generalized partially linear single index models abstract the typical generalized linear model for a regression of a response y on predictors x z has conditional mean function based on a linear combination of x z we generalize these models to have a nonparametric component replacing the linear combination t 0x t 0z by 0 t 0x t 0z where 0 is an unknown function we call these generalized partially linear single index models gplsim the models include the single index models which have 0 0 using local linear methods we propose estimates of the unknown parameters 0 0 and the unknown function 0 and obtain their asymptotic distributions examples illustrate the models and the proposed estimation methodology

initial boundary value problem for two dimensional viscous boussinesq equations we study the initial boundary value problem of two dimensional viscous boussinesq equations over a bounded domain with smooth boundary we show that the equations have a unique classical solution for h3 initial data and the no slip boundary condition in addition we show that the kinetic energy is uniformly bounded in time

global well posedness for a boussinesq navier stokes system with critical dissipation abstract in this paper we study a fractional diffusion boussinesq model which couples a navier stokes type equation with fractional diffusion for the velocity and a transport equation for the temperature we establish global well posedness results with rough initial data

doubly robust and efficient estimators for heteroscedastic partially linear single index models allowing high dimensional covariates we study the heteroscedastic partially linear single index model with an unspecified error variance function which allows for high dimensional covariates in both the linear and the single index components of the mean function we propose a class of consistent estimators of the parameters by using a proper weighting strategy an interesting finding is that the linearity condition which is widely assumed in the dimension reduction literature is not necessary for methodological or theoretical development it contributes only to the simplification of non optimal consistent estimation we also find that the performance of the usual weighted least square type of estimators deteriorates when the non parametric component is badly estimated however estimators in our family automatically provide protection against such deterioration in that the consistency can be achieved even if the baseline non parametric function is completely misspecified we further show that the most efficient estimator is a member of this family and can be easily obtained by using non parametric estimation properties of the estimators proposed are presented through theoretical illustration and numerical simulations an example on gender discrimination is used to demonstrate and to compare the practical performance of the estimators

nonlinear autoregressive model with stochastic volatility innovations semiparametric and bayesian approach abstract the first order nonlinear autoregressive model with the stochastic volatility as the model of dependent innovations is considered and a semiparametric method is proposed to estimate the unknown function optimal filtering technique based on sequential monte carlo perspective is used for estimation of the hidden log volatility in this model bayesian paradigm is applied for estimation of both the unknown parameters and hidden process using particle marginal metropolis hastings scheme furthermore an empirical application on simulated data and on the monthly excess returns of s p 500 index is presented to study the performance of the schemes implemented

the 2d boussinesq equations with logarithmically supercritical velocities abstract this paper investigates the global in time regularity of solutions to a system of equations that generalize the vorticity formulation of the 2d boussinesq navier stokes equations the velocity u in this system is related to the vorticity through the relations u and log i which reduces to the standard velocity vorticity relation when 0 when either 0 or 0 the velocity u is more singular the quasi velocity v determined by v satisfies an equation of very special structure this paper establishes the global regularity and uniqueness of solutions for the case when 0 and 0 in addition the vorticity is shown to be globally bounded in several functional settings such as l 2 for 0 in a suitable range

burmann expansion and test for additivity summary we propose a lagrange multiplier test for additivity based on the btirmann expansion of a conditional mean function the asymptotic null distribution of the test is shown to be x2 under some regularity conditions in contrast the lagrange multiplier test proposed by chen et al 1995 is based on the volterra expansion of the conditional mean function we discuss some desirable advantages of the bt rmann expansion over the volterra expansion for nonlinear time series modelling we also reported an empirical study which shows that in terms of empirical power the lagrange multiplier test motivated by the biirmann expansion outperforms the test of chen et al 1995 for the cases for which the lagrange multiplier test is designed for other cases for which none of the tests is specifically designed the empirical powers of the two tests are comparable finally we illustrated the use of the lagrange multiplier test with a blowfly experimental system

