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We study the $(k+1,k)$ diagonal map for $k=2,3,4,...$. We call this map $\Delta_k$. The map $\Delta_1$ is the pentagram map and $\Delta_k$ is a generalization. $\Delta_k$ does not preserve convexity, but we prove that $\Delta_k$ preserves a subset $B_k$ of certain star-shaped polygons which we call $k$-birds. The action of $\Delta_k$ on $B_k$ seems similar to the action of $\Delta_1$ on the space of convex polygons. We show that some classic geometric results about $\Delta_1$ generalize to this setting., Comment: 60 pages, computer experiment inspired but mostly traditional math

We introduce the crisscross and the cup, both of which are immersed $3$-twist polygonal paper Moebius band of aspect ratio $3$. We explain why these two objects are limits of smooth embedded paper Moebius bands having knotted boundary. We conjecture that any smooth embedded paper Moebius band with knotted boundary has aspect ratio greater than $3$. The crisscross is planar but the cup is not., Comment: This is, in some sense, a sequel to arXiv 2308.12641. However, the material here is independent from the earlier paper

A smooth twisted paper cylinder of aspect ratio $\lambda$ is an isometric embedding of a $1 \times \lambda$ cylinder into $\pmb{R}^3$ such that the images of the boundary components are linked. We prove that for such an object to exist we must have $\lambda>2$ and that this bound is sharp. We also show that any sequence of examples having aspect ratio converging to $2$ must converge, up to isometries, to a certain $4$-fold wrapping of a right-angled isosceles triangle., Comment: This paper is a sequel to my paper about the optimal paper Moebius band, arXiv:2308.12641. This version is the same as the previous one except that (1) I correct a misstatement about the uniqueness of the right isosceles cylinder and (2) I mention the connection to folded ribbon knots

In this paper we prove that a smooth embedded paper Moebius band must have aspect ratio greater than $\sqrt 3$. We also prove that any sequence of smooth embedded paper Moebius bands whose aspect ratio converges to $\sqrt 3$ must converge, up to isometry, to the famous triangular Moebius band. These results answer the minimum aspect ratio question discussed by W. Wunderlich in 1962 and prove the more specific conjecture of B Halpern and C. Weaver from 1977., Comment: This paper has been accepted for publication by the Annals of Mathematics. This is the final version, the one I am publishing. This differs from the previous version in two ways: I removed an appendix and I changed the typesetting to the journal's specs

The purpose of this paper is to unite two games, symplectic billiards and tiling billiards. The new game is called symplectic tiling billiards. I will prove a result about periodic orbits of symplectic tiling billiards in a very special case and then show how this result is related to planar linkages and hyperbolic geometry., Comment: This version is quite similar to the previous one. I eliminated a typo and also (in Sect 3.1) I added a remark describing a very nice observation of Jannik Westermann and some forthcoming joint work of mine and Jannik's that is based on this observation

This work rigorously verifies the phase transition in 5-point energy minimization first observed by Melnyk-Knop-Smith in 1977. More precisely, we prove that there is a constant S = [15+24/512,15+25/512] such that the triangular bi-pyramid is the energy minimizer with respect to the s-power law potential for all s in (0,S) and some pyramid with square base is the unique minimizer for all s in (S,15+512/25]. Taking s=1 gives another solution to Thomson's 5 electron problem from 1904. The work here is a simplification of my monograph from 6 years ago which also proved the main result. The version here is half as long as the original. Also, I wrote it in an experimental style which facilitates the verification process. As I explicitly indicate in the text, the proof can be broken up into 10 independent pieces, all less than 15 pages, which can be checked independently of the other pieces., Comment: 77 pages long. This is a computer assisted proof, and the code is available from my website as indicated in the text. I further shortened the proof (and removed typos). My goal is to make this thing as short and as easily verifiable as possible

We study the geometry of some proper 4-colorings of the vertices of sphere triangulations with degree sequence 6,...,6,2,2,2. Such triangulations are the simplest examples which have non-negative combinatorial curvature. The examples we construct, which are roughly extremal in some sense, are based on a novel geometric interpretation of continued fractions. We also present a conjectural sharp "isoperimetric inequality" for colorings of this kind of triangulation., Comment: This version is the same as the previous one, except that I edited it to remove typos and other little glitches

