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

topological properties of concept spaces full version based on the observation that the category of concept spaces with the positive information topology is equivalent to the category of countably based t 0 topological spaces we investigate further connections between the learning in the limit model of inductive inference and topology in particular we show that the texts or positive presentations of concepts in inductive inference can be viewed as special cases of the admissible representations of computable analysis we also show that several structural properties of concept spaces have well known topological equivalents in addition to topological methods we use algebraic closure operators to analyze the structure of concept spaces and we show the connection between these two approaches the goal of this paper is not only to introduce new perspectives to learning theorists but also to present the field of inductive inference in a way more accessible to domain theorists and topologists

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

efficient bootstrap simulation on donne deux idees pour renforcer une simulation la premiere est d equilibrer les echantillons simules la seconde est de faire un usage explicite des approximations qui ne demandent pas de simulation

set systems by reformulating a learning process of a set system l as a game between teacher and learner we define the order type of l to be the order type of the game tree if the tree is well founded the features of the order type of l diml in symbol are 1 we can represent any well quasi order wqo for short by the set system l of the upper closed sets of the wqo such that the maximal order type of the wqo is equal to diml 2 diml is an upper bound of the mind change complexity of l diml is defined iff l has a finite elasticity fe for short where according to computational learning theory if an indexed family of recursive languages has fe then it is learnable by an algorithm from positive data regarding set systems as subspaces of cantor spaces we prove that fe of set systems is preserved by any continuous function which is monotone with respect to the set inclusion by it we prove that finite elasticity is preserved by various nondeterministic language operators kleene closure shuffle closure union product intersection the monotone continuous functions represent nondeterministic computations if a monotone continuous function has a computation tree with each node followed by at most n immediate successors and the order type of a set system l is then the direct image of l is a set system of order type at most n adic diagonal ramsey number of furthermore we provide an order type preserving contravariant embedding from the category of quasi orders and finitely branching simulations between them into the complete category of subspaces of cantor spaces and monotone continuous functions having girard s linearity between them

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

locally efficient estimation of the quality adjusted lifetime distribution with right censored data and covariates summary zhao and tsiatis 1997 consider the problem of estimation of the distribution of the quality adjusted lifetime when the chronological survival time is subject to right censoring the quality adjusted lifetime is typically defined as a weighted sum of the times spent in certain states up until death or some other failure time they propose an estimator and establish the relevant asymptotics under the assumption of independent censoring in this paper we extend the data structure with a covariate process observed until the end of follow up and identify the optimal estimation problem because of the curse of dimensionality no globally efficient nonparametric estimators which have a good practical performance at moderate sample sizes exist given a correctly specified model for the hazard of censoring conditional on the observed quality of life and covariate processes we propose a closed form one step estimator of the distribution of the quality adjusted lifetime whose asymptotic variance attains the efficiency bound if we can correctly specify a lower dimensional working model for the conditional distribution of quality adjusted lifetime given the observed quality of life and covariate processes the estimator remains consistent and asymptotically normal even if this latter submodel is misspecified the practical performance of the estimators is illustrated with a simulation study we also extend our proposed one step estimator to the case where treatment assignment is confounded by observed risk factors so that this estimator can be used to test a treatment effect in an observational study

better bootstrap confidence intervals abstract we consider the problem of setting approximate confidence intervals for a single parameter in a multiparameter family the standard approximate intervals based on maximum likelihood theory can be quite misleading in practice tricks based on transformations bias corrections and so forth are often used to improve their accuracy the bootstrap confidence intervals discussed in this article automatically incorporate such tricks without requiring the statistician to think them through for each new application at the price of a considerable increase in computational effort the new intervals incorporate an improvement over previously suggested methods which results in second order correctness in a wide variety of problems in addition to parametric families bootstrap intervals are also developed for nonparametric situations

joint regression analysis of survival and quality adjusted survival summary in this paper a semiparametric bivariate linear regression model for survival and quality adjusted survival is investigated even with a parametric specification for the joint distribution maximum likelihood is not applicable because of induced informative censoring we propose inference procedures based on estimating functions the estimators are consistent and asymptotically normal hypothesis tests and confidence intervals may be constructed with easy to implement resampling techniques simultaneous regression modeling of survival and quality adjusted survival has not been studied formally our methodology gives parameter estimates that are highly interpretable in the context of a cost effectiveness analysis the usefulness of the proposal is illustrated with a breast cancer dataset