bias corrected confidence bands in nonparametric regression summary bias corrected confidence bands for general nonparametric regression models are considered we use local polynomial fitting to construct the confidence bands and combine the cross validation method and the plug in method to select the bandwidths related asymptotic results are obtained our simulations show that confidence bands constructed by local polynomial fitting have much better coverage than those constructed by using the nadaraya watson estimator the results are also applicable to nonparametric autoregressive time series models

global well posedness issues for the inviscid boussinesq system with yudovich s type data the present paper is dedicated to the study of the global existence for the inviscid two dimensional boussinesq system we focus on finite energy data with bounded vorticity and we find out that under quite a natural additional assumption on the initial temperature there exists a global unique solution no smallness conditions are imposed on the data the global existence issues for infinite energy initial velocity and for the benard system are also discussed

remarks on global regularity of the 2d boussinesq equations with fractional dissipation abstract in this paper we are interested in the study of the cauchy problem to the two dimensional 2d incompressible boussinesq equations with fractional dissipation by making use of the nonlinear lower bounds for the fractional laplacian established in constantin and vicol 2012 we establish the global regularity of the smooth solutions of the 2d boussinesq equations with a new range of fractional powers of the laplacian this result significantly improves the recent works of constantin and vicol 2012 and yang et al 2014

generalized 2d euler boussinesq equations with a singular velocity abstract this paper studies the global in time regularity problem concerning a system of equations generalizing the two dimensional incompressible boussinesq equations the velocity here is determined by the vorticity through a more singular relation than the standard biot savart law and involves a fourier multiplier operator the temperature equation has a dissipative term given by the fractional laplacian operator we establish the global existence and uniqueness of solutions to the initial value problem of this generalized boussinesq equations when the velocity is double logarithmically more singular than the one given by the biot savart law this global regularity result goes beyond the critical case in addition we recover a result of chae constantin and wu 8 when the initial temperature is set to zero

long time behavior of the two dimensional boussinesq equations without buoyancy diffusion abstract we study the global well posedness and stability instability of perturbations near a special type of hydrostatic equilibrium associated with the 2d boussinesq equations without buoyancy a k a thermal diffusion on a bounded domain subject to stress free boundary conditions the boundary of the domain is not necessarily smooth and may have corners such as in the case of rectangles we achieve three goals first we establish the global in time existence and uniqueness of large amplitude classical solutions efforts are made to reduce the regularity assumptions on the initial data second we obtain the large time asymptotics of the full nonlinear perturbation in particular we show that the kinetic energy and the first order derivatives of the velocity field converge to zero as time goes to infinity regardless of the magnitude of the initial data and the flow stratifies in the vertical direction in a weak topology third we prove the linear stability of the hydrostatic equilibrium t y satisfying t y 0 and the linear instability of periodic perturbations when t y 0 numerical simulations are supplemented to corroborate the analytical results and predict some phenomena that are not proved the authors are pleased to dedicate this paper to professor edriss saleh titi on the occasion of his 60th birthday professor titi s myriad research contributions including contributions to the problem constituting the focus of this work and leadership in the mathematical fluid dynamics community serve as an inspiration to his students collaborators and colleagues

global regularity of solutions of 2d boussinesq equations with fractional diffusion abstract the goal of this work is to study the boussinesq equations for an incompressible fluid in r 2 with diffusion modeled by fractional laplacian the existence the uniqueness and the regularity of solution has been proved

an adaptive estimation of dimension reduction space summary searching for an effective dimension reduction space is an important problem in regression especially for high dimensional data we propose an adaptive approach based on semiparametric models which we call the conditional minimum average variance estimation mave method within quite a general setting the mave method has the following advantages most existing methods must undersmooth the nonparametric link function estimator to achieve a faster rate of consistency for the estimator of the parameters than for that of the nonparametric function in contrast a faster consistency rate can be achieved by the mave method even without undersmoothing the nonparametric link function estimator the mave method is applicable to a wide range of models with fewer restrictions on the distribution of the covariates to the extent that even time series can be included because of the faster rate of consistency for the parameter estimators it is possible for us to estimate the dimension of the space consistently the relationship of the mave method with other methods is also investigated in particular a simple outer product gradient estimator is proposed as an initial estimator in addition to theoretical results we demonstrate the efficacy of the mave method for high dimensional data sets through simulation two real data sets are analysed by using the mave approach

