Topology and Order

We will discuss topologies as orders, orders on sets of topologies, and topologies on ordered sets. More specifically, we will discuss Alexandroff topologies as quasiorders, the lattice of topologies on a finite set, and partially ordered topological spaces. Some topological properties of Alexandroff spaces are characterized in terms of their order. Complementation in the lattice of topologies on a set and in the lattice of convex topologies on a partially ordered set will be discussed

Continued Radicals

A nested radical with terms $a_1, a_2, ldotsa_N$ is an expression of form $sqrt{a_N + cdots + sqrt{a_2 + sqrt{a_1}}}$. The limit as $N$ approaches infinity of such an expression, if it exists, is called a continued radical. We consider the set of real numbers $S(M)$ representable as a continued radical whose terms $a_1, a_2, ldots$ are all from a finite set $M$ of nonnegative real numbers. We give conditions on the set $M$ for $S(M)$ to be (a) an interval, and (b) homeomorphic to the Cantor set

The Algorithmic Composition of Classical Music through Data Mining

The desire to teach a computer how to algorithmically compose music has been a topic in the world of computer science since the 1950’s, with roots of computer-less algorithmic composition dating back to Mozart himself. One limitation of algorithmically composing music has been the difficulty of eliminating the human intervention required to achieve a musically homogeneous composition. We attempt to remedy this issue by teaching a computer how the rules of composition differ between the six distinct eras of classical music by having it examine a dataset of musical scores, rather than explicitly telling the computer the formal rules of composition. To pursue this automated composition process, we examined the intersectionality of algorithmic composition with the machine learning concept of classification. Using a Naïve Bayes classifier, the computer classifies pieces of classical music into their respective era based upon a number of attributes. It then attempts to recreate each of the six classical styles using a technique inspired by cellular automata. The success of this process is twofold determined by feeding composition samples into a number of classifiers, as well as analysis by studied musicians. We concluded that there is potential for further hybridization of classification and composition techniques

Genetic evidence for causal relationships between maternal obesity-related traits and birth weight

Importance  Neonates born to overweight or obese women are larger and at higher risk of birth complications. Many maternal obesity-related traits are observationally associated with birth weight, but the causal nature of these associations is uncertain.Objective  To test for genetic evidence of causal associations of maternal body mass index (BMI) and related traits with birth weight.Design, Setting, and Participants  Mendelian randomization to test whether maternal BMI and obesity-related traits are potentially causally related to offspring birth weight. Data from 30 487 women in 18 studies were analyzed. Participants were of European ancestry from population- or community-based studies in Europe, North America, or Australia and were part of the Early Growth Genetics Consortium. Live, term, singleton offspring born between 1929 and 2013 were included. Exposures  Genetic scores for BMI, fasting glucose level, type 2 diabetes, systolic blood pressure (SBP), triglyceride level, high-density lipoprotein cholesterol (HDL-C) level, vitamin D status, and adiponectin level.Main Outcome and Measure  Offspring birth weight from 18 studies.Results  Among the 30 487 newborns the mean birth weight in the various cohorts ranged from 3325 g to 3679 g. The maternal genetic score for BMI was associated with a 2-g (95% CI, 0 to 3 g) higher offspring birth weight per maternal BMI-raising allele (P = .008). The maternal genetic scores for fasting glucose and SBP were also associated with birth weight with effect sizes of 8 g (95% CI, 6 to 10 g) per glucose-raising allele (P = 7 × 10−14) and −4 g (95% CI, −6 to −2g) per SBP-raising allele (P = 1×10−5), respectively. A 1-SD ( ≈ 4 points) genetically higher maternal BMI was associated with a 55-g higher offspring birth weight (95% CI, 17 to 93 g). A 1-SD ( ≈ 7.2 mg/dL) genetically higher maternal fasting glucose concentration was associated with 114-g higher offspring birth weight (95% CI, 80 to 147 g). However, a 1-SD ( ≈ 10 mm Hg) genetically higher maternal SBP was associated with a 208-g lower offspring birth weight (95% CI, −394 to −21 g). For BMI and fasting glucose, genetic associations were consistent with the observational associations, but for systolic blood pressure, the genetic and observational associations were in opposite directions.Conclusions and Relevance  In this mendelian randomization study, genetically elevated maternal BMI and blood glucose levels were potentially causally associated with higher offspring birth weight, whereas genetically elevated maternal SBP was potentially causally related to lower birth weight. If replicated, these findings may have implications for counseling and managing pregnancies to avoid adverse weight-related birth outcomes

