Approximation Lemma Research Articles

Tree approximation in quasi-trees

n this paper, we investigate the geometric properties of quasi-trees and prove some equivalent criteria. We give a general construction of a tree that approximates the ends of a geodesic space, and use this to prove that every quasi-tree is (1,C) -quasi-isometric to a simplicial tree. As a consequence, we show that Gromov's tree approximation lemma for hyperbolic spaces (1987) can be improved in the case of quasi-trees to give a uniform approximation for any set of points, independent of cardinality. From this we show that having uniform tree approximation for finite subsets is equivalent to being able to uniformly approximate the entire space by a tree. As another consequence, we note that the boundary of a quasi-tree is isometric to the boundary of its approximating tree under a certain choice of visual metric, and that this gives a natural extension of the standard metric on the boundary of a tree.

The fourth-order total variation flow in $ \mathbb{R}^n $

<abstract><p>We define rigorously a solution to the fourth-order total variation flow equation in $ \mathbb{R}^n $. If $ n\geq3 $, it can be understood as a gradient flow of the total variation energy in $ D^{-1} $, the dual space of $ D^1_0 $, which is the completion of the space of compactly supported smooth functions in the Dirichlet norm. However, in the low dimensional case $ n\leq2 $, the space $ D^{-1} $ does not contain characteristic functions of sets of positive measure, so we extend the notion of solution to a larger space. We characterize the solution in terms of what is called the Cahn-Hoffman vector field, based on a duality argument. This argument relies on an approximation lemma which itself is interesting. We introduce a notion of calibrability of a set in our fourth-order setting. This notion is related to whether a characteristic function preserves its form throughout the evolution. It turns out that all balls are calibrable. However, unlike in the second-order total variation flow, the outside of a ball is calibrable if and only if $ n\neq2 $. If $ n\neq2 $, all annuli are calibrable, while in the case $ n = 2 $, if an annulus is too thick, it is not calibrable. We compute explicitly the solution emanating from the characteristic function of a ball. We also provide a description of the solution emanating from any piecewise constant, radially symmetric datum in terms of a system of ODEs.</p></abstract>

Mathcal {A}-caloric approximation and partial regularity for parabolic systems with Orlicz growth

We prove a new mathcal {A}-caloric approximation lemma compatible with an Orlicz setting. With this result, we establish a partial regularity result for parabolic systems of the type ut-diva(Du)=0.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} u_{t}- {{\\,\ extrm{div}\\,}}a(Du)=0. \\end{aligned}$$\\end{document}Here the growth of a is bounded by the derivative of an N-function {varphi }. The primary assumption for {varphi } is that t{varphi }''(t) and {varphi }'(t) are uniformly comparable on (0,infty ).

Interpolation inequalities for partial regularity

Abstract We propose two new direct methods for proving partial regularity of solutions of nonlinear elliptic or parabolic systems. The methods are based on two similar interpolation inequalities for solutions of linear systems with constant coefficient. The first results from an interpolation inequality of L p {L^{p}} norms in combination with an L p {L^{p}} estimate with low exponent p > 1 {p>1} . For the second, we provide a functional-analytic proof, that also sheds light upon the A-harmonic approximation lemma of Duzaar and Steffen. Both methods use a Caccioppoli inequality and avoid higher integrability. We illustrate the methods in detail for the case of a quasilinear elliptic system.

Limit profiles for reversible Markov chains

In a recent breakthrough, Teyssier (Ann Probab 48(5):2323–2343, 2020) introduced a new method for approximating the distance from equilibrium of a random walk on a group. He used it to study the limit profile for the random transpositions card shuffle. His techniques were restricted to conjugacy-invariant random walks on groups; we derive similar approximation lemmas for random walks on homogeneous spaces and for general reversible Markov chains. We illustrate applications of these lemmas to some famous problems: the k-cycle shuffle, sharpening results of Hough (Probab Theory Relat Fields 165(1–2):447–482, 2016) and Berestycki, Schramm and Zeitouni (Ann Probab 39(5):1815–1843, 2011), the Ehrenfest urn diffusion with many urns, sharpening results of Ceccherini-Silberstein, Scarabotti and Tolli (J Math Sci 141(2):1182–1229, 2007), a Gibbs sampler, which is a fundamental tool in statistical physics, with Binomial prior and hypergeometric posterior, sharpening results of Diaconis, Khare and Saloff-Coste (Stat Sci 23(2):151–178, 2008).

Exact and inexact decompositions of trade price indices

A reasonable concept for the true trade price index in situations where low-price countries capture market shares from high-price countries is the average price paid by importers for the same quality of good or service from all exporting countries. However, decompositions of trade price indices are usually inexact in the sense that the average price used as the underlying aggregator formula is not exactly reproduced. In this paper, we compare analytically exact and inexact decompositions of trade price indices, paying particular attention to the bias in aggregate inflation occurring from applying the first-order Taylor series approximation and not the quadratic approximation lemma to a geometric average price. Our calculations, using the Norwegian clothing industry as an illustration, reveal that the bias in aggregate inflation over the sample period of 1997–2016 is quite substantial and as much as 0.6 percentage point in some years. We therefore conclude that the quadratic approximation lemma should be used in practice to exactly reproduce the underlying aggregator formula.

