123 research outputs found
Infinitely many fast homoclinic solutions for some second-order nonautonomous systems
We investigate the existence of infinitely many fast homoclinic solutions for a class of second-order nonautonomous systems. Our main tools are based on the variant fountain theorem. A criterion guaranteeing that the second-order system has infinitely many fast homoclinic solutions is obtained. Recent results from the literature are generalized and significantly improved.Досліджєно існування нескінченної кількості швидких гомоклінічних розв'язків для класу неавтономних систем другого порядку. Наш основний метод базується на модифікації теореми про фонтан. Отримано критерій, що гарантує наявність нескінченної кількості швидких гомоклінічних розв'язків системи другого порядку. Узагальнено та значно покращено нещодавно опубліковані результати
Positive Periodic Solutions for Impulsive Functional Differential Equations with Infinite Delay and Two Parameters
We apply the Krasnoselskii's fixed point theorem to study the existence of multiple positive periodic solutions for a class of impulsive functional differential equations with infinite delay and two parameters. In particular, the presented criteria improve and generalize some related results in the literature. As an application, we study some special cases of systems, which have been studied extensively in the literature
Global Positive Periodic Solutions for Periodic Two-Species Competitive Systems with Multiple Delays and Impulses
A set of easily verifiable sufficient conditions are derived to guarantee the existence and the global stability of positive periodic solutions for two-species competitive systems with multiple delays and impulses, by applying some new analysis techniques. This improves and extends a series of the well-known sufficiency theorems in the literature about the problems mentioned previously
Existence and Stability of Positive Periodic Solutions for a Neutral Multispecies Logarithmic Population Model with Feedback Control and Impulse
We investigate a neutral multispecies logarithmic population model with feedback control and impulse. By applying the contraction mapping principle and some inequality techniques, a set of easily applicable criteria for the existence, uniqueness, and global attractivity of positive periodic solution are established. The conditions we obtained are weaker than the previously known ones and can be easily reduced to several special cases. We also give an example to illustrate the applicability of our results
Global Positive Periodic Solutions of Generalized n
By applying the fixed point theorem in a cone of Banach space, we obtain an easily verifiable necessary and sufficient condition for the existence of positive periodic solutions of two kinds of generalized n-species competition systems with multiple delays and impulses as follows:
xi′(t)=xi(t)[ai(t)-bi(t)xi(t)-∑j=1ncij(t)xi(t-τij(t))-∑j=1ndij(t)xj(t-γij(t))-∑j=1neij(t)∫-σij0fij(s)xj(t+s)ds], a.e., t>0,t≠tk, k∈Z+, i=1,2,…,n; xi(tk+)-xi(tk-)=θikxi(tk), i=1,2,…,n, k∈Z+; and xi′(t)=xi(t)[ai(t)-bi(t)xi(t)+∑j=1ncij(t)xi(t-τij(t))-∑j=1ndij(t)xj(t-γij(t))-∑j=1neij(t)∫-σij0fij(s)xj(t+s)ds], a.e., t>0, t≠tk, k∈Z+, i=1,2,…,n; xi(tk+)-xi(tk-)=θikxi(tk), i=1,2,…,n, k∈Z+. It improves and generalizes a series of the well-known sufficiency theorems in the literature about the problems mentioned previously
Reducibility of 1-D quantum harmonic oscillator with new unbounded oscillatory perturbations
Enlightened by Lemma 1.7 in \cite{LiangLuo2021}, we prove a similar lemma
which is based upon oscillatory integrals and Langer's turning point theory.
From it we show that the Schr{\"o}dinger equation can be
reduced in to an autonomous system for most values
of the frequency vector , where , and . The
functions and are analytic on
and will be chosen according to the value of . Comparing
with \cite{LiangLuo2021}, the novelty is that the phase functions of
oscillatory integral are more degenerate when .Comment: Journal of Dynamics and Differential Equation
Global Positive Periodic Solutions of Generalized n
We consider the following generalized n-species Lotka-Volterra type and Gilpin-Ayala type competition systems with multiple delays and impulses: xi′(t)=xi(t)[ai(t)-bi(t)xi(t)-∑j=1ncij(t)xjαij(t-ρij(t))-∑j=1ndij(t)xjβij(t-τij(t))-∑j=1neij(t)∫-ηij0kij(s)xjγij(t+s)ds-∑j=1nfij(t)∫-θij0Kij(ξ)xiδij(t+ξ)xjσij(t+ξ)dξ],a.e, t>0, t≠tk; xi(tk+)-xi(tk-)=hikxi(tk), i=1,2,…,n, k∈Z+. By applying the Krasnoselskii fixed-point theorem in a cone of Banach space, we derive some verifiable necessary and sufficient conditions for the existence of positive periodic solutions of the previously mentioned. As applications, some special cases of the previous system are examined and some earlier results are extended and improved
Enhanced phytoextraction of Pb and other metals from artificially contaminated soils through the combined application of EDTA and EDDS
Training Energy-Based Models with Diffusion Contrastive Divergences
Energy-Based Models (EBMs) have been widely used for generative modeling.
Contrastive Divergence (CD), a prevailing training objective for EBMs, requires
sampling from the EBM with Markov Chain Monte Carlo methods (MCMCs), which
leads to an irreconcilable trade-off between the computational burden and the
validity of the CD. Running MCMCs till convergence is computationally
intensive. On the other hand, short-run MCMC brings in an extra non-negligible
parameter gradient term that is difficult to handle. In this paper, we provide
a general interpretation of CD, viewing it as a special instance of our
proposed Diffusion Contrastive Divergence (DCD) family. By replacing the
Langevin dynamic used in CD with other EBM-parameter-free diffusion processes,
we propose a more efficient divergence. We show that the proposed DCDs are both
more computationally efficient than the CD and are not limited to a
non-negligible gradient term. We conduct intensive experiments, including both
synthesis data modeling and high-dimensional image denoising and generation, to
show the advantages of the proposed DCDs. On the synthetic data learning and
image denoising experiments, our proposed DCD outperforms CD by a large margin.
In image generation experiments, the proposed DCD is capable of training an
energy-based model for generating the Celab-A dataset, which is
comparable to existing EBMs
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