Ancient shrinking spherical interfaces in the Allen–Cahn flow

We consider the parabolic Allen–Cahn equation in ${\mathbb{R}}^n$, $n\ge 2$,

$u_t=\mathrm{\Delta }u+(1?u^2)u\text{ in }{\mathbb{R}}^n\times (?\infty ,0].$

We construct an ancient radially symmetric solution $u(x,t)$ with any given number k of transition layers between ?1 and +1. At main order they consist of k time-traveling copies of w with spherical interfaces distant $O(\log ?|t|)$ one to each other as $t\to ?\infty$. These interfaces are resemble at main order copies of the shrinking sphere ancient solution to mean the flow by mean curvature of surfaces: $|x|=\sqrt{?2(n?1)t}$. More precisely, if $w(s)$ denotes the heteroclinic 1-dimensional solution of $w^{″}+(1?w^2)w=0$$w(\pm \infty )=\pm 1$ given by $w(s)=\tanh ?\left(\frac{s}{\sqrt{2}}\right)$ we have

$u(x,t)\approx \sum _{j=1}^k{(?1)}^{j?1}w(|x|?{\rho }_j(t))?\frac{1}{2}(1+{(?1)}^k)\text{ as }t\to ?\infty$

where

${\rho }_j(t)=\sqrt{?2(n?1)t}+\frac{1}{\sqrt{2}}(j?\frac{k+1}{2})\log ?\left(\frac{|t|}{\log ?|t|}\right)+O(1),j=1,\dots ,k.$

     

On properties of random dissections and triangulations

In this work we study properties of random graphs that are drawn uniformly at random from the class consisting of biconnected outerplanar graphs, or equivalently dissections of large convex polygons. We obtain very sharp concentration results for the number of vertices of any given degree, and for the number of induced copies of a given fixed graph. Our method gives similar results for random graphs from the class of triangulations of convex polygons.     

Where galaxies really come from

The fundamental paradox of the incompatibility of the observed large-scale uniformity of the Universe with the fact that the age of the Universe is finite is overcome by the introduction of an initial period of superluminal expansion of space, called cosmic inflation. Inflation can also produce the small deviations from uniformity needed for the formation of structures in the Universe such as galaxies. This is achieved by the conjunction of inflation with the quantum vacuum, through the so-called particle production process. This mechanism is explained and linked with Hawking radiation of black holes. The nature of the particles involved is discussed and the case of using massive vector boson fields instead of scalar fields is presented, with emphasis on its distinct observational signatures. Finally, a particular implementation of these ideas is included, which can link the formation of galaxies, the standard model vector bosons and the observed galactic magnetic fields.     

Rumor spreading on random regular graphs and expanders

Broadcasting algorithms are important building blocks of distributed systems. In this work we investigate the typical performance of the classical and well‐studied push model. Assume that initially one node in a given network holds some piece of information. In each round, every one of the informed nodes chooses independently a neighbor uniformly at random and transmits the message to it. In this paper we consider random networks where each vertex has degree d ≥ 3, i.e., the underlying graph is drawn uniformly at random from the set of all d ‐regular graphs with n vertices. We show that with probability 1 ‐ o(1) the push model broadcasts the message to all nodes within (1 + o(1))C_d lnn rounds, where Particularly, we can characterize precisely the effect of the node degree to the typical broadcast time of the push model. Moreover, we consider pseudo‐random regular networks, where we assume that the degree of each node is very large. There we show that the broadcast time is (1 + o(1))Clnn with probability 1 ‐ o(1), where \begin{align*}C = \lim_{d\to\infty}C_d = \frac{1}{\ln2} + 1\end{align*}. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

Using Goffman's concepts to explore collaborative interaction processes in elementary school mathematics

The present paper describes the combined implementation of two approaches which are based on the work of Erving Goffman and focus on the analysis of the interactional participation structures. Our aim is to examine the ways that differing interpretations of the ongoing interactions may lead the students to adopt differing roles, which in turn affects the process of negotiating and establishing shared meanings. For this purpose we analyse the interaction processes in one ‘expert’ group within the ‘Jigsaw’ form of cooperation (Aronson, 1978 Aronson, E. 1978. The jigsaw classroom, Beverly Hills, CA: Sage.  [Google Scholar]), decomposing the participants’ roles according to their interactional status and their interpersonal effect. Our analysis points out the different roles that emerge in the process of collaborative interaction, and the influence of these roles on group achievement and individual learning possibilities.     