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

on the role of procrastination in machine learning the traditional model of inductive inference is enhanced to allow learning machines to procrastinate about how many trials they will need to complete an inference or about how accurate their solution will be hierarchies of classes of learnable phenomena represented by sets of recursive functions isomorphic to the structure of the constructive ordinals are revealed the existence of such hierarchies indicates a potential advantage of procrastination as a learning technique tradeoffs between the two types of procrastination are shown not to exist in general one of the new classes of sets of inferrible functions introduced in this paper turns out to have the somewhat unusual property of being closed under finite unions a new technique based on a hardest to lear set relative to a class is used

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

pairwise comparisons of trimmed means for two or more groups the paper takes up the problem of performing all pairwise comparisons amongj independent groups based on 20 trimmed means currently a method that stands out is the percentile t bootstrap method where the bootstrap is used to estimate the quantiles of a studentized maximum modulus distribution when all pairs of population trimmed means are equal however a concern is that in simulations the actual probability of one or more type i errors can drop well below the nominal level when sample sizes are small a practical issue is whether a method can be found that corrects this problem while maintaining the positive features of the percentile t bootstrap three new methods are considered here one of which achieves the desired goal another method which takes advantage of theoretical results by singh 1998 performs almost as well but is not recommended when the smallest sample size drops below 15 in some situations however it gives substantially shorter confidence intervals

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

mind change optimal learning of bayes net structure from dependency and independency data this paper analyzes the problem of learning the structure of a bayes net in the theoretical framework of gold s learning paradigm bayes nets are one of the most prominent formalisms for knowledge representation and probabilistic and causal reasoning we follow constraint based approaches to learning bayes net structure where learning is based on observed conditional dependencies and independencies between variables of interest e g the data are of the form x is dependent on y given any assignment to variables s or of the form x is independent of y given any assignment to variables s applying learning criteria in this model leads to the following results 1 the mind change complexity of identifying a bayes net graph over variables v from either dependency data or from independency data are v 2 the maximum number of edges 2 there is a unique fastest mind change optimal bayes net learner for either data type convergence speed is evaluated using gold s dominance notion of uniformly faster convergence for dependency data the optimal learner conjectures a graph if it is the unique bayes net pattern that satisfies the observed dependencies with a minimum number of edges and outputs no guess otherwise for independency data the optimal learner conjectures a graph if it is the unique bayes net pattern that satisfies the observed dependencies with a maximum number of edges and outputs no guess otherwise we investigate the complexity of computing the output of the fastest mind change optimal learner for either data type and show that each of these two problems is np hard assuming p rp to our knowledge these are the first np hardness results concerning the existence of a uniquely optimal bayes net structure

on estimating the distribution function for quality of life in cancer clinical trials summary the analysis of quality of life data collected at periodic intervals in a cancer clinical trial must account for those patients data that are missing because of death or administrative censoring we examine estimates of the cumulative distribution function of an interpolated area under a quality of life curve as summaries of the quality of life experience of a treat ment group a proposed estimator although biased offers an improvement over a kaplan meier estimator that assumes incorrectly that the censoring is noninformative modalities may be an area in which quality of life measurements can be particularly infor mative because patient s self reported quality of life may not correspond to investigators perceptions e g see sugarbaker et al 1982 many of the statistical issues involved with assessing quality of life have been discussed by cox et al 1992 we focus in this paper on an unresolved issue pertinent to cancer clin ical trials estimating the distribution function of area under the curve auc quality of life scores for a treatment group to put the issue in context we note that in a typical cancer clinical trial both the proportion of patients who die on study as well as the proportion of patients whose data are censored because the patient is alive at the time of analysis are not small if quality of life is measured at fixed times after a patient is entered onto the trial then one must account for the data missing for both reasons it may be possible to score the quality of life measurements so that a score of zero for a patient who is not alive at that time point is reasonable we assume this throughout we also assume that the quality of life data at a given time point can usefully be summarized by a small set of scores corre

