the steklov eigenvalue problem in a cuspidal domain in this paper we analyze the approximation by piecewise linear finite elements of a steklov eigenvalue problem in a plane domain with an external cusp this problem is not covered by the literature and its analysis requires a special treatment indeed we develop new trace theorems and we also obtain regularity results for the source counterpart moreover under appropriate assumptions on the meshes we present interpolation error estimates for functions in fractional sobolev spaces these estimates allow us to obtain appropriate convergence results of the source counterpart which in the context of the theory of compact operator are a fundamental tool in order to prove the convergence of the eigenpairs at the end we prove the convergence of the eigenpairs by using graded meshes and present some numerical tests

a comparison of homogeneity tests for regional frequency analysis 1 the assessment of regional homogeneity is a critical point in regional frequency analysis to this end many homogeneity tests have been proposed even though a general comparison among them is still lacking commonly used homogeneity tests based on l moments ratios are considered here in a comparison with two rank tests that do not rely on particular assumptions regarding the parent distribution the performance of these tests is assessed in a series of monte carlo simulation experiments in particular the power and type i error of each test are determined for different scale and shape parameters of the regional parent distributions the tests are also evaluated by varying the number of sites belonging to the region the series length the type of the parent distributions and the degree of heterogeneity we find that l moments based tests are more powerful when the samples are slightly skewed while the rank tests have better performances in case of high skewness on the basis of these findings we propose a simple method to guide the choice of the homogeneity test to be used for the different possible cases

k sample anderson darling tests abstract two k sample versions of an anderson darling rank statistic are proposed for testing the homogeneity of samples their asymptotic null distributions are derived for the continuous as well as the discrete case in the continuous case the asymptotic distributions coincide with the k 1 fold convolution of the asymptotic distribution for the anderson darling one sample statistic the quality of this large sample approximation is investigated for small samples through monte carlo simulation this is done for both versions of the statistic under various degrees of data rounding and sample size imbalances tables for carrying out these tests are provided and their usage in combining independent one or k sample anderson darling tests is pointed out the test statistics are essentially based on a doubly weighted sum of integrated squared differences between the empirical distribution functions of the individual samples and that of the pooled sample one weighting adjusts for the possibly different s

higher genus modular graph functions string invariants and their exact asymptotics the concept and the construction of modular graph functions are generalized from genus one to higher genus surfaces the integrand of the four graviton superstring amplitude at genus two provides a generating function for a special class of such functions a general method is developed for analyzing the behavior of modular graph functions under non separating degenerations in terms of a natural real parameter t for arbitrary genus the arakelov green function and the kawazumi zhang invariant degenerate to a laurent polynomial in t of degree 1 1 in the limit t to infty for genus two each coefficient of the low energy expansion of the string amplitude degenerates to a laurent polynomial of degree w w in t where w 2 is the degree of homogeneity in the kinematic invariants these results are exact to all orders in t up to exponentially suppressed corrections the non separating degeneration of a general class of modular graph functions at arbitrary genus is sketched and similarly results in a laurent polynomial in t of bounded degree the coefficients in the laurent polynomial are generalized modular graph functions for a punctured riemann surface of lower genus

second variation of zhang s λ invariant on the moduli space of curves we compute the second variation of the invariant recently introduced by s zhang on the complex moduli space mg of curves of genus g 2 using work of n kawazumi as a result we prove that 8g 4 is equal up to a constant to the invariant introduced some time ago by r hain and d reed we deduce some consequences for example we calculate the invariant for each hyperelliptic curve expressing it in terms of the petersson norm of the discriminant modular form 1 introduction recently independently s zhang 27 and n kawazumi 16 introduced a new interesting real valued function on the moduli space mg of complex curves of genus g 2 its value at a curve x m g is given as follows let h 0 x x be the space of holomorphic differentials on x equipped with the

global weak solutions for the two dimensional motion of several rigid bodies in an incompressible viscous fluid we consider the two dimensional motion of several non homogeneous rigid bodies immersed in an incompressible non homogeneous viscous fluid the fluid and the rigid bodies are contained in a fixed open bounded set of r 2 the motion of the fluid is governed by the navier stokes equations for incompressible fluids and the standard conservation laws of linear and angular momentum rule the dynamics of the rigid bodies the time variation of the fluid domain due to the motion of the rigid bodies is not known a priori so we deal with a free boundary value problem the main novelty here is the demonstration of the global existence of weak solutions for this problem more precisely the global character of the solutions we obtain is due to the fact that we do not need any assumption concerning the lack of collisions between several rigid bodies or between a rigid body and the boundary we give estimates of the velocity of the bodies when their mutual distance or the distance to the boundary tends to zero

finite element approximations in a non lipschitz domain part ii fil acosta gabriel universidad de buenos aires facultad de ciencias exactas y naturales departamento de matematica argentina

finite element approximation of spectral problems with neumann boundary conditions on curved domains this paper deals with the finite element approximation of the spectral problem for the laplace equation with neumann boundary conditions on a curved nonconvex domain convergence and optimal order error estimates are proved for standard piecewise linear continuous elements on a discrete polygonal domain h in tile framework of the abstract spectral approximation theory

