Regularity of pullback attractors for non-autonomous stochastic FitzHugh-Nagumo systems with additive noises on unbounded domains

In this paper, we prove the existences of pullback attractors in $L^{p}(\mathbb{R}^N)\times L^{2}(\mathbb{R}^N)$ for stochastic Fitzhugh-Nagumo system driven by both additive noises and deterministic non-autonomous forcings. The nonlinearity is polynomial like growth with exponent $p-1$. The asymptotic compactness for the cocycle in $L^{p}(\mathbb{R}^N)\times L^{2}(\mathbb{R}^N)$ is proved by using asymptotic a priori method, where the plus and minus signs of the nonlinearity at large value are not required

Popular differences and generalized Sidon sets

For a subset $A \subseteq [N]$, we define the representation function $r_{A-A}(d) := \#\{(a,a') \in A \times A : d = a - a'\}$ and define M_D(A) := \max_{1 \leq d 1. We study the smallest possible value of $M_D(A)$ as $A$ ranges over all possible subsets of $[N]$ with a given size. We give explicit asymptotic expressions with constant coefficients determined for a large range of $D$. We shall also see how this problem connects to a well-known problem about generalized Sidon sets.Comment: 15 pages, questions adde

Error Estimation of Numerical Solvers for Linear Ordinary Differential Equations

Solving Linear Ordinary Differential Equations (ODEs) plays an important role in many applications. There are various numerical methods and solvers to obtain approximate solutions. However, few work about global error estimation can be found in the literature. In this paper, we first give a definition of the residual, based on the piecewise Hermit interpolation, which is a kind of the backward-error of ODE solvers. It indicates the reliability and quality of numerical solution. Secondly, the global error between the exact solution and an approximate solution is the forward error and a bound of it can be given by using the backward-error. The examples in the paper show that our estimate works well for a large class of ODE models.Comment: 13 pages,6 figure

Zero-sum and nonzero-sum differential games without Isaacs condition

In this paper we study the zero-sum and nonzero-sum differential games with not assuming Isaacs condition. Along with the partition $\pi$ of the time interval $[0,T]$, we choose the suitable random non-anticipative strategy with delay to study our differential games with asymmetric information. Using Fenchel transformation, we prove that the limits of the upper value function $W^\pi$ and lower value function $V^\pi$ coincide when the mesh of partition $\pi$ tends to 0. Moreover, we give a characterization for the Nash equilibrium payoff (NEP, for short) of our nonzero-sum differential games without Isaacs condition, then we prove the existence of the NEP of our games. Finally, by considering all the strategies along with all partitions, we give a new characterization for the value of our zero-sum differential game with asymmetric information under some equivalent Isaacs condition.Comment: Juan Li gave a talk on this paper in the International Conference on Mathematical Control Theory-- In Memory of Professor Xunjing Li for His 80th Birthday(16-19 July 2015, Sichuan University, Chengdu

Regularity of pullback attractors and equilibria for a stochastic non-autonomous reaction-diffusion equations perturbed by a multiplicative noise

In this paper, a standard about the existence and upper semi-continuity of pullback attractors in the non-initial space is established for some classes of non-autonomous SPDE. This pullback attractor, which is the omega-limit set of the absorbing set constructed in the initial space, is completely determined by the asymptotic compactness of solutions in both the initial and non-initial spaces. As applications, the existences and upper semi-continuity of pullback attractors in $H^1(\mathbb{R}^N)$ are proved for stochastic non-autonomous reaction-diffusion equation driven by a multiplicative noise. Finally we show that under some additional conditions the cocycle admits a unique equilibrium.Comment: 35pages. arXiv admin note: substantial text overlap with arXiv:1411.774

A robust version of Freiman's 3k−43k-4 Theorem and applications

We prove a robust version of Freiman's $3k - 4$ theorem on the restricted sumset $A+_{\Gamma}B$, which applies when the doubling constant is at most $\tfrac{3+\sqrt{5}}{2}$ in general and at most $3$ in the special case when $A = -B$. As applications, we derive robust results with other types of assumptions on popular sums, and structure theorems for sets satisfying almost equalities in discrete and continuous versions of the Riesz-Sobolev inequality.Comment: 16 pages, references update

Exploiting Multiple Access in Clustered Millimeter Wave Networks: NOMA or OMA?

