Sums and differences of correlated random sets

Many fundamental questions in additive number theory (such as Goldbach's conjecture, Fermat's last theorem, and the Twin Primes conjecture) can be expressed in the language of sum and difference sets. As a typical pair of elements contributes one sum and two differences, we expect that $|A-A| > |A+A|$ for a finite set $A$. However, in 2006 Martin and O'Bryant showed that a positive proportion of subsets of $\{0, \dots, n\}$ are sum-dominant, and Zhao later showed that this proportion converges to a positive limit as $n \to \infty$. Related problems, such as constructing explicit families of sum-dominant sets, computing the value of the limiting proportion, and investigating the behavior as the probability of including a given element in $A$ to go to zero, have been analyzed extensively. We consider many of these problems in a more general setting. Instead of just one set $A$, we study sums and differences of pairs of \emph{correlated} sets $(A,B)$. Specifically, we place each element $a \in \{0,\dots, n\}$ in $A$ with probability $p$, while $a$ goes in $B$ with probability $\rho_1$ if $a \in A$ and probability $\rho_2$ if $a \not \in A$. If $|A+B| > |(A-B) \cup (B-A)|$, we call the pair $(A,B)$ a \emph{sum-dominant $(p,\rho_1, \rho_2)$-pair}. We prove that for any fixed $\vec{\rho}=(p, \rho_1, \rho_2)$ in $(0,1)^3$, $(A,B)$ is a sum-dominant $(p,\rho_1, \rho_2)$-pair with positive probability, and show that this probability approaches a limit $P(\vec{\rho})$. Furthermore, we show that the limit function $P(\vec{\rho})$ is continuous. We also investigate what happens as $p$ decays with $n$, generalizing results of Hegarty-Miller on phase transitions. Finally, we find the smallest sizes of MSTD pairs.Comment: Version 1.0, 19 pages. Keywords: More Sum Than Difference sets, correlated random variables, phase transitio

Sets Characterized by Missing Sums and Differences in Dilating Polytopes

A sum-dominant set is a finite set $A$ of integers such that $|A+A| > |A-A|$. As a typical pair of elements contributes one sum and two differences, we expect sum-dominant sets to be rare in some sense. In 2006, however, Martin and O'Bryant showed that the proportion of sum-dominant subsets of $\{0,\dots,n\}$ is bounded below by a positive constant as $n\to\infty$. Hegarty then extended their work and showed that for any prescribed $s,d\in\mathbb{N}_0$, the proportion $\rho^{s,d}_n$ of subsets of $\{0,\dots,n\}$ that are missing exactly $s$ sums in $\{0,\dots,2n\}$ and exactly $2d$ differences in $\{-n,\dots,n\}$ also remains positive in the limit. We consider the following question: are such sets, characterized by their sums and differences, similarly ubiquitous in higher dimensional spaces? We generalize the integers in a growing interval to the lattice points in a dilating polytope. Specifically, let $P$ be a polytope in $\mathbb{R}^D$ with vertices in $\mathbb{Z}^D$, and let $\rho_n^{s,d}$ now denote the proportion of subsets of $L(nP)$ that are missing exactly $s$ sums in $L(nP)+L(nP)$ and exactly $2d$ differences in $L(nP)-L(nP)$. As it turns out, the geometry of $P$ has a significant effect on the limiting behavior of $\rho_n^{s,d}$. We define a geometric characteristic of polytopes called local point symmetry, and show that $\rho_n^{s,d}$ is bounded below by a positive constant as $n\to\infty$ if and only if $P$ is locally point symmetric. We further show that the proportion of subsets in $L(nP)$ that are missing exactly $s$ sums and at least $2d$ differences remains positive in the limit, independent of the geometry of $P$. A direct corollary of these results is that if $P$ is additionally point symmetric, the proportion of sum-dominant subsets of $L(nP)$ also remains positive in the limit.Comment: Version 1.1, 23 pages, 7 pages, fixed some typo

