We show that, under finitely many ergodicity assumptions, any multicorrelation sequence defined by invertible measure-preserving $\mathbb {Z}^d$ -actions with multivariable integer polynomial iterates is the sum of a nilsequence and a nullsequence, extending a recent result of the second author. To this end, we develop a new seminorm bound estimate for multiple averages by improving the results in a previous work of the first, third, and fourth authors. We also use this approach to obtain new criteria for joint ergodicity of multiple averages with multivariable polynomial iterates on ${\mathbb Z}^{d}$ -systems.

Let $m\in \mathbb{N}$ and $\mathbf{X}=(X,{\mathcal{X}},\unicode[STIX]{x1D707},(T_{\unicode[STIX]{x1D6FC}})_{\unicode[STIX]{x1D6FC}\in \mathbb{R}^{m}})$ be a measure-preserving system with an $\mathbb{R}^{m}$-action. We say that a Borel measure $\unicode[STIX]{x1D708}$ on $\mathbb{R}^{m}$ is weakly equidistributed for $\mathbf{X}$ if there exists $A\subseteq \mathbb{R}$ of density 1 such that, for all $f\in L^{\infty }(\unicode[STIX]{x1D707})$, we have

$$\begin{eqnarray}\lim _{t\in A,t\rightarrow \infty }\int _{\mathbb{R}^{m}}f(T_{t\unicode[STIX]{x1D6FC}}x)\,d\unicode[STIX]{x1D708}(\unicode[STIX]{x1D6FC})=\int _{X}f\,d\unicode[STIX]{x1D707}\end{eqnarray}$$

$\unicode[STIX]{x1D707}$

$x\in X$

$W(\mathbf{X})$

$\unicode[STIX]{x1D6FC}\in \mathbb{R}^{m}$

$\mathbb{R}$

$(T_{t\unicode[STIX]{x1D6FC}})_{t\in \mathbb{R}}$

$\unicode[STIX]{x1D708}$

$\mathbb{R}^{m}$

$\mathbf{X}$

$\unicode[STIX]{x1D708}(W(\mathbf{X})+\unicode[STIX]{x1D6FD})=0$

$\unicode[STIX]{x1D6FD}\in \mathbb{R}^{m}$

$\unicode[STIX]{x1D708}$

$\mathbb{R}^{m}$

$\unicode[STIX]{x1D708}(\ell )=0$

$\ell$

$\mathbb{R}^{m}$

Let $(X,{\mathcal{B}},\unicode[STIX]{x1D707},T)$ be an ergodic measure-preserving system, let $A\in {\mathcal{B}}$ and let $\unicode[STIX]{x1D716}>0$. We study the largeness of sets of the form

$$\begin{eqnarray}S=\{n\in \mathbb{N}:\unicode[STIX]{x1D707}(A\cap T^{-f_{1}(n)}A\cap T^{-f_{2}(n)}A\cap \cdots \cap T^{-f_{k}(n)}A)>\unicode[STIX]{x1D707}(A)^{k+1}-\unicode[STIX]{x1D716}\}\end{eqnarray}$$

$\{f_{1},\ldots ,f_{k}\}$

$f_{i}:\mathbb{N}\rightarrow \mathbb{N}$

$k\leq 3$

$f_{i}(n)=if(n)$

$S$

$f(n)=q(p_{n})$

$q\in \mathbb{Z}[x]$

$q(1)$

$q(-1)=0$

$p_{n}$

$n$

$f$

$T^{q}$

$q\in \mathbb{N}$

$r\in \mathbb{Z}$

$S$

$f(n)=qn+r$

$f_{i}(n)=a_{i}n$

$a_{i}$

$S$

$k\geq 4$

$k=3$

$S$

$f_{i}$

There has been widespread recent interest in self-assembly and synthesis of porphyrin and its derivatives-based ordered arrays aiming to emulate natural light-harvesting processes and energy storage. However, technologies that leverage the structural advantages of individual porphyrins have not been fully realized and have been limited by available synthesis methods. This article provides general perspectives on porphyrin and derivative chemistry, and discussions on surfactant-assisted cooperative self-assembly using amphiphilic surfactants and functional porphyrins and derivatives. The cooperative self-assembly amplifies the intrinsic advantages of individual porphyrins by engineering them into well-defined one-dimensional–three-dimensional (1D–3D) nanostructures. Surfactant-assisted self-assembly of amphiphilic surfactants and porphyrins has been utilized to form well-defined “micelle-like” nanostructures. Driven by intermolecular interactions, subsequent nucleation and growth confined within these nanostructures lead to the formation of 1D–3D ordered optically and electrically active nanomaterials with structure and function on multiple length scales.