a consistent estimator for the distribution of quality adjusted survival time summary quality adjusted survival analysis is a new approach to therapy evaluation in clinical trials it has received much attention recently because of its ability to take patients quality of life into consideration in this paper we present a method that enables us to calculate the survival distribution of quality adjusted lifetime using martingale theory for counting processes we can show that our estimator is asymptotically consistent normally distributed and its asymptotic variance estimate can be obtained analytically simulation experiments are conducted to compare our estimator with the true underlying distribution for two cases that are of practical importance

probabilistic computability and choice we study the computational power of randomized computations on infinite objects such as real numbers in particular we introduce the concept of a las vegas computable multi valued function which is a function that can be computed on a probabilistic turing machine that receives a random binary sequence as auxiliary input the machine can take advantage of this random sequence but it always has to produce a correct result or to stop the computation after finite time if the random advice is not successful with positive probability the random advice has to be successful we characterize the class of las vegas computable functions in the weihrauch lattice with the help of probabilistic choice principles and weak weak k nig s lemma among other things we prove an independent choice theorem that implies that las vegas computable functions are closed under composition in a case study we show that nash equilibria are las vegas computable while zeros of continuous functions with sign changes cannot be computed on las vegas machines however we show that the latter problem admits randomized algorithms with weaker failure recognition mechanisms the last mentioned results can be interpreted such that the intermediate value theorem is reducible to the jump of weak weak k nig s lemma but not to weak weak k nig s lemma itself these examples also demonstrate that las vegas computable functions form a proper superclass of the class of computable functions and a proper subclass of the class of non deterministically computable functions we also study the impact of specific lower bounds on the success probabilities which leads to a strict hierarchy of classes in particular the classical technique of probability amplification fails for computations on infinite objects we also investigate the dependency on the underlying probability space besides cantor space we study the natural numbers the euclidean space and baire space

on the intrinsic complexity of learning a new view of learning is presented the basis of this view is a natural notion of reduction we prove completeness and relative difficulty results an infinite hierarchy of intrinsically more and more difficult to learn concepts is presented our results indicate that the complexity notion captured by our new notion of reduction differs dramatically from the traditional studies of the complexity of the algorithms performing learning tasks

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

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

extended admissibility we give a new definition of admissible representations which allows to handle also non countably based topological spaces in the framework of type 2 theory of effectivity we prove that admissible representations x y of topological spaces x y have the desirable property that every partial function f x y is continuously realizable with respect to x y if and only if f is sequentially continuous furthermore we characterize the class of the spaces having an admissible representation many interesting operators creating new topological spaces from old ones are shown to preserve the property of having an admissible representation in particular the class of sequential spaces with admissible representations turns out to be cartesian closed thus a reasonable computability theory is possible on important non countably based spaces

biometrika centenary survival analysis although papers relating to survival analysis have appeared in biometrika since the very first issue the current emphasis on the topic in this journal is fairly new especially as it relates to applications in biostatistics this survey traces the development of survival analysis throughout the twentieth century as reflected in the pages of biometrika focusing primarily on work published since 1980

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

generalized notions of mind change complexity gold introduced the notion of learning in the limit where a class s is learnable iff there is a recursive machine m which reads the course of values of a function f and converges to a program for f whenever f is in s an important measure for the speed of convergence in this model is the quantity of mind changes before the onset of convergence the oldest model is to consider a constant bound on the number of mind changes m makes on any input function such a bound is referred here as type 1 later this was generalized to a bound of type 2 where a counter ranges over constructive ordinals and is counted down at every mind change although ordinal bounds permit the inference of richer concept classes than constant bounds they still are a severe restriction therefore the present work introduces two more general approaches to bounding mind changes these are based on counting by going down in a linearly ordered set type 3 and on counting by going down in a partially ordered set type 4 in both cases the set must not contain infinite descending recursive sequences these four types of mind changes yield a hierarchy and there are identifiable classes that cannot be learned with the most general mind change bound of type 4 it is shown that existence of type 2 bound is equivalent to the existence of a learning algorithm which converges on every also nonrecursive input function and the existence of type 4 is shown to be equivalent to the existence of a learning algorithm which converges on every recursive function a partial characterization of type 3 yields a result of independent interest in recursion theory the interplay between mind change complexity and choice of hypothesis space is investigated it is established that for certain concept classes a more expressive hypothesis space can sometimes reduce mind change complexity of learning these classes the notion of mind change bound for behaviourally correct learning is indirectly addressed by employing the above four types to restrict the number of predictive errors of commission in finite error next value learning nv a model equivalent to behaviourally correct learning again natural characterizations for type 2 and type 4 bounds are derived their naturalness is further illustrated by characterizing them in terms of branches of uniformly recursive families of binary trees

