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We study the properties of finite graphs in which the ball of radius $r$ around each vertex induces a graph isomorphic to some fixed graph $F$. This is a natural extension of the study of regular graphs, and of the study of graphs of constant link. We focus on the case where $F$ is $\mathbb{L}^d$, the $d$-dimensional square lattice. We obtain a characterisation of all the finite graphs in which the ball of radius $3$ around each vertex is isomorphic to the ball of radius $3$ in $\mathbb{L}^d$, for each integer $d \geq 3$. These graphs have a very rigidly proscribed global structure, much more so than that of $(2d)$-regular graphs. (They may be viewed as quotient lattices of $\mathbb{L}^d$ in various compact orbifolds.) In the $d=2$ case, our methods yield new proofs of structure theorems of Thomassen and of M\'arquez, de Mier, Noy and Revuelta, and also yield short, `algebraic' restatements of these theorems. Our proofs use a mixture of techniques and results from combinatorics, algebraic topology and group theory., Comment: 23 pages, shortened proofs, one new figure

We prove an isoperimetric inequality for conjugation-invariant sets of size $k$ in $S_n$, showing that these necessarily have edge-boundary considerably larger than some other sets of size $k$ (provided $k$ is small). Specifically, let $T_n$ denote the Cayley graph on $S_n$ generated by the set of all transpositions. We show that if $A \subset S_n$ is a conjugation-invariant set with $|A| = pn! \leq n!/2$, then the edge-boundary of $A$ in $T_n$ has size at least $$c \cdot \frac {\log_2 (\tfrac 1{p})}{\log_2 \log_2 (\tfrac 2{p})}\cdot n \cdot |A|,$$ where $c$ is an absolute constant. (This is sharp up to an absolute constant factor, when $p = \Theta(1/s!)$ for any $s \in \{1,2,...,n\}$.) It follows that if $p = n^{-\Theta(1)}$, then the edge-boundary of a conjugation-invariant set of measure $p$ is necessarily a factor of $\Omega(\log n / \log \log n)$ larger than the minimum edge-boundary over all sets of measure $p$., Comment: 20 pages. We have added an Appendix containing a proof of (2), which in the previous version was left as an exercise for the reader. We have also added a discussion of what happens for subsets of $S_n$ which are invariant under conjugation by some transitive subgroup of $S_n$. In addition, some minor typos have been corrected

We demonstrate the emission of indistinguishable photons along a semiconductor chip originating from carrier recombination in an InAs quantum dot. The emitter is integrated in the waveguiding region of a photonic crystal structure, allowing for on-chip light propagation. We perform a Hong-Ou-Mandel-type of experiment with photons collected from the exit of the waveguide and we observe two-photon interference under continuous wave excitation. Our results pave the way for the integration of quantum emitters in advanced photonic quantum circuits., Comment: 10 pages, 3 figures

We demonstrate the emission and routing of single photons along a semiconductor chip originating from carrier recombination in an actively positioned InAs quantum dot. Device scale arrays of quantum dots are formed by a two step regrowth process. We precisely locate the propagating region of a unidirectional photonic crystal waveguide with respect to the quantum dot nucleation site. Under pulsed optical excitation, the multiphoton emission probability from the exit of the waveguide is 12 \pm 5 % before any background correction. Our results are a major step towards the deterministic integration of a quantum emitter with the waveguiding components of photonic quantum circuits., Comment: 8 pages, 4 figures

The CHOICES Project funded by the Institute of Education Sciences (IES), U.S. Department of Education, addresses the need for ready access to information for parents, students, school, and community agency personnel regarding transitional and community support programs. At this time we have created two databases (student information and community information), five applications (student game, student survey, parent, teacher, community questionnaires) to populate those databases, and a web site to host them in addition to transition-related content. [This article was published in: Proceedings of the Society for Information Technology & Teacher Education International Conference (SITE) March, 2013.]

We show that if $A \subset [k]^n$, then $A$ is $\epsilon$-close to a junta depending upon at most $\exp(O(|\partial A|/(k^{n-1}\epsilon)))$ coordinates, where $\partial A$ denotes the edge-boundary of $A$ in the $\ell^1$-grid. This is sharp up to the value of the absolute constant in the exponent. This result can be seen as a generalisation of the Junta theorem for the discrete cube, from [E. Friedgut, Boolean functions with low average sensitivity depend on few coordinates, Combinatorica 18 (1998), 27-35], or as a characterization of large subsets of the $\ell^1$-grid whose edge-boundary is small. We use it to prove a result on the structure of Lipschitz functions between two discrete tori; this can be seen as a discrete, quantitative analogue of a recent result of Austin [T. Austin, On the failure of concentration for the $\ell^{\infty}$-ball, preprint]. We also prove a refined version of our junta theorem, which is sharp in a wider range of cases., Comment: 29 pages. A mistake in Example 2 (pointed out by an anonymous referee) has now been corrected

