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65
Semiconcave Functions, HamiltonJacobi Equations, and Optimal Control
, 2003
"... otion of semiconvexity. The answer is that this choice better fits minimization, as opposed to maximization, and this is the formulation we have adopted for the optimization problems of interest here. Interest in semiconcave functions was initially motivated by research on nonlinear partial di#eren ..."
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Cited by 52 (2 self)
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otion of semiconvexity. The answer is that this choice better fits minimization, as opposed to maximization, and this is the formulation we have adopted for the optimization problems of interest here. Interest in semiconcave functions was initially motivated by research on nonlinear partial di#erential equations. In fact, it was exactly in classes of semiconcave functions that the first global existence and uniqueness results were obtained for HamiltonJacobiBellman equations, see Douglis [67] and Kruzhkov [97, 98, 100]. Afterwards, more powerful uniqueness theo M. Dibdin, Blood rain, Faber and Faber, London, 1999. iii iv ries, such as viscosity solutions and minimax solutions, were developed. Nevertheless, semiconcavity maintains its importance even in modern PDE theory, as the maximal type of regularity that can be expected for certain nonlinear problems. As such, it has been investigated in the modern textbooks on HamiltonJacobi equations by Lions [107], Bardi and Capuzzo D
An IV Model of Quantile Treatment Effects
 Econometrica
, 2001
"... Headnote.The ability of quantile regression models to characterize the heterogeneous impact of variables on different points of an outcome distribution makes them appealing in many economic applications. However, in observational studies, the variables of interest (e.g. education, prices) are ofte ..."
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Cited by 39 (2 self)
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Headnote.The ability of quantile regression models to characterize the heterogeneous impact of variables on different points of an outcome distribution makes them appealing in many economic applications. However, in observational studies, the variables of interest (e.g. education, prices) are often endogenous, making conventional quantile regression inconsistent and hence inappropriate for recovering the causal effects of these variables on the quantiles of economic outcomes. In order to address this problem, we develop a model of quantile treatment effects (QTE) in the presence of endogeneity and obtain conditions for identification of the QTE without functional form assumptions. The principal feature of the model is the imposition of conditions which restrict the evolution of ranks across treatment states. This feature allows us to overcome the endogeneity problem and recover the true QTE through the use of instrumental variables. The proposed model can also be equivalently viewed as a structural simultaneous equation model with nonadditive errors, where QTE can be interpreted as the structural quantile effects (SQE). Key Words: endogeneity, quantile regression, simultaneous equations, instrumental regression, identification, nonlinear model, monotone likelihood ratio, bounded completeness,
Rough solution of the Einstein constraint equations
 URL (cited on 15 February 2005): http://arXiv.org/abs/grqc/0405088
, 2004
"... Abstract. We construct low regularity solutions of the vacuum Einstein constraint equations. In particular, on 3manifolds we obtain solutions with metrics in Hs loc with s> 3 2. The theory of maximal asymptotically Euclidean solutions of the constraint equations descends completely the low regulari ..."
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Cited by 26 (0 self)
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Abstract. We construct low regularity solutions of the vacuum Einstein constraint equations. In particular, on 3manifolds we obtain solutions with metrics in Hs loc with s> 3 2. The theory of maximal asymptotically Euclidean solutions of the constraint equations descends completely the low regularity setting. Moreover, every rough, maximal, asymptotically Euclidean solution can be approximated in an appropriate topology by smooth solutions. These results have application in an existence theorem for rough solutions of the Einstein evolution equations. 1.
Instrumental variable quantile regression: A robust inference approach
 Journal of Econometrics
, 2008
"... Quantile regression is an increasingly important tool that estimates the conditional quantiles of a response Y given a vector of regressors D. It usefully generalizes Laplace’s median regression and can be used to measure the effect of covariates not only in the center of a distribution, but also in ..."
