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September 2021 , Volume 29 , Issue 4

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Note on coisotropic Floer homology and leafwise fixed points
Fabian Ziltener
2021, 29(4): 2553-2560 doi: 10.3934/era.2021001 +[Abstract](661) +[HTML](317) +[PDF](313.5KB)
Abstract:

For an adiscal or monotone regular coisotropic submanifold \begin{document}$ N $\end{document} of a symplectic manifold I define its Floer homology to be the Floer homology of a certain Lagrangian embedding of \begin{document}$ N $\end{document}. Given a Hamiltonian isotopy \begin{document}$ \varphi = ( \varphi^t) $\end{document} and a suitable almost complex structure, the corresponding Floer chain complex is generated by the \begin{document}$ (N, \varphi) $\end{document}-contractible leafwise fixed points. I also outline the construction of a local Floer homology for an arbitrary closed coisotropic submanifold.

Results by Floer and Albers about Lagrangian Floer homology imply lower bounds on the number of leafwise fixed points. This reproduces earlier results of mine.

The first construction also gives rise to a Floer homology for a Boothby-Wang fibration, by applying it to the circle bundle inside the associated complex line bundle. This can be used to show that translated points exist.

Classification and simulation of chaotic behaviour of the solutions of a mixed nonlinear Schrödinger system
Riadh Chteoui, Abdulrahman F. Aljohani and Anouar Ben Mabrouk
2021, 29(4): 2561-2597 doi: 10.3934/era.2021002 +[Abstract](796) +[HTML](357) +[PDF](1477.82KB)
Abstract:

In this paper, we study a couple of NLS equations characterized by mixed cubic and super-linear sub-cubic power laws. Classification as well as existence and uniqueness of the steady state solutions have been investigated. Numerical simulations have been also provided illustrating graphically the theoretical results. Such simulations showed that possible chaotic behaviour seems to occur and needs more investigations.

Global stability of traveling waves for a spatially discrete diffusion system with time delay
Ting Liu and Guo-Bao Zhang
2021, 29(4): 2599-2618 doi: 10.3934/era.2021003 +[Abstract](700) +[HTML](337) +[PDF](386.34KB)
Abstract:

This article deals with the global stability of traveling waves of a spatially discrete diffusion system with time delay and without quasi-monotonicity. Using the Fourier transform and the weighted energy method with a suitably selected weighted function, we prove that the monotone or non-monotone traveling waves are exponentially stable in \begin{document}$ L^\infty(\mathbb{R})\times L^\infty(\mathbb{R}) $\end{document} with the exponential convergence rate \begin{document}$ e^{-\mu t} $\end{document} for some constant \begin{document}$ \mu>0 $\end{document}.

On the universal $ \alpha $-central extensions of the semi-direct product of Hom-preLie algebras
Bing Sun, Liangyun Chen and Yan Cao
2021, 29(4): 2619-2636 doi: 10.3934/era.2021004 +[Abstract](643) +[HTML](331) +[PDF](406.05KB)
Abstract:

We study Hom-actions, semidirect product and describe the relation between semi-direct product extensions and split extensions of Hom-preLie algebras. We obtain the functorial properties of the universal \begin{document}$ \alpha $\end{document}-central extensions of \begin{document}$ \alpha $\end{document}-perfect Hom-preLie algebras. We give that a derivation or an automorphism can be lifted in an \begin{document}$ \alpha $\end{document}-cover with certain constraints. We provide some necessary and sufficient conditions about the universal \begin{document}$ \alpha $\end{document}-central extension of the semi-direct product of two \begin{document}$ \alpha $\end{document}-perfect Hom-preLie algebras.

Tori can't collapse to an interval
Sergio Zamora
2021, 29(4): 2637-2644 doi: 10.3934/era.2021005 +[Abstract](597) +[HTML](303) +[PDF](394.75KB)
Abstract:

Here we prove that under a lower sectional curvature bound, a sequence of Riemannian manifolds diffeomorphic to the standard \begin{document}$ m $\end{document}-dimensional torus cannot converge in the Gromov–Hausdorff sense to a closed interval.

The proof is done by contradiction by analyzing suitable covers of a contradicting sequence, obtained from the Burago–Gromov–Perelman generalization of the Yamaguchi fibration theorem.

