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关键词： Laplacian   Robin boundary conditions   eigenvalues   Polya's conjecture  

摘要： We investigate the question of whether the eigenvalues of the Laplacian with Robin boundary conditions can satisfy inequalities of the same type as those in Polya's conjecture for the Dirichlet and Neumann Laplacians and, if so, what form these inequalities should take. Motivated in part by Polya's original approach and in part by recent analogous works treating the Dirichlet and Neumann Laplacians, we consider rectangles and unions of rectangles and show that for these two families of domains, for any fixed positive value alpha of the boundary parameter, Polya-type inequalities do indeed hold, albeit with an exponent smaller than that of the corresponding Weyl asympotics for a fixed domain. We determine the optimal exponents in both cases, showing that they are different in the two situations. Our approach to proving these results includes a characterization of the corresponding extremal domains for the kth eigenvalue in regions of the (k, alpha)-plane, which in turn supports recent conjectures on the nature of the extrema among all bounded domains.

关键词： 量子力学  

ISBN: (纸本)9787564197025

摘要： 本书从量子光学的基本理论出发，介绍了原子与光子相互作用的基本原理和处理方法，分析了不同空间模式、不同统计性质的光子对量子信息处理的影响。

关键词： Quantum pseudodot   Thermodynamic properties   Spin orbit interaction  

摘要： We investigate the effect of spin orbit interaction (SOI) on thermodynamics quantities of a quantum pseudodot. The energy levels have been derived and thermal properties are evaluated using the Tsallis formalism. Compared to BG, Tsallis formalism consists in evaluating the thermodynamic quantities only on the accessible states. The results show that spin orbit interaction (SOI) has a great effect on entropy since it increases the number of accessible states causing the splitting on internal energy. Our results also show that chemical potential influences the spin-states in a quantum pseudodot system and creates splitting of some thermodynamic quantities. SOI and chemical potential are prominent parameters to modulate the spin alignment and perform the thermodynamics of electronic devices.

关键词： Fractional Laplacian   Schrodinger equation   gauge  

摘要： Let Omega subset of R-n be an open set, where n >= 2. Suppose omega is a locally finite Borelmeasure on Omega. For alpha is an element of (0, 2), define the fractional Laplacian (-Delta)(alpha/2) via the Fourier transform on R-n, and let G be the corresponding Green's operator of order alpha on Omega. Define T (u) = G(u omega). If parallel to T parallel to(L2(omega)-> L2(omega)) < 1, we obtain a representation for the unique weak solution u in the homogeneous Sobolev space L-0(alpha/2,2) (Omega) of (-Delta)(alpha/2) u = u omega + nu on Omega, u = 0 on Omega(c), for nu in the dual Sobolev space L-(alpha/2,2)(Omega). If Omega is a bounded C-1,C-1 domain, this representation yields matching exponential upper and lower pointwise estimates for the solution when nu = chi(Omega). These estimates are used to study the existence of a solution u(1) (called the "gauge") of the integral equation u(1) = 1 + G(u(1)omega) corresponding to the problem (-Delta)(alpha/2)u = u omega on Omega, u >= 0 on Omega, u = 1 on Omega(c). We show that if parallel to T parallel to < 1, then u(1) always exists if 0 < alpha < 1. For 1 <= alpha < 2, a solution exists if the norm of T is sufficiently small. We also show that the condition parallel to T parallel to < 1 does not imply the existence of a solution if 1 < alpha < 2. The condition parallel to T parallel to <= 1 is necessary for the existence of u(1) for all 0 < alpha = 2.

关键词： Dimensionality reduction   Clustering methods   Clustering algorithms   Principal component analysis   Convergence   Numerical models   Symmetric matrices   Constrained clustering   convergence   convex relaxation   dissimilarity propagation (DP)   Karush-Kuhn-Tucker (KKT)  

摘要： In this article, we propose a novel model for constrained clustering, namely, the dissimilarity propagation-guided graph-Laplacian principal component analysis (DP-GLPCA). By fully utilizing a limited number of weakly supervisory information in the form of pairwise constraints, the proposed DP-GLPCA is capable of capturing both the local and global structures of input samples to exploit their characteristics for excellent clustering. More specifically, we first formulate a convex semisupervised low-dimensional embedding model by incorporating a new dissimilarity regularizer into GLPCA (i.e., an unsupervised dimensionality reduction model), in which both the similarity and dissimilarity between low-dimensional representations are enforced with the constraints to improve their discriminability. An efficient iterative algorithm based on the inexact augmented Lagrange multiplier is designed to solve it with the global convergence guaranteed. Furthermore, we innovatively propose to propagate the cannot-link constraints (i.e., dissimilarity) to refine the dissimilarity regularizer to be more informative. The resulting DP model is iteratively solved, and we also prove that it can converge to a Karush-Kuhn-Tucker point. Extensive experimental results over nine commonly used benchmark data sets show that the proposed DP-GLPCA can produce much higher clustering accuracy than state-of-the-art constrained clustering methods. Besides, the effectiveness and advantage of the proposed DP model are experimentally verified. To the best of our knowledge, it is the first time to investigate DP, which is contrast to existing pairwise constraint propagation that propagates similarity. The code is publicly available at https://***/jyh-learning/DP-GLPCA.