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Cited by 21 (4 self)
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Quantile regression is an increasingly important tool that estimates the conditional quantiles of a response Y given a vector of regressors D. It usefully generalizes Laplace’s median regression and can be used to measure the effect of covariates not only in the center of a distribution, but also in the upper and lower tails. For the linear quantile model defined by Y = D ′ γ(U) where D ′ γ(U) is strictly increasing in U and U is a standard uniform variable independent of D, quantile regression allows estimation of quantile specific covariate effects γ(τ) for τ ∈ (0, 1). In this paper, we propose an instrumental variable quantile regression estimator that appropriately modifies the conventional quantile regression and recovers quantilespecific covariate effects in an instrumental variables model defined by Y = D ′ α(U) where D ′ α(U) is strictly increasing in U and U is a uniform variable that may depend on D but is independent of a set of instrumental variables Z. The proposed estimator and inferential procedures are computationally convenient in typical applications and can be carried out using software available for conventional quantile regression. In addition, the proposed estimation procedure gives rise to a convenient inferential procedure that is naturally robust to weak identification. The use of the proposed estimator and testing procedure is illustrated through two empirical examples.
On Two InteriorPoint Mappings for Nonlinear Semidefinite Complementarity Problems
 Mathematics of Operations Research
, 1997
"... Extending our previous work Monteiro and Pang (1996), this paper studies properties of two fundamental mappings associated with the family of interiorpoint methods for solving monotone nonlinear complementarity problems over the cone of symmetric positive semidefinite matrices. The first of these m ..."
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Cited by 20 (6 self)
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Extending our previous work Monteiro and Pang (1996), this paper studies properties of two fundamental mappings associated with the family of interiorpoint methods for solving monotone nonlinear complementarity problems over the cone of symmetric positive semidefinite matrices. The first of these maps lead to a family of new continuous trajectories which include the central trajectory as a special case. These trajectories completely "fill up" the set of interior feasible points of the problem in the same way as the weighted central paths do the interior of the feasible region of a linear program. Unlike the approach based on the theory of maximal monotone maps taken by Shida and Shindoh (1996) and Shida, Shindoh, and Kojima (1995), our approach is based on the theory of local homeomorphic maps in nonlinear analysis. Key words: interior point methods, mixed nonlinear complementarity problems, generalized complementarity problems, maximal monotonicity, monotone mappings, continuous traj...
Periodic Solutions of Nonlinear Wave Equations with General Nonlinearities
"... We prove the existence of small amplitude periodic solutions, with strongly irrational frequency # close to one, for completely resonant nonlinear wave equations. We provide multiplicity results for both monotone and nonmonotone nonlinearities. For # close to one we prove the existence of a large nu ..."
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Cited by 17 (3 self)
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We prove the existence of small amplitude periodic solutions, with strongly irrational frequency # close to one, for completely resonant nonlinear wave equations. We provide multiplicity results for both monotone and nonmonotone nonlinearities. For # close to one we prove the existence of a large number N# of 2#/#periodic in time solutions u 1 , . . . , un , . . . , uN : N# 1. The minimal period of the nth solution un is proved to be 2#/n#. The proofs are based on a LyapunovSchmidt reduction and variational arguments.
Heat kernels on metricmeasure spaces and an application to semilinear elliptic equations
 Trans. Amer. Math. Soc
, 2003
"... Abstract. We consider a metric measure space (M, d,µ) andaheat kernel pt(x, y) on M satisfying certain upper and lower estimates, which depend on two parameters α and β. We show that under additional mild assumptions, these parameters are determined by the intrinsic properties of the space (M, d, µ) ..."
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Cited by 13 (4 self)
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Abstract. We consider a metric measure space (M, d,µ) andaheat kernel pt(x, y) on M satisfying certain upper and lower estimates, which depend on two parameters α and β. We show that under additional mild assumptions, these parameters are determined by the intrinsic properties of the space (M, d, µ). Namely, α is the Hausdorff dimension of this space, whereas β, called the walk dimension, is determined via the properties of the family of Besov spaces W σ,2 on M. Moreover, the parameters α and β are related by the inequalities 2 ≤ β ≤ α +1. We prove also the embedding theorems for the space W β/2,2, and use them to obtain the existence results for weak solutions to semilinear elliptic equations on M of the form −Lu + f(x, u) =g(x), where L is the generator of the semigroup associated with pt. The framework in this paper is applicable for a large class of fractal domains, including the generalized Sierpiński carpet in Rn. 1.
