo without boundary which is smoothly imbedded in r let mo be represented locally by a diffeomorphism

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

a classification theorem for helfrich surfaces in this paper we study the functional mathcal w lambda 1 lambda 2 which is the sum of the willmore energy lambda 1 weighted surface area and lambda 2 weighted volume for surfaces immersed in mathbb r 3 this coincides with the helfrich functional with zero spontaneous curvature our main result is a complete classification of all smooth immersed critical points of the functional with lambda 1 ge 0 and small l 2 norm of tracefree curvature with no assumption on the growth of the curvature in l 2 at infinity this not only improves the gap lemma due to kuwert and schatzle for willmore surfaces immersed in mathbb r 3 but also implies the non existence of critical points of the functional satisfying the energy condition for which the surface area and enclosed volume are positively weighted

foliations of asymptotically flat manifolds by surfaces of willmore type the goal of this paper is to establish the existence of a foliation of the asymptotic region of an asymptotically flat manifold with positive mass by surfaces which are critical points of the willmore functional subject to an area constraint equivalently these surfaces are critical points of the geroch hawking mass thus our result has applications in the theory of general relativity

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 multiplicity result for the yamabe problem on sn abstract we prove a multiplicity result for the yamabe problem on the manifold s g where g is a perturbation of the standard metric g 0 of s n solutions are found by variational methods via an abstract perturbation result

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

singular perturbations of mean curvature flow we consider a regularization method for mean curvature now of a submanifold of arbitrary codimension in the euclidean space through higher order equations we prove that the regularized problems converge to the mean curvature flow for all times before the first singularity

variational perturbative methods and bifurcation of bound states from the essential spectrum this paper consists of two main parts the first deals with a perturbative method in critical point theory and can be seen as the generalisation and completion of some earlier results the second part is concerned with applications of the abstract setup to the existence of bound states of a class of elliptic differential equations that branch off from the infimum of the essential spectrum

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

inference of the youden index and associated threshold using empirical likelihood for quantiles the youden index is a widely used measure in the framework of medical diagnostics where the effectiveness of a biomarker screening marker or predictor for classifying a disease status is studied when the biomarker is continuous it is important to determine the threshold or cut off point to be used in practice for the discrimination between diseased and healthy populations we introduce two methods aimed at estimating the youden index and its associated threshold the first one is a modified version of a recent approach based on the delta method and the second one is based on the adjusted empirical likelihood for quantiles in the setting of a two sample problem we also include cis for both of them in the simulation study we compare both methods under different scenarios finally a real example of prostatic cancer well known in the literature is analysed to provide the reader with a better understanding of the new methodology copyright 2011 john wiley sons ltd

phase field approximations of the willmore functional and flow we discuss in this paper phase field approximations of the willmore functional and the associated mathrm l 2 l2 flow after recollecting known results on the approximation of the willmore energy and its mathrm l 1 l1 relaxation we derive the expression of the flows associated with various approximations and we show their behavior by formal arguments based on matched asymptotic expansions we introduce an accurate numerical scheme whose local convergence is proved to describe with more details the behavior of two flows the classical and the flow associated with an approximation model due to mugnai we propose a series of numerical simulations in 2d and 3d to illustrate their behavior in both smooth and singular situations

parametric approximation of willmore flow and related geometric evolution equations we present various variational approximations of willmore flow in mathbb r d d 2 3 as well as the classic willmore flow we also consider variants that are a volume preserving and b volume and area preserving the latter evolution law is the so called helfrich flow in addition we consider motion by gaus curvature the presented fully discrete schemes are easy to solve as they are linear at each time level and they have good properties with respect to the distribution of mesh points finally we present numerous numerical experiments including simulations for energies appearing in the modeling of biological cell membranes

critical weak immersed surfaces within sub manifolds of the teichmuller space abstract we prove that the critical points of various energies such as the area the willmore energy the frame energy for tori etc among possibly branched immersions constrained to evolve within a smooth sub manifold of the teichmuller space satisfy the corresponding constrained euler lagrange equation we deduce that critical points of the willmore energy or the frame energy for tori are smooth analytic surfaces away possibly from isolated branched points under the condition that either the genus is at most 2 or if the sub manifold does not intersect the subspace of hyper elliptic points using a compactness result from 19 we can conclude that each closed sub manifold of the teichmuller space including points under the previous assumptions possess a possibly branched smooth willmore minimizer satisfying the conformal constrained willmore equation

joint confidence region estimation for area under roc curve and youden index in the field of diagnostic studies the area under the roc curve auc serves as an overall measure of a biomarker diagnostic test s accuracy youden index defined as the overall correct classification rate minus one at the optimal cut off point is another popular index for continuous biomarkers of binary disease status although researchers mainly evaluate the diagnostic accuracy using auc for the purpose of making diagnosis youden index provides an important and direct measure of the diagnostic accuracy at the optimal threshold and hence should be taken into consideration in addition to auc furthermore auc and youden index are generally correlated in this paper we initiate the idea of evaluating diagnostic accuracy based on auc and youden index simultaneously as the first step toward this direction this paper only focuses on the confidence region estimation of auc and youden index for a single marker we present both parametric and non parametric approaches for estimating joint confidence region of auc and youden index we carry out extensive simulation study to evaluate the performance of the proposed methods in the end we apply the proposed methods to a real data set copyright 2013 john wiley sons ltd