special finite element methods for a class of second order elliptic problems with rough coefficients in this paper the approximate solution of a class of second order elliptic equations with rough coefficients is considered problems of the type considered arise in the analysis of unidirectional composites where the coefficients represent the properties of the material several methods for this class of problems are presented and it is shown that they have the same accuracy as usual methods have for problems with smooth coefficients the methods are referred to as special finite element methods because they are of finite element type but employ special shape functions chosen to accurately model the unknown solution

the maximum angle condition for mixed and nonconforming elements application to the stokes equations for the lagrange interpolation it is known that optimal order error estimates hold for elements satisfying the maximum angle condition the objective of this paper is to obtain similar results for the raviart thomas interpolation arising in the analysis of mixed methods we prove that optimal order error estimates hold under the maximum angle condition for this interpolation both in two and three dimensions and moreover that this condition is indeed necessary to have these estimates error estimates for the mixed approximation of second order elliptic problems and for the nonconforming piecewise linear approximation of the stokes equations are derived from our results

finite element approximations in a nonlipschitz domain in this paper we analyze the approximation by standard piecewise linear finite elements of a nonhomogeneous neumann problem in a cuspidal domain since the domain is not lipschitz many of the results on sobolev spaces which are fundamental in the usual error analysis do not apply therefore we need to work with weighted sobolev spaces and to develop some new theorems on traces and extensions we show that in the domain considered here suboptimal order can be obtained with quasi uniform meshes even when the exact solution is in h 2 and we prove that the optimal order with respect to the number of nodes can be recovered by using appropriate graded meshes

improving the rate of convergence of high order finite elements on polygons and domains with cusps let u and uv v be the solution and respectively the discrete solution of the non homogeneous dirichlet problem u f on u 0 for any m and any bounded polygonal domain we provide a construction of a new sequence of finite dimensional subspaces vn such that where f hm 1 is arbitrary and c is a constant that depends only on and not on n we do not assume u hm 1 the existence of such a sequence of subspaces was first proved in a ground breaking paper by babuska 8 our method is different it is based on the homogeneity properties of sobolev spaces with weights and the well posedness of non homogeneous dirichlet problem in suitable sobolev spaces with weights for which we provide a new proof and which is a substitute of the usual shift theorems for boundary value problems in domains with smooth boundary our results extended right away to domains whose boundaries have conical points we also indicate some of the changes necessary to deal with domains with cusps our numerical computation are in agreement with our theoretical results

the asymptotics of the arakelov geen s function and faltings delta invariant we study the behavior of the arakelov green s function and faltings delta invariant on degenerating riemann surfaces

l moment diagrams should replace product moment diagrams it is well known that product moment ratio estimators of the coefficient of variation c skewness and kurtosis exhibit substantial bias and variance for the small n 100 samples normally encountered in hydrologic applications consequently l moment ratio estimators termed l coefficient of variation 2 l skewness 3 and l kurtosis 4 are now advocated because they are nearly unbiased for all underlying distributions the advantages of l moment ratio estimators over product moment ratio estimators are not limited to small samples monte carlo experiments reveal that product moment estimators of c and are also remarkably biased for extremely large samples n 1000 from highly skewed distributions a case study using large samples n 5000 of average daily streamflow in massachusetts reveals that conventional moment diagrams based on estimates of product moments c and reveal almost no information about the distributional properties of daily streamflow whereas l moment diagrams based on estimators of 2 3 and 4 enabled us to discriminate among alternate distributional hypotheses

the combined effect of numerical integration and approximation of the boundary in the finite element method for eigenvalue problems the paper deals with the finite element analysis of second order elliptic eigenvalue problems when the approximate domains omega h are not subdomains of the original domain omega subset r 2 and when at the same time numerical integration is used for computing the involved bilinear forms the considerations are restricted to piecewise linear approximations the optimum rate of convergence o h 2 for approximate eigenvalues is obtained provided that a quadrature formula of first degree of precision is used in the case of a simple exact eigenvalue the optimum rate of convergence o h for approximate eigenfunctions in the h 1 omega h norm is proved while in the l 2 omega h norm an almost optimum rate of convergence i e near to o h 2 is achieved in both cases a quadrature formula of first degree of precision is used quadrature formulas with degree of precision equal to zero are also analyzed and in the case when the exact eigenfunctions belong only to h 1 omega the convergence without the rate of convergence is proved in the case of a multiple exact eigenvalue the approximate eigenfunctions are compard in contrast to standard considerations with linear combinations of exact eigenfunctions with coefficients not depending on the mesh parameter h