We show that a set $A \subset \{0,1\}^{n}$ with edge-boundary of size at most $|A| (\log_{2}(2^{n}/|A|) + \epsilon)$ can be made into a subcube by at most $(2 \epsilon/\log_{2}(1/\epsilon))|A|$ additions and deletions, provided $\epsilon$ is less than an absolute constant. We deduce that if $A \subset \{0,1\}^{n}$ has size $2^{t}$ for some $t \in \mathbb{N}$, and cannot be made into a subcube by fewer than $\delta |A|$ additions and deletions, then its edge-boundary has size at least $|A| \log_{2}(2^{n}/|A|) + |A| \delta \log_{2}(1/\delta) = 2^{t}(n-t+\delta \log_{2}(1/\delta))$, provided $\delta$ is less than an absolute constant. This is sharp whenever $\delta = 1/2^{j}$ for some $j \in \{1,2,\ldots,t\}$., Comment: Typos corrected in Conjectures 12 and 13

We prove that for $n$ sufficiently large, if $A$ is a family of permutations of $\{1,2,\ldots,n\}$ with no two permutations in $\mathcal{A}$ agreeing exactly once, then $|\mathcal{A}| \leq (n-2)!$, with equality holding only if $\mathcal{A}$ is a coset of the stabilizer of 2 points. We also obtain a Hilton-Milner type result, namely that if $\mathcal{A}$ is such a family which is not contained within a coset of the stabilizer of 2 points, then it is no larger than the family $\{\sigma \in S_{n}:\ \sigma(1)=1,\sigma(2)=2,\ \#\{\textrm{fixed points of}\sigma \geq 5\} \neq 1\} \cup \{(1\ 3)(2\ 4),(1\ 4)(2\ 3),(1\ 3\ 2\ 4),(1\ 4\ 2\ 3)\}$. We conjecture that for $t \in \mathbb{N}$, and for $n$ sufficiently large depending on $t$, if $\mathcal{A}$ is family of permutations of $\{1,2,\ldots,n\}$ with no two permutations in $\mathcal{A}$ agreeing exactly $t-1$ times, then $|\mathcal{A}| \leq (n-t)!$, with equality holding only if $\mathcal{A}$ is a coset of the stabilizer of $t$ points. This can be seen as a permutation analogue of a conjecture of Erd\H{o}s on families of $k$-element sets with a forbidden intersection, proved by Frankl and F\"uredi in [P. Frankl and Z. F\"uredi, Forbidding Just One Intersection, Journal of Combinatorial Theory, Series A, Volume 39 (1985), pp. 160-176]., Comment: 26 pages

A recently published paper [arXiv:1206.5417 ]showed strong incommensurate diffraction peaks in EuTiO3 around zone boundary R-points. We wish to convey that in our samples, and, it seems, at least another case from the literature, these peaks are absent., Comment: 1-2 pages 1 figure

We give lower bounds on the maximum possible girth of an $r$-uniform, $d$-regular hypergraph with at most $n$ vertices, using the definition of a hypergraph cycle due to Berge. These differ from the trivial upper bound by an absolute constant factor (viz., by a factor of between $3/2+o(1)$ and $2 +o(1)$). We also define a random $r$-uniform `Cayley' hypergraph on $S_n$ which has girth $\Omega (n^{1/3})$ with high probability, in contrast to random regular $r$-uniform hypergraphs, which have constant girth with positive probability., Comment: A hole in the proof of Theorem 8 was recently pointed out by Eberhard [10]. In this version, we repair the hole, giving a slightly weaker bound. We use a very similar method to that of Eberhard in [10], where a slightly weaker variant of [13, Theorem 3] is proved; our original proof contained a similar hole to the original proof of [13, Theorem 3]

We prove that a balanced Boolean function on $S_{n}$ whose Fourier transform is highly concentrated on the first two irreducible representations of $S_{n}$, is close in structure to a dictatorship, a function which is determined by the image or pre-image of a single element. As a corollary, we obtain a stability result concerning extremal isoperimetric sets in the Cayley graph on $S_{n}$ generated by the transpositions. Our proof works in the case where the expectation of the function is bounded away from $0$ and $1$. In contrast, [Ellis, D., Filmus, Y., Friedgut, E., A quasi-stability result for dictatorships in $S_{n}$, Combinatorica 35 (2015), pp. 573-618] deals with Boolean functions of expectation O(1/n) whose Fourier transform is highly concentrated on the first two irreducible representations of $S_{n}$. These need not be close to dictatorships; rather, they must be close to a union of a constant number of cosets of point-stabilizers., Comment: 29 pages. Conjecture 1 in Section 5 has been adjusted. The Introduction has been corrected and expanded