Collapsing of connected sums and the eigenvalues of the Laplacian

We study the behavior of the eigenvalues of the Laplacian acting on functions when one side of a connected sum of two closed Riemannian manifolds collapses to a point. We prove that the eigenvalues converge to those of the limit space, by using the method of Anné and Colbois. From this, we obtain a gluing theorem for the eigenvalues.     

On the distributions of Laplacian eigenvalues versus node degrees in complex networks

In this paper, the important issue of Laplacian eigenvalue distributions is investigated through theory-guided extensive numerical simulations, for four typical complex network models, namely, the ER random-graph networks, WS and NW small-world networks, and BA scale-free networks. It is found that these four types of complex networks share some common features, particularly similarities between the Laplacian eigenvalue distributions and the node degree distributions.     

The Integrated Density of States of the Random Graph Laplacian

We analyse the density of states of the random graph Laplacian in the percolating regime. A symmetry argument and knowledge of the density of states in the nonpercolating regime allows us to isolate the density of states of the percolating cluster (DSPC) alone, thereby eliminating trivially localised states due to finite subgraphs. We derive a nonlinear integral equation for the integrated DSPC and solve it with a population dynamics algorithm. We discuss the possible existence of a mobility edge and give strong evidence for the existence of discrete eigenvalues in the whole range of the spectrum.     

Random walk of polarized light in turbid media

We study the propagation of polarized light in turbid media as a random walk of vector photons. Both propagation and polarization directions of light are found to isotropize, following a power law of the number of scattering events. The characteristic length scale governing light isotropization and linear depolarization, the isotropization length , is derived using the exact Mie scattering for spherical particles. A simple relation is obtained for Rayleigh-Gans scatterers where is the transport mean free path and is the mean cosine of scattering angles.     

On an elaboration of M. Kac's theorem concerning eigenvalues of the Laplacian in a region with randomly distributed small obstacles

We removem-balls of centersw ₁,...,w _m with the same radius α/m from a bounded domain Ω inR ³ with smooth boundary γ. Let μ _k (α/m;w(m)) denote thek-th eigenvalue of the Laplacian in Ω/m-balls under the Dirichlet condition. We consider μ _k (α/m;w(m)) as a random variable on a probability space (w ₁,...,w _m)∈Ω × ... × Ω and we examine a precise behaviour of μ _k (α/m;w(m)) asm → ∞. We give an elaboration of. M. Kac's theorem.     

Improving Network Representation Learning via Dynamic Random Walk,Self-Attention and Vertex Attributes-Driven Laplacian Space Optimization

Network data analysis is a crucial method for mining complicated object interactions. In recent years, random walk and neural-language-model-based network representation learning (NRL) approaches have been widely used for network data analysis. However, these NRL approaches suffer from the following deficiencies: firstly, because the random walk procedure is based on symmetric node similarity and fixed probability distribution, the sampled vertices’ sequences may lose local community structure information; secondly, because the feature extraction capacity of the shallow neural language model is limited, they can only extract the local structural features of networks; and thirdly, these approaches require specially designed mechanisms for different downstream tasks to integrate vertex attributes of various types. We conducted an in-depth investigation to address the aforementioned issues and propose a novel general NRL framework called dynamic structure and vertex attribute fusion network embedding, which firstly defines an asymmetric similarity and h-hop dynamic random walk strategy to guide the random walk process to preserve the network’s local community structure in walked vertex sequences. Next, we train a self-attention-based sequence prediction model on the walked vertex sequences to simultaneously learn the vertices’ local and global structural features. Finally, we introduce an attributes-driven Laplacian space optimization to converge the process of structural feature extraction and attribute feature extraction. The proposed approach is exhaustively evaluated by means of node visualization and classification on multiple benchmark datasets, and achieves superior results compared to baseline approaches.     

Fractional Laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency power-law dependency

Frequency-dependent attenuation typically obeys an empirical power law with an exponent ranging from 0 to 2. The standard time-domain partial differential equation models can describe merely two extreme cases of frequency-independent and frequency-squared dependent attenuations. The otherwise nonzero and nonsquare frequency dependency occurring in many cases of practical interest is thus often called the anomalous attenuation. In this study, a linear integro-differential equation wave model was developed for the anomalous attenuation by using the space-fractional Laplacian operation, and the strategy is then extended to the nonlinear Burgers equation. A new definition of the fractional Laplacian is also introduced which naturally includes the boundary conditions and has inherent regularization to ease the hypersingularity in the conventional fractional Laplacian. Under the Szabo's smallness approximation, where attenuation is assumed to be much smaller than the wave number, the linear model is found consistent with arbitrary frequency power-law dependency.     