Fragility Analysis of Space Reinforced Concrete Frame Structures with Structural Irregularity in Plan

Because significant damages to structures having structural irregularity in their plans were repeatedly observed during many past earthquakes, there have been great research efforts to evaluate their seismic vulnerability. Although most of the previous studies used simplified structural representations such as one-dimensional or two-dimensional models in the fragility analysis of plan-irregular structures, simple analytical models could not represent true seismic behavior from the complicated nonlinear coupling between lateral and torsional responses as the degree of irregularity increased. For space structures with high irregularity, more realistic representations such as three-dimensional models are needed for proper seismic assessment. However, the use of computationally expensive models is not practically feasible with existing approaches of fragility analysis. Thus, in this study, a different approach is adopted that can produce vulnerability curves efficiently, even with a three-dimensional model. In this approach, an integrated computational framework is established that combines reliability analysis and structural analysis. This enables evaluation of the limit-state faction without constructing its explicit formula, and the failure probability is calculated with the first-order reliability method (FORM) to deal with the computational challenge. Under the integrated framework, this study investigates the seismic vulnerability of space reinforced concrete frame structures with varying plan irregularity. Material uncertainty is considered, and more representative seismic fragility curves are derived with their three-dimensional analytical models. The effectiveness of the adopted approach is discussed, and the significant effect of structural irregularity on seismic vulnerability is highlighted

Methyl 9-diethyl­amino-2,2-bis­(4-meth­oxy­phen­yl)-2H-benzo[h]chromene-5-carboxyl­ate

In the title compound, C31H29NO5, the methyl carboxyl­ate and dimethyl­amino groups on the naphtho­pyran group are almost coplanar with the naphtho­pyran ring system [r.m.s. deviations = 0.08 (2) and 0.161 (2) Å, respectively]. The dihedral angle between the methyl carboxyl­ate and dimethyl­amino groups is 4.9 (1)°. The pyran ring has an envelope conformation with the quaternary C atom out of plane by 0.4739 (13) Å. The meth­oxy­phenyl substituent forms a dihedral angle of 16.6 (1)° with the plane of the benzene ring, while the other meth­oxy­phenyl group is almost coplanar, making a dihedral angle of 1.4 (1)°

N,N-Diethyl-4-[9-meth­oxy-6-(4-methoxy­phen­yl)-5-methyl-2-phenyl-2H-benzo[h]chromen-2-yl]aniline

In the title compound, C38H37NO3, the pyran ring has an envelope conformation with the quaternary Cq atom as the flap atom. The dihedral angle formed between the meth­oxy­phenyl group and the naphthalene ring system is 67.32 (6)°. The ethyl­amino groups lie to the same side of the plane through the phenyl ring and form dihedral angles of 84.6 (3) and 75.8 (2)° with it

Environmental tobacco smoke and children's health

Passive exposure to tobacco smoke significantly contributes to morbidity and mortality in children. Children, in particular, seem to be the most susceptible population to the harmful effects of environmental tobacco smoke (ETS). Paternal smoking inside the home leads to significant maternal and fetal exposure to ETS and may subsequently affect fetal health. ETS has been associated with adverse effects on pediatric health, including preterm birth, intrauterine growth retardation, perinatal mortality, respiratory illness, neurobehavioral problems, and decreased performance in school. A valid estimation of the risks associated with tobacco exposure depends on accurate measurement. Nicotine and its major metabolite, cotinine, are commonly used as smoking biomarkers, and their levels can be determined in various biological specimens such as blood, saliva, and urine. Recently, hair analysis was found to be a convenient, noninvasive technique for detecting the presence of nicotine exposure. Because nicotine/cotinine accumulates in hair during hair growth, it is a unique measure of long-term, cumulative exposure to tobacco smoke. Although smoking ban policies result in considerable reductions in ETS exposure, children are still exposed significantly to tobacco smoke not only in their homes but also in schools, restaurants, child-care settings, cars, buses, and other public places. Therefore, more effective strategies and public policies to protect preschool children from ETS should be consolidated