Sharp regularity for the inhomogeneous porous medium equation

We show that locally bounded solutions of the inhomogeneous porous medium equation $$u_{t} - {\rm div} \left( m |u|^{m-1} \nabla u \right) = f \in L^{q,r}, \quad m >1 ,$$ are locally H\"older continuous, with exponent $$\gamma =\min \left\{ \frac{\alpha_{0}^-}{m}, \frac{[(2q - n)r -2q]}{q[(mr - (m-1)]} \right\},$$ where $\alpha_{0}$ denotes the optimal H\"older exponent for solutions of the homogeneous case. The proof relies on an approximation lemma and geometric iteration in the appropriate intrinsic scaling.

Gradient regularity for a singular parabolic equation in non-divergence form

In this paper we consider viscosity solutions of a class of non-homogeneous singular parabolic equations \begin{document}$ \partial_t u-|Du|^\gamma\Delta_p^N u = f, $\end{document} where $ -1<\gamma<0 $, $ 1<p<\infty $, and $ f $ is a given bounded function. We establish interior Hölder regularity of the gradient by studying two alternatives: The first alternative uses an iteration which is based on an approximation lemma. In the second alternative we use a small perturbation argument.

A sparse regular approximation lemma

We introduce a new variant of Szemeredi's regularity which we call the sparse regular approximation lemma (SRAL). The input to this is a graph $G$ of edge density $p$ and parameters $\epsilon, \delta$, where we think of $\delta$ as a constant. The goal is to construct an $\epsilon$-regular partition of $G$ while having the freedom to add/remove up to $\delta |E(G)|$ edges. As we show here, this weaker variant of the regularity already suffices for proving the graph removal and the hypergraph regularity lemma, which are two of the main applications of the (standard) regularity lemma. This of course raises the following question: can one obtain quantitative bounds for SRAL that are significantly better than those associated with the regularity lemma? Our first result answers the above question affirmatively by proving an upper bound for SRAL given by a tower of height $O(\log 1/p)$. This allows us to reprove Fox's upper bound for the graph removal lemma. Our second result is a matching lower bound for SRAL showing that a tower of height $\Omega(\log 1/p)$ is unavoidable. We in fact prove a more general multicolored lower bound which is essential for proving lower bounds for the hypergraph regularity lemma.

On the existence of invariant tori in non-conservative dynamical systems with degeneracy and finite differentiability

In this paper, we establish a KAM-theorem about the existenceof invariant tori in non-conservative dynamical systems with finitely differentiable vector fields and multiple degeneracies under the assumption that theintegrable part is finitely differentiable with respect to parameters, instead ofthe usual assumption of analyticity. We prove these results by constructingapproximation and inverse approximation lemmas in which all functions arefinitely differentiable in parameters.

Refinements of the holonomic approximation lemma

The holonomic approximation lemma of Eliashberg and Mishachev is a powerful tool in the philosophy of the $h-$principle. By carefully keeping track of the quantitative geometry behind the holonomic approximation process, we establish several refinements of this lemma. Gromov's idea from convex integration of working one pure partial derivative at a time is central to the discussion. We give applications of our results to flexible symplectic and contact topology.

On locally essentially bounded divergence measure fields and sets of locally finite perimeter

Abstract Chen, Torres and Ziemer ([9], 2009) proved the validity of generalized Gauss–Green formulas and obtained the existence of interior and exterior normal traces for essentially bounded divergence measure fields on sets of finite perimeter using an approximation theory through sets with a smooth boundary. However, it is known that the proof of a crucial approximation lemma contained a gap. Taking inspiration from a previous work of Chen and Torres ([7], 2005) and exploiting ideas of Vol’pert ([29], 1985) for essentially bounded fields with components of bounded variation, we present here a direct proof of generalized Gauss–Green formulas for essentially bounded divergence measure fields on sets of finite perimeter which includes the existence and essential boundedness of the normal traces. Our approach appears to be simpler since it does not require any special approximation theory for the domains and it relies only on the Leibniz rule for divergence measure fields. This freedom allows one to localize the constructions and to derive more general statements in a natural way.

Parabolic Lipschitz truncation and caloric approximation

We develop an improved version of the parabolic Lipschitz truncation, which allows qualitative control of the distributional time derivative and the preservation of zero boundary values. As a consequence, we establish a new caloric approximation lemma. We show that almost p-caloric functions are close to p-caloric functions. The distance is measured in terms of spatial gradients as well as almost uniformly in time. Both results are extended to the setting of Orlicz growth.

Large Deviations of the Threshold Estimator of Integrated (Co-)Volatility Vector in the Presence of Jumps

Recently considerable interest has been paid to the estimation problem of the realized volatility and covolatility by using high-frequency data of financial price processes in financial econometrics. Threshold estimation is one of the useful techniques in the inference for jump-type stochastic processes from discrete observations. In this paper, we adopt the threshold estimator introduced by Mancini (Scand Actuar J 1:42–52, 2004) where only the variations under a given threshold function are taken into account. The purpose of this work is to investigate large and moderate deviations for the threshold estimator of the integrated variance–covariance vector. This paper is an extension of the previous work in Djellout et al. (Stoch Process Appl 1–35, 2017), where the problem has been studied in the absence of a jump component. We will use the approximation lemma to prove large and moderate deviations results. As the reader can expect, we obtain the same results as in the case without jump.

