A Nontrivial Homotopy Element of Order p2 Detected by the Classical Adams Spectral Sequence

Let p be an odd prime. The authors detect a nontrivial element ã _p of order p² in the stable homotopy groups of spheres by the classical Adams spectral sequence. It is represented by \(a_0^{p - 2} h_1 \in Ext_A^{p - 1,pq + p - 2} (\mathbb{Z}/p,\mathbb{Z}/p)\) in the E₂-term of the ASS and meanwhile p · ã ^p is the first periodic element α ^p .     

Radial symmetry for systems of fractional Laplacian

In this paper, we consider systems of fractional Laplacian equations in Rnwith nonlinear terms satisfying some quite general structural conditions. These systems were categorized critical and subcritical cases. We show that there is no positive solution in the subcritical cases, and we classify all positive solutions ui in the critical cases by using a direct method of moving planes introduced in Chen-Li-Li [11] and some new maximum principles in Li-Wu-Xu [27].     

Doubly nonlocal Cahn–Hilliard equations

We consider a doubly nonlocal nonlinear parabolic equation which describes phase-segregation of a two-component material in a bounded domain. This model is a more general version than the recent nonlocal Cahn–Hilliard equation proposed by Giacomin and Lebowitz [26], such that it reduces to the latter under certain conditions. We establish well-posedness results along with regularity and long-time results in the case when the interaction between the two levels of nonlocality is strong-to-weak.     

Counting arithmetical structures on paths and cycles

Let $G$ be a finite, connected graph. An arithmetical structure on $G$ is a pair of positive integer vectors $\mathbf{d},\mathbf{r}$ such that $(diag\left(\mathbf{d}\right)?A)\mathbf{r}=0$, where $A$ is the adjacency matrix of $G$. We investigate the combinatorics of arithmetical structures on path and cycle graphs, as well as the associated critical groups (the torsion part of the cokernels of the matrices $(diag\left(\mathbf{d}\right)?A)$). For paths, we prove that arithmetical structures are enumerated by the Catalan numbers, and we obtain refined enumeration results related to ballot sequences. For cycles, we prove that arithmetical structures are enumerated by the binomial coefficients $\left(\genfrac{}{}{0ex}{}{2n?1}{n?1}\right)$, and we obtain refined enumeration results related to multisets. In addition, we determine the critical groups for all arithmetical structures on paths and cycles.     

Fast second-order accurate difference schemes for time distributed-order and Riesz space fractional diffusion equations

The aim of this paper is to develop fast second-order accurate difference schemes for solving one- and two-dimensional time distributed-order and Riesz space fractional diffusion equations. We adopt the same measures for one- and two-dimensional problems as follows: we first transform the time distributed-order fractional diffusion problem into the multi-term time-space fractional diffusion problem with the composite trapezoid formula. Then, we propose a second-order accurate difference scheme based on the interpolation approximation on a special point to solve the resultant problem. Meanwhile, the unconditional stability and convergence of the new difference scheme in $L_2$-norm are proved. Furthermore, we find that the discretizations lead to a series of Toeplitz systems which can be efficiently solved by Krylov subspace methods with suitable circulant preconditioners. Finally, numerical results are presented to show the effectiveness of the proposed difference methods and demonstrate the fast convergence of our preconditioned Krylov subspace methods.     

On the asymptotic behavior of solutions of certain integro-differential equations

The authors present conditions under which every positive solution $x(t)$ of the integro--differential equation $x^{\prime \prime }(t)=a(t)+\int_{c}^{t}(t-s)^{\alpha-1}[e(s)+k(t,s)f(s,x(s))]ds, \quad c>1, \ \alpha >0,$ satisfies $x(t)=O(tA(t))\textrm{ as }t\rightarrow \infty,$ i.e, $\limsup_{t\rightarrow \infty }\frac{x(t)}{tA(t)}<\infty, \textrm{where} \ A(t)=\int_{c}^{t}a(s)ds.$ From the results obtained, they derive a technique that can be applied to some related integro--differential equations that are equivalent to certain fractional differential equations of Caputo type of any order.     

Fractional Hamiltonian systems with positive semi-definite matrix

We study the existence of solutions for the following fractional Hamiltonian systems $$ \left\{ \begin{array}{ll} - _tD^{\alpha}_{\infty}(_{-\infty}D^{\alpha}_{t}u(t))-\lambda L(t)u(t)+\nabla W(t,u(t))=0,\\[0.1cm] u\in H^{\alpha}(\mathbb{R},\mathbb{R}^n), \end{array} \right. ~~~~~~~~~~~~~~~~~(FHS)_\lambda $$ where $\alpha\in (1/2,1)$, $t\in \mathbb{R}$, $u\in \mathbb{R}^n$, $\lambda>0$ is a parameter, $L\in C(\mathbb{R},\mathbb{R}^{n^2})$ is a symmetric matrix, $W\in C^1(\mathbb{R} \times \mathbb{R}^n,\mathbb{R})$. Assuming that $L(t)$ is a positive semi-definite symmetric matrix, that is, $L(t)\equiv 0$ is allowed to occur in some finite interval $T$ of $\mathbb{R}$, $W(t,u)$ satisfies some superquadratic conditions weaker than Ambrosetti-Rabinowitz condition, we show that (FHS)$_\lambda$ has a solution which vanishes on $\mathbb{R}\setminus T$ as $\lambda \to \infty$, and converges to some $\tilde{u}\in H^{\alpha}(\R, \R^n)$. Here, $\tilde{u}\in E_{0}^{\alpha}$ is a solution of the Dirichlet BVP for fractional systems on the finite interval $T$. Our results are new and improve recent results in the literature even in the case $\alpha =1$.     

Spectral properties of the massless relativistic quartic oscillator

An explicit solution of the spectral problem of the non-local Schrödinger operator obtained as the sum of the square root of the Laplacian and a quartic potential in one dimension is presented. The eigenvalues are obtained as zeroes of special functions related to the fourth order Airy function, and closed formulae for the Fourier transform of the eigenfunctions are derived. These representations allow to derive further spectral properties such as estimates of spectral gaps, heat trace and the asymptotic distribution of eigenvalues, as well as a detailed analysis of the eigenfunctions. A subtle spectral effect is observed which manifests in an exponentially tight approximation of the spectrum by the zeroes of the dominating term in the Fourier representation of the eigenfunctions and its derivative.     

Fractional Sobolev inequalities associated with singular problems

In this paper we study Sobolev‐type inequalities associated with singular problems for the fractional p‐Laplacian operator in a bounded domain of , .