$ C^* $-algebras associated with asymptotic equivalence relations defined by hyperbolic toral automorphisms
Kengo Matsumoto
2021, 29(4): 2645-2656 doi: 10.3934/era.2021006 +[Abstract](557) +[HTML](313) +[PDF](316.12KB)
Abstract:

We study the \begin{document}$ C^* $\end{document}-algebras of the étale groupoids defined by the asymptotic equivalence relations for hyperbolic automorphisms on the two-dimensional torus. The algebras are proved to be isomorphic to four-dimensional non-commutative tori by an explicit numerical computation. The ranges of the unique tracial states of its \begin{document}$ K_0 $\end{document}-groups of the \begin{document}$ C^* $\end{document}-algebras are described in terms of the hyperbolic matrices of the automorphisms on the torus.

Telescoping method, summation formulas, and inversion pairs Special Issues
Qing-Hu Hou and Yarong Wei
2021, 29(4): 2657-2671 doi: 10.3934/era.2021007 +[Abstract](645) +[HTML](337) +[PDF](346.35KB)
Abstract:

Based on Gosper's algorithm, we present an approach to the telescoping of general sequences. Along this approach, we propose a summation formula and a bibasic extension of Ma's inversion formula. From the formulas, we are able to derive several hypergeometric and elliptic hypergeometric identities.

Automorphism group and twisted modules of the twisted Heisenberg-Virasoro vertex operator algebra Special Issues
Hongyan Guo
2021, 29(4): 2673-2685 doi: 10.3934/era.2021008 +[Abstract](650) +[HTML](311) +[PDF](337.68KB)
Abstract:

We first determine the automorphism group of the twisted Heisenberg-Virasoro vertex operator algebra \begin{document}$ V_{\mathcal{L}}(\ell_{123},0) $\end{document}. Then, for any integer \begin{document}$ t>1 $\end{document}, we introduce a new Lie algebra \begin{document}$ \mathcal{L}_{t} $\end{document}, and show that \begin{document}$ \sigma_{t} $\end{document}-twisted \begin{document}$ V_{\mathcal{L}}(\ell_{123},0) $\end{document}(\begin{document}$ \ell_{2} = 0 $\end{document})-modules are in one-to-one correspondence with restricted \begin{document}$ \mathcal{L}_{t} $\end{document}-modules of level \begin{document}$ \ell_{13} $\end{document}, where \begin{document}$ \sigma_{t} $\end{document} is an order \begin{document}$ t $\end{document} automorphism of \begin{document}$ V_{\mathcal{L}}(\ell_{123},0) $\end{document}. At the end, we give a complete list of irreducible \begin{document}$ \sigma_{t} $\end{document}-twisted \begin{document}$ V_{\mathcal{L}}(\ell_{123},0) $\end{document}(\begin{document}$ \ell_{2} = 0 $\end{document})-modules.

On $ n $-slice algebras and related algebras Special Issues
Jin-Yun Guo, Cong Xiao and Xiaojian Lu
2021, 29(4): 2687-2718 doi: 10.3934/era.2021009 +[Abstract](494) +[HTML](325) +[PDF](541.55KB)
Abstract:

The \begin{document}$ n $\end{document}-slice algebra is introduced as a generalization of path algebra in higher dimensional representation theory. In this paper, we give a classification of \begin{document}$ n $\end{document}-slice algebras via their \begin{document}$ (n+1) $\end{document}-preprojective algebras and the trivial extensions of their quadratic duals. One can always relate tame \begin{document}$ n $\end{document}-slice algebras to the McKay quiver of a finite subgroup of \begin{document}$ \mathrm{GL}(n+1, \mathbb C) $\end{document}. In the case of \begin{document}$ n = 2 $\end{document}, we describe the relations for the \begin{document}$ 2 $\end{document}-slice algebras related to the McKay quiver of finite Abelian subgroups of \begin{document}$ \mathrm{SL}(3, \mathbb C) $\end{document} and of the finite subgroups obtained from embedding \begin{document}$ \mathrm{SL}(2, \mathbb C) $\end{document} into \begin{document}$ \mathrm{SL}(3,\mathbb C) $\end{document}.

Local well-posedness of perturbed Navier-Stokes system around Landau solutions
Jingjing Zhang and Ting Zhang
2021, 29(4): 2719-2739 doi: 10.3934/era.2021010 +[Abstract](516) +[HTML](263) +[PDF](413.85KB)
Abstract:

For the incompressible Navier-Stokes system, when initial data are uniformly locally square integral, the local existence of solutions has been obtained. In this paper, we consider perturbed system and show that perturbed solutions of Landau solutions to the Navier-Stokes system exist locally under \begin{document}$ L^q_{\text{uloc}} $\end{document}-perturbations, \begin{document}$ q\geq 2 $\end{document}. Furthermore, when \begin{document}$ q\geq 3, $\end{document} the solution is well-posed. Precisely, we give the explicit formula of the pressure term.