Partial Outer Convexification for Traffic Light Optimization in Road Networks

We consider the problem of computing optimal traffic light programs for urban road intersections using traffic flow conservation laws on networks. Based on a partial outer convexification approach, which has been successfully applied in the area of mixed-integer optimal control for systems of ordinary or differential algebraic equations, we develop a computationally tractable two-stage solution heuristic. The two-stage approach consists of the solution of a (smoothed) nonlinear programming problem with dynamic constraints and a reconstruction mixed-integer linear program without dynamic constraints. The two-stage approach is founded on a discrete approximation lemma for partial outer convexification, whose grid-independence properties for (smoothed) conservation laws are investigated. We use the two-stage approach to compute traffic light programs for two scenarios on different discretizations and demonstrate that the solution candidates cannot be improved in a reasonable amount of time by global state-of...

Partial Regularity for Holonomic Minimisers of Quasiconvex Functionals

We prove partial regularity for local minimisers of certain strictly quasiconvex integral functionals, over a class of Sobolev mappings into a compact Riemannian manifold, to which such mappings are said to be holonomically constrained. Our approach uses the lifting of Sobolev mappings to the universal covering space, the connectedness of the covering space, an application of Ekeland’s variational principle and a certain tangential $${\mathbb{A}}$$ -harmonic approximation lemma obtained directly via a Lipschitz approximation argument. This allows regularity to be established directly on the level of the gradient. Several applications to variational problems in condensed matter physics with broken symmetries are also discussed, in particular those concerning the superfluidity of liquid helium-3 and nematic liquid crystals.

Partial regularity for subquadratic parabolic systems under controllable growth conditions

In this paper, we prove a regularity result for weak solutions away from singular set of parabolic systems with subquadratic growth condition of the type∂tu(z)−diva(z,u,Du)=B(z,u,Du)for z=(x,t), where the structure function a satisfies standard ellipticity and growth condition with polynomial growth rate 2nn+2<p<2. The inhomogeneity B satisfies the controllable growth condition|B(z,u,Du)|≤λ(1+|u|p⁎−1+|Du|p(1−1p⁎)), where p⁎=(n+2)pn+2−p, if n+2≠p. Here p⁎ can be any exponent if n+2=p. The proof is based on the A-caloric approximation lemma to parabolic systems with subquadratic growth.

Quasilinear elliptic equations on convex domains

We show global regularity for p-Laplacian type elliptic equations on convex domains with W1,q(q≥p>1) boundary data. The maximal function, Vitali covering lemma and approximation (or compactness method) are the main techniques in the proof. To derive the approximation lemma, we need W1,∞ estimates for the corresponding homogeneous equations, where the convexity of the domain plays an important role.

On the properness of an impulsive control extension of dynamic optimization problems

This article investigates the properness, or well-posedness, of impulsive extension of a conventional optimal control problem. This includes both well-posedness of the solution to impulsive control systems arising as result of an impulsive extension of ordinary differential systems, and existence theorems. Well-posedness in the classic Cauchy sense is proved. Approximation lemmas that guarantee sensitivity to small perturbations in control variables are obtained. Filippov type existence theorems are established. A model example is provided to show the relevance of the impulsive controls problems which are under study.

On a lemma of Jacques-Louis Lions and its relation to other fundamental results

Let Ω be a domain in RN, i.e., a bounded and connected open subset of RN with a Lipschitz-continuous boundary ∂Ω, the set Ω being locally on the same side of ∂Ω. A fundamental lemma, due to Jacques-Louis Lions, provides a characterization of the space L2(Ω), as the space of all distributions on Ω whose gradient is in the space H−1(Ω). This lemma, which provides in particular a short proof of a crucial inequality due to J. Nečas, is also a key for proving other basic results, such as, among others, the surjectivity of the divergence operator acting from H01(Ω) into L02(Ω), a “weak” form of the Poincaré lemma or a “simplified version” of de Rham theorem, each of which provides sufficient conditions insuring that a vector field in H−1(Ω) is the gradient of a function in L2(Ω). The main objective of this paper is to establish an “equivalence theorem”, which asserts that J.L. Lions lemma is in effect equivalent to a number of other fundamental properties, which include in particular the ones mentioned above. The key for proving this theorem is a specific “approximation lemma”, itself one of these equivalent results, which appears to be new to the best of our knowledge. Some of these equivalent properties can be given an independent, i.e., “direct”, proof, such as for instance the constructive proof by M.E. Bogovskii of the surjectivity of the divergence operator. Therefore, the proof of any one of such properties provides, by way of our equivalence theorem, a means of proving J.L. Lions lemma, the known “direct” proofs of which for a general domain are notoriously difficult.