mind change complexity of inferring unbounded unions of restricted pattern languages from positive data this paper shows that the mind change complexity of inferring from positive data the class of unbounded unions of languages of regular patterns with constant segment length bound is of the form w w a b assuming that the patterns are defined over a finite alphabet containing at least two elements here a and b are natural numbers and we give tight bounds on their values based on the length of the constant segments and the size of the alphabet of the pattern languages this is to the authors knowledge the first time a natural class of languages has been shown to be inferable with mind change complexity above w w the proof uses the notion of closure operators on a class of languages and also uses the order type of well partial orderings to obtain a mind change bound the inference algorithm presented can be easily applied to a wide range of classes of languages finally we show an interesting connection between proof theory and mind change complexity

the synthesis of language learners abstract an index for an r e class of languages by definition is a procedure which generates a sequence of grammars defining the class an index for an indexed family of languages by definition is a procedure which generates a sequence of decision procedures defining the family studied is the metaproblem of synthesizing from indices for r e classes and for indexed families of languages various kinds of language learners for the corresponding classes or families indexed many positive results as well as some negative results are presented regarding the existence of such synthesizers the negative results essentially provide lower bounds for the positive results the proofs of some of the positive results yield as pleasant corollaries subset principle or tell tale style characterizations for the learnability of the corresponding classes or families indexed for example the indexed families of recursive languages that can be behaviorally correctly identified from positive data are surprisingly characterized by angluin s condition 2 the subset principle for circumventing overgeneralization

mind change complexity of learning logic programs the present paper motivates the study of mind change complexity for learning minimal models of length bounded logic programs it establishes ordinal mind change complexity bounds for learnability of these classes both from positive facts and from positive and negative facts building on angluin s notion of finite thickness and wright s work on finite elasticity shinohara defined the property of bounded finite thickness to give a sufficient condition for learnability of indexed families of computable languages from positive data this paper shows that an effective version of shinohara s notion of bounded finite thickness gives sufficient conditions for learnability with ordinal mind change bound both in the context of learnability from positive data and for learnability from complete both positive and negative data let be a notation for the first limit ordinal then it is shown that if a language defining framework yields a uniformly decidable family of languages and has effective bounded finite thickness then for each natural number m 0 the class of languages defined by formal systems of length m is identifiable in the limit from positive data with a mind change bound of m is identifiable in the limit from both positive and negative data with an ordinal mind change bound of m the above sufficient conditions are employed to give an ordinal mind change bound for learnability of minimal models of various classes of length bounded prolog programs including shapiro s linear programs arimura and shinohara s depth bounded linearly covering programs and krishna rao s depth bounded linearly moded programs it is also noted that the bound for learning from positive data is tight for the example classes considered

the two sample problem with induced dependent censorship summary induced dependent censorship is a general phenomenon in health service evaluation studies in which a measure such as quality adjusted survival time or lifetime medical cost is of interest we investigate the two sample problem and propose two classes of nonparametric tests based on consistent estimation of the survival function for each sample the two classes of test statistics examine the cumulative weighted difference in hazard functions and in survival functions we derive a unified asymptotic null distribution theory and inference procedure the tests are applied to trial v of the international breast cancer study group and show that long duration chemotherapy significantly improves time without symptoms of disease and toxicity of treatment as compared with the short duration treatment simulation studies demonstrate that the proposed tests with a wide range of weight choices perform well under moderate sample sizes

