Line Patterns in Free Groups

We study line patterns in a free group by considering the topology of the decomposition space, a quotient of the boundary at infinity of the free group related to the line pattern. We show that the group of quasi-isometries preserving a line pattern in a free group acts by isometries on a related space if and only if there are no cut pairs in the decomposition space.Comment: 35 pages, 22 figures, PDFLatex; v2. finite index requires extra hypothesis; v3. 37 pages, 24 figures: updated references and add example in Section 6.3 of a rigid pattern for which the free group is not finite index in the group of pattern preserving quasi-isometries; v4. 40 pages, 26 figures: improved exposition and add example in Section 6.4 of a rigid pattern whose cube complex is not a tre

Принцип разума

Academician Vladimir Ivanovich Vernadsky, and his contemporary, Albert Einstein, situated the summation of their greatest scientific achievements within that Riemannian concept of dynamics which is traced, formally, in modern science, from Gottfried Leibniz’s 1690s resurrection of that concept of dynamis known to the Classical Greek of the Pythagoreans and Plato. As Einstein emphasized, the relevance of this for the presently known foundations of competent modern science, is expressed in that uniquely original discovery of the general principle of gravitation by Johannes Kepler, as in Kepler’s The Harmonies of the World. When our attention is turned to include the subject of certain related, deeper implications concerning the human mind, implications which are prompted from within Vernadsky’s treatment of the Noösphere, a certain, implicitly very important, but presently still controversial question is posed. That subject is to be identified as a topic within the framework of a unified field theory. Albert Einstein posed the question, and Academician Vernadsky, whether one presumes that he knew it, or not, supplied a crucial clue which leads in the direction of the solution. That is the subject here.Академик Владимир Иванович Вернадский и его современник Альберт Эйнштейн рассматривали свои величайшие научные достижения в рамках римановского понятия динамики, которое явно прослеживается в современной науке от возрождения Готфридом Лейбницем понятия «дюнамис», известного еще в классическом греческом языке Платона и последователей Пифагора. Как отмечал сам Эйнштейн, его значение для известных в настоящее время базовых положений современной науки проявляется в оригинальном открытии общего принципа тяготения Иоганнесом Кеплером, как это было изложено в работе последнего «Гармония мира» (Harmonices Mundi). Обращаясь к предмету определенных смежных, более глубоких последствий, связанных с человеческим разумом и вытекающих из учения Вернадского о ноосфере, нам приходится иметь дело с очень важным, но все еще спорным вопросом

Slender-ribbon theory

Ribbons are long narrow strips possessing three distinct material length scales (thickness, width, and length) which allow them to produce unique shapes unobtainable by wires or filaments. For example when a ribbon has half a twist and is bent into a circle it produces a M\"obius strip. Significant effort has gone into determining the structural shapes of ribbons but less is know about their behavior in viscous fluids. In this paper we determine, asymptotically, the leading-order hydrodynamic behavior of a slender ribbon in Stokes flows. The derivation, reminiscent of slender-body theory for filaments, assumes that the length of the ribbon is much larger than its width, which itself is much larger than its thickness. The final result is an integral equation for the force density on a mathematical ruled surface, termed the ribbon plane, located inside the ribbon. A numerical implementation of our derivation shows good agreement with the known hydrodynamics of long flat ellipsoids, and successfully captures the swimming behavior of artificial microscopic swimmers recently explored experimentally. We also study the asymptotic behavior of a ribbon bent into a helix, that of a twisted ellipsoid, and we investigate how accurately the hydrodynamics of a ribbon can be effectively captured by that of a slender filament. Our asymptotic results provide the fundamental framework necessary to predict the behavior of slender ribbons at low Reynolds numbers in a variety of biological and engineering problems.This research was funded in part by the European Union through a Marie Curie CIG Grant and the Cambridge Trusts.This is the author accepted manuscript. The final version is available from American Institute of Physics via http://dx.doi.org/10.1063/1.493856

A nonaspherical cell-like 2-dimensional simply connected continuum and related constructions

