Close-Enough Calculations: Quick and Accurate Approximations

This chapter describes techniques to estimate division and multiplication by certain numbers. The purpose is to help in test taking. By knowing some ways to approximate answers rapidly, some potential answers can be eliminated, enhancing the chance of choosing the correct answer from the smaller number of remaining options. It presents easy ways to determine whether a number is divisible by 11, 17, and 19. It introduces sphenic numbers, cannonball (or square pyramidal numbers) in relation to the Kepler Conjecture, as well as the Kaprekar number. Examples are drawn from the Azerbaijan Grand Prix and two triple dead heats in dog racing.

The Airborne and Gastrointestinal Coronavirus SARS-COV-2 Pathways

Since there is not a clear consensus about the possibility for COVID-2 to be an airborne disease, exists a controversy regarding the need to use surgical masks to prevent its spread. Here, using the Kepler conjecture for ideal packaging, the number of virions of different sizes that can be accommodated inside droplets was calculated and are proportional to the 3rd potency of the droplet/virion diameter. The differences between particles of 5 um and 100 μm are around four orders of magnitude, explaining why the airborne spread is much more difficult but still possible. There is no solid evidence yet that the airborne coronaviruses may reach enough concentration to infect, but this may be the case under certain circumstances. The WHO partially recognizes now this fact in a warning to health workers (from my point of view too late, as it was the declaration of a pandemic). Another issue is whether the virus stays infective in aerosols generated from patients. This has not been directly proved yet except with artificial aerosols, but there are no reasons why the virus cannot remain in the air and be infective if the viral charge and time of exposure are enough. We must also consider whether the virus can infect the intestine; there are some signs in this sense. Finally, and most importantly, we need to reduce interactions by using surgical masks to flatten the curve, leave the quarantine and avoid a rebound. For cultural reasons, a social distance of 2 meters (2M) is extremely hard to manage. Surgical masks do reduce the interactions in conditions of proximity and, therefore, help to “flatten the curve”. The WHO and CDC “laissez-faire” on this matter do not help and we are running out of time. Anticipated actions, such as the use of surgical masks for the general population, are critical.

The Airborne and Gastrointestinal Coronavirus SARS-COV-2 Pathways

Since there is not a clear consensus about the possibility for COVID-2 to be an airborne disease, a controversy also exists regarding the need to use surgical masks to prevent its spreading. Here, using the Kepler conjecture for ideal packaging, the number of virions of different sizes that can be accommodated inside droplets was calculated and are proportional to the 3rd potency of the droplet/virion diameter. The differences between particles of 5 um and 100 μm are around four orders of magnitude, explaining why the airborne spread is much more difficult but still possible. There is no solid evidence yet that the airborne coronaviruses may reach enough concentration to infect, but in certain circumstances, this may be true. The WHO partially recognizes now this fact in a warning to health workers (from my point of view too late as the pandemic declaration). Another issue is if the virus stays infective in aerosols generated from patients. This has not been directly probed yet except with artificial aerosols, but there are no reasons by which the virus cannot remain in the air and be infective if the viral charge and time of exposure are enough. Another issue is if the virus can infect the intestine; there are some signs in this sense. Finally, and most importantly, to flatten the curve, leave the quarantine, and avoid a rebound, we need to reduce the interactions by using surgical masks. For cultural reasons, a social distance of 2 meters (2M) is extremely hard to manage. Surgical masks do the task of reducing the interactions in conditions of proximity and, therefore, help to “flatten the curve”. The WHO and CDC “laissez-faire” in this matter does not help and we are running out of time. Anticipated actions, such as the use of surgical masks for the general population, are critical.

External Tools for the Formal Proof of the Kepler Conjecture

The Kepler conjecture asserts that no packing of congruent balls in three-dimensional Euclidean space has density greater than that of the familiar cannonball arrangement. The proof of the Kepler conjecture was announced in 1998, but it went several years without publication because of the lingering doubts of referees about the correctness of the proof. In response to these publication hurdles, the Flyspeck project seeks to give a complete formal proof of the Kepler conjecture using the proof assistant HOL Light.The original proof of the Kepler relies on long computer calculations, and these calculations present special formalization challenges. A major part of the Flyspeck project requires the integration of external computational tools with the proof assistant. Some of these external tools are the GNU linear programming kit, AMPL (a modeling language for mathematical programming), Mathematica calculations, nonlinear optimization, and custom code in C++, C, C#, Java, and Objective Caml.Earlier work by A. Solovyev has implemented efficient linear programming in HOL Light. This talk will include a description of his more recent work that automates the link between linear programming and the Flyspeck project.

A FORMAL PROOF OF THE KEPLER CONJECTURE

This article describes a formal proof of the Kepler conjecture on dense sphere packings in a combination of the HOL Light and Isabelle proof assistants. This paper constitutes the official published account of the now completed Flyspeck project.

Ultrahigh-pressure polyamorphism in GeO2 glass with coordination number >6

Knowledge of pressure-induced structural changes in glasses is important in various scientific fields as well as in engineering and industry. However, polyamorphism in glasses under high pressure remains poorly understood because of experimental challenges. Here we report new experimental findings of ultrahigh-pressure polyamorphism in GeO2 glass, investigated using a newly developed double-stage large-volume cell. The Ge–O coordination number (CN) is found to remain constant at ∼6 between 22.6 and 37.9 GPa. At higher pressures, CN begins to increase rapidly and reaches 7.4 at 91.7 GPa. This transformation begins when the oxygen-packing fraction in GeO2 glass is close to the maximal dense-packing state (the Kepler conjecture = ∼0.74), which provides new insights into structural changes in network-forming glasses and liquids with CN higher than 6 at ultrahigh-pressure conditions.