Modeling of polarization effects on n-GaN/i-InGaN/p-Gan solar cells with ultrathin GaN interlayers

We report the numerical study of n-GaN/i-InGaN/p-GaN solar cells on Ga-face substrates with thin GaN interlayers present in the intrinsic InGaN region. These interlayers have recently been shown to significantly increase the crystal quality of thick InGaN layers $(>\!\!\!120\,\text{ nm})$ . We find that tunneling is efficient in n-i-p structures having interlayers $\le \! 1.5\,\text{ nm}$ thick if polarization charges are sufficiently screened. If left unscreened, the large polarization charges naturally formed at the heterointerfaces degrades n-i-p performance, at a given interlayer thickness, because polarization charges increase the distance that carriers must tunnel. Simulations identify favorable parameter ranges.     

One‐pot Synthesis of Fused Dipyranocoumarins from Dihydroxycoumarins and Propargyl Chlorides under Microwave Irradiation

Dipetalolactone and 4‐methyldipetalolactone are prepared in excellent yield by a one‐pot tandem propargylation/Claisen rearrangement/cyclization reaction of the corresponding 5,7‐dihydroxycoumarins with 3‐chloro‐3‐methylbut‐1‐yne in the presence of Cs₂CO₃ under microwave irradiation. The analogous reactions of propargyl chloride with esculetins or 5,7‐dihydroxycoumarins led to dipropargyloxy derivatives. The later by treatment with gold nanoparticles supported on TiO₂ or BF₃.Et₂O in N,N‐dimethylformamide (DMF) under microwave irradiation resulted in very good to excellent yield to the corresponding fused dipyranocoumarins. The reactions of esculetins with 3‐chloro‐3‐methylbut‐1‐yne gave mainly exomethylene fused dioxino[g]coumarins.     

Spontaneous Resolution by Crystallization of an Octanuclear Iron(III) Complex Using Only Racemic Reagents

The P and M enantiomers of the octanuclear [Fe₈(μ₄‐O)₄(μ‐4‐Cl‐pz)₁₂Cl₄] complex, having T symmetry, were resolved by temporary substitution of chloride ligands by racemic 4‐^sBu‐phenolates and subsequent crystallization, where the (S)‐ and (R)‐phenolates coordinate selectively to the M and P complexes, respectively. The complexes were characterized by circular dichroism analysis and X‐ray structure determination. This work constitutes a rare example of enantiomeric recognition resulting in spontaneous resolution upon crystallization.     

Unprecedented Five‐Coordinate Iron(IV) Imides Generate Divergent Spin States Based on the Imide R‐Groups

Three five‐coordinate iron(IV) imide complexes have been synthesized and characterized. These novel structures have disparate spin states on the iron as a function of the R‐group attached to the imide, with alkyl groups leading to low‐spin diamagnetic (S=0) complexes and an aryl group leading to an intermediate‐spin (S=1) complex. The different spin states lead to significant differences in the bonding about the iron center as well as the spectroscopic properties of these complexes. Mössbauer spectroscopy confirmed that all three imide complexes are in the iron(IV) oxidation state. The combination of diamagnetism and ¹⁵N labeling allowed for the first ¹⁵N NMR resonance recorded on an iron imide. Multi‐reference calculations corroborate the experimental structural findings and suggest how the bonding is distinctly different on the imide ligand between the two spin states.     

Monte Carlo studies of supersymmetric matrix quantum mechanics with sixteen supercharges at finite temperature

We present the first Monte Carlo results for supersymmetric matrix quantum mechanics with 16 supercharges at finite temperature. The recently proposed nonlattice simulation enables us to include the effects of fermionic matrices in a transparent and reliable manner. The internal energy nicely interpolates the weak coupling behavior obtained by the high temperature expansion, and the strong coupling behavior predicted from the dual black-hole geometry. The Polyakov line asymptotes at low temperature to a characteristic behavior for a deconfined theory, suggesting the absence of a phase transition. These results provide highly nontrivial evidence for the gauge-gravity duality.