For any measure-preserving system $(X,{\mathcal{B}},\unicode[STIX]{x1D707},T_{1},\ldots ,T_{d})$ with no commutativity assumptions on the transformations $T_{i},$$1\leq i\leq d,$ we study the pointwise convergence of multiple ergodic averages with iterates of different growth coming from a large class of sublinear functions. This class properly contains important subclasses of Hardy field functions of order zero and of Fejér functions, i.e., tempered functions of order zero. We show that the convergence of the single average, via an invariant property, implies the convergence of the multiple one. We also provide examples of sublinear functions which are, in general, bad for convergence on arbitrary systems, but good for uniquely ergodic systems. The case where the fastest function is linear is addressed as well, and we provide, in all the cases, an explicit formula of the limit function.

We show that for every ergodic measure-preserving system $(X,{\mathcal{X}},\unicode[STIX]{x1D707},S,T)$ with commuting transformations $S$ and $T$, the average

$$\begin{eqnarray}\frac{1}{N^{3}}\mathop{\sum }_{i,j,k=0}^{N-1}f_{0}(S^{j}T^{k}x)f_{1}(S^{i+j}T^{k}x)f_{2}(S^{j}T^{i+k}x)\end{eqnarray}$$

$\unicode[STIX]{x1D707}$

$x\in X$

$N\rightarrow \infty$

$f_{0},f_{1},f_{2}\in L^{\infty }(\unicode[STIX]{x1D707})$

$(X,{\mathcal{X}},\unicode[STIX]{x1D707},S,T)$

$$\begin{eqnarray}\frac{1}{N}\mathop{\sum }_{i=0}^{N-1}f_{1}(S^{i}x)f_{2}(T^{i}x)\end{eqnarray}$$

$\unicode[STIX]{x1D707}$

$x\in X$

$N\rightarrow \infty$

$f_{1},f_{2}\in L^{\infty }(\unicode[STIX]{x1D707})$

We show that any subset $A\subset \mathbb{N}$ with positive upper Banach density contains the pattern $\{m,m+[n{\it\alpha}],\dots ,m+k[n{\it\alpha}]\}$, for some $m\in \mathbb{N}$ and $n=p-1$ for some prime $p$, where ${\it\alpha}\in \mathbb{R}\setminus \mathbb{Q}$. Making use of the Furstenberg correspondence principle, we do this by proving an associated recurrence result in ergodic theory along the shifted primes. We also prove the convergence result for the associated averages along primes and indicate other applications of these methods.

(Zr62Cu15.4Ni12.6) (x = 6–12) in situ glassy composites containing uniformly distributed Ta-rich particles were prepared by arc-melting and copper mould casting. The results show that addition of 6–10 at.% Ta to Zr62Cu15.4Ni12.6Al10 results in dissolution of 2.4 to 4.6 at.% Ta in the glassy matrix, which promotes glass-forming ability, and the remaining Ta precipitates out as body-centered cubic (BCC) Ta-rich particles dispersed on the glassy matrix. The critical diameters for the composites with 6, 8, and 10 at.% Ta are 7, 7, and 6 mm, respectively. At 12 at.% Ta addition, the glass-forming ability is dramatically reduced because of the precipitation of secondary dendritic Ta-rich particles and other nanocrystallites from melts during copper mould casting. Also, owing to the solid-liquid reaction during induction heating, some Ta-rich particles formed in the master alloys will redissolve into the glassy matrix, resulting in a smaller volume fraction of Ta-rich particles in the as-cast glassy rods than that of the corresponding ingots. The glassy matrix composites exhibit enhanced plastic strain of about 7.5 to 22.5% at room temperature. The optimum Ta content in the glassy alloys is determined to be 10 at.%, which corresponds to the highest ultimate stress of 2220 MPa and the largest plastic strain of 22.5%. The plastic strain increases with increasing volume fraction of in situ BCC Ta-rich particles. This is apparently ascribed to the impedance of Ta-rich particles to shear bands. Ta-rich particles seed the initiation of multiple shear bands and block the shear band propagation, leading to intensive multiplication and bifurcation of shear bands.