In this paper we will give a short and direct proof that Wolfgang Kuehnel's 9-vertex triangulation of the complex projective plane really is the complex projective plane. The idea of our proof is to recall the trisection of the complex projective plane into 3 bi-disks and then to see this trisection inside a symmetry-breaking subdivision of the triangulation. Following the basic proof, we will elaborate on the construction., Comment: This is the version that will appear as an article in the Mathematical Intelligencer. I revised the paper according to the many helpful comments of a referee who had a supernatural understanding of this complex

The evolute of a curve is the envelope of its normals. In this note we consider a projectively natural discrete analog of this construction: we define projective perpendicular bisectors of the sides of a polygon in the projective plane, and study the map that sends a polygon to the new polygon formed by the projective perpendicular bisectors of its sides. We consider this map acting on the moduli space of projective polygons. We analyze the case of pentagons; the moduli space is 2-dimensional in this case. The second iteration of the map has one integral whose level curves are cubic curves, and the transformation on these level curves is conjugated to the map $x\mapsto -4x$ mod 1. We also present the results of an experimental study in the case of hexagons.

This note gives a very succinct conceptual proof of Brahmagupta's formula for cyclic quadrilaterals, and then a long discussion about how the proof is inspired by ideas from modern mathematics., Comment: This version is more polished. I added another picture and also another reference. Fixed a few typos

In this paper I will establish a special case of a conjecture that intertwines the deep diagonal pentagram maps and Poncelet polygons. The special case is that of the 3-diagonal map acting on affine equivalence classes of centrally symmetric octagons. This is the simplest case that goes beyond an analysis of elliptic curves. The proof involves establishing that the map is Arnold-Liouville integrable in this case, and then exploring the Lagrangian surface foliation in detail., Comment: This paper is a fairly substantial revision of the previous version. I revised it for publication. The new version is generally cleaner and has fewer typos. Also, the endgame of the proof is simpler

In this paper I will explain a rigidity conjecture that intertwines the deep diagonal pentagram maps and Poncelet polygons. I will also establish a simple case of the conjecture, the one involving the $3$-diagonal map on a convex $8$-gon with $4$-fold rotational symmetry. This case involves a textbook analysis of a pencil of elliptic curves., Comment: This version is the same as the previous one, except that I remove all the experimental observations about a more general case. I have now proved the more general case and posted it on the arXiv as a companion paper

The Spectrometer/Telescope for Imaging X-rays (STIX) is the HXR instrument onboard Solar Orbiter designed to observe solar flares over a broad range of flare sizes, between 4-150 keV. We report the first STIX observations of microflares recorded during the instrument commissioning phase in order to investigate the STIX performance at its detection limit. This first result paper focuses on the temporal and spectral evolution of STIX microflares occuring in the AR12765 in June 2020, and compares the STIX measurements with GOES/XRS, SDO/AIA, and Hinode/XRT. For the observed microflares of the GOES A and B class, the STIX peak time at lowest energies is located in the impulsive phase of the flares, well before the GOES peak time. Such a behavior can either be explained by the higher sensitivity of STIX to higher temperatures compared to GOES, or due to the existence of a nonthermal component reaching down to low energies. The interpretation is inconclusive due to limited counting statistics for all but the largest flare in our sample. For this largest flare, the low-energy peak time is clearly due to thermal emission, and the nonthermal component seen at higher energies occurs even earlier. This suggests that the classic thermal explanation might also be favored for the majority of the smaller flares. In combination with EUV and SXR observations, STIX corroborates earlier findings that an isothermal assumption is of limited validity. Future diagnostic efforts should focus on multi-wavelength studies to derive differential emission measure distributions over a wide range of temperatures to accurately describe the energetics of solar flares. Commissioning observations confirm that STIX is working as designed. As a rule of thumb, STIX detects flares as small as the GOES A class. For flares above the GOES B class, detailed spectral and imaging analyses can be performed., Comment: 19 pages, 11 figures

