modified signed log likelihood ratio summary the signed log likelihood ratio r for a one dimensional interest parameter can be modified as r r r log u r so that r is asymptotically standard normally distributed with error of order o n 3 2 u being interpretable as a test statistic a new more direct derivation of this result is given which leads to a more explicit expression for u and provides a better handle on the error term the relation of the tail area approximation determined by r to a generalization of the lugannani rice approximation is discussed and the applicability of r to the analysis of residual variation is indicated

a theorem of real analysis and its application to free boundary problems in this note we show how comparison theorems for solutions to equations with non smooth coefficients yield in a very simple fashion some free boundary regularity theorems in cases where there is a priori geometrical information for instance when one knows that the solution u is monotone in some given direction

perron s method for hamilton jacobi equations on considere l existence des solutions d equations aux derivees partielles non lineaires scalaires d ordre 1 f x u du 0 dans ou est un sous ensemble ouvert de r n f r r n r est continue u r est l inconnue

bootstrap bartlett adjustment in seemingly unrelated regression abstract seemingly unrelated regression sur is a method introduced by zellner 1962 for estimating several regression equations simultaneously a common and important problem in econometrics this article considers hypothesis testing in sur using the log likelihood ratio llr test although the asymptotic distribution of this statistic is well known substantial departures occur in samples of the size commonly employed by economists significance levels given by the use of this asymptotic distribution are too low sometimes by an order of magnitude leading to the rejection of too many true null hypotheses a common approach to this problem is to use a bartlett 1937 adjustment to the llr statistic in which the test statistic is multiplied by a factor derived from second order asymptotics before it is referred to the asymptotic 2 q distribution this article proposes that the bartlett adjustment factor be computed using efron s bootstrap 1979 1982 in the cases examined this leads to more accu

implicit function theorem over free groups abstract we introduce the notion of a regular quadratic equation and a regular ntq system over a free group we prove the results that can be described as implicit function theorems for algebraic varieties corresponding to regular quadratic and ntq systems we will also show that the implicit function theorem is true only for these varieties in algebraic geometry such results would be described as lifting solutions of equations into generic points from the model theoretic view point we claim the existence of simple skolem functions for particular formulas over free groups proving these theorems we describe in details a new version of the makanin razborov process for solving equations in free groups we also prove a weak version of the implicit function theorem for ntq systems which is one of the key results in the solution of the tarski s problems about the elementary theory of a free group we call it the parametrization theorem

the structure of the free boundary for lower dimensional obstacle problems we study the regularity of the free surface in boundary obstacle problems we show that near a non degenerate point the free boundary is a c 1 a n 2 dimensional surface in r n 1

nonlinear neumann boundary conditions for quasilinear degenerate elliptic equations and applications abstract we prove comparison results between viscosity sub and supersolutions of degenerate elliptic and parabolic equations associated to possibly nonlinear neumann boundary conditions these results are obtained under more general assumptions on the equation in particular the dependence in the gradient of the solution and they allow applications to quasilinear possibly singular elliptic or parabolic equations one of the main applications is the extension of the so called level set approach for equations set in bounded domains with nonlinear neumann boundary conditions in such a framework the level set approach provides a weak notion for the motion of hypersurfaces with curvature dependent velocities and a prescribed contact angle at the boundary

some existence results for variational inequalities with nonlocal fractional operators abstract in this paper we consider the following nonlocal fractional variational inequality u x 0 s u a e in u v u x 0 s u v u 2 f x u x u x v x u x d x a a a a a a a a a a a a a a a a a a a a a a a a a a a a a for any v x 0 s v a e in where r n is a smooth bounded open set with continuous boundary s 0 1 n 2 s is a real parameter f is function with subcritical growth 0 s 2 and is the obstacle function as it is well known the dependence of the nonlinearity f on the term u makes non variational the nature of this problem using an iterative technique and a penalization method we get the existence of a nontrivial nonnegative solution for the problem under consideration performing the mountain pass theorem this result can be seen as the extension of known existence theorem for variational inequalities driven by the laplace operator or more general uniformly elliptic operators to the nonlocal fractional setting

properties of sufficiency and statistical tests 1 in a previous paper dealing with the importance of properties of sufficiency in the statistical theory of small samples attention was mainly confined to the theory of estimation in the present paper the structure of small sample tests whether these are related to problems of estimation and fiducial distributions or are of the nature of tests of goodness of fit is considered further

