Full Laplacian Research Articles

3D anisotropic Navier–Stokes equations in : stability and large-time behaviour

The study on the large-time behaviour of solutions to the 3D incompressible anisotropic Navier–Stokes (ANS) equations is very recent. Powerful tools designed for the Navier–Stokes equations with full Laplacian dissipation such as the Fourier splitting method no longer apply to the case when there is only horizontal dissipation. For the whole space , as , solutions of the ANS equations converge to the trivial solution and the convergence rate is algebraic. This paper is devoted to the case when the spatial domain Ω is . Our results reveal that the large-time behaviour for is quite different from that for . We show that any small initial velocity field leads to a unique global solution u that remains small in . More importantly, as , the velocity field u converges to a nontrivial steady state. The first two components of the steady state are given by the horizontal average of the first two components of u 0 while the third component vanishes. In addition, this convergence is exponentially fast.

Large-scale non-negative subspace clustering based on Nyström approximation

Large-scale subspace clustering usually drops the requirements of the full similarity matrix and Laplacian matrix but constructs the anchor affinity matrix and uses matrix approximation methods to reduce the clustering complexity. However, the existing anchor affinity matrix calculation methods only consider the global structure of data. Moreover, directly using the anchor affinity matrix to approximate the full Laplacian matrix cannot guarantee the best low-rank approximation, which affects the clustering results. To address these problems, this paper proposes a large-scale non-negative subspace clustering method based on Nyström approximation. Firstly, we modify the objective function of the anchor affinity matrix by adding the local structure term, taking into account both the local and global data characteristics for better affinity learning. Secondly, a matrix iterative update rule is derived to optimize the objective function according to the non-negative Lagrangian relaxation, and its rationality and convergence are proved. Finally, two effective Laplacian matrix decomposition methods based on Nyström approximation are designed to obtain more accurate eigenvectors to improve the clustering quality. The proposed algorithms are tested on various benchmark datasets. The experimental results show that our methods have competitive performance compared with state-of-the-art large-scale clustering algorithms.

NonUniqueness in Law for Two-Dimensional Navier--Stokes Equations with Diffusion Weaker than a Full Laplacian

We study the two-dimensional Navier-Stokes equations forced by random noise with a diffusive term generalized via a fractional Laplacian that has a positive exponent strictly less than one. Because intermittent jets are inherently three-dimensional, we instead adapt the theory of intermittent form of the two-dimensional stationary flows to the stochastic approach presented by Hofmanov$\acute{\mathrm{a}}$, Zhu $\&$ Zhu (2019, arXiv:1912.11841 [math.PR]) and prove its non-uniqueness in law.

Non-uniqueness in law for the Boussinesq system forced by random noise

Non-uniqueness in law for three-dimensional Navier-Stokes equations forced by random noise was established recently in Hofmanov$\acute{\mathrm{a}}$ et al. (2019, arXiv:1912.11841 [math.PR]). The purpose of this work is to prove non-uniqueness in law for the Boussinesq system forced by random noise. Diffusion within the equation of its temperature scalar field has a full Laplacian and the temperature scalar field can be initially smooth.

Deep learning super-diffusion in multiplex networks

Complex network theory has shown success in understanding the emergent and collective behavior of complex systems Newman 2010 Networks: An Introduction (Oxford: Oxford University Press). Many real-world complex systems were recently discovered to be more accurately modeled as multiplex networks Bianconi 2018 Multilayer Networks: Structure and Function (Oxford: Oxford University Press); Boccaletti et al 2014 Phys. Rep. 544 1–122; Lee et al 2015 Eur. Phys. J. B 88 48; Kivelä et al 2014 J. Complex Netw. 2 203–71; De Domenico et al 2013 Phys. Rev. X 3 041022—in which each interaction type is mapped to its own network layer; e.g. multi-layer transportation networks, coupled social networks, metabolic and regulatory networks, etc. A salient physical phenomena emerging from multiplexity is super-diffusion: exhibited by an accelerated diffusion admitted by the multi-layer structure as compared to any single layer. Theoretically super-diffusion was only known to be predicted using the spectral gap of the full Laplacian of a multiplex network and its interacting layers. Here we turn to machine learning (ML) which has developed techniques to recognize, classify, and characterize complex sets of data. We show that modern ML architectures, such as fully connected and convolutional neural networks (CNN), can classify and predict the presence of super-diffusion in multiplex networks with 94.12% accuracy. Such predictions can be done in situ, without the need to determine spectral properties of a network.

