Moduli space actions on the Hochschild Co-Chains of a Frobenius algebra I: Cell Operads

This is the first of two papers in which we prove that a cell model of the moduli space of curves with marked points and tangent vectors at the marked points acts on the Hochschild co--chains of a Frobenius algebra. We also prove that a there is dg--PROP action of a version of Sullivan Chord diagrams which acts on the normalized Hochschild co-chains of a Frobenius algebra. These actions lift to operadic correlation functions on the co--cycles. In particular, the PROP action gives an action on the homology of a loop space of a compact simply--connected manifold. In this first part, we set up the topological operads/PROPs and their cell models. The main theorems of this part are that there is a cell model operad for the moduli space of genus $g$ curves with $n$ punctures and a tangent vector at each of these punctures and that there exists a CW complex whose chains are isomorphic to a certain type of Sullivan Chord diagrams and that they form a PROP. Furthermore there exist weak versions of these structures on the topological level which all lie inside an all encompassing cyclic (rational) operad.Comment: 50 pages, 7 figures. Newer version has minor changes. Some material shifted. Typos and small things correcte

Chains

Chains is a poem that was inspired by the events surrounding the Steubenville Rape Case, and it is my interpretation of what the victim could have been feeling. The poem was written as a way for me to try to understand how something like this could have happened

Falling chains

The one-dimensional fall of a folded chain with one end suspended from a rigid support and a chain falling from a resting heap on a table is studied. Because their Lagrangians contain no explicit time dependence, the falling chains are conservative systems. Their equations of motion are shown to contain a term that enforces energy conservation when masses are transferred between subchains. We show that Cayley's 1857 energy nonconserving solution for a chain falling from a resting heap is incorrect because it neglects the energy gained when a transferred link leaves a subchain. The maximum chain tension measured by Calkin and March for the falling folded chain is given a simple if rough interpretation. Other aspects of this falling folded chain are briefly discussed.Comment: 9 pages, 1 figure; the Abstract has been shortened, three paragraphs have been re-written for greater clarity, and textual improvements have been made throughout the paper; to be published by the Am. J. Physic

Tracking chains revisited

The structure ${\cal C}_2:=(1^\infty,\le,\le_1,\le_2)$, introduced and first analyzed in Carlson and Wilken 2012 (APAL), is shown to be elementary recursive. Here, $1^\infty$ denotes the proof-theoretic ordinal of the fragment $\Pi^1_1$-$\mathrm{CA}_0$ of second order number theory, or equivalently the set theory $\mathrm{KPl}_0$, which axiomatizes limits of models of Kripke-Platek set theory with infinity. The partial orderings $\le_1$ and $\le_2$ denote the relations of $\Sigma_1$- and $\Sigma_2$-elementary substructure, respectively. In a subsequent article we will show that the structure ${\cal C}_2$ comprises the core of the structure ${\cal R}_2$ of pure elementary patterns of resemblance of order $2$. In Carlson and Wilken 2012 (APAL) the stage has been set by showing that the least ordinal containing a cover of each pure pattern of order $2$ is $1^\infty$. However, it is not obvious from Carlson and Wilken 2012 (APAL) that ${\cal C}_2$ is an elementary recursive structure. This is shown here through a considerable disentanglement in the description of connectivity components of $\le_1$ and $\le_2$. The key to and starting point of our analysis is the apparatus of ordinal arithmetic developed in Wilken 2007 (APAL) and in Section 5 of Carlson and Wilken 2012 (JSL), which was enhanced in Carlson and Wilken 2012 (APAL) specifically for the analysis of ${\cal C}_2$.Comment: The text was edited and aligned with reference [10], Lemma 5.11 was included (moved from [10]), results unchanged. Corrected Def. 5.2 and Section 5.3 on greatest immediate $\le_1$-successors. Updated publication information. arXiv admin note: text overlap with arXiv:1608.0842

Magnetically hindered chain formation in transition-metal break junctions

Based on first-principles calculations, we demonstrate that magnetism impedes the formation of long chains in break junctions. We find a distinct softening of the binding energy of atomic chains due to the creation of magnetic moments that crucially reduces the probability of successful chain formation. Thereby, we are able to explain the long standing puzzle why most of the transition-metals do not assemble as long chains in break junctions and provide thus an indirect evidence that in general suspended atomic chains in transition-metal break junctions are magnetic.Comment: 5 pages, 3 figure