Maximal antichains of minimum size

Let $n\geqslant 4$ be a natural number, and let $K$ be a set $K\subseteq [n]:={1,2,...,n}$. We study the problem to find the smallest possible size of a maximal family $\mathcal{A}$ of subsets of $[n]$ such that $\mathcal{A}$ contains only sets whose size is in $K$, and $A\not\subseteq B$ for all ${A,B}\subseteq\mathcal{A}$, i.e. $\mathcal{A}$ is an antichain. We present a general construction of such antichains for sets $K$ containing 2, but not 1. If $3\in K$ our construction asymptotically yields the smallest possible size of such a family, up to an $o(n^2)$ error. We conjecture our construction to be asymptotically optimal also for $3\not\in K$, and we prove a weaker bound for the case $K={2,4}$. Our asymptotic results are straightforward applications of the graph removal lemma to an equivalent reformulation of the problem in extremal graph theory which is interesting in its own right.Comment: fixed faulty argument in Section 2, added reference

Minimizing the regularity of maximal regular antichains of 2- and 3-sets

Let $n\geqslant 3$ be a natural number. We study the problem to find the smallest $r$ such that there is a family $\mathcal{A}$ of 2-subsets and 3-subsets of $[n]=\{1,2,...,n\}$ with the following properties: (1) $\mathcal{A}$ is an antichain, i.e. no member of $\mathcal A$ is a subset of any other member of $\mathcal A$, (2) $\mathcal A$ is maximal, i.e. for every $X\in 2^{[n]}\setminus\mathcal A$ there is an $A\in\mathcal A$ with $X\subseteq A$ or $A\subseteq X$, and (3) $\mathcal A$ is $r$-regular, i.e. every point $x\in[n]$ is contained in exactly $r$ members of $\mathcal A$. We prove lower bounds on $r$, and we describe constructions for regular maximal antichains with small regularity.Comment: 7 pages, updated reference

Assessment of the environmental toxicity and carcinogenicity of tungsten-based shot.

The toxicity of elemental tungsten released from discharged shot was assessed against previous studies that established a 1% toxic threshold for soil organisms. Extremely heavy theoretical shot loadings of 69,000 shot/ha were used to generate estimated environmental concentrations (EEC) for two brands of tungsten-based shot containing 51% and 95% tungsten. The corresponding tungsten EEC values were 6.5–13.5 mg W/kg soil, far below the 1% toxic threshold. The same shot loading in water produced tungsten EEC values of 2.1–4.4 mg W/L, levels that are not toxic under experimental conditions. Pure tungsten has not been shown to exhibit carcinogenic properties when ingested or embedded in animal tissues, but nickel, with which it is often alloyed, has known carcinogenicity. Given the large number of waterfowl that carry shot embedded in their body, it is advisable to screen lead shot substitutes for their carcinogenic potential through intra-muscular implantation

Minimum Weight Flat Antichains of Subsets

Building on classical theorems of Sperner and Kruskal-Katona, we investigate antichains $\mathcal F$ in the Boolean lattice $B_n$ of all subsets of $[n]:=\{1,2,\dots,n\}$, where $\mathcal F$ is flat, meaning that it contains sets of at most two consecutive sizes, say $\mathcal F=\mathcal{A}\cup\mathcal{B}$, where $\mathcal{A}$ contains only $k$-subsets, while $\mathcal{B}$ contains only $(k-1)$-subsets. Moreover, we assume $\mathcal{A}$ consists of the first $m$ $k$-subsets in squashed (colexicographic) order, while $\mathcal{B}$ consists of all $(k-1)$-subsets not contained in the subsets in $\mathcal{A}$. Given reals $\alpha,\beta>0$, we say the weight of $\mathcal F$ is $\alpha\cdot|\mathcal{A}|+\beta\cdot|\mathcal{B}|$. We characterize the minimum weight antichains $\mathcal F$ for any given $n,k,\alpha,\beta$, and we do the same when in addition $\mathcal F$ is a maximal antichain. We can then derive asymptotic results on both the minimum size and the minimum Lubell function

