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Register a new userThrowing a Sofa Through the Window
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Added by
Dan Halperin
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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We study several variants of the problem of moving a convex polytope $K$, with $n$ edges, in three dimensions through a flat rectangular (and sometimes more general) window. Specifically: $bullet$ We study variants where the motion is restricted to translations only, discuss situations where such a motion can be reduced to sliding (translation in a fixed direction), and present efficient algorithms for those variants, which run in time close to $O(n^{8/3})$. $bullet$ We consider the case of a `gate (an unbounded window with two parallel infinite edges), and show that $K$ can pass through such a window, by any collision-free rigid motion, if and only if it can slide through it. $bullet$ We consider arbitrary compact convex windows, and show that if $K$ can pass through such a window $W$ (by any motion) then $K$ can slide through a gate of width equal to the diameter of $W$. $bullet$ We study the case of a circular window $W$, and show that, for the regular tetrahedron $K$ of edge length $1$, there are two thresholds $1 > delta_1approx 0.901388 > delta_2approx 0.895611$, such that (a) $K$ can slide through $W$ if the diameter $d$ of $W$ is $ge 1$, (b) $K$ cannot slide through $W$ but can pass through it by a purely translational motion when $delta_1le d < 1$, (c) $K$ cannot pass through $W$ by a purely translational motion but can do it when rotations are allowed when $delta_2 le d < delta_1$, and (d) $K$ cannot pass through $W$ at all when $d < delta_2$. $bullet$ Finally, we explore the general setup, where we want to plan a general motion (with all six degrees of freedom) for $K$ through a rectangular window $W$, and present an efficient algorithm for this problem, with running time close to $O(n^4)$.
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The architecture of exoplanetary systems is often different from the solar system, with some exoplanets being in close orbits around their host stars and having orbital periods of only a few days. In analogy to interactions between stars in close binary systems, one may expect interactions between the star and the exoplanet as well. From theoretical considerations, effects on the host star through tidal and magnetic interaction with the exoplanet are possible; for the exoplanet, some interesting implications are the evaporation of the planetary atmosphere and potential effects on the planetary magnetism. In this review, several possible interaction pathways and their observational prospects and existing evidence are discussed. A particular emphasis is put on observational opportunities for these kinds of effects in the high-energy regime.
What initial trajectory angle maximizes the arc length of an ideal projectile? We show the optimal angle, which depends neither on the initial speed nor on the acceleration of gravity, is the solution x to a surprising transcendental equation: csc(x) = coth(csc(x)), i.e., x = arccsc(y) where y is the unique positive fixed point of coth. Numerically, $x approx 0.9855 approx 56.47^circ$. The derivation involves a nice application of differentiation under the integral sign.
The experimental issue of the search for new particles of unknown mass poses the challenge of exploring a wide interval to look for the usual signatures represented by excess of events above the background. A side effect of such a broad range quest is that the traditional significance calculations valid for signals of known location are no more applicable when such an information is missing. In this note the specific signal search approach via observation windows sliding over the range of interest is considered; in the assumptions of known background and of fixed width of the exploring windows the statistical implications of such a search scheme are described, with special emphasis on the correct significance assessment for a claimed discovery.
It is well-known that the Continuum Hypothesis (CH) is independent of the other axioms of Zermelo-Fraenkel set theory with choice (ZFC). This raises the question of whether an intuitive justification exists for CH as an additional axiom, or conversely whether it is more intuitive to deny CH. Freilings Axiom of Symmetry (AS) is one candidate for an intuitively justifiable axiom that, when appended to ZFC, is equivalent to the denial of CH. The intuition relies on a probabalistic argument, usually cast in terms of throwing random darts at the real line, and has been defended by researchers as well as popular writers. In this note, the intuitive argument is reviewed. Following William Abram, it is suggested that while accepting CH leads directly to a counterexample to AS, this is not necessarily fatal to our intuition. Instead, we suggest, it serves to alert us to the error in a naive intuition that leaps too readily from the near-certainty of individual events to near-certainty of a joint event.
The simplest possibility to explain the baryon asymmetry of the Universe is to assume that radiation is created asymmetrically between baryons and anti-baryons after the inflation. We propose a new mechanism of this kind where CP-violating flavor oscillations of left-handed leptons in the reheating era distribute the lepton asymmetries partially into the right-handed neutrinos while net asymmetry is not created. The asymmetry stored in the right-handed neutrinos is later washed out by the lepton number violating decays, and it ends up with the net lepton asymmetry in the Standard Model particles, which is converted into the baryon asymmetry by the sphaleron process. This scenario works for a range of masses of the right-handed neutrinos while no fine-tuning among the masses is required. The reheating temperature of the Universe can be as low as $O(10)$~TeV if we assume that the decays of inflatons in the perturbative regime are responsible for the reheating. For the case of the reheating via the dissipation effects, the reheating temperature can be as low as $O(100)$~GeV.
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