Stability and optimal decay for the 3D Navier-Stokes equations with horizontal dissipation

Stability and large-time behavior are essential properties of solutions to many partial differential equations (PDEs) and play crucial roles in many practical applications. When there is full Laplacian, many techniques such as the Fourier splitting method have been created to obtain the large-time decay rates. However, when a PDE is anisotropic and involves only partial dissipation, these methods no longer apply and no effective approach is currently available. This paper aims at the stability and large-time behavior of the 3D anisotropic Navier-Stokes equations. We present a systematic approach to obtain the optimal decay rates of the stable solutions emanating from a small data. We establish that, if the initial velocity is small in the Sobolev space H4(R3)∩Hh−σ(R3), then the anisotropic Navier-Stokes equations have a unique global solution, and the solution and its first-order derivatives all decay at the optimal rates. Here Hh−σ with σ>0 denotes a Sobolev space with negative horizontal index.

Global regularity results for the 2D Boussinesq equations and micropolar equations with partial dissipation

This paper establishes the global regularity of the two-dimensional (2D) Boussinesq equations and micropolar equations with partial dissipation. Our first result is the global regularity of the 2D Boussinesq equations with fractional vertical dissipation in the horizontal velocity, horizontal dissipation in the vertical velocity and zero thermal diffusion, which is shown by taking advantage of the nice structure of the 2D Boussinesq equations and several refined commutator estimates. The second goal of this paper is to consider a system of the 2D incompressible micropolar equations with vertical dissipation in the horizontal velocity equation, horizontal dissipation in the vertical velocity equation and the fractional Λα dissipation in the micro-rotation velocity. In order to overcome the difficulty caused by the lack of full Laplacian diffusion in the velocity equations, we fully exploit the nice structure of the corresponding equations to show that this equations with arbitrarily small α>0 always possesses a unique global classical solution.

Regularity results for the 2D critical Oldroyd-B model in the corotational case

AbstractThis paper studies the regularity results of classical solutions to the two-dimensional critical Oldroyd-B model in the corotational case. The critical case refers to the full Laplacian dissipation in the velocity or the full Laplacian dissipation in the non-Newtonian part of the stress tensor. Whether or not their classical solutions develop finite time singularities is a difficult problem and remains open. The object of this paper is two-fold. Firstly, we establish the global regularity result to the case when the critical case occurs in the velocity and a logarithmic dissipation occurs in the non-Newtonian part of the stress tensor. Secondly, when the critical case occurs in the non-Newtonian part of the stress tensor, we first present many interesting global a priori bounds, then establish a conditional global regularity in terms of the non-Newtonian part of the stress tensor. This criterion comes naturally from our approach to obtain a global L∞-bound for the vorticity ω.

The Ricci curvature in noncommutative geometry

Motivated by the local formulae for asymptotic expansion of heat kernels in spectral geometry, we propose a definition of Ricci curvature in noncommutative settings. The Ricci operator of an oriented closed Riemannian manifold can be realized as a spectral functional, namely the functional defined by the zeta function of the full Laplacian of the de Rham complex, localized by smooth endomorphisms of the cotangent bundle and their trace. We use this formulation to introduce the Ricci functional in a noncommutative setting and in particular for curved noncommutative tori. This Ricci functional uniquely determines a density element, called the Ricci density, which plays the role of the Ricci operator. The main result of this paper provides an explicit computation of the Ricci density when the conformally flat geometry of the noncommutative two torus is encoded by the modular de Rham spectral triple.

