Showing papers on "Bounded function published in 2014"

On asymptotically optimal confidence regions and tests for high-dimensional models

Abstract: We propose a general method for constructing confidence intervals and statistical tests for single or low-dimensional components of a large parameter vector in a high-dimensional model. It can be easily adjusted for multiplicity taking dependence among tests into account. For linear models, our method is essentially the same as in Zhang and Zhang [J. R. Stat. Soc. Ser. B Stat. Methodol. 76 (2014) 217–242]: we analyze its asymptotic properties and establish its asymptotic optimality in terms of semiparametric efficiency. Our method naturally extends to generalized linear models with convex loss functions. We develop the corresponding theory which includes a careful analysis for Gaussian, sub-Gaussian and bounded correlated designs.

Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates

Abstract: This is the first of two articles dealing with the equation ( − Δ ) s v = f ( v ) in R n , with s ∈ ( 0 , 1 ) , where ( − Δ ) s stands for the fractional Laplacian — the infinitesimal generator of a Levy process. This equation can be realized as a local linear degenerate elliptic equation in R + n + 1 together with a nonlinear Neumann boundary condition on ∂ R + n + 1 = R n . In this first article, we establish necessary conditions on the nonlinearity f to admit certain type of solutions, with special interest in bounded increasing solutions in all of R . These necessary conditions (which will be proven in a follow-up paper to be also sufficient for the existence of a bounded increasing solution) are derived from an equality and an estimate involving a Hamiltonian — in the spirit of a result of Modica for the Laplacian. Our proofs are uniform as s ↑ 1 , establishing in the limit the corresponding known results for the Laplacian. In addition, we study regularity issues, as well as maximum and Harnack principles associated to the equation.

The Optimal Hard Threshold for Singular Values is $4/\sqrt {3}$

Abstract: We consider recovery of low-rank matrices from noisy data by hard thresholding of singular values, in which empirical singular values below a threshold λ are set to 0. We study the asymptotic mean squared error (AMSE) in a framework, where the matrix size is large compared with the rank of the matrix to be recovered, and the signal-to-noise ratio of the low-rank piece stays constant. The AMSE-optimal choice of hard threshold, in the case of n-by-n matrix in white noise of level σ, is simply (4/√3)√nσ ≈ 2.309√nσ when σ is known, or simply 2.858 · y med when σ is unknown, where y med is the median empirical singular value. For nonsquare, m by n matrices with m ≠ n the thresholding coefficients 4/√3 and 2.858 are replaced with different provided constants that depend on m/n. Asymptotically, this thresholding rule adapts to unknown rank and unknown noise level in an optimal manner: it is always better than hard thresholding at any other value, and is always better than ideal truncated singular value decomposition (TSVD), which truncates at the true rank of the low-rank matrix we are trying to recover. Hard thresholding at the recommended value to recover an n-by-n matrix of rank r guarantees an AMSE at most 3 nrσ 2 . In comparison, the guarantees provided by TSVD, optimally tuned singular value soft thresholding and the best guarantee achievable by any shrinkage of the data singular values are 5 nrσ 2 , 6 nrσ 2 , and 2 nrσ 2 , respectively. The recommended value for hard threshold also offers, among hard thresholds, the best possible AMSE guarantees for recovering matrices with bounded nuclear norm. Empirical evidence suggests that performance improvement over TSVD and other popular shrinkage rules can be substantial, for different noise distributions, even in relatively small n.

Metric measure spaces with Riemannian Ricci curvature bounded from below

Abstract: In this paper, we introduce a synthetic notion of Riemannian Ricci bounds from below for metric measure spaces (X,d,m) which is stable under measured Gromov– Hausdorff convergence and rules out Finsler geometries. It can be given in terms of an enforcement of the Lott, Sturm, and Villani geodesic convexity condition for the entropy coupled with the linearity of the heat flow. Besides stability, it enjoys the same tensorization, global-to-local, and local-to-global properties. In these spaces, which we call RCD(K,∞) spaces, we prove that the heat flow (which can be equivalently characterized either as the flow associated to the Dirichlet form, or as the Wasserstein gradient flow of the entropy) satisfies Wasserstein contraction estimates and several regularity properties, in particular Bakry–Emery estimates and the L∞-Lip Feller regularization. We also prove that the distance induced by the Dirichlet form coincides with d, that the local energy measure has density given by the square of Cheeger’s relaxed slope, and, as a consequence, that the underlying Brownian motion has continuous paths. All these results are obtained independently of Poincare and doubling assumptions on the metric measure structure and therefore apply also to spaces which are not locally compact, as the infinite-dimensional ones.

