Abstract:
In this work, pure and different metal ions doped ZnO thin films were obtained by a facile electrochemical deposition process. Different morphologies of ZnO, such as nanoplates, nanoparticles, as well as dense film can be obtained by doping Cu^{2+}, In^{3+}, and Al^{3+}, respectively. Besides, the electrical properties of ZnO were also dependent on the doping ions. In this work, only pure ZnO shows resistive switching characteristics, indicating that the defects in ZnO is a key role in inducing resistive switching behaviour.

Abstract:
We show that if $G$ and $H$ are finitely generated groups whose Hilbert compression exponent is positive, then so is the Hilbert compression exponent of the wreath $G \wr H$. We also prove an analogous result for coarse embeddings of wreath products. In the special case $G=\Z$, $H=\Z \wr \Z$ our result implies that the Hilbert compression exponent of $\Z\wr (\Z\wr\Z)$ is at least 1/4, answering a question posed as to whether it has positive Hilbert compression exponent.

Abstract:
We give lower bound estimates for the macroscopic scale of coarse differentiability of Lipschitz maps from a Carnot group with the Carnot-Carath\'{e}odory metric $(G,\dcc)$ to a few different classes of metric spaces. Using this result, we derive lower bound estimates for quantitative nonembeddability of Lipschitz embeddings of $G$ into a metric space $(X,d_X)$ if $X$ is either an Alexandrov space with nonpositive or nonnegative curvature, a superreflexive Banach space, or another Carnot group that does not admit a biLipschitz homomorphic embedding of $G$. For the same targets, we can further give lower bound estimates for the biLipschitz distortion of every embedding $f : B(n) \to X$, where B(n) is the ball of radius $n$ of a finitely generated nonabelian torsion-free nilpotent group $G$. We also prove an analogue of Bourgain's discretization theorem for Carnot groups and show that Carnot groups have nontrivial Markov convexity. These give the first examples of metric spaces that have nontrivial Markov convexity but cannot biLipschitzly embed into Banach spaces of nontrivial Markov convexity.

Abstract:
We compute the Markov convexity invariant of the continuous infinite dimensional Heisenberg group $\mathbb{H}_\infty$ to show that it is Markov 4-convex and cannot be Markov $p$-convex for any $p < 4$. As Markov convexity is a biLipschitz invariant and Hilbert space is Markov 2-convex, this gives a different proof of the classical theorem of Pansu and Semmes that the Heisenberg group does not admit a biLipschitz embedding into any Euclidean space. The Markov convexity lower bound will follow from exhibiting an explicit embedding of Laakso graphs $G_n$ into $\mathbb{H}_\infty$ that has distortion at most $C n^{1/4} \sqrt{\log n}$. We use this to show that if $X$ is a Markov $p$-convex metric space, then balls of the discrete Heisenberg group $\mathbb{H}(\mathbb{Z})$ of radius $n$ embed into $X$ with distortion at least some constant multiple of $$\frac{(\log n)^{\frac{1}{p}-\frac{1}{4}}}{\sqrt{\log \log n}}.$$ Finally, we show that Markov 4-convexity does not give the optimal distortion for embeddings of binary trees $B_m$ into $\mathbb{H}_\infty$ by showing that the distortion is on the order of $\sqrt{\log m}$.

Abstract:
Let $f : G \to H$ be a Lipschitz map between two Carnot groups. We show that if $B$ is ball of $G$, then there exists a subset $Z \subset B$, whose image in $H$ under $f$ has small Hausdorff content, such that $B \backslash Z$ can be decomposed into a controlled number of pieces, the restriction of $f$ on each of which is quantitatively biLipschitz. This extends a result of \cite{meyerson}, which proved the same result, but with the restriction that $G$ has an appropriate discretization. We provide an example of a Carnot group not admitting such a discretization.

Abstract:
The mechanical properties of polymer ultrathin films are usually different from those of their counterparts in bulk. Understanding the effect of thickness on the mechanical properties of these films is crucial for their applications. However, it is a great challenge to measure their elastic modulus experimentally with in situ heating. In this study, a thermodynamic model for temperature- (T) and thickness (h)-dependent elastic moduli of polymer thin films Ef(T,h) is developed with verification by the reported experimental data on polystyrene (PS) thin films. For the PS thin films on a passivated substrate, Ef(T,h) decreases with the decreasing film thickness, when h is less than 60 nm at ambient temperature. However, the onset thickness (h*), at which thickness Ef(T,h) deviates from the bulk value, can be modulated by T. h* becomes larger at higher T because of the depression of the quenching depth, which determines the thickness of the surface layer δ.

Abstract:
We show that if a subset $K$ in the Heisenberg group (endowed with the Carnot-Carath\'{e}odory metric) is contained in a rectifiable curve, then it satisfies a modified analogue of Peter Jones's geometric lemma. This is a quantitative version of the statement that a finite length curve has a tangent at almost every point. This condition complements that of \cite{FFP} except a power 2 is changed to a power 4. Two key tools that we use in the proof are a geometric martingale argument like that of \cite{Schul-TSP} as well as a new curvature inequality in the Heisenberg group.

Abstract:
We show that a sufficient condition for a subset $E$ in the Heisenberg group (endowed with the Carnot-Carath\'{e}odory metric) to be contained in a rectifiable curve is that it satisfies a modified analogue of Peter Jones's geometric lemma. Our estimates improve on those of \cite{FFP}, by replacing the power $2$ of the Jones-$\beta$-number with any power $r<4$. This complements (in an open ended way) our work \cite{Li-Schul-beta-leq-length}, where we showed that such an estimate was necessary, but with $r=4$.

Abstract:
We characterize $n$-rectifiable metric measure spaces as those spaces that admit a countable Borel decomposition so that each piece has positive and finite $n$-densities and one of the following: is an $n$-dimensional Lipschitz differentiability space; has $n$-independent Alberti representations; satisfies David's condition for an $n$-dimensional chart. The key tool is an iterative grid construction which allows us to show that the image of a ball with a high density of curves from the Alberti representations under a chart map contains a large portion of a uniformly large ball and hence satisfies David's condition. This allows us to apply previously known "biLipschitz pieces" results on the charts.

Abstract:
We introduce a notion of connecting points in a metric measure space by Alberti representations and show that it is equivalent to the space satisfying the differentiability theory of Cheeger for Lipschitz maps into Banach spaces with the Radon-Nikodym property. We then prove that this is also equivalent to satisfying an asymptotic non-homogeneous Poincar\'e inequality. Finally we show that this non-homogeneous Poincar\'e inequality is improved to a non-asymptotic version under taking Gromov-Hausdorff tangents and use this improved form to derive quasiconvexity of the tangents.