Lyapunov Center Theorem For Some Nonlinear PDEs: A Simple Proof
 Ann. Scuola Norm. Sup. Pisa Cl. Sci
, 1999
"... . We give a simple proof of existence of small oscillations in some nonlinear partial dierential equations. The proof is based on the Lyapunov{Schmidt decomposition and the contraction mapping principle; the linear frequencies ! j are assumed to satisfy a Diophantine type nonresonance condition (of ..."
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Cited by 13 (3 self)
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. We give a simple proof of existence of small oscillations in some nonlinear partial dierential equations. The proof is based on the Lyapunov{Schmidt decomposition and the contraction mapping principle; the linear frequencies ! j are assumed to satisfy a Diophantine type nonresonance condition (of the kind of the rst Melnikov condition) slightly stronger than the usual one. If ! j j d with d > 1, such Diophantine condition will be proved to have full measure in a sense specied below; if d = 1, we will prove that the condition is satised in a set of zero measure. Applications to nonlinear beam equations and to nonlinear wave equations with Dirichlet boundary condition are given. The result also applies to more general systems and boundary conditions (e.g. periodic). MSC: 35B10, 35B32, 34C15 1. Introduction In this paper we study the extension of the Lyapunov center theorem on the existence of nonlinear small oscillations to some partial dierential equations. The result we are...
An Interior Point Potential Reduction Method for Constrained Equations
, 1995
"... We study the problem of solving a constrained system of nonlinear equations by a combination of the classical damped Newton method for (unconstrained) smooth equations and the recent interior point potential reduction methods for linear programs, linear and nonlinear complementarity problems. In gen ..."
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Cited by 11 (3 self)
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We study the problem of solving a constrained system of nonlinear equations by a combination of the classical damped Newton method for (unconstrained) smooth equations and the recent interior point potential reduction methods for linear programs, linear and nonlinear complementarity problems. In general, constrained equations provide a unified formulation for many mathematical programming problems, including complementarity problems of various kinds and the KarushKuhnTucker systems of variational inequalities and nonlinear programs. Combining ideas from the damped Newton and interior point methods, we present an iterative algorithm for solving a constrained system of equations and investigate its convergence properties. Specialization of the algorithm and its convergence analysis to complementarity problems of various kinds and the KarushKuhnTucker systems of variational inequalities are discussed in detail. We also report the computational results of the implementation of the algo...
General InteriorPoint Maps and Existence of Weighted Paths for Nonlinear Semidefinite Complementarity Problems
, 1999
"... Extending the previous work of Monteiro and Pang (1998), this paper studies properties of fundamental maps that can be used to describe the central path of the monotone nonlinear complementarity problems over the cone of symmetric positive semidefinite matrices. Instead of focusing our attention on ..."
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Cited by 10 (2 self)
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Extending the previous work of Monteiro and Pang (1998), this paper studies properties of fundamental maps that can be used to describe the central path of the monotone nonlinear complementarity problems over the cone of symmetric positive semidefinite matrices. Instead of focusing our attention on a specific map as was done in the approach of Monteiro and Pang (1998), this paper considers a general form of a fundamental map and introduces conditions on the map that allow us to extend the main results of Monteiro and Pang (1998) to this general map. Each fundamental map leads to a family of "weighted" continuous trajectories which include the central trajectory as a special case. Hence, for complementarity problems over the cone of symmetric positive semidefinite matrices, the notion of weighted central path depends on the fundamental map used to represent the central path. Key words: interior point methods, mixed nonlinear complementarity problems, generalized complementarity problem...