a semiparametric method for estimating nonlinear autoregressive model with dependent errors abstract the first order nonlinear autoregressive model is considered and a semiparametric method is proposed to estimate regression function in the presented model dependent errors are defined as first order autoregressive ar 1 the conditional least squares method is used for parametric estimation and the nonparametric kernel approach is applied to estimate regression adjustment in this case some asymptotic behaviors and simulated results for the semiparametric method are presented furthermore the method is applied for the financial data in iran s tejarat bank

identifiability of single index models and additive index models summary we provide a proof for the identifiability for both single index models and partially linear single index models assuming only the continuity of the regression function a condition much weaker than the differentiability conditions assumed in the existing literature our discussion is then extended to the identifiability of the additive index models

small global solutions to the damped two dimensional boussinesq equations abstract the two dimensional 2d incompressible euler equations have been thoroughly investigated and the resolution of the global in time existence and uniqueness issue is currently in a satisfactory status in contrast the global regularity problem concerning the 2d inviscid boussinesq equations remains widely open in an attempt to understand this problem we examine the damped 2d boussinesq equations and study how damping affects the regularity of solutions since the damping effect is insufficient in overcoming the difficulty due to the vortex stretching we seek unique global small solutions and the efforts have been mainly devoted to minimizing the smallness assumption by positioning the solutions in a suitable functional setting more precisely the homogeneous besov space b 1 1 we are able to obtain a unique global solution under a minimal smallness assumption

varying index coefficient models it has been a long history of using interactions in regression analysis to investigate alterations in covariate effects on response variables in this article we aim to address two kinds of new challenges arising from the inclusion of such high order effects in the regression model for complex data the first kind concerns a situation where interaction effects of individual covariates are weak but those of combined covariates are strong and the other kind pertains to the presence of nonlinear interactive effects directed by low effect covariates we propose a new class of semiparametric models with varying index coefficients which enables us to model and assess nonlinear interaction effects between grouped covariates on the response variable as a result most of the existing semiparametric regression models are special cases of our proposed models we develop a numerically stable and computationally fast estimation procedure using both profile least squares method and local fitting we establish both

on the profiles of locally self similar solutions for the 2d inviscid boussinesq equations abstract in this paper we investigate the locally self similar solutions of the 2d inviscid boussinesq equations and mainly focus on the possible velocity and temperature blowup profiles we derive the formula of the pressure profiles in terms of the velocity and temperature profiles in the case that the possible nontrivial profiles have non decaying spatial asymptotics then using the local energy inequality of the profiles and iterative method we prove some nonexistence results and obtain the energy behavior concerning these possible profiles

a law of the logarithm for kernel density estimators in this paper we derive a law of the logarithm for the maximal deviation between a kernel density estimator and the true underlying density function extensions to higher derivatives are included the results are applied to get optimal window widths with respect to almost sure uniform convergence

global regularity results for the 2d boussinesq equations with vertical dissipation abstract the two dimensional 2d incompressible boussinesq equations model geophysical fluids and play an important role in the study of the raleigh bernard convection mathematically this 2d system retains some key features of the 3d navier stokes and euler equations such as the vortex stretching mechanism the issue of whether the 2d boussinesq equations always possess global in time classical solutions can be difficult when there is only partial dissipation or no dissipation at all this paper obtains the global regularity for two partial dissipation cases and proves several global a priori bounds for two other prominent partial dissipation cases these results take us one step closer to a complete resolution of the global regularity issue for all the partial dissipation cases involving the 2d boussinesq equations

on polynomial exponential equations let m k be the set of absolute values of k normalized such that they extend the standard or a p adic absolute value of q let s c m k be the subset consisting of the archimedean absolute values as well as of the non archimedean absolute values v e m k having i ol 4 1 for some ej thus s is a finite set an s unit is an element e k having i 1 1 for every v s then in particular the o and therefore every a txe with x e z is an s unit therefore 1 2 is related to the s unit equation