Transitions between 4-intersection values of planar regions

[EN] If A(t) and B(t) are subsets of the Euclidean plane which are continuously morphing, we investigate the question of whether they may morph directly from being disjoint to overlapping so that the boundary and interior of A(t) both intersect the boundary and interior of B(t) without first passing through a state in which only their boundaries intersect. More generally, we consider which 4-intersection values---binary 4-tuples specifying whether the boundary and interior of A(t) intersect the boundary and interior of B(t)---are adjacent to which in the sense that one may morph into the other without passing through a third value. The answers depend on what forms the regions A(t) and B(t) are allowed to assume and on the definition of continuous morphing of the sets.Bell, K.; Richmond, T. (2017). Transitions between 4-intersection values of planar regions. Applied General Topology. 18(1):183-202. doi:10.4995/agt.2017.6716.SWORD183202181C. Adams and R. Franzosa, Introduction to Topology: Pure and Applied, Pearson Prentice Hall, Upper Saddle River, NJ, 2008.Chen, J., Li, C., Li, Z., & Gold, C. (2001). A Voronoi-based 9-intersection model for spatial relations. International Journal of Geographical Information Science, 15(3), 201-220. doi:10.1080/13658810151072831Clementini, E., Sharma, J., & Egenhofer, M. J. (1994). Modelling topological spatial relations: Strategies for query processing. Computers & Graphics, 18(6), 815-822. doi:10.1016/0097-8493(94)90007-8Egenhofer, M. J., & Al-Taha, K. K. (1992). Reasoning about gradual changes of topological relationships. Lecture Notes in Computer Science, 196-219. doi:10.1007/3-540-55966-3_12Egenhofer‡, M. J., Clementini, E., & di Felice, P. (1994). Research Paper. International journal of geographical information systems, 8(2), 129-142. doi:10.1080/02693799408901990EGENHOFER, M. J., & FRANZOSA, R. D. (1991). Point-set topological spatial relations. International journal of geographical information systems, 5(2), 161-174. doi:10.1080/02693799108927841M. Egenhofer and R. Franzosa, On equivalence of topological relations, International Journal for Geographical Information Systems 8, no. 6 (1994), 133-152. https://doi.org/10.1016/0022-247X(85)90246-XFrancaviglia, S., Lechicki, A., & Levi, S. (1985). Quasi-uniformization of hyperspaces and convergence of nets of semicontinuous multifunctions. Journal of Mathematical Analysis and Applications, 112(2), 347-370. doi:10.1016/0022-247x(85)90246-xNedas, K. A., Egenhofer, M. J., & Wilmsen, D. (2007). Metric details of topological line–line relations. International Journal of Geographical Information Science, 21(1), 21-48. doi:10.1080/13658810600852164Roy, A. J., & Stell, J. G. (2001). Spatial relations between indeterminate regions. International Journal of Approximate Reasoning, 27(3), 205-234. doi:10.1016/s0888-613x(01)00033-0SMITH, T. R., & PARK, K. K. (1992). Algebraic approach to spatial reasoning. International journal of geographical information systems, 6(3), 177-192. doi:10.1080/0269379920890190

Second harmonic generation from thin slabs in the discrete dipole approach

The nonlinear optical response of thin Si slabs is calculated using a discrete dipole approach. The s-polarized second harmonic response as a function of the angle of incidence appears to be in reasonable agreement with experimental results. The p-polarized SHG shows a high sensitivity for the shape of the polarizability profile