Fuzzy Neural Network-Based Adaptive Control for a Class of Uncertain Nonlinear Stochastic Systems

Abstract: This paper studies an adaptive tracking control for a class of nonlinear stochastic systems with unknown functions. The considered systems are in the nonaffine pure-feedback form, and it is the first to control this class of systems with stochastic disturbances. The fuzzy-neural networks are used to approximate unknown functions. Based on the backstepping design technique, the controllers and the adaptation laws are obtained. Compared to most of the existing stochastic systems, the proposed control algorithm has fewer adjustable parameters and thus, it can reduce online computation load. By using Lyapunov analysis, it is proven that all the signals of the closed-loop system are semiglobally uniformly ultimately bounded in probability and the system output tracks the reference signal to a bounded compact set. The simulation example is given to illustrate the effectiveness of the proposed control algorithm.

On the Efficacy of Nonlinear Control in Uncertain Linear

Abstract: We consider a class of linear dynamical systems containing uncertain elements and subject to uncertain inputs, and for which either uncertain state or output is available. We construct a feedback control, utilizing measured state or estimated state, which guarantees that every system response is ultimately bounded within a certain neighborhood of the zero state. Performance resulting from use of this control is compared with that due to the use of purely linear feedback control.

Bounded biharmonic weights for real-time deformation

Abstract: Changing an object's shape is a basic operation in computer graphics, necessary for transforming raster images, vector graphics, geometric models, and animated characters. The fastest approaches for such object deformation involve linearly blending a small number of given affine transformations, typically each associated with bones of an internal skeleton, vertices of an enclosing cage, or a collection of loose point handles. Unfortunately, linear blending schemes are not always easy to use because they may require manually painting influence weights or modeling closed polyhedral cages around the input object. Our goal is to make the design and control of deformations simpler by allowing the user to work freely with the most convenient combination of handle types. We develop linear blending weights that produce smooth and intuitive deformations for points, bones, and cages of arbitrary topology. Our weights, called bounded biharmonic weights, minimize the Laplacian energy subject to bound constraints. Doing so spreads the influences of the handles in a shape-aware and localized manner, even for objects with complex and concave boundaries. The variational weight optimization also makes it possible to customize the weights so that they preserve the shape of specified essential object features. We demonstrate successful use of our blending weights for real-time deformation of 2D and 3D shapes.

Fourier spectral methods for fractional-in-space reaction-diffusion equations

Abstract: Fractional differential equations are becoming increasingly used as a powerful modelling approach for understanding the many aspects of nonlocality and spatial heterogeneity. However, the numerical approximation of these models is demanding and imposes a number of computational constraints. In this paper, we introduce Fourier spectral methods as an attractive and easy-to-code alternative for the integration of fractional-in-space reaction-diffusion equations described by the fractional Laplacian in bounded rectangular domains of \(\mathbb {R}^n\). The main advantages of the proposed schemes is that they yield a fully diagonal representation of the fractional operator, with increased accuracy and efficiency when compared to low-order counterparts, and a completely straightforward extension to two and three spatial dimensions. Our approach is illustrated by solving several problems of practical interest, including the fractional Allen–Cahn, FitzHugh–Nagumo and Gray–Scott models, together with an analysis of the properties of these systems in terms of the fractional power of the underlying Laplacian operator.