damped infinite energy solutions of the 3d euler and boussinesq equations abstract we revisit a family of infinite energy solutions of the 3d incompressible euler equations proposed by gibbon et al 9 and shown to blowup in finite time by constantin 6 by adding a damping term to the momentum equation we examine how the damping coefficient can arrest this blowup further we show that similar infinite energy solutions of the inviscid 3d boussinesq system with damping can develop a singularity in finite time as long as the damping effects are insufficient to arrest the undamped 3d euler blowup in the associated damped 3d euler system

inference on the primary parameter of interest with the aid of dimension reduction estimation summary as high dimensional data become routinely available in applied sciences sufficient dimension reduction has been widely employed and its research has received considerable attention however with the majority of sufficient dimension reduction methodology focusing on the dimension reduction step complete analysis and inference after dimension reduction have yet to receive much attention we couple the strategy of sufficient dimension reduction with a flexible semiparametric model we concentrate on inference with respect to the primary variables of interest and we employ sufficient dimension reduction to bring down the dimension of the regression effectively extensive simulations demonstrate the efficacy of the method proposed and a real data analysis is presented for illustration

asymptotics for a censored generalized linear model with unknown link function for censored response variable against projected co variable a generalized linear model with an unknown link function can cover almost all existing models under censorship its special cases include the accelerated failure time model with censored data such a model in the uncensored case is called the single index model in econometrics in this paper we systematically study the asymptotic properties we derive the central limit theorem and the law of the iterated logarithm for an estimator of the direction parameter we also obtain the optimal convergence rate of an estimator of the unknown link function in the model

semiparametric estimates of the relation between weather and electricity sales abstract a nonlinear relationship between electricity sales and temperature is estimated using a semiparametric regression procedure that easily allows linear transformations of the data this accommodates introduction of covariates timing adjustments due to the actual billing schedules and serial correlation the procedure is an extension of smoothing splines with the smoothness parameter estimated from minimization of the generalized cross validation criterion introduced by craven and wahba 1979 estimates are presented for residential sales for four electric utilities and are compared with models that represent the weather using only heating and cooling degree days or with piecewise linear splines

global regularity for the two dimensional anisotropic boussinesq equations with vertical dissipation this paper establishes the global in time existence of classical solutions to the two dimensional anisotropic boussinesq equations with vertical dissipation when only vertical dissipation is present there is no direct control on the horizontal derivatives and the global regularity problem is very challenging to solve this problem we bound the derivatives in terms of the l infty norm of the vertical velocity v and prove that v l r with 2 leqq r infty does not grow faster than sqrt r log r at any time as r increases a delicate interpolation inequality connecting v l infty and v l r then yields the desired global regularity

a maximum principle applied to quasi geostrophic equations we study the initial value problem for dissipative 2d quasi geostrophic equations proving local existence global results for small initial data in the super critical case decay of lp norms and asymptotic behavior of viscosity solution in the critical case our proofs are based on a maximum principle valid for more general flows

optimal smoothing in single index models single index models generalize linear regression they have applications to a variety of fields such as discrete choice analysis in econometrics and dose response models in biometrics where high dimensional regression models are often employed single index models are similar to the first step of projection pursuit regression a dimension reduction method in both cases the orientation vector can be estimated root n consistently even if the unknown univariate function or nonparametric link function is assumed to come from a large smoothness class however as we show in the present paper the similarities end there in particular the amount of smoothing necessary for root n consistent orientation estimation is very different in the two cases we suggest a simple empirical rule for selecting the bandwidth appropriate to single index models this rule is studied in a small simulation study and an application in binary response models

large time decay of solutions to the boussinesq system with fractional dissipation abstract in this paper we extend the results of brandolese and schonbek 2 to the boussinesq system with fractional dissipation let and represent the fractional laplacian dissipation in the velocity and the temperature equations respectively we show that if the initial data u 0 0 l 2 l 1 l 2 then t c 1 t d 2 and u t c 1 t max 0 1 d 2 if d 2 u t c l n 2 t if d 2 if we additionally assume 0 0 and 0 l 1 1 then t c 1 t d 2 2 and u t 0 as t comparing 2 we don t need to assume that 0 1 is sufficiently small when 0 d 1 2