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

computational parametric willmore flow with spontaneous curvature and area difference elasticity effects a new stable continuous in time semidiscrete parametric finite element method for willmore flow is introduced the approach allows for spontaneous curvature and area difference elasticity effects which are important for many applications in particular in the context of membranes the method extends ideas from dziuk and the present authors to obtain an approximation that allows for a tangential redistribution of mesh points which typically leads to better mesh properties moreover we consider volume and surface area preserving variants of these schemes and in particular we obtain stable approximations of helfrich flow we also discuss fully discrete variants and present several numerical computations

the large genus limit of the infimum of the willmore energy the willmore energy of a surface immersed into bbb r n is the integral of its squared mean curvature it is known that the infimum beta n p of the willmore energy among all closed oriented surfaces of genus p in bbb r n is attained by a smooth embedded surface and that 4 pi beta n p 8 pi for p geq 1 we show that beta n p converges to 8 pi as p goes to infinity

generalized confidence intervals abstract the definition of a confidence interval is generalized so that problems such as constructing exact confidence regions for the difference in two normal means can be tackled without the assumption of equal variances under certain conditions the extended definition is shown to preserve a repeated sampling property that a practitioner expects from exact confidence intervals the proposed procedure is also applied to the problem of constructing confidence intervals for the difference in two exponential means and for variance components in mixed models a repeated sampling property of generalized p values is also given with this characterization one can carry out fixed level tests of parameters of continuous distributions on the basis of generalized p values finally pratt s paradox is revisited and a procedure that resolves the paradox is given

willmore spheres in compact riemannian manifolds abstract the paper is devoted to the variational analysis of the willmore and other l 2 curvature functionals among immersions of 2 dimensional surfaces into a compact riemannian m manifold m m h with m 2 the goal of the paper is two fold on one hand we give the right setting for doing the calculus of variations including minmax methods of such functionals for immersions into manifolds and on the other hand we prove the existence results for possibly branched willmore spheres under various constraints prescribed homotopy class prescribed area or under curvature assumptions for m m to this aim using the integrability by compensation theory we first establish the regularity for the critical points of such functionals we then prove a rigidity theorem concerning the relation between cmc and willmore spheres then we prove that for every non null 2 homotopy class there exists a representative given by a lipschitz map from the 2 sphere into m m realizing a connected family of conformal smooth possibly branched area constrained willmore spheres as explained in the introduction this comes as a natural extension of the minimal immersed spheres in homotopy class constructed by sacks and uhlenbeck 1981 in 38 in situations when they do not exist moreover for every a 0 we minimize the willmore functional among connected families of weak possibly branched immersions of the 2 sphere having prescribed total area equal to a and we prove full regularity for the minimizer finally under a mild curvature condition on m m h we minimize the sum of the area with the square of the l 2 norm of the second fundamental form among weak possibly branched immersions of the two spheres and we prove the regularity of the minimizer

some results about the existence of critical points for the willmore functional using a perturbative approach it is shown existence and multiplicity of critical points for the willmore functional in ambient manifold mathbb r 3 g epsilon where g epsilon is a metric close and asymptotic to the euclidean one with the same technique it is proved a non existence result in general riemannian manifold m g of dimension three

diffraction by a two dimensional traction free elastic wedge this paper addresses the canonical two dimensional problem of diffraction of the elastic plane wave by a traction free straight edged wedge made of an isotropic solid its solution can be used in applications to model diffraction from curved surface breaking cracks with a curvature that is small compared to a wavelength we use the fourier transform to obtain a semianalytical solution conduct internal checks crossvalidate the code based on this scheme with the one based on the sommerfeld integral and describe experimental validation of the codes finally we draw attention to high sensitivity of the backscatter diffraction coefficients to the poisson ratio

lifespan theorem for simple constrained surface diffusion flows abstract we consider closed immersed hypersurfaces in r 3 and r 4 evolving by a special class of constrained surface diffusion flows this class of constrained flows includes the classical surface diffusion flow in this paper we present a lifespan theorem for these flows which gives a positive lower bound on the time for which a smooth solution exists and a small upper bound on the total curvature during this time the hypothesis of the theorem is that the surface is not already singular in terms of concentration of curvature this turns out to be a deep property of the initial manifold as the lower bound on maximal time obtained depends precisely upon the concentration of curvature of the initial manifold in l 2 for m 2 immersed in r 3 and additionally on the concentration in l 3 for m 3 immersed in r 4 this is stronger than a previous result on a different class of constrained surface diffusion flows as here we obtain an improved lower bound on maximal time a better estimate during this period and eliminate any assumption on the area of the evolving hypersurface

a numerical method for simulation dynamics of incompressible lipid membranes in viscous fluid a numerical scheme is presented for modeling dynamics of incompressible elastic lipid membranes imbedded into viscous fluid a new elliptic equation for tension in the membrane implying its local incompressibility is derived the membranes dynamics is approximated in a semi implicit way the lattice boltzmann method is used to approximate the fluid flow forces acting on the fluid from the lipid membrane are implemented using the immersed boundary method by peskin the method is illustrated by examples of axisymmetric membranes with deformations and flows typical for experiments with lipid vesicles and nanotubes where deformations and flows can be modulated by external forces applied to the membrane