Strong Jumps and Lagrangians of Non-Uniform Hypergraphs

The hypergraph jump problem and the study of Lagrangians of uniform hypergraphs are two classical areas of study in the extremal graph theory. In this paper, we refine the concept of jumps to strong jumps and consider the analogous problems over non-uniform hypergraphs. Strong jumps have rich topological and algebraic structures. The non-strong-jump values are precisely the densities of the hereditary properties, which include the Tur\'an densities of families of hypergraphs as special cases. Our method uses a generalized Lagrangian for non-uniform hypergraphs. We also classify all strong jump values for $\{1,2\}$-hypergraphs.Comment: 19 page

Boolean algebras and Lubell functions

Let $2^{[n]}$ denote the power set of $[n]:=\{1,2,..., n\}$. A collection \B\subset 2^{[n]} forms a $d$-dimensional {\em Boolean algebra} if there exist pairwise disjoint sets $X_0, X_1,..., X_d \subseteq [n]$, all non-empty with perhaps the exception of $X_0$, so that \B={X_0\cup \bigcup_{i\in I} X_i\colon I\subseteq [d]}. Let $b(n,d)$ be the maximum cardinality of a family \F\subset 2^X that does not contain a $d$-dimensional Boolean algebra. Gunderson, R\"odl, and Sidorenko proved that $b(n,d) \leq c_d n^{-1/2^d} \cdot 2^n$ where $c_d= 10^d 2^{-2^{1-d}}d^{d-2^{-d}}$. In this paper, we use the Lubell function as a new measurement for large families instead of cardinality. The Lubell value of a family of sets \F with \F\subseteq \tsupn is defined by h_n(\F):=\sum_{F\in \F}1/{{n\choose |F|}}. We prove the following Tur\'an type theorem. If \F\subseteq 2^{[n]} contains no $d$-dimensional Boolean algebra, then h_n(\F)\leq 2(n+1)^{1-2^{1-d}} for sufficiently large $n$. This results implies $b(n,d) \leq C n^{-1/2^d} \cdot 2^n$, where $C$ is an absolute constant independent of $n$ and $d$. As a consequence, we improve several Ramsey-type bounds on Boolean algebras. We also prove a canonical Ramsey theorem for Boolean algebras.Comment: 10 page

Americans’ support for college financial aid for prisoners depends on how the benefits are described

Allowing prisoners to access college-level education reduces the risk of reoffending and can save state governments money because of the reduction in future crime. But Americans, by contrast, often oppose providing prisoners such benefits. With the recent overturning of the ban on prisoners’ receiving Pell Grants, Travis M. Johnston and Kevin H. Wozniak look at how the public reacts to prison education reform, depending on how its benefits are described. They find that when the benefits to the incarcerated and to society are noted, Americans were more likely to support providing college aid to prisoners

Integrated Analysis and Application of Reservoir Models to Early Permian Detrital Carbonate Deposits, Midland Basin, Texas

A 3-D seismic volume, wireline logs and core data were integrated to determine the spatial distribution of porous reservoirs within the Wolfcampian-Leonardian detrital carbonate slope and basin strata in Glasscock County, Texas. A 3-D seismic amplitude volume was used to construct a seismic facies analysis of the detrital carbonate section, and generated attribute volumes helped identify detrital carbonate depositional trends, as well as establish a potential correlation between thick detrital carbonate intervals and associated amplitude response. Eight lithofacies were identified in core and were subsequently classified into three main facies: debris flow, grain flow/turbidite, and basinal shale. A facies type log was then created, which was used to supervise the creation of facies logs within other wells to ultimately use in the creation of a 3-D facies model. Cross sections through the study area show an increase in bathymetric relief beginning in Wolfcampian time and continuing through the Leonardian. Detrital carbonate deposition increases dramatically during the Leonardian, consisting of large gravity flows deposited basinward in a northwest-southeast linear trend, rapidly thinning basinward. Individual flows are discontinuous and bounded by basinal shale facies. Four seismic facies were identified within the interval of interest using a structurally smoothed attribute volume, while an RMS amplitude attribute volume provided a correlation between high RMS amplitude values and detrital carbonate thickness. A high RMS amplitude value corresponding to the debris flow facies was extracted from the RMS attribute volume in the form of a seismic geobody. Two facies models and one porosity model were generated by using upscaled values from the gamma ray, total porosity, and lithofacies logs, which were applied over areas with the densest well control. Although the facies model populated from upscaled GR values was useful in stratigraphic interpretation, it is determined that the models should be applied over areas with denser well spacing in order to provide a more accurate and geologically viable subsurface model

Convective transport of macromolecules in gels

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 1999.Includes bibliographical references (leaves 141-144).by Scott Travis Johnston.Ph.D