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

estimation of the conformal factor under bounded willmore energy we prove for closed orientable surfaces in mathbb r 3 with willmore energy less that 8 pi delta and whose conformal structures are compactly contained in moduli space that after applying appropriate mobius transformations the conformal factors between the induced metrics and conformal metrics of constant curvature are uniformly bounded by constants depending only on delta 0 the genus of the surfaces and the compact subset of the moduli space secondly for a given sequence of closed orientable surfaces as above we prove that the conformal factor remains bounded without applying mobius transformations if and only if no topology is lost similar estimates hold in higher codimension

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

prepivoting test statistics a bootstrap view of asymptotic refinements abstract approximate tests for a composite null hypothesis about a parameter may be obtained by referring a test statistic to an estimated critical value either asymptotic theory or bootstrap methods can be used to estimate the desired quantile the simple asymptotic test a refers the test statistic to a quantile of its asymptotic null distribution after unknown parameters have been estimated the bootstrap approach used here is based on the concept of prepivoting prepivoting is the transformation of a test statistic by the cdf of its bootstrap null distribution the simple bootstrap test b refers the prepivoted test statistic to a quantile of the uniform 0 1 distribution under regularity conditions the bootstrap test b has a smaller asymptotic order of error in level than does the asymptotic test a provided that the asymptotic null distribution of the test statistic does not depend on unknown parameters in the contrary case both a and b have the same order of level error certain class

mind change efficient learning this paper studies efficient learning with respect to mind changes our starting point is the idea that a learner that is efficient with respect to mind changes minimizes mind changes not only globally in the entire learning problem but also locally in subproblems after receiving some evidence formalizing this idea leads to the notion of strong mind change optimality we characterize the structure of language classes that can be identified with at most mind changes by some learner not necessarily effective a language class l is identifiable with mind changes iff the accumulation order of l is at most accumulation order is a classic concept from point set topology we show that accumulation order is related to other established notions of structural complexity such as thickness and intrinsic complexity to aid the construction of learning algorithms we show that the characteristic property of strongly mind change optimal learners is that they output conjectures languages with maximal accumulation order we illustrate the theory by describing strongly mind change optimal learners for various problems such as identifying linear subspaces one variable patterns and fixed length patterns

two guidelines for bootstrap hypothesis testing two guidelines for nonparametric bootstrap hypothesis testing are highlighted the first recommends that resampling be done in a way that reflects the null hypothesis even when the true hypothesis is distant from the null the second guideline argues that bootstrap hypothesis tests should employ methods that are already recognized as having good features in the closely related problem of confidence interval construction violation of the first guideline can seriously reduce the power of a test sometimes this reduction is spectacular since it is most serious when the null hypothesis is grossly in error the second guideline is of some importance when the conclusion of a test is equivocal it has no direct bearing on power but improves the level accuracy of a test

boundary regularity and the dirichlet problem for harmonic maps in a previous paper 10 we developed an interior regularity theory for energy minimizing harmonic maps into riemannian manifolds in the first two sections of this paper we prove boundary regularity for energy minimizing maps with prescribed dirichlet boundary condition we show that such maps are regular in a full neighborhood of the boundary assuming appropriate regularity on the manifolds the boundary and the data the reader may refer to theorem 2 7 for a statement of the precise result it is not surprising that the boundary regularity is actually stronger than the partial regularity we obtained for the interior this is due to the fact that there are no nontrivial smooth harmonic maps from hemispheres s which map the boundary s 9s to a point i j n 2 and is analogous to the fact that in certain cases we were able to obtain complete regularity in the interior many authors have worked on boundary regularity for this general type of problem we mention hildebrandt and widman 5 and hamilton 4 as having obtained important results specifically for harmonic maps morrey had obtained the boundary regularity for domain dimension n 2 in conjunction with his investigation of the plateau problem in riemannian manifolds 8 in 3 of this paper we observe that the direct method gives solvability of the dirichlet problem under reasonable hypotheses on the manifolds we give as an application an amusing proof of a theorem of sacks and uhlenbeck 9 on the existence of minimal 2 spheres representing the second homotopy group of a manifold the same method gives smooth harmonic representations for k n for a certain class of manifolds n these are characterized by the nonexistence of lower dimensional harmonic spheres whose homogeneous extensions are minimal see proposition 3 4 in the last section of the paper we discuss approximation of l maps by smooth maps we give a simple example of an l map from the three dimensional ball to the two sphere which is not an l limit of continuous maps we