optimal regularity for the signorini problem and its free boundary we will show optimal regularity for minimizers of the signorini problem for the lame system in particular if mathbf u u 1 u 2 dots u n in w 1 2 b 1 mathbb r n minimizes begin aligned j mathbf u int b 1 nabla mathbf u nabla bot mathbf u 2 lambda hbox div mathbf u 2 end aligned in the convex set begin aligned k left mathbf u u 1 u 2 dots u n in w 1 2 b 1 mathbb r n u n ge 0 quad hbox on quad pi right left mathbf u f in c infty partial b 1 quad hbox on quad partial b 1 right end aligned where lambda ge 0 say then mathbf u in c 1 1 2 b 1 2 moreover the free boundary given by begin aligned gamma mathbf u partial x u n x 0 x n 0 cap b 1 end aligned will be a c 1 alpha graph close to points where mathbf u is not degenerate similar results have been know before for scalar partial differential equations see for instance 5 6 the novelty of this approach is that it does not rely on maximum principle methods and is therefore applicable to systems of equations

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

irreducible affine varieties over a free group i irreducibility of quadratic equations and nullstellensatz the object of this paper which consists of two parts is to describe irreducible varieties over free groups and to characterize finitely generated fully residually free groups we prove that any variety over a free group f can be defined by a finite number of systems of equations s 1 in triangular form where quadratic words play the role of leading terms algebraically irreducible varieties are exactly the varieties whose coordinate groups are fully residually f the crucial point of the classification of fully residually free groups is to prove that the coordinate groups of irreducible varieties are embeddable into lyndon s free z x group f since every finitely generated fully residually free group is a free factor of the coordinate group of an irreducible variety and the group f is fully residually f we obtain a characterization of finitely generated fully residually free groups as subgroups of f the group f and its subgroups have been studied extensively during the last several years in particular every finitely generated subgroup of f hence every finitely generated fully residually free group can be obtained from free abelian groups of finite rank by finitely many free products with amalgamation and hnn extensions of the type where amalgamated and associated subgroups are free abelian of finite rank in particular this implies that every finitely generated fully residually free group is finitely presented there are three parts to this paper algebraic geometry over free groups the theory of free exponential groups and makanin razborov s machinery to deal with equations over free groups 11 14 13 the algebraic geometry approach has been shown to be very usefull in dealing with equations over groups it provides necessary topological means and a method to transcribe geometric notions into pure group theoretic language following baumslag myasnikov remeslennikov 1 we use the standard algebraic geometry notions such as variety zariski topology irreducibility of varieties radicals and coordinate groups some of the ideas of the algebraic geometry approach go back to r lyndon 9 e rips j stallings 16

a direct monte carlo approach for bayesian analysis of the seemingly unrelated regression model abstract computationally efficient methods for bayesian analysis of seemingly unrelated regression sur models are described and applied that involve the use of a direct monte carlo dmc approach to calculate bayesian estimation and prediction results using diffuse or informative priors this dmc approach is employed to compute bayesian marginal posterior densities moments intervals and other quantities using data simulated from known models and also using data from an empirical example involving firms sales the results obtained by the dmc approach are compared to those yielded by the use of a markov chain monte carlo mcmc approach it is concluded from these comparisons that the dmc approach is worthwhile and applicable to many sur and other problems

obstacle problems for integro differential operators higher regularity of free boundaries abstract we study the higher regularity of free boundaries in obstacle problems for integro differential operators our main result establishes that once free boundaries are c 1 then they are c this completes the study of regular points initiated in 5 in order to achieve this we need to establish optimal boundary regularity estimates for solutions to linear nonlocal equations in c k domains these new estimates are the core of our paper and extend previously known results by grubb for k and by the second author and serra for k 1

higher regularity of the free boundary in the elliptic signorini problem abstract in this paper we study the higher regularity of the free boundary for the elliptic signorini problem by using a partial hodograph legendre transformation we show that the regular part of the free boundary is real analytic the first complication in the study is the invertibility of the hodograph transform which is only c 0 1 2 which can be overcome by studying the precise asymptotic behavior of the solutions near regular free boundary points the second and main complication in the study is that the equation satisfied by the legendre transform is degenerate however the equation has a subelliptic structure and can be viewed as a perturbation of the baouendi grushin operator by using the l p theory available for that operator we can bootstrap the regularity of the legendre transform up to real analyticity which implies the real analyticity of the free boundary