New stability results for Einstein scalar gravity

We consider asymptotically anti de Sitter gravity coupled to a scalar field with mass slightly above the Breitenlohner-Freedman bound. This theory admits a large class of consistent boundary conditions characterized by an arbitrary function $W$. An important open question is to determine which $W$ admit stable ground states. It has previously been shown that the total energy is bounded from below if $W$ is bounded from below and the bulk scalar potential $V(\phi)$ admits a suitable superpotential. We extend this result and show that the energy remains bounded even in some cases where $W$ can become arbitrarily negative. As one application, this leads to the possibility that in gauge/gravity duality, one can add a double trace operator with negative coefficient to the dual field theory and still have a stable vacuum

Effect of Light-Cure Initiation Time on Polymerization Efficiency and Orthodontic Bond Strength with a Resin-Modified Glass-Ionomer

Objectives –  The polymerization and acid–base reactions in resin-modified glass-ionomers (RMGI) are thought to compete with and inhibit one another. To examine the effect of visible light-cure (VLC) delay on the polymerization efficiency and orthodontic bond strength of a dual-cured RMGI. Setting and Sample Population –  The Orthodontics Graduate Program at Marquette University. An in vitro study utilizing 72 freshly extracted human bicuspid teeth. Materials and Methods –  A RMGI light-cured immediately, 2.5, 5, or 10 min after mixing comprised the experimental groups. Isothermal and dynamic temperature scan differential scanning calorimetry (DSC) analysis of the RMGI was performed to determine extents of VLC polymerization and acid–base reaction exotherms. Human premolars (n = 18/group) were bonded with the RMGI. Shear bond strength and adhesive remnant index (ARI) scores were determined. Results –  Differential scanning calorimetry results showed the 10-min-delay RMGI group experienced significantly (p \u3c 0.05) lower VLC polymerization compared with the other groups. Acid–base reaction exotherms were undetected in all groups except the 10-min delay group. No significant differences (p \u3e 0.05) were noted among the groups for mean shear bond strength. A chi-square test showed no significant difference (p = 0.428) in ARI scores between groups. Conclusions –  Delay in light-curing may reduce polymerization efficiency and alter the structure of the RMGI, but orthodontic shear bond strength does not appear to be compromised

Astrophysics datamining in the classroom: Exploring real data with new software tools and robotic telescopes

Within the efforts to bring frontline interactive astrophysics and astronomy to the classroom, the Hands on Universe (HOU) developed a set of exercises and platform using real data obtained by some of the most advanced ground and space observatories. The backbone of this endeavour is a new free software Web tool - Such a Lovely Software for Astronomy based on Image J (Salsa J). It is student-friendly and developed specifically for the HOU project and targets middle and high schools. It allows students to display, analyze, and explore professionally obtained astronomical images, while learning concepts on gravitational dynamics, kinematics, nuclear fusion, electromagnetism. The continuous evolving set of exercises and tutorials is being completed with real (professionally obtained) data to download and detailed tutorials. The flexibility of the Salsa J platform tool enables students and teachers to extend the exercises with their own observations. The software developed for the HOU program has been designed to be a multi-platform, multi-lingual experience for image manipulation and analysis in the classroom. Its design enables easy implementation of new facilities (extensions and plugins), minimal in-situ maintenance and flexibility for exercise plugin. Here, we describe some of the most advanced exercises about astrophysics in the classroom, addressing particular examples on gravitational dynamics, concepts currently introduced in most sciences curricula in middle and high schools.Comment: 10 pages, 12 images, submitted to the special theme issue Using Astronomy and Space Science Research in Physics Courses of the American Journal of Physic

Sizes of flat maximal antichains of subsets

This is the second of two papers investigating for which positive integers $m$ there exists a maximal antichain of size $m$ in the Boolean lattice $B_n$ (the power set of $[n]:=\{1,2,\dots,n\}$, ordered by inclusion). In the first part, the sizes of maximal antichains have been characterized. Here we provide an alternative construction with the benefit of showing that almost all sizes of maximal antichains can be obtained using antichains containing only $l$-sets and $(l+1)$-sets for some $l$

