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In the present paper we deal with the existence of multiple solutions for a quasilinear elliptic problem involving an arbitrary perturbation. Our approach, based on an abstract result of Ricceri, combines truncation arguments with Moser-type iteration technique.

The authors prove the long time stability of KAM tori (thus quasi-periodic solutions) for nonlinear Schrödinger equation \\sqrt{-1}\\, u_{t}=u_{xx}-M_{\\xi}u+\\varepsilon|u|^2u, subject to Dirichlet boundary conditions u(t,0)=u(t,\\pi)=0, where M_{\\xi} is a real Fourier multiplier. More precisely, they show that, for a typical Fourier multiplier M_{\\xi}, any solution with the initial datum in the \\delta-neighborhood of a KAM torus still stays in the 2\\delta-neighborhood of the KAM torus for a polynomial long time such as |t|\\leq \\delta^{-\\mathcal{M}} for any given \\mathcal M with 0\\leq \\mathcal{M}\\leq C(\\varepsilon), where C(\\varepsilon) is a constant depending on \\varepsilon and C(\\varepsilon)\\rightarrow\\infty as \\varepsilon\\rightarrow0.

A chaotic map, which is realized on finite precision device, such as computer, will suffer dynamical degradation. Such chaotic maps cannot be regarded as rigorous chaos anymore, since their chaotic characteristics are degraded, and naturally, these kinds of chaotic maps are not secure enough for cryptographic use. Therefore, in this paper, a novel perturbation method is proposed to reduce the dynamical degradation of digital chaotic maps. Once the state is repeated during the iteration, the parameter and state are both perturbed to make the state jump out from a cycle. This method is convenient to implement without any external sources and can be used for different kinds of digital chaotic maps. The most widely used logistic map is used as an example to prove the effectiveness of this method. Several numerical experiments are provided to prove the effectiveness of this method. Under the same precision, the number of iterations when entering a cycle and the period of the improved map are greater than those of the original one. The complexity analysis shows that the improved map can get an ideal complexity level under a lower precision. All these results prove that this perturbed method can greatly improve the dynamical characteristics of original chaotic map and is competitive with other remedies. Furthermore, we improve this method by using a variable perturbation, where the perturbation is affected according to the number of iteration steps. Numerical experiments further prove that this improved perturbation method has a better performance in suppressing dynamical degradation.

This paper is concerned with the interplay between statistical asymmetry and spectral methods. Suppose we are interested in estimating a rank-1 and symmetric matrix M* ∈ ℝ n×n , yet only a randomly perturbed version M is observed. The noise matrix M − M* is composed of independent (but not necessarily homoscedastic) entries and is, therefore, not symmetric in general. This might arise if, for example, when we have two independent samples for each entry of M* and arrange them in an asymmetric fashion. The aim is to estimate the leading eigenvalue and the leading eigenvector of M*. We demonstrate that the leading eigenvalue of the data matrix M can be O(√n) times more accurate (up to some log factor) than its (unadjusted) leading singular value of M in eigenvalue estimation. Moreover, the eigendecomposition approach is fully adaptive to heteroscedasticity of noise, without the need of any prior knowledge about the noise distributions. In a nutshell, this curious phenomenon arises since the statistical asymmetry automatically mitigates the bias of the eigenvalue approach, thus eliminating the need of careful bias correction. Additionally, we develop appealing nonasymptotic eigenvector perturbation bounds; in particular, we are able to bound the perturbation of any linear function of the leading eigenvector of M (e.g., entrywise eigenvector perturbation). We also provide partial theory for the more general rank-r case. The takeaway message is this: arranging the data samples in an asymmetric manner and performing eigendecomposition could sometimes be quite beneficial.

Recovering low-rank structures via eigenvector perturbation analysis is a common problem in statistical machine learning, such as in factor analysis, community detection, ranking, matrix completion, among others. While a large variety of bounds are available for average errors between empirical and population statistics of eigenvectors, few results are tight for entrywise analyses, which are critical for a number of problems such as community detection. This paper investigates entrywise behaviors of eigenvectors for a large class of random matrices whose expectations are low rank, which helps settle the conjecture in Abbe, Bandeira and Hall (2014) that the spectral algorithm achieves exact recovery in the stochastic block model without any trimming or cleaning steps. The key is a first-order approximation of eigenvectors under the ℓ ∞ norm: u k ≈ A u k * λ k * , where {uk } and { u k * } are eigenvectors of a random matrix A and its expectation 𝔼A, respectively. The fact that the approximation is both tight and linear in A facilitates sharp comparisons between uk and u k * . In particular, it allows for comparing the signs of uk and u k * even if ‖ u k − u k * ‖ ∞ is large. The results are further extended to perturbations of eigenspaces, yielding new ℓ ∞- type bounds for synchronization (ℤ₂-spikedWigner model) and noisy matrix completion.

Matter evolved under the influence of gravity from minuscule density fluctuations. Nonperturbative structure formed hierarchically over all scales and developed non-Gaussian features in the Universe, known as the cosmic web. To fully understand the structure formation of the Universe is one of the holy grails of modern astrophysics. Astrophysicists survey large volumes of the Universe and use a large ensemble of computer simulations to compare with the observed data to extract the full information of our own Universe. However, to evolve billions of particles over billions of years, even with the simplest physics, is a daunting task. We build a deep neural network, the Deep Density Displacement Model (D³M), which learns from a set of prerun numerical simulations, to predict the nonlinear large-scale structure of the Universe with the Zel’dovich Approximation (ZA), an analytical approximation based on perturbation theory, as the input. Our extensive analysis demonstrates that D³M outperforms the second-order perturbation theory (2LPT), the commonly used fast-approximate simulation method, in predicting cosmic structure in the nonlinear regime. We also show that D³M is able to accurately extrapolate far beyond its training data and predict structure formation for significantly different cosmological parameters. Our study proves that deep learning is a practical and accurate alternative to approximate 3D simulations of the gravitational structure formation of the Universe.