We consider the curve shortening flow applied to a class of figure-eight curves: those with dihedral symmetry, convex lobes, and a monotonicity assumption on the curvature. We prove that when (non-conformal) linear transformations are applied to the solution so as to keep the bounding box the unit square, the renormalized limit converges to a quadrilateral which we call a bowtie. Along the way we prove that suitably chosen arcs of our evolving curves, when suitably rescaled, converge to the Grim Reaper Soliton under the flow. Our Grim Reaper Theorem is an analogue of a theorem of S. Angenent, which is proven in the locally convex case., Comment: Our paper was accepted to the Journal of Differential Geometry, and this version is the final version we sent to the journal. It agrees with the previous version except that we corrected a few typos and also shortened the proof of the Migration Lemma in the last chapter

Let $X$ be the regular dodecahedron, equipped with its intrinsic path metric. Given $p \in X$ let $G(p)=-q$ where $q$ is the point on $X$ which maximizes the distance to $p$. (Generically, $G$ is single-valued.) We give a complete description of the map $G$ and as a consequence show that the $\omega$-limit set of $G$ is the $1$-skeleton of a subdivision of $X$ into $180$ convex quadrilaterals. $G$ is a piecewise bi-quadratic map, and each algebraic piece is defined by a straight line construction involving a rhombus. The rhombi involved have the same shapes as the ones in the Penrose tiling. Our proof is computer-assisted but rigorous., Comment: 64 pages, computer assisted proof. I am disappointed at the length and complexity of the proof, and I don't know if I will try to publish this paper and thereby inflict it on a referee. However, I think it is worth having this result, and some proof, on the record

Let $A=\sqrt 3$. There are two main conjectures about paper Moebius bands. First, a smooth embedded paper Moebius band must have aspect ratio at least A. Second, any sequence of smooth embedded paper Moebius bands having aspect ratio converging to A converges, in the Hausdorff topology and up to isometries, to an equilateral triangle of semiperimeter A. We will reduce these conjectures to 10 statements about the positivity of certain explicit piecewise algebraic expressions., Comment: This paper is completely superseded by my new paper, "The Optimal Paper Moebius Band" (arXiv:2308.12641) The new paper solves the conjecture that the now withdrawn paper made partial progress towards and thereby renders the withdrawn paper irrelevant

This paper gives another proof of the key lemma in my recent paper which solves the optimal paper Moebius band conjecture of Halpern and Weaver, namely Lemma T. The proof here is longer but it offers more geometric intuition about what is going on., Comment: This paper is entirely superseded by my newer paper, arXiv:2308.12641. Now that I have thought a lot about the T-pattern lemma proved in this paper, I think that the long proof here is not worth reading

Several aspects of early language skills, including parent-report measures of vocabulary, phoneme discrimination, speech segmentation, and speed of lexical access predict later childhood language outcomes. To date, no studies have examined the long-term predictive validity of novel word learning. We examined whether individual differences in novel word learning at 21 months predict later childhood receptive vocabulary outcomes rather than generalized cognitive abilities. Twenty-eight 21-month-olds were taught novel words using a modified version of the Intermodal Preferential Looking Paradigm. Seventeen children (range 7-10 years) returned to participate in a longitudinal follow-up. Novel word learning in infancy uniquely accounted for 22% of the variance in childhood receptive vocabulary but did not predict later childhood visuospatial ability or non-verbal IQ. These results suggest that the ability to associate novel sound patterns to novel objects, an index of the PROCESS of word learning, may be especially important for long-term language mastery. [This is the in press version of an article published in "Journal of Child Language." For the published version of this article, see EJ1218484.]