extreme rainstorms comparing regional envelope curves to stochastically generated events 1 the depth duration envelope curves ddecs are regional upper bounds on observed rainfall maxima for several durations recently a probabilistic interpretation has been proposed in the literature in order to associate a recurrence interval t to the ddecs and consequently to retrieve point rainfall quantiles for ungauged sites alternatively extreme rainfall quantiles can be retrieved from long synthetic rainfall series obtained with stochastic rainfall generators calibrated to local time series of rainfall events while ddecs are sensitive to outliers and data errors the stochastic rainfall generator performance is affected by the limited record lengths used for calibration the objective of this study is to assess the reliability of the two alternative methods by verifying if they give consistent results for a wide study region in austria relative to previous studies we propose some generalizations of the ddec procedure in order to better represent the austrian data the comparison of rainfall quantiles estimated with the two methods for large t shows an excellent agreement for intermediate durations from 1 to 6 h while the agreement worsen for very short 15 min and long 24 h durations the results are scrupulously analyzed and discussed in light of the exceptionality of rainfall events that set the regional envelopes and the characteristics of the stochastic generator used our study points out that the combined use of these regional and local methods can be very useful for estimating reliable point rainfall quantiles associated with large t within regions where many rain gauges are available but with limited record lengths

finite element approximation of spectral acoustic problems on curved domains this paper deals with the finite element approximation of the displacement formulation of the spectral acoustic problem on a curved non convex two dimensional domain convergence and error estimates are proved for raviart thomas elements on a discrete polygonal domain h inline equation in the framework of the abstract spectral approximation theory similar results have been previously proved only for polygonal domains numerical tests confirming the theoretical results are reported

bosonization on higher genus riemann surfaces we prove the equivalence between certain fermionic and bosonic theories in two spacetime dimensions the theories have fields of arbitrary spin on compact surfaces with any number of handles global considerations require that we add new topological terms to the bosonic action the proof that our prescription is correct relies on methods of complex algebraic geometry

a nonconforming finite element method for a two dimensional curl curl and grad div problem a numerical method for a two dimensional curl curl and grad div problem is studied in this paper it is based on a discretization using weakly continuous p 1 vector fields and includes two consistency terms involving the jumps of the vector fields across element boundaries optimal convergence rates up to an arbitrary positive epsilon in both the energy norm and the l 2 norm are established on graded meshes the theoretical results are confirmed by numerical experiments

the relationship between monte carlo estimators of heterogeneity and error for daily to monthly time steps in a small minnesota precipitation gauge network precipitation quantile estimates are used in engineering agriculture and a variety of other disciplines index flood regional frequency methods pool normalized gauge data in the case of homogeneity among the constituent gauges of the region unitless regional quantile estimates are outputted and rescaled at each gauge because violation of the homogeneity hypothesis is a major component of quantile estimation error in regional frequency analysis heterogeneity estimators should be reasonable proxies of the error of quantile estimation in this study three monte carlo heterogeneity statistics tested in hosking and wallis 1997 are plotted against monte carlo estimates of quantile error for all five or more gauge regionalizations in a 12 gauge network in the twin cities region of minnesota upper tail quantiles with nonexceedance probabilities of 0 75 and above are examined at time steps ranging from daily to monthly a linear relationship between heterogeneity and error estimates is found and quantified using pearson s r score two of hosking and wallis s 1997 heterogeneity measures incorporating the coefficient of variation in one case and additionally the skewness in the other are found to be reasonable proxies for quantile error at the l moment ratio values characterizing these data this result in addition to confirming the utility of a commonly used coefficient of variation based heterogeneity statistic provides evidence for the utility of a heterogeneity measure that incorporates skewness information

nonhomogeneous neumann problem for the poisson equation in domains with an external cusp abstract in this work we analyze the existence and regularity of the solution of a nonhomogeneous neumann problem for the poisson equation in a plane domain with an external cusp in order to prove that there exists a unique solution in h 1 using the lax milgram theorem we need to apply a trace theorem since is not a lipschitz domain the standard trace theorem for h 1 does not apply in fact the restriction of h 1 functions is not necessarily in l 2 so we introduce a trace theorem by using weighted sobolev norms in under appropriate assumptions we prove that the solution of our problem is in h 2 and we obtain an a priori estimate for the second derivatives of the solution

the p and h p versions of the finite element method basic principles and properties in the classical form of the finite element method called the hversion piecewise polynomials of fixed degree p are used and the mesh size h is decreased for accuracy in this paper we discuss the fundamental theoretical ideas behind the relatively recent p version and h p version in the p version a fixed mesh is used and p is allowed to increase the h p version combines both approaches the authors describe and explain the basic properties and characteristics of these newer versions especially in areas where their behavior is significantly different from that of the h version simplified proofs of key concepts are included and computational illustrations of several results are provided a benchmark comparison between the various versions in included

probabilistic behavior of a regional envelope curve 1 a regional envelope curve rec summarizes the current bound on our experience of extreme floods in a region although recs are available for many regions of the world their traditional deterministic interpretation limits the use of the curves for design purposes as magnitude but not frequency of extreme flood events can be quantified a probabilistic interpretation of a rec is introduced via an estimate of its exceedance probability the effect of intersite correlation on the exceedance probability is discussed monte carlo experiments are performed to assess the performance of the estimate of the rec s exceedance probability generalized curves are presented for assessing the impact of regional intersite cross correlation on the likelihood of exceeding an observed rec we also discuss how future developments in the theory of records may improve our ability to quantify the information content of a rec