Stress-energy tensors and the Lichnerowicz Laplacian

To a critical point of a variational problem, we associate a divergence-free symmetric 2-tensor, called the stress-energy tensor. We calculate the Laplacian of this object as defined by Lichnerowicz. This has the property that it commutes with the divergence provided the Ricci curvature is covariantly constant. We deduce relations between different stress-energy tensors, discuss growth formulae and harmonic maps between spheres.     

Random walk particle tracking simulations of non-Fickian transport in heterogeneous media

Derivations of continuum nonlocal models of non-Fickian (anomalous) transport require assumptions that might limit their applicability. We present a particle-based algorithm, which obviates the need for many of these assumptions by allowing stochastic processes that represent spatial and temporal random increments to be correlated in space and time, be stationary or non-stationary, and to have arbitrary distributions. The approach treats a particle trajectory as a subordinated stochastic process that is described by a set of Langevin equations, which represent a continuous time random walk (CTRW). Convolution-based particle tracking (CBPT) is used to increase the computational efficiency and accuracy of these particle-based simulations. The combined CTRW–CBPT approach enables one to convert any particle tracking legacy code into a simulator capable of handling non-Fickian transport.     

Eigenfunctions and eigenvalues of the Wigner operator

It can be proved, that the Wigner operator, which results from the Quantum-Mechanical foundation of Bopp, accepts as eigenvalues the differences of the eigenvalues of two equivalent Schrödinger equations. The eigenfunctions result with the help of a Fourier transform in the phase space of the corresponding eigenfunctions of the Schrödinger equations.     

Quantum tunneling and negative eigenvalues

In the path-integral approach to the decay of a metastable state by quantum tunneling, the tunneling process is dominated by a solution to the imaginary-time equations of motion, called the bounce. In all known cases, the second variational derivative of the euclidean action at the bounce has one and only one negative eigenvalue. This note explains this phenomenon by showing it is an inevitable feature of the bounce for a wide class of systems. This class includes a set of particles interacting through potentials obeying some mild technical restrictions, and also theories of interacting scalar and gauge fields. There may exist solutions in other ways like bounces and which have more than one negative eigenvalue, but, even if they do exist, they have nothing to do with tunneling.     

Critical sets of eigenfunctions of the Laplacian

We give an upper bound for the (n−1)$(n-1)$-dimensional Hausdorff measure of the critical set of eigenfunctions of the Laplacian on compact analytic n$n$-dimensional Riemannian manifolds. This is the analog of a result on the nodal set of eigenfunctions by H. Donnelly and C. Fefferman.     

First and second derivatives of the eigenvalues of the hamiltonian

A concise general matrix formalism for deriving expressions for derivatives of eigenvalues of the hamiltonian is proposed. The formulae can be used for calculations of the electric and magnetic moments, polarizability, magnetic susceptibility, nuclear magnetic shielding, force constants, dipolemoment derivatives, etc. of different molecules, and also of other properties.     

Distribution of eigenvalues for the modular group

The two-point correlation functions of energy levels for free motion on the modular domain, both with periodic and Dirichlet boundary conditions, are explicitly computed using a generalization of the Hardy-Littlewood method. It is shown that in the limit of small separations they show an uncorrelated behaviour and agree with the Poisson distribution but they have prominent number-theoretical oscillations at larger scale. The results agree well with numerical simulations.     

Operator method and eigenvalues of the triple-well anharmonic oscillator

Using an algebraic technique based on the Lie algebra of the group SO(2, 1) the eigenvalues of the triple-well anharmonic oscillator are calculated for all ranges of the coupling constant. Approximate analytic formulas for the eigenvalues up to second order are given. The results obtained are compared with those of other authors.     

Analytical computation of the eigenvalues and eigenvectors in DT-MRI

In this paper a noniterative algorithm to be used for the analytical determination of the sorted eigenvalues and corresponding orthonormalized eigenvectors obtained by diffusion tensor magnetic resonance imaging (DT-MRI) is described. The algorithm uses the three invariants of the raw water spin self-diffusion tensor represented by a 3 x 3 positive definite matrix and certain math functions that do not require iteration. The implementation requires a positive definite mask to preserve the physical meaning of the eigenvalues. This algorithm can increase the speed of eigenvalue/eigenvector calculations by a factor of 5-40 over standard iterative Jacobi or singular-value decomposition techniques. This approach may accelerate the computation of eigenvalues, eigenvalue-dependent metrics, and eigenvectors especially when having high-resolution measurements with large numbers of slices and large fields of view.