A multiscale stochastic criminal behavior model under a hybrid scheme Special Issues
Chuntian Wang and Yuan Zhang
2021, 29(4): 2741-2753 doi: 10.3934/era.2021011 +[Abstract](474) +[HTML](281) +[PDF](1163.89KB)
Abstract:

Crime in urban environment is a major social problem nowadays. As such, many efforts have been made to develop mathematical models for this type of crime. The pioneering work [M. B. Short, M. R. D'Orsogna, V. B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi and L. B. Chayes, Math. Models Methods Appl. Sci., 18, (2008), pp. 1249-1267] establishes an agent-based human-environment interaction model of criminal behavior for residential burglary, where aggregate pattern formation of "hotspots" is quantitatively studied for the first time. Potential offenders are assumed to interact with environment according to well-known criminology and sociology notions. However long-term simulations for the coupled dynamics are computationally costly due to all components evolving on slow time scales. In this paper, we introduce a new-generation criminal behavior model with separated spatio-temporal scales for the agent actions and the environment parameter reactions. The computational cost is reduced significantly, while the essential stochastic features of the pioneering model are preserved. Moreover, the separation of scales brings the model into the theoretical framework of piecewise deterministic Markov processes (PDMP). A martingale approach is applicable which will be useful to analyze both stochastic and statistical features of the model in subsequent studies.

An a posteriori error estimator based on least-squares finite element solutions for viscoelastic fluid flows Special Issues
Hsueh-Chen Lee and Hyesuk Lee
2021, 29(4): 2755-2770 doi: 10.3934/era.2021012 +[Abstract](741) +[HTML](298) +[PDF](4771.42KB)
Abstract:

We present a posteriori error estimator strategies for the least-squares finite element method (LS) to approximate the exponential Phan-Thien-Tanner (PTT) viscoelastic fluid flows. The error estimator provides adaptive mass weights and mesh refinement criteria for improving LS solutions using lower-order basis functions and a small number of elements. We analyze an a priori error estimate for the first-order linearized LS system and show that the estimate is supported by numerical results. The LS approach is numerically tested for a convergence study and then applied to the flow past a slot channel. Numerical results verify that the proposed approach improves numerical solutions and resolves computational difficulties related to the presence of corner singularities and limitations arising from the exorbitant number of unknowns.

Graded post-Lie algebra structures and homogeneous Rota-Baxter operators on the Schrödinger-Virasoro algebra Special Issues
Pengliang Xu and Xiaomin Tang
2021, 29(4): 2771-2789 doi: 10.3934/era.2021013 +[Abstract](531) +[HTML](290) +[PDF](346.41KB)
Abstract:

In this paper, we characterize the graded post-Lie algebra structures on the Schrödinger-Virasoro Lie algebra. Furthermore, as an application, we obtain the all homogeneous Rota-Baxter operator of weight \begin{document}$ 1 $\end{document} on the Schrödinger-Virasoro Lie algebra.

Averaging principle on infinite intervals for stochastic ordinary differential equations Special Issues
David Cheban and Zhenxin Liu
2021, 29(4): 2791-2817 doi: 10.3934/era.2021014 +[Abstract](602) +[HTML](260) +[PDF](410.94KB)
Abstract:

In contrast to existing works on stochastic averaging on finite intervals, we establish an averaging principle on the whole real axis, i.e. the so-called second Bogolyubov theorem, for semilinear stochastic ordinary differential equations in Hilbert space with Poisson stable (in particular, periodic, quasi-periodic, almost periodic, almost automorphic etc) coefficients. Under some appropriate conditions we prove that there exists a unique recurrent solution to the original equation, which possesses the same recurrence property as the coefficients, in a small neighborhood of the stationary solution to the averaged equation, and this recurrent solution converges to the stationary solution of averaged equation uniformly on the whole real axis when the time scale approaches zero.

Ergodic measures of intermediate entropy for affine transformations of nilmanifolds Special Issues
Wen Huang, Leiye Xu and Shengnan Xu
2021, 29(4): 2819-2827 doi: 10.3934/era.2021015 +[Abstract](510) +[HTML](270) +[PDF](330.89KB)
Abstract:

In this paper we study ergodic measures of intermediate entropy for affine transformations of nilmanifolds. We prove that if an affine transformation \begin{document}$ \tau $\end{document} of nilmanifold has a periodic point, then for every \begin{document}$ a\in[0, h_{top}(\tau)] $\end{document} there exists an ergodic measure \begin{document}$ \mu_a $\end{document} of \begin{document}$ \tau $\end{document} such that \begin{document}$ h_{\mu_a}(\tau) = a $\end{document}.

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5 Year Impact Factor: 1.833

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