We prove the existence of a 2-dimensional nonaspherical simply connected cell-like Peano continuum (the space itself was constructed in one of our earlier papers). We also indicate some relations between this space and the well-known Griffiths' space from the 1950's

Comparison of microscale sealed vessel pyrolysis (MSSVpy) and hydropyrolysis (Hypy) for the characterisation of extant and sedimentary organic matter

Microscale sealed vessel pyrolysis (MSSVpy) and catalytic hydropyrolysis (Hypy) combined with gas chromatography-mass spectrometry have emerged in recent years as useful and versatile organic analytical and characterisation methods. Both now commercially available, these pyrolysis methods complement traditional flash pyrolysis analysis which can be limited by excessive degradation or inadequate chromatographic resolution of pyrolysates of high structural polarity. To assess the versatility and merits of these two pyrolysis methods they were separately applied to several organic samples reflecting different thermal maturities. This comparison revealed many product similarities, but also several important features unique to each

Peak reduction technique in commutative algebra

The "peak reduction" method is a powerful combinatorial technique with applications in many different areas of mathematics as well as theoretical computer science. It was introduced by Whitehead, a famous topologist and group theorist, who used it to solve an important algorithmic problem concerning automorphisms of a free group. Since then, this method was used to solve numerous problems in group theory, topology, combinatorics, and probably in some other areas as well. In this paper, we give a survey of what seems to be the first applications of the peak reduction technique in commutative algebra and affine algebraic geometry.Comment: survey; 10 page

Pre-Settlement Vegetation at Casey\u27s Paha State Preserve, Iowa

Paha are loess-capped ridges standing 10-30 m above the surrounding plain of the Iowan Surface. Although Iowa was almost entirely covered with prairie and wetlands just prior to Euro-American settlement, the paha are believed to have been forested based on soil types and on early vegetation maps. The objective of this study was to find evidence that paha were forested by measuring the δ13C value of humin, the fraction of soil organic matter that is insoluble in acid and base. Previous work has shown that humin retains the δ13C signature of vegetation on a 1000-year time scale, as opposed to the more mobile and soluble humic and fulvic acids that reflect the δ13C signature of more recent vegetation. Soil samples were obtained from Casey\u27s Paha State Preserve in Tama County from four locations at depths ranging from 5-85 cm. Carbonates were removed with 1.0 M HCl and humic and fulvic acids were removed by repeated application of 0.5 M NaOH. The δ13C values of the humin fraction (-22.031% to -24.358%) were within or slightly above the upper range of δ13C values for woody vegetation (-23% to -34%) and well below the range for prairie grasses (-9% to -17%). Although it has been suggested that prairie fires bypassed the paha or that perched water tables maintained the forest, we suggest that the paha forests resulted from activity by Native Americans

Probing small-x parton densities in proton- proton (-nucleus) collisions in the very forward direction

We present calculations of several pp scattering cross sections with potential applications at the LHC. Significantly large rates for momentum fraction, x, as low as 10^-7 are obtained, allowing for possible extraction of quark and gluon densities in the proton and nuclei down to these small x values provided a detector with good acceptance at maximal rapidities is used.Comment: 14 pages, LaTeX, 12 figures, uses revtex.st

Representational dissimilarity metric spaces for stochastic neural networks

Quantifying similarity between neural representations -- e.g. hidden layer activation vectors -- is a perennial problem in deep learning and neuroscience research. Existing methods compare deterministic responses (e.g. artificial networks that lack stochastic layers) or averaged responses (e.g., trial-averaged firing rates in biological data). However, these measures of deterministic representational similarity ignore the scale and geometric structure of noise, both of which play important roles in neural computation. To rectify this, we generalize previously proposed shape metrics (Williams et al. 2021) to quantify differences in stochastic representations. These new distances satisfy the triangle inequality, and thus can be used as a rigorous basis for many supervised and unsupervised analyses. Leveraging this novel framework, we find that the stochastic geometries of neurobiological representations of oriented visual gratings and naturalistic scenes respectively resemble untrained and trained deep network representations. Further, we are able to more accurately predict certain network attributes (e.g. training hyperparameters) from its position in stochastic (versus deterministic) shape space