The Pohozaev identity for the fractional Laplacian

Abstract: In this paper we prove the Pohozaev identity for the semilinear Dirichlet problem \({(-\Delta)^s u =f(u)}\) in \({\Omega, u\equiv0}\) in \({{\mathbb R}^n\backslash\Omega}\) Here, \({s\in(0,1)}\) , (−Δ)s is the fractional Laplacian in \({\mathbb{R}^n}\) , and Ω is a bounded C1,1 domain To establish the identity we use, among other things, that if u is a bounded solution then \({u/\delta^s|_{\Omega}}\) is Cα up to the boundary ∂Ω, where δ(x) = dist(x,∂Ω) In the fractional Pohozaev identity, the function \({u/\delta^s|_{\partial\Omega}}\) plays the role that ∂u/∂ν plays in the classical one Surprisingly, from a nonlocal problem we obtain an identity with a boundary term (an integral over ∂Ω) which is completely local As an application of our identity, we deduce the nonexistence of nontrivial solutions in star-shaped domains for supercritical nonlinearities

1-Bit matrix completion

Abstract: The problem of recovering a matrix from an incomplete sampling of its entries—also known as matrix completion—arises in a wide variety of practical situations, including collaborative filtering, system identification, sensor localization, rank aggregation, and many more. While many of these applications have a relatively long history, recent advances in the closely related field of compressed sensing have enabled a burst of progress in the last few years, and we now have a strong base of theoretical results concerning matrix completion. A typical result from this literature is that a generic d × d matrix of rank r can be exactly recovered from O(r dpolylog(d)) randomly chosen entries. Similar results can be established in the case of noisy observations and approximately low-rank matrices. See [1] and references therein for further details. Although these results are quite impressive, there is an important gap between the statement of the problem as considered in the matrix completion literature and many of the most common applications discussed therein. As an example, consider collaborative filtering and the now-famous “Netflix problem.” In this setting, we assume that there is some unknown matrix whose entries each represent a rating for a particular user on a particular movie. Since any user will rate only a small subset of possible movies, we are only able to observe a small fraction of the total entries in the matrix, and our goal is to infer the unseen ratings from the observed ones. If the rating matrix is low-rank, then this would seem to be the exact problem studied in the matrix completion literature. However, there is a subtle difference: the theory developed in this literature generally assumes that observations consist of (possibly noisy) continuous-valued entries of the matrix, whereas in the Netflix problem the observations are “quantized” to the set of integers between 1 and 5. If we believe that it is possible for a user’s true rating for a particular movie to be, for example, 4.5, then we must account for the impact of this “quantization noise” on our recovery. Of course, one could potentially treat quantization simply as a form of bounded noise, but this is somewhat unsatisfying because the ratings aren’t just quantized — there are also hard limits placed on the minimum and maximum allowable ratings. (Why should we suppose that a movie given a rating of 5 could not have a true underlying rating of 6 or 7 or 10?) The inadequacy of standard matrix completion techniques in dealing with this effect is particularly pronounced when we consider recommender systems where each rating consists of a single bit representing a positive or negative rating (consider for example rating music on Pandora, the relevance of advertisements on Hulu, or posts on Reddit or MathOverflow). Similar situations arise in nearly every application that has been proposed for matrix completion, including the analysis of incomplete survey data, the recovery of pairwise distance matrices (multidimensional scaling), quantum state tomography, and many others. In such cases, the assumptions made in the existing theory of matrix completion do not apply, standard algorithms are ill-posed, and a new theory is required. In this work we describe the approach we take in [1] to extend the theory of matrix completion to the case of 1-bit observations. We consider a statistical model for such data where a binary output is generated according to a probability distribution which is parameterized by the corresponding entry of the unknown matrix M . The central question we ask is: “Given observations of this form, can we recover the underlying matrix?” Several new challenges arise when trying to develop a theory for 1-bit matrix completion. First, matrix completion is in some sense a more challenging problem than compressed sensing. Specifically, some additional difficulty arises because the set of low-rank matrices is “coherent” with single entry measurements—there will always be certain (sparse) low-rank matrices that we cannot hope to recover without essentially sampling every entry of the matrix. The typical way to deal with this possibility is to consider a reduced set of lowrank matrices by placing restrictions on the entry-wise maximum of the matrix or its singular vectors—informally, we require that the matrix is not too “spiky”. However, we introduce an entirely new dimension of ill-posedness by restricting ourselves to 1-bit observations. An example of this is described in detail in [1] and shows that in the case where we simply observe Y = sign(M), the problem of recovering M is illposed. To see this, let M = uv∗ for any vectors u,v ∈ R, and for simplicity assume that there are no zero entries in u or v. Now let ũ and ṽ be any vectors with the same sign pattern as u and v respectively. It is apparent that either M or M = ũṽ will yield the same observations Y , and thus M and M are indistinguishable. Note that while it is obvious that this 1-bit measurement process will destroy any information we have regarding the scaling of M , this ill-posedness remains even if we knew something about the scaling a priori (such as the Frobenius norm of M ). For any given set of observations, there will always be radically different possible matrices that are all consistent with observed measurements. After considering this example, the problem might seem hopeless. However, an interesting surprise is that when we add noise to the problem (that is, when we observe a subset of the matrix Y = sign(M + Z) where Z 6= 0 is an appropriate stochastic matrix) the picture completely changes—this noise has a “dithering” effect and the problem becomes well-posed. In fact, we will show that in this setting we can sometimes recover M to the same degree of accuracy that is possible when given access to completely unquantized measurements! In particular, under appropriate conditions, O(rd) measurements are sufficient to accurately recover M . We will provide an overview of these results and discuss a number of practical applications.