We study singular perturbations of optimal stochastic control problems and differential games arising in the dimension reduction of system with multiple time scales. We analyze the uniform convergence of the value functions via the associated Hamilton-Jacobi-Bellman-Isaacs equations, in the framework of viscosity solutions. The crucial properties of ergodicity and stabilization to a constant that the Hamiltonian must possess are formulated as differential games with ergodic cost criteria. They are studied under various different assumptions and with PDE as well as control-theoretic methods. We construct also an explicit example where the convergence is not uniform. Finally we give some applications to the periodic homogenization of Hamilton-Jacobi equations with non-coercive Hamiltonian and of some degenerate parabolic PDEs.

Recent advances in multiplexed single‐cell transcriptomics experiments facilitate the high‐throughput study of drug and genetic perturbations. However, an exhaustive exploration of the combinatorial perturbation space is experimentally unfeasible. Therefore, computational methods are needed to predict, interpret, and prioritize perturbations. Here, we present the compositional perturbation autoencoder (CPA), which combines the interpretability of linear models with the flexibility of deep‐learning approaches for single‐cell response modeling. CPA learns to in silico predict transcriptional perturbation response at the single‐cell level for unseen dosages, cell types, time points, and species. Using newly generated single‐cell drug combination data, we validate that CPA can predict unseen drug combinations while outperforming baseline models. Additionally, the architecture's modularity enables incorporating the chemical representation of the drugs, allowing the prediction of cellular response to completely unseen drugs. Furthermore, CPA is also applicable to genetic combinatorial screens. We demonstrate this by imputing in silico 5,329 missing combinations (97.6% of all possibilities) in a single‐cell Perturb‐seq experiment with diverse genetic interactions. We envision CPA will facilitate efficient experimental design and hypothesis generation by enabling in silico response prediction at the single‐cell level and thus accelerate therapeutic applications using single‐cell technologies. Synopsis The compositional perturbation autoencoder (CPA) is a deep learning model for predicting the transcriptomic responses of single cells to single or combinatorial treatments from drugs and genetic manipulations. CPA can be trained on highly multiplexed, single‐cell experiments with thousands of conditions to predict unmeasured phenotypes (e.g., specific dose responses). It can generalize to predict responses to small molecules never seen in the training by adding priors on chemical space. Validations using a newly generated combinatorial drug perturbation dataset demonstrate the accuracy of CPA in predicting unseen drug combinations. CPA is also applicable to genetic combinatorial screens, as shown by imputing in silico 5,329 missing combinations in a single‐cell perturb‐seq experiment with diverse genetic interactions. Graphical Abstract The compositional perturbation autoencoder (CPA) is a deep learning model for predicting the transcriptomic responses of single cells to single or combinatorial treatments from drugs and genetic manipulations.

Land conversion, climate change and species invasions are contributing to the widespread emergence of novel ecosystems, which demand a shift in how we think about traditional approaches to conservation, restoration and environmental management. They are novel because they exist without historical precedents and are self-sustaining. Traditional approaches emphasizing native species and historical continuity are challenged by novel ecosystems that deliver critical ecosystems services or are simply immune to practical restorative efforts. Some fear that, by raising the issue of novel ecosystems, we are simply paving the way for a more laissez-faire attitude to conservation and restoration. Regardless of the range of views and perceptions about novel ecosystems, their existence is becoming ever more obvious and prevalent in today's rapidly changing world. In this first comprehensive volume to look at the ecological, social, cultural, ethical and policy dimensions of novel ecosystems, the authors argue these altered systems are overdue for careful analysis and that we need to figure out how to intervene in them responsibly. This book brings together researchers from a range of disciplines together with practitioners and policy makers to explore the questions surrounding novel ecosystems. It includes chapters on key concepts and methodologies for deciding when and how to intervene in systems, as well as a rich collection of case studies and perspective pieces. It will be a valuable resource for researchers, managers and policy makers interested in the question of how humanity manages and restores ecosystems in a rapidly changing world. A companion website with additional resources is available at www.wiley.com/go/hobbs/ecosystems [http://www.wiley.com/go/hobbs/ecosystems]

Purpose This study aims that very lately, Mohand transform is introduced to solve the ordinary and partial differential equations (PDEs). In this paper, the authors modify this transformation and associate it with a further analytical method called homotopy perturbation method (HPM) for the fractional view of Newell–Whitehead–Segel equation (NWSE). As Mohand transform is restricted to linear obstacles only, as a consequence, HPM is used to crack the nonlinear terms arising in the illustrated problems. The fractional derivatives are taken into the Caputo sense. Design/methodology/approach The specific objective of this study is to examine the problem which performs an efficient role in the form of stripe orders of two dimensional systems. The authors achieve the multiple behaviors and properties of fractional NWSE with different positive integers. Findings The main finding of this paper is to analyze the fractional view of NWSE. The obtain results perform very good in agreement with exact solution. The authors show that this strategy is absolutely very easy and smooth and have no assumption for the constriction of this approach. Research limitations/implications This paper invokes these two main inspirations: first, Mohand transform is associated with HPM, secondly, fractional view of NWSE with different positive integers. Practical implications In this paper, the graph of approximate solution has the excellent promise with the graphs of exact solutions. Social implications This paper presents valuable technique for handling the fractional PDEs without involving any restrictions or hypothesis. Originality/value The authors discuss the fractional view of NWSE by a Mohand transform. The work of the present paper is original and advanced. Significantly, to the best of the authors’ knowledge, no such work has yet been published in the literature.