Q2Q_2-free families in the Boolean lattice

For a family $\mathcal{F}$ of subsets of [n]=\{1, 2, ..., n} ordered by inclusion, and a partially ordered set P, we say that $\mathcal{F}$ is P-free if it does not contain a subposet isomorphic to P. Let $ex(n, P)$ be the largest size of a P-free family of subsets of [n]. Let $Q_2$ be the poset with distinct elements a, b, c, d, a<b, c<d; i.e., the 2-dimensional Boolean lattice. We show that $2N -o(N) \leq ex(n, Q_2)\leq 2.283261N +o(N),$ where $N = \binom{n}{\lfloor n/2 \rfloor}$. We also prove that the largest $Q_2$-free family of subsets of [n] having at most three different sizes has at most 2.20711N members.Comment: 18 pages, 2 figure

Development of a three-dimensional two-fluid code with transient neutronic feedback for LWR applications

The development of a three-dimensional coupled neutronics/thermalhydraulics code for LWR safety analysis has been initiated. The transient neutronics code QUANDRY has been joined to the two-fluid thermal-hydraulics code THERMIT with the appropriate feedback mechanisms modeled. A literature review of the existing coupled neutronics/thermal-hydraulics codes is presented. It indicates that all of the known codes have limitations in their neutronic and/or thermal-hydraulic models which limit their generality of application and accuracy. It was also found that a tandem coupling scheme was most often employed and generally performed well. A detailed steady-state and transient coupling scheme based on the tandem technique was devised, taking into account the important operational characteristics of QUANDRY and THERMIT. The two codes were combined and the necessary programming modifications were performed to allow steady-state calculations with feedback. A simple steady-state sample problem was produced for the purpose of testing and debugging the coupled code

Investigation of the Reversible Hysteresis Effect in Hexagonal Metal Single Crystals and the MAX Phases

Hexagonal close packed (hcp) materials are abundant in nature, and are of great technological importance since they are used in many applications. When cyclically loaded some hcp solids outline fully and spontaneously reversible stress-strain hysteresis loops. To date, the micromechanical origin of these loops is unknown. To shed light on the subject, a spherical nanoindenter was repeatedly indented - up to 50 times in the same location - into Mg, Zn and Ti3SiC2 single crystals of various orientations, followed by select, post-indentation transmission electron microscope (TEM) cross sectional analysis. Based on the totality of the results, the energy dissipated per unit volume per cycle in the hexagonal metals can be related to the bowing out - and back - of geometrically necessary dislocations - in most cases in the form of low angle kink boundaries (LAKBs) - through statistically stored ones. Kinks were observed after indentations normal to the basal planes in Mg and Zn and also when indented normal to the (101 ̅1) and (101 ̅2) planes in Zn. When indented parallel to the basal planes in Zn, if hysteresis loops formed at all, they were insignificant in area. When (101 ̅0) planes in Mg were indented, tensile twins formed. The most probable explanation for the energy dissipated in this direction is the growth and contraction of these twins. In the case of Ti3SiC2, hysteresis loops were observed even in absence of kink boundaries. No direct evidence for twins or non-basal slip was found nor has been reported in literature. Evidence presented in this study supports the existence of a new type of defect in bulk layered solids known as ripplocations - which combine features of dislocations and surface ripples - that are able to explain the phenomena observed in this study on Ti3SiC2 in ways conventional dislocations cannot explain. It is the migration of these ripplocations that are believed to cause energy dissipation in Ti3SiC2. The energy dissipation due to ripplocations was found to be higher than the energy dissipation due to dislocations, which may offer a possible signature to distinguish between the two. However, the simplest method to distinguish between ripplocations and dislocations is to load the basal planes edge-on under a spherical indenter as carried out here. The formation of cracks, normal to the basal planes, are the unequivocal signature of ripplocations.Ph.D., Materials Science and Engineering -- Drexel University, 201

Application of a three-dimensional prognostic model during the ETEX real-time modeling exercise: Evaluation of results

Increases in computing capabilities and ready access to large-scale model output make it possible to employ advanced three-dimensional prognostic models to forecast the long-range transport of toxic or radioactive gases for emergency response. The Savannah River Technology Center (SRTC) of the U.S. Department of Energy`s Savannah River Site demonstrated this during the European Tracer EXperiment (ETEX). ETEX, conducted in the Fall of 1994, provided an opportunity to evaluate the performance of models for long-range atmospheric pollutant transport and dispersion. A comparison of SRTC forecast results for the first ETEX experiment with measured surface tracer gas concentrations shows that the predicted plume is transported too quickly and surface concentrations are low. However, modeling studies show that the forecast performance is significantly improved if convective parameterization is not employed

Lower Bounds for the Graph Homomorphism Problem

The graph homomorphism problem (HOM) asks whether the vertices of a given $n$-vertex graph $G$ can be mapped to the vertices of a given $h$-vertex graph $H$ such that each edge of $G$ is mapped to an edge of $H$. The problem generalizes the graph coloring problem and at the same time can be viewed as a special case of the $2$-CSP problem. In this paper, we prove several lower bound for HOM under the Exponential Time Hypothesis (ETH) assumption. The main result is a lower bound $2^{\Omega\left( \frac{n \log h}{\log \log h}\right)}$. This rules out the existence of a single-exponential algorithm and shows that the trivial upper bound $2^{{\mathcal O}(n\log{h})}$ is almost asymptotically tight. We also investigate what properties of graphs $G$ and $H$ make it difficult to solve HOM$(G,H)$. An easy observation is that an ${\mathcal O}(h^n)$ upper bound can be improved to ${\mathcal O}(h^{\operatorname{vc}(G)})$ where $\operatorname{vc}(G)$ is the minimum size of a vertex cover of $G$. The second lower bound $h^{\Omega(\operatorname{vc}(G))}$ shows that the upper bound is asymptotically tight. As to the properties of the "right-hand side" graph $H$, it is known that HOM$(G,H)$ can be solved in time $(f(\Delta(H)))^n$ and $(f(\operatorname{tw}(H)))^n$ where $\Delta(H)$ is the maximum degree of $H$ and $\operatorname{tw}(H)$ is the treewidth of $H$. This gives single-exponential algorithms for graphs of bounded maximum degree or bounded treewidth. Since the chromatic number $\chi(H)$ does not exceed $\operatorname{tw}(H)$ and $\Delta(H)+1$, it is natural to ask whether similar upper bounds with respect to $\chi(H)$ can be obtained. We provide a negative answer to this question by establishing a lower bound $(f(\chi(H)))^n$ for any function $f$. We also observe that similar lower bounds can be obtained for locally injective homomorphisms.Comment: 19 page

Anti-Pasch optimal packings with triples

It is shown that for v ≠ 6, 7, 10, 11, 12, 13, there exists an optimal packing with triples on v points that contains no Pasch configurations. Furthermore, for all v ≡ 5 (mod 6), there exists a pairwise balanced design of order v, whose blocks are all triples apart from a single quintuple, and that has no Pasch configurations amongst its triples