a bartlett adjustment to the likelihood ratio test for a system of equations abstract a bartlett adjustment to be applied to the likelihood ratio test statistic when testing parameter restrictions in a system of equations is derived as special cases results are also obtained for testing hypotheses on the mean of a normal variate linear restrictions in the linear model restrictions in a nonlinear model and linear restrictions in the multivariate linear model

new monotonicity formulas and the optimal regularity in the signorini problem with variable coefficients abstract we study the interior signorini or lower dimensional obstacle problem for a uniformly elliptic divergence form operator l div a x with lipschitz continuous coefficients our main result states that similarly to what happens when l the variational solution has the optimal interior regularity c loc 1 1 2 m when m is a codimension one flat manifold which supports the obstacle we achieve this by proving some new monotonicity formulas for an appropriate generalization of the celebrated almgren s frequency functional

regularity estimates for the solution and the free boundary of the obstacle problem for the fractional laplacian we use a characterization of the fractional laplacian as a dirichlet to neumann operator for an appropriate differential equation to study its obstacle problem we write an equivalent characterization as a thin obstacle problem in this way we are able to apply local type arguments to obtain sharp regularity estimates for the solution and study the regularity of the free boundary

regularity for the nonlinear signorini problem abstract we study the regularity of the solution to a fully nonlinear version of the thin obstacle problem in particular we prove that the solution is c 1 for some small 0 this extends a result of luis caffarelli of 1979 our proof relies on new estimates up to the boundary for fully nonlinear equations with neumann boundary data developed recently by the authors

the structure of the free boundary in the fully nonlinear thin obstacle problem abstract we study the regularity of the free boundary in the fully nonlinear thin obstacle problem our main result establishes that the free boundary is c 1 near any regular point this extends to the fully nonlinear setting the celebrated result of athanasopoulos caffarelli salsa 1 the proofs we present here are completely independent from those in 1 and do not rely on any monotonicity formula furthermore an interesting and novel feature of our proofs is that we establish the regularity of the free boundary without classifying blow ups a priori they could be non homogeneous and or non unique we do not classify blow ups but only prove that they are 1d on x n 0

optimal regularity and structure of the free boundary for minimizers in cohesive zone models we study optimal regularity and free boundary for minimizers of an energy functional arising in cohesive zone models for fracture mechanics under smoothness assumptions on the boundary conditions and on the fracture energy density we show that minimizers are c 1 1 2 and that near non degenerate points the fracture set is c 1 alpha for some alpha in 0 1

on the fine structure of the free boundary for the classical obstacle problem in the classical obstacle problem the free boundary can be decomposed into regular and singular points as shown by caffarelli in his seminal papers caffarelli in acta math 139 155 184 1977 j fourier anal appl 4 383 402 1998 regular points consist of smooth hypersurfaces while singular points are contained in a stratified union of c 1 manifolds of varying dimension in two dimensions this c 1 result has been improved to c 1 alpha by weiss invent math 138 23 50 1999 in this paper we prove that for n 2 singular points are locally contained in a c 2 curve in higher dimension n ge 3 we show that the same result holds with c 1 1 manifolds or with countably many c 2 manifolds up to the presence of some anomalous points of higher codimension in addition we prove that the higher dimensional stratum is always contained in a c 1 alpha manifold thus extending to every dimension the result in weiss 1999 we note that in terms of density decay estimates for the contact set our result is optimal in addition for n ge 3 we construct examples of very symmetric solutions exhibiting linear spaces of anomalous points proving that our bound on their hausdorff dimension is sharp

a simple general formula for tail probabilities for frequentist and bayesian inference summary we describe a simple general formula for approximating the p value for testing a scalar parameter in the presence of nuisance parameters the formula covers both frequentist and bayesian contexts and does not require explicit nuisance parameterisation implementation is discussed in terms of computer algebra packages examples are given and the relationship to barndorff nielsen s approximation is discussed