large time decay and growth for solutions of a viscous boussinesq system in this paper we analyze the decay and the growth for large time of weak and strong solutions to the three dimensional viscous boussinesq system we show that generic solutions blow up as t in the sense that the energy and the l norms of the velocity field grow to infinity for large time for 1 p 3 in the case of strong solutions we provide sharp estimates both from above and from below and explicit asymptotic profiles we also show that solutions arising from u0 0 with zero mean for the initial temperature 0 have a special behavior as x or t tends to infinity contrarily to the generic case their energy dissipates to zero for large time

global well posedness for the 2d boussinesq system with anisotropic viscosity and without heat diffusion abstract we establish global existence and uniqueness theorems for the two dimensional non diffusive boussinesq system with anisotropic viscosity acting only in the horizontal direction which arises in ocean dynamics models global well posedness for this system was proven by danchin and paicu however an additional smoothness assumption on the initial density was needed to prove uniqueness they stated that it is not clear whether uniqueness holds without this additional assumption the present work resolves this question and we establish uniqueness without this additional assumption furthermore the proof provided here is more elementary we use only tools available in the standard theory of sobolev spaces and without resorting to para product calculus we use a new approach by defining an auxiliary stream function associated with the density analogous to the stream function associated with the vorticity in 2d incompressible euler equations then we adapt some of the ideas of yudovich for proving uniqueness for 2d euler equations

regularity criterion of the 2d benard equations with critical and supercritical dissipation abstract in this paper we investigate the cauchy problem for the two dimensional 2d incompressible benard equations on the one hand we prove global in time existence of smooth solutions to the 2d benard equations with critical dissipation on the other hand we establish several regularity criteria involving temperature for 2d benard equations with supercritical dissipation

semiparametric gee analysis in partially linear single index models for longitudinal data in this article we study a partially linear single index model for longitudinal data under a general framework which includes both the sparse and dense longitudinal data cases a semiparametric estimation method based on a combination of the local linear smoothing and generalized estimation equations gee is introduced to estimate the two parameter vectors as well as the unknown link function under some mild conditions we derive the asymptotic properties of the proposed parametric and nonparametric estimators in different scenarios from which we find that the convergence rates and asymptotic variances of the proposed estimators for sparse longitudinal data would be substantially different from those for dense longitudinal data we also discuss the estimation of the covariance or weight matrices involved in the semiparametric gee method furthermore we provide some numerical studies including monte carlo simulation and an empirical application to illustrate our methodology and theory

on the regularity criteria of the 2d boussinesq equations with partial dissipation abstract in this paper we consider the two dimensional 2d boussinesq equations with partial dissipation the issue of whether the 2d boussinesq equations always possess global in time classical solutions can be difficult when there is only partial dissipation therefore the goal of this paper is to establish some regularity criteria of the corresponding systems

semiparametric least squares sls and weighted sls estimation of single index models abstract for the class of single index models i construct a semiparametric estimator of coefficients up to a multiplicative constant that exhibits 1 n consistency and asymptotic normality this class of models includes censored and truncated tobit models binary choice models and duration models with unobserved individual heterogeneity and random censoring i also investigate a weighting scheme that achieves the semiparametric efficiency bound

a constrained single index regression for estimating interactions between a treatment and covariates we consider a single index regression model uniquely constrained to estimate interactions between a set of pretreatment covariates and a treatment variable on their effects on a response variable in the context of analyzing data from randomized clinical trials we represent interaction effect terms of the model through a set of treatment specific flexible link functions on a linear combination of the covariates a single index subject to the constraint that the expected value given the covariates equals 0 while leaving the main effects of the covariates unspecified we show that the proposed semiparametric estimator is consistent for the interaction term of the model and that the efficiency of the estimator can be improved with an augmentation procedure the proposed single index regression provides a flexible and interpretable modeling approach to optimizing individualized treatment rules based on patients data measured at baseline as illustrated by simulation examples and an application to data from a depression clinical trial

global well posedness of the 2d boussinesq equations with fractional laplacian dissipation abstract as a continuation of the previous work 48 in this paper we focus on the cauchy problem of the two dimensional 2d incompressible boussinesq equations with fractional laplacian dissipation we give an elementary proof of the global regularity of the smooth solutions of the 2d boussinesq equations with a new range of fractional powers of the laplacian the argument is based on the nonlinear lower bounds for the fractional laplacian established in 13 consequently this result significantly improves the recent works 13 45 48