Differentially Private Empirical Risk Minimization: Efficient Algorithms and Tight Error Bounds

Abstract: In this paper, we initiate a systematic investigation of differentially private algorithms for convex empirical risk minimization. Various instantiations of this problem have been studied before. We provide new algorithms and matching lower bounds for private ERM assuming only that each data point's contribution to the loss function is Lipschitz bounded and that the domain of optimization is bounded. We provide a separate set of algorithms and matching lower bounds for the setting in which the loss functions are known to also be strongly convex. Our algorithms run in polynomial time, and in some cases even match the optimal non-private running time (as measured by oracle complexity). We give separate algorithms (and lower bounds) for $(\epsilon,0)$- and $(\epsilon,\delta)$-differential privacy; perhaps surprisingly, the techniques used for designing optimal algorithms in the two cases are completely different. Our lower bounds apply even to very simple, smooth function families, such as linear and quadratic functions. This implies that algorithms from previous work can be used to obtain optimal error rates, under the additional assumption that the contributions of each data point to the loss function is smooth. We show that simple approaches to smoothing arbitrary loss functions (in order to apply previous techniques) do not yield optimal error rates. In particular, optimal algorithms were not previously known for problems such as training support vector machines and the high-dimensional median.

Continuous Bounded Cohomology of Locally Compact Groups

Abstract: The purpose of this monograph is (a) to lay the foundations for a conceptual approach to bounded cohomology; (b) to harvest the resulting applications in rigidity theory. Of central importance is the new interplay between measure theory, amenability, Banach representations on one hand, with the homological apparatus on the other hand. The applications obtained in this text include rigidity for actions on Teichmuller spaces and homeomorphisms of the circle. The main tools include Poisson boundaries for random walks, spectral sequences, Zimmer-amenability, cohomological induction.

Global convergence of splitting methods for nonconvex composite optimization

Abstract: We consider the problem of minimizing the sum of a smooth function $h$ with a bounded Hessian, and a nonsmooth function. We assume that the latter function is a composition of a proper closed function $P$ and a surjective linear map $\cal M$, with the proximal mappings of $\tau P$, $\tau > 0$, simple to compute. This problem is nonconvex in general and encompasses many important applications in engineering and machine learning. In this paper, we examined two types of splitting methods for solving this nonconvex optimization problem: alternating direction method of multipliers and proximal gradient algorithm. For the direct adaptation of the alternating direction method of multipliers, we show that, if the penalty parameter is chosen sufficiently large and the sequence generated has a cluster point, then it gives a stationary point of the nonconvex problem. We also establish convergence of the whole sequence under an additional assumption that the functions $h$ and $P$ are semi-algebraic. Furthermore, we give simple sufficient conditions to guarantee boundedness of the sequence generated. These conditions can be satisfied for a wide range of applications including the least squares problem with the $\ell_{1/2}$ regularization. Finally, when $\cal M$ is the identity so that the proximal gradient algorithm can be efficiently applied, we show that any cluster point is stationary under a slightly more flexible constant step-size rule than what is known in the literature for a nonconvex $h$.