关键词： Laplacian solver   Distributed algorithm   Queueing networks   Random walks   Random spanning trees  

摘要： We use queueing networks to present a new approach to solving Laplacian systems. This marks a significant departure from the existing techniques, mostly based on graph-theoretic constructions and sampling. Our distributed solver works for a large and important class of Laplacian systems that we call "one-sink" Laplacian systems. Specifically, our solver can produce solutions for systems of the form Lx = b where exactly one of the coordinates of b is negative. Our solver is a distributed algorithm that takes (O) over tilde (t(hit) (d) over cap (max)) time (where (O) over tilde hides poly log n factors) to produce an approximate solution where t(hit) is the worst-case hitting time of the random walk on the graph, which is Theta(n) for a large set of important graphs, and (d) over cap (max) is the maximum degree of the graph. The class of one-sink Laplacians includes the important voltage computation problem and allows us to compute the effective resistance between nodes in a distributed setting. As a result, our Laplacian solver can be used to adapt the approach by Kelner and Madry (2009) to give the first distributed algorithm to compute approximate random spanning trees efficiently.

关键词： 35B27   35P05   47A10   band   Bethe–Sommerfeld hypothesis   Laplacian   oscillating boundary  

摘要： In this paper, we consider the Laplace operator in a flat band whose lower boundary periodically oscillates under the Dirichlet boundary condition. The period and the amplitude of oscillations are two independent small parameters. The main result obtained in the paper is the absence of internal lacunas in the lower part of the spectrum of the operator for sufficiently small period and amplitude. We obtain explicit upper estimates of the period and amplitude in the form of constraints with specific numerical constants. The length of the lower part of the spectrum, in which the absence of lacunas is guaranteed, is also expressed explicitly in terms of the period function and the amplitude. © 2021, Springer Science+Business Media, LLC, part of Springer Nature.

关键词： Electroencephalogram(EEG)   Cross task   Visual cognitive load   Bidirectional Long Short-Term Memory (BLSTM)  

摘要： Cognitive workload assessment is important for optimizing multitasking abilities in the current information-rich world. A few good research studies assess cross task classification where a model trained for one task can efficiently handle other tasks. Building a reliable model for the estimation of cross task cognitive workload using electroencephalogram (EEG) signal is challenging as brain dynamics differ for various activities. The present study explores the wide visual aspects for inducing two levels of cognitive workload in the two self-design tasks. The present study addresses temporal dynamics of EEG signal using time windowing, time segment smoothing, and formation of variable duration time frames. The present study computes features from the statistical, morphological and non-linear domain for variable duration segments considering four clinical sub-bands. The neighbourhood component feature selection algorithm identifies the topmost common features of both tasks. Binary classification of cross task utilizes deep structure employing bidirectional long short-term memory (BLSTM) and LSTM. The result shows that the deep recurrent neural network (RNN) achieves the highest classification accuracy of 92.8% for the cross task in comparison to the traditional classifiers. The deep RNN also outperforms the existing literature studies for cross task classification. Result also reveals a significant improvement in classification accuracy with the usage of handcrafted features when supported by proper modelling of temporal dynamics of EEG signal. The present study exploits the capability of deep recurrent structure in handling long term dependencies of non-stationary signal and proves to be very efficient.

关键词： Graded mesh   Finite element method   Matrix transfer technique   Irregular domains  

摘要： In this paper, we propose a novel numerical technique to 2D multi-term time and space fractional Bloch-Torrey equations defined on an irregular convex domain. First, we consider the problem with space integral Laplacian operator. We present the semi discrete and fully-discrete schemes by using the L1 formula on a temporal graded mesh and an unstructured-mesh Galerkin finite element method (FEM) based on the matrix transfer technique (MTT). Moreover, the unconditional stability and convergence of the schemes are discussed and theoretically proved. Then we extend the technique and develop a numerical scheme for the case of space fractional Laplacian operator. In addition, the recent L1+ formula is utilized to improving the temporal accuracy. Finally, three numerical examples in rectangular, elliptical and human brain-like irregular domains are given to demonstrate the accuracy and efficiency of the proposed numerical scheme. (C) 2021 Elsevier B.V. All rights reserved.

关键词： Quantum phase transitions   Out-of-equilibrium quantum dynamics   Dissipative mechanisms   Dynamic scaling at quantum transitions  

摘要： The many-body physics at quantum phase transitions shows a subtle interplay between quantum and thermal fluctuations, emerging in the low-temperature limit. In this review, we first give a pedagogical introduction to the equilibrium behavior of systems in that context, whose scaling framework is essentially developed by exploiting the quantum-to-classical mapping and the renormalization-group theory of critical phenomena at continuous phase transitions. Then we specialize to protocols entailing the out-of-equilibrium quantum dynamics, such as instantaneous quenches and slow passages across quantum transitions. These are mostly discussed within dynamic scaling frameworks, obtained by appropriately extending the equilibrium scaling laws. We review phenomena at first-order quantum transitions as well, whose peculiar scaling behaviors are characterized by an extreme sensitivity to the boundary conditions, giving rise to exponentials or power laws for the same bulk system. In the last part, we cover aspects related to the effects of dissipative interactions with an environment, through suitable generalizations of the dynamic scaling at quantum transitions. The presentation is limited to issues related to, and controlled by, the quantum transition developed by closed many-body systems, treating the dissipation as a perturbation of the critical regimes, as for the temperature at the zero-temperature quantum transition. We focus on the physical conditions giving rise to a nontrivial interplay between critical modes and various dissipative mechanisms, generally realized when the involved mechanism excites only the low-energy modes of the quantum transitions. (C) 2021 Elsevier B.V. All rights reserved.