model checking in regression via dimension reduction lack of fit checking for parametric and semiparametric models is essential in reducing misspecification the efficiency of most existing model checking methods drops rapidly as the dimension of the covariates increases we propose to check a model by projecting the fitted residuals along a direction that adapts to the systematic departure of the residuals from the desired pattern consistency of the method is proved for parametric and semiparametric regression models a bootstrap implementation is also discussed simulation comparisons with several existing methods are made suggesting that the proposed methods are more efficient than the existing methods when the dimension increases air pollution data from chicago are used to illustrate the procedure

global regularity for the 2d boussinesq equations with partial viscosity terms abstract in this paper we prove the global in time regularity for the 2d boussinesq system with either the zero diffusivity or the zero viscosity we also prove that as diffusivity viscosity tends to zero the solutions of the fully viscous equations converge strongly to those of zero diffusion viscosity equations our result for the zero diffusion system in particular solves the problem no 3 posed by moffatt in r l ricca ed kluwer academic publishers dordrecht the netherlands 2001 pp 3 10

on extended partially linear single index models aiming to explore the relation between the response y and the stochastic explanatory vector variable x beyond the linear approximation we consider the single index model which is a well known approach in multidimensional cases specifically we extend the partially linear single index model to take the form y 0 t x o 0 0 t x e where e is a random variable with ee 0 and var e 2 unknown 0 and o are unknown parametric vectors and o is an unknown real function the model is also applicable to nonlinear time series analysis in this paper we propose a procedure to estimate the model and prove some related asymptotic results both simulated and real data are used to illustrate the results

the 2d boussinesq equations with vertical viscosity and vertical diffusivity this paper aims at the global regularity of classical solutions to the 2d boussinesq equations with vertical dissipation and vertical thermal diffusion we prove that the l r norm of the vertical velocity v for any 1 0 would guarantee the global regularity of classical solutions

global well posedness for a class of 2d boussinesq systems with fractional dissipation abstract the incompressible boussinesq equations not only have many applications in modeling fluids and geophysical fluids but also are mathematically important the well posedness and related problem on the boussinesq equations have recently attracted considerable interest this paper examines the global regularity issue on the 2d boussinesq equations with fractional laplacian dissipation and thermal diffusion attention is focused on the case when the thermal diffusion dominates we establish the global well posedness for the 2d boussinesq equations with a new range of fractional powers of the laplacian

distribution of integral division points on the algebraic torus let k be a number field with algebraic closure overline k and let s be a finite set of places of k containing all the infinite ones let it gamma 0 be a finitely generated subgroup of mathbb g textup m overline k and let it gamma subset mathbb g textup m overline k be the division group attached to it gamma 0 here is an illustration of what we will prove in this article fix a proper closed subinterval i of 0 infty and a nonzero effective divisor d on mathbb g textup m which is not the translate of any torsion divisor on the algebraic torus mathbb g textup m by any point of it gamma with height belonging to i then we prove a statement which easily implies that the set of integral division points on mathbb g textup m with height near i i e the set of points of it gamma with standard absolute logarithmic weil height in j which are s integral on mathbb g textup m relative to d is finite for some fixed subinterval j of 0 infty properly containing i we propose a conjecture on the nondensity of integral division points on semi abelian varieties with prescribed height values which generalizes some previously known conjectures as well as this finiteness result for mathbb g textup m finally we also propose an analogous version for a dynamical system on mathbb p 1

generating functions fibonacci numbers and rational knots abstract we describe rational knots with any of the possible combinations of the properties a chirality non positivity non fiberedness and unknotting number one or higher and determine exactly their number for a given number of crossings in terms of their generating functions we show in particular how fibonacci numbers occur in the enumeration of fibered achiral and unknotting number one rational knots then we show how to enumerate rational knots of given crossing number depending on genus and or signature this allows to determine the asymptotical average value of these invariants among rational knots we give also an application to the enumeration of lens spaces