Weak and viscosity solutions of the fractional laplace equation

Abstract: Aim of this paper is to show that weak solutions of the following fractional Laplacian equation ( ( ) s u = f in u = g in R n n are also continuous solutions (up to the boundary) of this problem in the viscosity sense. Here s2 (0; 1) is a xed parameter, is a bounded, open subset of R n (n > 1) with C 2 -boundary, and ( ) s is the fractional Laplacian operator, that may be dened as

CBMC – C Bounded Model Checker

Abstract: CBMC implements bit-precise bounded model checking for C programs and has been developed and maintained for more than ten years. CBMC verifies the absence of violated assertions under a given loop unwinding bound. Other properties, such as SV-COMP’s ERROR labels or memory safety properties are reduced to assertions via automated instrumentation. Only recently support for efficiently checking concurrent programs, including support for weak memory models, has been added. Thus, CBMC is now capable of finding counterexamples in all of SV-COMP’s categories. As back end, the competition submission of CBMC uses MiniSat 2.2.0.

Relativistic, model-independent, three-particle quantization condition

Abstract: We present a generalization of Luescher's relation between the finite-volume spectrum and scattering amplitudes to the case of three particles. We consider a relativistic scalar field theory in which the couplings are arbitrary aside from a Z2 symmetry that removes vertices with an odd number of particles. The theory is assumed to have two-particle phase shifts that are bounded by \pi/2 in the regime of elastic scattering. We determine the spectrum of the finite-volume theory from the poles in the odd-particle-number finite-volume correlator, which we analyze to all orders in perturbation theory. We show that it depends on the infinite-volume two-to-two K-matrix as well as a nonstandard infinite-volume three-to-three K-matrix. A key feature of our result is the need to subtract physical singularities in the three-to-three amplitude and thus deal with a divergence-free quantity. This allows our initial, formal result to be truncated to a finite dimensional determinant equation. At present, the relation of the three-to-three K-matrix to the corresponding scattering amplitude is not known, although previous results in the non-relativistic limit suggest that such a relation exists.

Learning without Concentration

Abstract: We obtain sharp bounds on the estimation error of the Empirical Risk Minimization procedure, performed in a convex class and with respect to the squared loss, without assuming that class members and the target are bounded functions or have rapidly decaying tails.Rather than resorting to a concentration-based argument, the method used here relies on a “small-ball” assumption and thus holds for classes consisting of heavy-tailed functions and for heavy-tailed targets.The resulting estimates scale correctly with the “noise level” of the problem, and when applied to the classical, bounded scenario, always improve the known bounds.

Adaptive Fuzzy Control for a Class of Nonlinear Discrete-Time Systems With Backlash

Abstract: An adaptive fuzzy controller design is studied for uncertain nonlinear systems in this paper. The considered systems are of the discrete-time form in a triangular structure and include the backlash and the external disturbance. By using the prediction function of future states, the systems are transformed into an n-step ahead predictor. The fuzzy logic systems (FLSs) are used to approximate the unknown functions, unknown backlash, and backlash inversion, respectively. A discrete-time tuning algorithm is developed to estimate the optimal fuzzy parameters. Compared with the previous works for the discrete-time systems with backlash, the main contributions of the paper are that 1) the rigorous restriction for the functional estimation error is removed, and 2) the external disturbance is bounded, but the bound is not required to be known. A novel controller and the adaptation laws are constructed by using the discrete Taylor series expansion and the difference Lyapunov analysis, and thus, those limitations in the previous works are overcome. It is proven that all the signals in the closed-loop system are bounded and that the system output can be to follow the reference signal to a bounded compact set. A simulation example is provided to illustrate the effectiveness of the proposed approach.