Global Regularity and Time Decay for the 2D Magnetohydrodynamic Equations with Fractional Dissipation and Partial Magnetic Diffusion

This paper focuses on a system of the 2D magnetohydrodynamic (MHD) equations with the kinematic dissipation given by the fractional operator $$(-\Delta )^\alpha $$ and the magnetic diffusion by partial Laplacian. We are able to show that this system with any $$\alpha >0$$ always possesses a unique global smooth solution when the initial data is sufficiently smooth. In addition, we make a detailed study on the large-time behavior of these smooth solutions and obtain optimal large-time decay rates. Since the magnetic diffusion is only partial here, some classical tools such as the maximal regularity property for the 2D heat operator can no longer be applied. A key observation on the structure of the MHD equations allows us to get around the difficulties due to the lack of full Laplacian magnetic diffusion. The results presented here are the sharpest on the global regularity problem for the 2D MHD equations with only partial magnetic diffusion.

Global regularity of the 2D magnetic Bénard system with partial dissipation

In this paper, we consider the Cauchy problem of the two-dimensional (2D) magnetic Bénard system with partial dissipation. On the one hand, we obtain the global regularity of the 2D magnetic Bénard system with zero thermal conductivity. The main difficulty is the zero thermal conductivity. To bypass this difficulty, we exploit the structure of the coupling system about the vorticity and the temperature and use the Maximal $L_t^{p}L_x^{q}$ regularity for the heat kernel. On the other hand, we also establish the global regularity of the 2D magnetic Bénard system with horizontal dissipation, horizontal magnetic diffusion and with either horizontal or vertical thermal diffusivity. This settles the global regularity issue unsolved in the previous works. Additionally, in the Appendix, we also show that with a full Laplacian for the diffusive term of the magnetic field and half of the full Laplacian for the temperature field, the global regularity result holds true as long as the power $\alpha$ of the fractional Laplacian dissipation for the velocity field is positive.

Local estimate about schrödinger maximal operator on H-type groups

Let Δ be full Laplacian on H-type group G. Then for every compact set D ⊆ G, a local estimate of the Schrodinger maximal operator holds, that is, ∫Dsup0<t<1|eitΔf(x)|2dx∼<‖f‖Hs2, s>12.We also show that the above inequality fails when s < 1/4.

Global regularity of generalized magnetic Benard problem

We study the magnetic Bénard problem in two‐dimensional space with generalized dissipative and diffusive terms, namely, fractional Laplacians and logarithmic supercriticality. Firstly, we show that when the diffusive term for the magnetic field is a full Laplacian, the solution initiated from data sufficiently smooth preserves its regularity as long as the power of the fractional Laplacians for the dissipative term of the velocity field and the diffusive term of the temperature field adds up to 1. Secondly, we show that with zero dissipation for the velocity field and a full Laplacian for the diffusive term of the temperature field, the global regularity result also holds when the diffusive term for the magnetic field consists of the fractional Laplacian with its power strictly bigger than 1. Finally, we show that with no diffusion from the magnetic and the temperature fields, the global regularity result remains valid as long as the dissipation term for the velocity field has its strength at least at the logarithmically supercritical level. These results represent various extensions of previous work on both Boussinesq and magnetohydrodynamics systems. Copyright © 2016 John Wiley &amp; Sons, Ltd.

Regularity results on the Leray-alpha magnetohydrodynamics systems

We study certain generalized Leray-alpha magnetohydrodynamics systems. We show that the solution pairs of velocity and magnetic fields to this system in two-dimension preserve their initial regularity in two cases: dissipation logarithmically weaker than a full Laplacian and zero diffusion, zero dissipation and diffusion logarithmically weaker than a full Laplacian. These results extend previous results in Zhou and Fan (2011). Moreover, we show that for a certain three-dimensional Leray-alpha magnetohydrodynamics system, sufficient condition of regularity may be reduced to a horizontal gradient or a partial derivative in just one direction of the magnetic field, reducing components from the results in Fan and Ozawa (2009).

