"This book is concerned with the application of methods from dynamical systems and bifurcation theories to the study of nonlinear oscillations. Chapter 1 provides a review of basic results in the theory of dynami...
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ISBN:
(数字)9781461211402
ISBN:
(纸本)9780387908199;9781461270201
"This book is concerned with the application of methods from dynamical systems and bifurcation theories to the study of nonlinear oscillations. Chapter 1 provides a review of basic results in the theory of dynamical systems, covering both ordinary differential equations and discrete mappings. Chapter 2 presents 4 examples from nonlinear oscillations. Chapter 3 contains a discussion of the methods of local bifurcation theory for flows and maps, including center manifolds and normal forms. Chapter 4 develops analytical methods of averaging and perturbation theory. Close analysis of geometrically defined two-dimensional maps with complicated invariant sets is discussed in chapter 5. Chapter 6 covers global homoclinic and heteroclinic bifurcations. The final chapter shows how the global bifurcations reappear in degenerate local bifurcations and ends with several more models of physical problems which display these behaviors." #;#1 "An attempt to make research tools concerning `strange attractors' developed in the last 20 years available to applied scientists and to make clear to research mathematicians the needs in applied works. Emphasis on geometric and topological solutions of differential equations. Applications mainly drawn from nonlinear oscillations." #;#2
In this paper, we present results on ordered representations of data in which different dimensions have different degrees of importance. To learn these representations we introduce nested dropout, a procedure for stoc...
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ISBN:
(纸本)9781634393973
In this paper, we present results on ordered representations of data in which different dimensions have different degrees of importance. To learn these representations we introduce nested dropout, a procedure for stochastically removing coherent nested sets of hidden units in a neural network. We first present a sequence of theoretical results for the special case of a semi-linear autoencoder. We rigorously show that the application of nested dropout enforces identifiability of the units, which leads to an exact equivalence with PCA. We then extend the algorithm to deep models and demonstrate the relevance of ordered representations to a number of applications. Specifically, we use the ordered property of the learned codes to construct hash-based data structures that permit very fast retrieval, achieving retrieval in time logarithmic in the database size and independent of the dimensionality of the representation. This allows codes that are hundreds of times longer than currently feasible for retrieval. We therefore avoid the diminished quality associated with short codes, while still performing retrieval that is competitive in speed with existing methods. We also show that ordered representations are a promising way to learn adaptive compression for efficient online data reconstruction. Copyright 2014 by the author(s).
In this paper, we propose a new algorithm for solving the split common fixed point problem for infinite families of demicontractive mappings. Strong convergence of the proposed method is established under suitable con...
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Modeling the transformation of biomass into biogas is complex, because it involves a nonlinear and coupled set of ordinary differential equations. Thus, obtaining an analytical-numerical solution becomes attractive fo...
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Photonic devices rarely provide both elaborate spatial control and sharp spectral control over an incoming *** optical metasurfaces,for example,the localized modes of individual meta-units govern the wavefront shape o...
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Photonic devices rarely provide both elaborate spatial control and sharp spectral control over an incoming *** optical metasurfaces,for example,the localized modes of individual meta-units govern the wavefront shape over a broad bandwidth,while nonlocal lattice modes extended over many unit cells support high quality-factor ***,we experimentally demonstrate nonlocal dielectric metasurfaces in the near-infrared that offer both spatial and spectral control of light,realizing metalenses focusing light exclusively over a narrowband resonance while leaving off-resonant frequencies *** devices attain this functionality by supporting a quasi-bound state in the continuum encoded with a spatially varying geometric *** leverage this capability to experimentally realize a versatile platform for multispectral wavefront shaping where a stack of metasurfaces,each supporting multiple independently controlled quasi-bound states in the continuum,molds the optical wavefront distinctively at multiple wavelengths and yet stay transparent over the rest of the *** a platform is scalable to the visible for applications in augmented reality and transparent displays.
作者：
Baras, John S.Applied Mathematics
Statistics and Scientific Computation Program Institute for Systems Research University of Maryland College Park United States
Networked systems are ubiquitous. A taxonomy of networked systems includes infrastructure and communication networks, social and economic networks, biological networks and biological swarms, robotic swarms, and severa...
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Unsupervised methods for dimensionality reduction of neural activity and behavior have provided unprecedented insights into the underpinnings of neural information processing. One popular approach involves the recurre...
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作者：
Kim, JaeukTorquato, SalvatorePrinceton Materials Institute
Department of Physics Department of Chemistry Princeton University PrincetonNJ08544 United States Department of Chemistry
Department of Physics Princeton Materials Institute Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States
Disordered stealthy hyperuniform (SHU) packings are an emerging class of exotic amorphous two-phase materials endowed with novel optical, transport, and mechanical properties. Such packings of identical spheres have b...
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Disordered stealthy hyperuniform (SHU) packings are an emerging class of exotic amorphous two-phase materials endowed with novel optical, transport, and mechanical properties. Such packings of identical spheres have been created from SHU ground-state point patterns via a modified collective-coordinate optimization scheme that includes a soft-core repulsion, besides the standard "stealthy" pair potential. To explore maximal ranges of the packing fraction , we investigate the distributions of minimum pair distances as well as nearest-neighbor distances of ensembles of SHU point patterns without and with soft-core repulsions in the first three space dimensions as a function of the stealthiness parameter χ and number of particles N within a hypercubic simulation box under periodic boundary conditions. Within the disordered regime (χ max(χ, d), decrease to zero on average as N increases if there are no soft-core repulsions. By contrast, the inclusion of soft-core repulsions results in very large max(χ, d) independent of N, reaching up to max(χ, d) = 1.0, 0.86, 0.63 in the zero-χ limit and decreasing to max(χ, d) = 1.0, 0.67, 0.47 at χ = 0.45 for d = 1, 2, 3, respectively. We obtain explicit formulas for max(χ, d) as functions of χ and N for a given value of d in both cases with and without soft-core repulsions. In two and three dimensions, our soft-core SHU ground-state packings for small χ become configurationally very close to the corresponding jammed hard-particle packings created by fast compression algorithms, as measured by their pair statistics. As χ increases beyond 0.20, the packings form fewer contacts and linear polymer-like chains as χ tends to 1/2. The resulting structure factors S(k) and pair correlation functions g2(r) reveal that soft-core repulsions significantly alter the short- and intermediate-range correlations in the SHU ground states. We show that the degree of large-scale order of the soft-core SHU ground states increases as χ increases from 0 to
In earlier papers, 2π-periodic spectral data windows have been used in spectral estimation of discrete-time random fields having finite second-order moments. In this paper, we show that 2π-periodic spectral windows ...
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In earlier papers, 2π-periodic spectral data windows have been used in spectral estimation of discrete-time random fields having finite second-order moments. In this paper, we show that 2π-periodic spectral windows can also be used to construct estimates of the spectral density of a homoge-neous symmetric α-stable discrete-time random field. These fields do not have second-order moments if 0 < α < 2. We construct an estimate of the spectrum, calculate the asymptotic mean and variance, and prove weak consistency of our estimate.
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