A Quantum Approximate Optimization Algorithm Applied to a Bounded Occurrence Constraint Problem

Abstract: We apply our recent Quantum Approximate Optimization Algorithm to the combinatorial problem of bounded occurrence Max E3LIN2. The input is a set of linear equations each of which contains exactly three boolean variables and each equation says that the sum of the variables mod 2 is 0 or is 1. Every variable is in no more than D equations. A random string will satisfy 1/2 of the equations. We show that the level one QAOA will efficiently produce a string that satisfies $\left(\frac{1}{2} + \frac{1}{101 D^{1/2}\, l n\, D}\right)$ times the number of equations. A recent classical algorithm achieved $\left(\frac{1}{2} + \frac{constant}{D^{1/2}}\right)$. We also show that in the typical case the quantum computer will output a string that satisfies $\left(\frac{1}{2}+ \frac{1}{2\sqrt{3e}\, D^{1/2}}\right)$ times the number of equations.

Online Synchronous Approximate Optimal Learning Algorithm for Multi-Player Non-Zero-Sum Games With Unknown Dynamics

Abstract: In this paper, we develop an online synchronous approximate optimal learning algorithm based on policy iteration to solve a multiplayer nonzero-sum game without the requirement of exact knowledge of dynamical systems. First, we prove that the online policy iteration algorithm for the nonzero-sum game is mathematically equivalent to the quasi-Newton's iteration in a Banach space. Then, a model neural network is established to identify the unknown continuous-time nonlinear system using input-output data. For each player, a critic neural network and an action neural network are used to approximate its value function and control policy, respectively. Our algorithm only needs to tune the weights of critic neural networks, so there will be less computational complexity during the learning process. All the neural network weights are updated online in real-time, continuously and synchronously. Furthermore, the uniform ultimate bounded stability of the closed-loop system is proved based on Lyapunov approach. Finally, two simulation examples are given to demonstrate the effectiveness of the developed scheme.

Constrained Consensus in Unbalanced Networks With Communication Delays

Abstract: In this note, a constrained consensus problem is studied for multi-agent systems in unbalanced networks in the presence of communication delays. Here each agent needs to lie in a closed convex constraint set while reaching a consensus. The communication graphs are directed, dynamically changing, and not necessarily balanced and only the union of the graphs is assumed to be strongly connected among each time interval of a certain bounded length. The analysis is performed based on an undelayed equivalent system that is composed of a linear main body and an error auxiliary. To tackle the loss of symmetry caused by unbalanced graphs and communication delays, a novel approach is proposed. The idea is to estimate the distance from each agent to the intersection set of all agents' constraint sets based on the properties of the projection on convex sets so as to show consensus convergence by contradiction. It is shown that the error auxiliary vanishes as time evolves and the linear main body converges to a vector with an exponential rate as a separate system. It is also shown that the communication delays do not affect the consensus stability and constrained consensus is reached even if the communication delays are arbitrarily bounded. Finally, a numerical example is included to illustrate the obtained theoretical results.

Balancing Covariates via Propensity Score Weighting

Abstract: Covariate balance is crucial for unconfounded descriptive or causal comparisons. However, lack of balance is common in observational studies. This article considers weighting strategies for balancing covariates. We define a general class of weights---the balancing weights---that balance the weighted distributions of the covariates between treatment groups. These weights incorporate the propensity score to weight each group to an analyst-selected target population. This class unifies existing weighting methods, including commonly used weights such as inverse-probability weights as special cases. General large-sample results on nonparametric estimation based on these weights are derived. We further propose a new weighting scheme, the overlap weights, in which each unit's weight is proportional to the probability of that unit being assigned to the opposite group. The overlap weights are bounded, and minimize the asymptotic variance of the weighted average treatment effect among the class of balancing weights. The overlap weights also possess a desirable small-sample exact balance property, based on which we propose a new method that achieves exact balance for means of any selected set of covariates. Two applications illustrate these methods and compare them with other approaches.