regularity results of the thin obstacle problem for the p x laplacian abstract we study thin obstacle problems involving the energy functional with p x growth we prove higher integrability and holder regularity for the gradient of minimizers of the thin obstacle problems under the assumption that the variable exponent p x is holder continuous

algebraic geometry over groups i algebraic sets and ideal theory abstract the object of this paper which is the first in a series of three is to lay the foundations of the theory of ideals and algebraic sets over groups

regularity of the obstacle problem for a fractional power of the laplace operator given a function and s 0 1 we will study the solutions of the following obstacle problem u in r n s u 0i nr n s u x 0 for those x such that u x x lim x u x 0 we show that when is c 1 s or smoother the solution u is in the space c 1 for every s in the case where the contact set u is convex we prove the optimal regularity result u c 1 s when is only c 1 for a s we prove that our solution u is c 1 for every c 2006 wiley periodicals inc

on systems of equations over free products of groups using an analogue of the makanin razborov diagrams we give a description of the solution set of systems of equations over an equationally noetherian free product of groups g equivalently we give a parametrisation of the set hom h g of all homomorphisms from a finitely generated group h to g furthermore we show that every algebraic set over g can be decomposed as a union of finitely many images of algebraic sets of ntq systems if the universal horn theory of g the theory of quasi identities is decidable then our constructions are effective 2010 elsevier inc

highly accurate likelihood analysis for the seemingly unrelated regression problem the linear and nonlinear seemingly unrelated regression problem with general error distribution is analyzed using recent likelihood theory that arguably provides the definitive distribution for assessing a scalar parameter this involves implicit but well defined conditioning and marginalization for determining intrinsic measures of departure highly accurate p values are obtained for the key difference between two regression coefficients of central interest the p value gives the statistical position of the data with respect to the key parameter the theory and the results indicate that this methodology provides substantial improvement on first order likelihood procedures both in distributional accuracy and in precise measurement of the key parameter

symmetry and non existence of positive solutions for fractional p laplacian systems abstract chen and li 2018 developed maximum principles for the fractional p laplacian which carry out the direct method of moving planes to the fractional p laplacian equation in this paper we generalize their method to the system case and obtain a symmetry result for the fractional p laplacian system on the whole space we also consider the system on the upper half space and obtain a non existence result

some new monotonicity formulas and the singular set in the lower dimensional obstacle problem we construct two new one parameter families of monotonicity formulas to study the free boundary points in the lower dimensional obstacle problem the first one is a family of weiss type formulas geared for points of any given homogeneity and the second one is a family of monneau type formulas suited for the study of singular points we show the uniqueness and continuous dependence of the blowups at singular points of given homogeneity this allows to prove a structural theorem for the singular set our approach works both for zero and smooth non zero lower dimensional obstacles the study in the latter case is based on a generalization of almgren s frequency formula first established by caffarelli salsa and silvestre

regression analysis nonlinear or nonnormal simple and accurate p values from likelihood analysis we develop simple approximations for the p values to use with regression models having linear or nonlinear parameter structure and normal or nonnormal error distribution computer iteration then gives confidence intervals both frequentist and bayesian versions are given the approximations are derived from recent developments in likelihood analysis and have third order accuracy also for very small and medium sized samples the accuracy can typically be high the likelihood basis of the procedure seems to provide the grounds for this general accuracy examples are discussed and simulations record the distributional accuracy

on the regularity of the non dynamic parabolic fractional obstacle problem abstract in the class of the non dynamic fractional obstacle problems of parabolic type it is shown existence uniqueness apriori bounds of the solution and optimal regularity of the space derivatives of the solution furthermore at free boundary points of positive parabolic density it is proven that the time derivative of the solution is holder continuous at such free boundary points space time regularity of the corresponding free boundary is obtained for any fraction s 0 1

quadratic equations in hyperbolic groups are np complete we prove that in a torsion free hyperbolic group the length of the value of each variable in a minimal solution of a quadratic equation q 1 is bounded by njqj 4 for an orientable equation and by njqj 16 for a non orientable equation where jqj is the length of the equation and the constant n can be computed we show that the problem whether a quadratic equation in has a solution is np complete and that there is a pspace algorithm for solving arbitrary equations in we also give a slightly larger bound for minimal solutions of quadratic equations in a toral relatively hyperbolic group