Blowup solutions and their blowup rates for parabolic equations with non-standard growth conditions

This paper concerns classical solutions for homogeneous Dirichlet problem of parabolic equations coupled via exponential sources involving variable exponents. We first establish blowup criteria for positive solutions. And then, for radial solutions, we obtain optimal classification for simultaneous and non-simultaneous blowup, which is represented by using the maxima of the involved variable exponents. At last, all kinds of blowup rates are determined for both simultaneous and non-simultaneous blowup solutions

Non-simultaneous blow-up of n components for nonlinear parabolic systems

AbstractThis paper deals with non-simultaneous and simultaneous blow-up for radially symmetric solution (u1,u2,…,un) to heat equations coupled via nonlinear boundary ∂ui∂η=uipiui+1qi+1 (i=1,2,…,n). It is proved that there exist suitable initial data such that ui (i∈{1,2,…,n}) blows up alone if and only if qi+1<pi. All of the classifications on the existence of only two components blowing up simultaneously are obtained. We find that different positions (different values of k, i, n) of ui−k and ui leads to quite different blow-up rates. It is interesting that different initial data lead to different blow-up phenomena even with the same requirements on exponent parameters. We also propose that ui−k,ui−k+1,…,ui (i∈{1,2,…,n},k∈{0,1,2,…,n−1}) blow up simultaneously while the other ones remain bounded in different exponent regions. Moreover, the blow-up rates and blow-up sets are obtained

State Estimation for Discrete-Time Fuzzy Cellular Neural Networks with Mixed Time Delays

This paper is concerned with the exponential state estimation problem for a class of discrete-time fuzzy cellular neural networks with mixed time delays. The main purpose is to estimate the neuron states through available output measurements such that the dynamics of the estimation error is globally exponentially stable. By constructing a novel Lyapunov-Krasovskii functional which contains a triple summation term, some sufficient conditions are derived to guarantee the existence of the state estimator. The linear matrix inequality approach is employed for the first time to deal with the fuzzy cellular neural networks in the discrete-time case. Compared with the present conditions in the form of M-matrix, the results obtained in this paper are less conservative and can be checked readily by the MATLAB toolbox. Finally, some numerical examples are given to demonstrate the effectiveness of the proposed results

Learning Distortion Invariant Representation for Image Restoration from A Causality Perspective

In recent years, we have witnessed the great advancement of Deep neural networks (DNNs) in image restoration. However, a critical limitation is that they cannot generalize well to real-world degradations with different degrees or types. In this paper, we are the first to propose a novel training strategy for image restoration from the causality perspective, to improve the generalization ability of DNNs for unknown degradations. Our method, termed Distortion Invariant representation Learning (DIL), treats each distortion type and degree as one specific confounder, and learns the distortion-invariant representation by eliminating the harmful confounding effect of each degradation. We derive our DIL with the back-door criterion in causality by modeling the interventions of different distortions from the optimization perspective. Particularly, we introduce counterfactual distortion augmentation to simulate the virtual distortion types and degrees as the confounders. Then, we instantiate the intervention of each distortion with a virtual model updating based on corresponding distorted images, and eliminate them from the meta-learning perspective. Extensive experiments demonstrate the effectiveness of our DIL on the generalization capability for unseen distortion types and degrees. Our code will be available at https://github.com/lixinustc/Causal-IR-DIL.Comment: Accepted by CVPR202

Common Diffusion Noise Schedules and Sample Steps are Flawed

We discover that common diffusion noise schedules do not enforce the last timestep to have zero signal-to-noise ratio (SNR), and some implementations of diffusion samplers do not start from the last timestep. Such designs are flawed and do not reflect the fact that the model is given pure Gaussian noise at inference, creating a discrepancy between training and inference. We show that the flawed design causes real problems in existing implementations. In Stable Diffusion, it severely limits the model to only generate images with medium brightness and prevents it from generating very bright and dark samples. We propose a few simple fixes: (1) rescale the noise schedule to enforce zero terminal SNR; (2) train the model with v prediction; (3) change the sampler to always start from the last timestep; (4) rescale classifier-free guidance to prevent over-exposure. These simple changes ensure the diffusion process is congruent between training and inference and allow the model to generate samples more faithful to the original data distribution

Open Knowledge Base Canonicalization with Multi-task Unlearning

The construction of large open knowledge bases (OKBs) is integral to many applications in the field of mobile computing. Noun phrases and relational phrases in OKBs often suffer from redundancy and ambiguity, which calls for the investigation on OKB canonicalization. However, in order to meet the requirements of some privacy protection regulations and to ensure the timeliness of the data, the canonicalized OKB often needs to remove some sensitive information or outdated data. The machine unlearning in OKB canonicalization is an excellent solution to the above problem. Current solutions address OKB canonicalization by devising advanced clustering algorithms and using knowledge graph embedding (KGE) to further facilitate the canonicalization process. Effective schemes are urgently needed to fully synergise machine unlearning with clustering and KGE learning. To this end, we put forward a multi-task unlearning framework, namely MulCanon, to tackle machine unlearning problem in OKB canonicalization. Specifically, the noise characteristics in the diffusion model are utilized to achieve the effect of machine unlearning for data in OKB. MulCanon unifies the learning objectives of diffusion model, KGE and clustering algorithms, and adopts a two-step multi-task learning paradigm for training. A thorough experimental study on popular OKB canonicalization datasets validates that MulCanon achieves advanced machine unlearning effects

Simulating collective behavior in the movement of immigrants by using a spatial prisoner’s dilemma with move option

The movement of immigrants is simulated by using a spatial Prisoner’s Dilemma (PD) with move option. We explore the effect of collective behavior in an evolutionary migrating dynamics. Simulation results show that immigrants adopting collective strategy perform better and thus gain higher survival rate than those not. This research suggests that the clustering of immigrants promotes cooperation

Initial Velocity Effect on Acceleration Fall of a Spherical Particle through Still Fluid

A spherical particle’s acceleration fall through still fluid was investigated analytically and experimentally using the Basset-Boussinesq-Oseen equation. The relationship between drag coefficient and Reynolds number was studied, and various parameters in the drag coefficient equation were obtained with respect to the small, medium, and large Reynolds number zones. Next, some equations were used to derive the finite fall time and distance equations in terms of certain assumptions. A simple experiment was conducted to measure the fall time and distance for a spherical particle falling through still water. Sets of experimental data were used to validate the relationship between fall velocity, time, and distance. Finally, the initial velocity effect on the total fall time and distance was discussed with different terminal Reynolds numbers, and it was determined that the initial velocity plays a more important role in the falling motion for small terminal Reynolds numbers than for large terminal Reynolds number scenarios