Hydrogen bond rearrangements and the motion of charge defects in water viewed using multidimensional ultrafast infrared spectroscopy

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemistry, February 2010."December 2009." Cataloged from PDF version of thesis. Vita.Includes bibliographical references.Compared with other molecular liquids, water is highly structured due to its ability to form up to four hydrogen bonds to its nearest neighbors, resulting in a tetrahedral network of molecules. However, this network is highly dynamic, exhibiting fluctuations and rearrangements that take place on femtosecond to picosecond time scales. The transport of excess protons and proton holes in water makes exclusive use of water's hydrogen bonding network. Compared to ions of similar size and charge density, both hydronium and hydroxide ions exhibit anomalously high diffusion rates due to the fact that water molecules that neighbor these ions can undergo a proton transfer reaction with the ion. This allows the structural diffusion of the ion to occur apart from the displacement of individual water molecules. In this thesis we adopt a joint experimental and theoretical approach to characterize the fluctuations of water's hydrogen bonding network and how these fluctuations act to drive the structural diffusion of the aqueous hydroxide ion. The experimental data that we present consists of a series of ultrafast nonlinear infrared spectroscopies, in particular two-dimensional infrared spectroscopy (2D IR), applied to the O-H stretching transition of a dilute solution of HOD in NaOD/D20. The frequency of the O-H stretch, (OH, is highly sensitive to the configuration of its hydrogen bonding partner. 2D IR spectroscopy allows us to measure rapid shifts in OH that reveal time scales for changes in the local environment of the HOD molecule.(cont.) The calculation of 2D IR spectra from molecular dynamics simulations then allows us to make a direct connection between the results of our experiments and the underlying dynamics of the system that drive both hydrogen bond exchange and the structural diffusion of the hydroxide ion. 2D IR spectra recorded for dilute HOD in D20 show a strong asymmetry, preferentially broadening in the frequency region indicative of strained or broken hydrogen bonds, indicating that these configurations are unstable and quickly return to a hydrogen bond. The time scale over which the 2D spectra broaden, ~60 fs, is similar to the librational period of water and suggests that molecules exchange hydrogen bonding partners though rapid, large amplitude rotations. Molecular dynamics simulations find that the transition state for hydrogen bond exchange resembles a bifurcated hydrogen bond. In roughly half of the examined exchange events, a second solvation shell water molecule inserts across the breaking hydrogen bond. This suggests that hydrogen bond rearrangements are tied to the restructuring of a water molecule's solvation shell. Upon the addition of NaOD to HOD/D20 solution, a large absorption continuum appears to the low frequency side of the O-H stretch due to the formation of strong hydrogen bonds between HOD molecules and OD ions. At early waiting times, 2D IR spectra show large, offdiagonal intensity in this frequency range that rapidly relaxes within ~110 fs.(cont.) Modeling using an empirical valence bond simulation (MS-EVB) model of aqueous NaOH suggests that as the 0-H stretching potential symmetrizes during proton transfer events, overtone transitions of the shared proton contribute strongly to 2D spectra. The rapid loss of offdiagonal intensity results from the spectral sweeping of these vibrational overtones as the solvent modulates the motion of the shared proton. The collective electric field of the solvent is found to be an appropriate reaction coordinate for the formation and modulation of shared proton states. Over picosecond waiting times, spectral features appear in the 2D IR spectra that are indicative of the exchange of population between OH~ ions and HOD molecules due to proton transfer. The construction of a spectral fitting model gives a lower bound of 3 ps for this exchange. Calculations of structural parameters following proton exchange using the MS-EVB simulation model suggest that the observed exchange process corresponds to the formation and breakage of hydrogen bonds donated by the HOD/OD~ pair formed as a result of the proton transfer. A full description of the structural diffusion of the hydroxide ion requires both a description of the local hydrogen bonding structure of the ion as well as the dielectric fluctuations of the surrounding solvent.by Sean T. Roberts.Ph.D