Spatio-temporal filtering of thermal video sequences for heart rate estimation

In this paper, a novel method for non-contact measurement of heart rate using thermal imaging was proposed. Thermal videos were recorded from subjects’ faces. The measurements are performed on three different areas: the whole face, the upper half of the face and the supraorbital region. A tracker was used to track these regions to make the algorithm invulnerable to the subject's motion. After tracking, the videos were spatially filtered using a full Laplacian pyramid decomposition to increase the signal to noise ratio; next, the video frames were successively temporally filtered using an ideal bandpass filter for extracting the thermal variations caused by blood circulation. Finally, the heart rate was calculated by using two methods including zero crossing and Fast Fourier Transform. For evaluating the results, the complement of absolute normalized difference (CAND) index was used which was introduced by Pavlidis. This index was 99.42% in the best case and 92.472% in average for 22 subjects. These results showed a growth in CAND index in comparison with previous work. Zerocrossing outperformed FFT because of the nonstationary nature of thermal signals. Another benefit of our method is that, the videos are taken from the face unlike most of the studies that take it from the neck and Carotid. Neck and carotid are less accessible than faces. Finally, the optimum ROI for estimating the heart rate from face was identified.

Logarithmically extended global regularity result of Lans-alpha MHD system in two-dimensional space

We study the two-dimensional generalized Lans alpha magnetohydrodynamics system. We show that the solution pairs of velocity and magnetic fields to this system preserve their initial regularity in two cases: dissipation logarithmically weaker than a full Laplacian and zero diffusion, zero dissipation and diffusion logarithmically weaker than a full Laplacian.

Semiclassical limits of the Schrödinger kernels on the $$h$$ h -Heisenberg group

We consider the Heisenberg group with the parameter \(h\) in the group multiplication, which tends to the Euclidean space when we take the limit as \(h\) tends to \(0\). We construct the Schrodinger kernels of the sub-Laplacian and the full Laplacian on the \(h\)-Heisenberg group and compute the limits of the Schrodinger kernels when \(h\) tends to \(0\).

Strichartz Inequalities for the Wave Equation with the Full Laplacian on H-Type Groups

We generalize the dispersive estimates and Strichartz inequalities for the solution of the wave equation related to the full Laplacian on H-type groups, by means of Besov spaces defined by a Littlewood-Paley decomposition related to the spectral of the full Laplacian. The dimension of the center on those groups ispand we assume thatp&gt;1. A key point consists in estimating the decay in time of theL∞norm of the free solution. This requires a careful analysis due also to the nonhomogeneous nature of the full Laplacian.

Observability and coarse graining of consensus dynamics through the external equitable partition

Using the intrinsic relationship between the external equitable partition (EEP) and the spectral properties of the graph Laplacian, we characterize convergence and observability properties of consensus dynamics on networks. In particular, we establish the relationship between the original consensus dynamics and the associated consensus of the quotient graph under varied initial conditions, and characterize the asymptotic convergence to the synchronization manifold under nonuniform input signals. We also show that the EEP with respect to a node can reveal nodes in the graph with an increased rate of asymptotic convergence to the consensus value, as characterized by the second smallest eigenvalue of the quotient Laplacian. Finally, we show that the quotient graph preserves the observability properties of the full graph and how the inheritance by the quotient graph of particular aspects of the eigenstructure of the full Laplacian underpins the observability and convergence properties of the system.

Bayesian lasso binary quantile regression

In this paper, a Bayesian hierarchical model for variable selection and estimation in the context of binary quantile regression is proposed. Existing approaches to variable selection in a binary classification context are sensitive to outliers, heteroskedasticity or other anomalies of the latent response. The method proposed in this study overcomes these problems in an attractive and straightforward way. A Laplace likelihood and Laplace priors for the regression parameters are proposed and estimated with Bayesian Markov Chain Monte Carlo. The resulting model is equivalent to the frequentist lasso procedure. A conceptional result is that by doing so, the binary regression model is moved from a Gaussian to a full Laplacian framework without sacrificing much computational efficiency. In addition, an efficient Gibbs sampler to estimate the model parameters is proposed that is superior to the Metropolis algorithm that is used in previous studies on Bayesian binary quantile regression. Both the simulation studies and the real data analysis indicate that the proposed method performs well in comparison to the other methods. Moreover, as the base model is binary quantile regression, a much more detailed insight in the effects of the covariates is provided by the approach. An implementation of the lasso procedure for binary quantile regression models is available in the R-package bayesQR.