In this paper, we introduce a clustered millimeter wave network with non-orthogonal multiple access (NOMA), where the base station (BS) is located at the center of each cluster and all users follow a Poisson Cluster Process. To provide a realistic directional beamforming, an actual antenna pattern is deployed at all BSs. We provide a nearest-random scheme, in which near user is the closest node to the corresponding BS and far user is selected at random, to appraise the coverage performance and universal throughput of our system. Novel closed-form expressions are derived under a loose network assumption. Moreover, we present several Monte Carlo simulations and numerical results, which show that: 1) NOMA outperforms orthogonal multiple access regarding the system rate; 2) the coverage probability is proportional to the number of possible NOMA users and a negative relationship with the variance of intra-cluster receivers; and 3) an optimal number of the antenna elements is existed for maximizing the system throughput.Comment: This paper has been accepted by IEEE International Conference on Communications (ICC), May, USA, 2018. Please cite the format version of this pape

Modeling and Analysis of MmWave Communications in Cache-enabled HetNets

In this paper, we consider a novel cache-enabled heterogeneous network (HetNet), where macro base stations (BSs) with traditional sub-6 GHz are overlaid by dense millimeter wave (mmWave) pico BSs. These two-tier BSs, which are modeled as two independent homogeneous Poisson Point Processes, cache multimedia contents following the popularity rank. High-capacity backhauls are utilized between macro BSs and the core server. A maximum received power strategy is introduced for deducing novel algorithms of the success probability and area spectral efficiency (ASE). Moreover, Monte Carlo simulations are presented to verify the analytical conclusions and numerical results demonstrate that: 1) the proposed HetNet is an interference-limited system and it outperforms the traditional HetNets; 2) there exists an optimal pre-decided rate threshold that contributes to the maximum ASE; and 3) 73 GHz is the best mmWave carrier frequency regarding ASE due to the large antenna scale.Comment: This paper has been accepted by IEEE International Conference on Communications (ICC), May, USA, 2018. Please cite the format version of this pape

Optimal Solution of Linear Ordinary Differential Equations by Conjugate Gradient Method

Solving initial value problems and boundary value problems of Linear Ordinary Differential Equations (ODEs) plays an important role in many applications. There are various numerical methods and solvers to obtain approximate solutions represented by points. However, few work about optimal solution to minimize the residual can be found in the literatures. In this paper, we first use Hermit cubic spline interpolation at mesh points to represent the solution, then we define the residual error as the square of the L2 norm of the residual obtained by substituting the interpolation solution back to ODEs. Thus, solving ODEs is reduced to an optimization problem in curtain solution space which can be solved by conjugate gradient method with taking advantages of sparsity of the corresponding matrix. The examples of IVP and BVP in the paper show that this method can find a solution with smaller global error without additional mesh points.Comment: 9 pages,6 figure

Strong (L2,Lγ∩H01)(L^2,L^\gamma\cap H_0^1)-continuity in initial data of nonlinear reaction-diffusion equation in any space dimension

In this paper, we study the continuity in initial data of a classical reaction-diffusion equation with arbitrary $p>2$ order nonlinearity and in any space dimension $N\geq 1$. It is proved that the weak solutions can be $(L^2, L^\gamma\cap H_0^1)$-continuous in initial data for any $\gamma\geq 2$ (independent of the physical parameters of the system), i.e., can converge in the norm of any $L^\gamma\cap H_0^1$ as the corresponding initial values converge in $L^2$. Applying this to the global attractor we find that, with external forcing only in $L^2$, the attractor $\mathscr{A}$ attracts bounded subsets of $L^2$ in the norm of any $L^\gamma\cap H_0^1$, and that every translation set $\mathscr{A}-z_0$ of $\mathscr{A}$ for any $z_0 \in \mathscr{A}$ is a finite dimensional compact subset of $L^\gamma\cap H_0^1$. The main technique we employ is a combination of the mathematical induction and a decomposition of the nonlinearity by which the continuity result is strengthened to $(L^2, L^\gamma \cap H_0^1)$-continuity and, since interpolation inequalities are avoided, the restriction on space dimension is removed