Data-Driven Neuro-Optimal Temperature Control of Water–Gas Shift Reaction Using Stable Iterative Adaptive Dynamic Programming

Abstract: In this paper, a novel data-driven stable iterative adaptive dynamic programming (ADP) algorithm is developed to solve optimal temperature control problems for water–gas shift (WGS) reaction systems. According to the system data, neural networks (NNs) are used to construct the dynamics of the WGS system and solve the reference control, respectively, where the mathematical model of the WGS system is unnecessary. Considering the reconstruction errors of NNs and the disturbances of the system and control input, a new stable iterative ADP algorithm is developed to obtain the optimal control law. The convergence property is developed to guarantee that the iterative performance index function converges to a finite neighborhood of the optimal performance index function. The stability property is developed to guarantee that each of the iterative control laws can make the tracking error uniformly ultimately bounded (UUB). NNs are developed to implement the stable iterative ADP algorithm. Finally, numerical results are given to illustrate the effectiveness of the developed method.

KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation

Abstract: We prove the existence of small amplitude quasi-periodic solutions for quasi-linear and fully nonlinear forced perturbations of the linear Airy equation. For Hamiltonian or reversible nonlinearities we also prove their linear stability. The key analysis concerns the reducibility of the linearized operator at an approximate solution, which provides a sharp asymptotic expansion of its eigenvalues. For quasi-linear perturbations this cannot be directly obtained by a KAM iteration. Hence we first perform a regularization procedure, which conjugates the linearized operator to an operator with constant coefficients plus a bounded remainder. These transformations are obtained by changes of variables induced by diffeomorphisms of the torus and pseudo-differential operators. At this point we implement a Nash–Moser iteration (with second order Melnikov non-resonance conditions) which completes the reduction to constant coefficients.

Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces

Abstract: In this paper, the fully parabolic Keller-Segel system \begin{equation}\label{problemAbstract}\left\{\begin{array}{ll} u_t=\Delta u- abla\cdot(u abla v), &(x,t)\in \Omega\times (0,T),\\ v_t=\Delta v-v+u, &(x,t)\in\Omega\times (0,T),\\ \end{array}\right.\tag{$\star$} \end{equation} is considered under Neumann boundary conditions in a bounded domain $\Omega\subset\mathbb{R}^n$ with smooth boundary, where $n\ge 2$. We derive a smallness condition on the initial data in optimal Lebesgue spaces which ensure global boundedness and large time convergence. More precisely, we shall show that one can find $\varepsilon_0>0$ such that for all suitably regular initial data $(u_0,v_0)$ satisfying $\|u_0\|_{L^{\frac{n}{2}}(\Omega)}<\varepsilon_0$ and $\| abla v_0\|_{L^{n}(\Omega)}<\varepsilon_0$, the above problem possesses a global classical solution which is bounded and converges to the constant steady state $(m,m)$ with $m:=\frac{1}{|\Omega|}\int_\Omega u_0$. Our approach allows us to furthermore study a general chemotaxis system with rotational sensitivity in dimension 2, which is lacking the natural energy structure associated with ($\star$). For such systems, we prove a global existence and boundedness result under corresponding smallness conditions on the initially present total mass of cells and the chemical gradient.

Adaptive neural tracking control for a class of stochastic nonlinear systems

Abstract: SUMMARY This paper investigates the problem of adaptive neural control design for a class of single-input single-output strict-feedback stochastic nonlinear systems whose output is an known linear function. The radial basis function neural networks are used to approximate the nonlinearities, and adaptive backstepping technique is employed to construct controllers. It is shown that the proposed controller ensures that all signals of the closed-loop system remain bounded in probability, and the tracking error converges to an arbitrarily small neighborhood around the origin in the sense of mean quartic value. The salient property of the proposed scheme is that only one adaptive parameter is needed to be tuned online. So, the computational burden is considerably alleviated. Finally, two numerical examples are used to demonstrate the effectiveness of the proposed approach. Copyright © 2012 John Wiley & Sons, Ltd.

On the uniform rectifiability of AD-regular measures with bounded Riesz transform operator: the case of codimension 1

Abstract: We prove that if μ is a d-dimensional Ahlfors-David regular measure in \({\mathbb{R}^{d+1}}\) , then the boundedness of the d-dimensional Riesz transform in L2(μ) implies that the non-BAUP David–Semmes cells form a Carleson family. Combined with earlier results of David and Semmes, this yields the uniform rectifiability of μ.