In this paper we show that a smoothly and locally isometrically embedded Moebius band has aspect ratio at least $\sqrt 3-(1/26)$. (The actual bound, an algebraic number that arises in an optimization problem, is a tiny bit better.) Our bound improves significantly on the previous known lower bound of $\pi/2$., Comment: This paper is entirely superseded by my new paper "The Optimal Paper Moebius Band" (arXiv:2308.12641). This withdrawn paper makes partial progress on the conjecture that the new paper proves in full. Also, the new paper corrects an error in this withdrawn paper

Graph-based extractive document summarization relies on the quality of the sentence similarity graph. Bag-of-words or tf-idf based sentence similarity uses exact word matching, but fails to measure the semantic similarity between individual words or to consider the semantic structure of sentences. In order to improve the similarity measure between sentences, we employ off-the-shelf deep embedding features and tf-idf features, and introduce a new text similarity metric. An improved sentence similarity graph is built and used in a submodular objective function for extractive summarization, which consists of a weighted coverage term and a diversity term. A Transformer based compression model is developed for sentence compression to aid in document summarization. Our summarization approach is extractive and unsupervised. Experiments demonstrate that our approach can outperform the tf-idf based approach and achieve state-of-the-art performance on the DUC04 dataset, and comparable performance to the fully supervised learning methods on the CNN/DM and NYT datasets.

On any compact space one can consider the map which sends a point to the set of points farthest from this point. In nice cases, there is just a single point farthest from a given point and so by restricting the domain slightly one can form a dynamical system on the space based on this map. We give a complete characterization of this map, including explicit formulas, when the metric space is the regular octahedron equipped with its intrinsic flat cone metric., Comment: This version of the paper is being published, as is, in the Journal of Experimental math. It is very similar to the previous version, except that I corrected some typos and mildly simplied the argument in Chapter 3

Let Sol be the $3$-dimensional solvable Lie group whose underlying space is $\mathbb{R}^3$ and whose left-invariant Riemannian metric is given by $$e^{-2z} dx^2 + e^{2z} dy^2 + dz^2.$$ Building on previous joint work with Matei Coiculescu, which characterizes the cut locus in Sol, we prove that the sphere of radius r in sol has area at most $20 \pi e^r$ provided that r is sufficiently large. This estimate is sharp up to a factor of 10, Comment: The previous version of the paper had a bound of $611 e^r$. In this version, I improve the constant from $611$ to $20 \pi$. This is an order of magnitude better. The proof is essentially the same. I am just more careful with the estimates in a few places

Maximum Entropy is an image reconstruction method conceived to image a sparsely occupied field of view and therefore particularly appropriate to achieve super-resolution effects. Although widely used in image deconvolution, this method has been formulated in radio astronomy for the analysis of observations in the spatial frequency domain, and an Interactive Data Language (IDL) code has been implemented for image reconstruction from solar X-ray Fourier data. However, this code relies on a non-convex formulation of the constrained optimization problem addressed by the Maximum Entropy approach and this sometimes results in unreliable reconstructions characterized by unphysical shrinking effects. This paper introduces a new approach to Maximum Entropy based on the constrained minimization of a convex functional. In the case of observations recorded by the Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI), the resulting code provides the same super-resolution effects of the previous algorithm, while working properly also when that code produces unphysical reconstructions. Results are also provided of testing the algorithm with synthetic data simulating observations of the Spectrometer/Telescope for Imaging X-rays (STIX) in Solar Orbiter. The new code is available in the {\em{HESSI}} folder of the Solar SoftWare (SSW)tree.

We prove some recent experimental observations of D. Reznik concerning periodic billiard orbits in ellipses. For example, the sum of cosines of the angles of a periodic billiard polygon remains constant in the one-parameter family of such polygons (that exist due to the Poncelet porism). In our proofs, we use geometric and complex analytic methods., Comment: 12 pages, 9 figures

In this article, we study polygonal symplectic billiards. We provide new results, some of which are inspired by numerical investigations. In particular, we present several polygons for which all orbits are periodic. We demonstrate their properties and derive various conjectures using two numerical implementations., Comment: 23 pages, 24 figures

Let Sol be the three-dimensional solvable Lie group equipped with its standard left-invariant Riemannian metric. We give a precise description of the cut locus of the identity, and a maximal domain in the Lie algebra on which the Riemannian exponential map is a diffeomorphism. As a consequence, we prove that the metric spheres in Sol are topological spheres, and we characterize their singular points almost exactly., Comment: 36 pages, 6 figures. This version is a revision. We revised it according to the referee reports from our Geometry&Topology submission of the first version. The exposition is somewhat cleaner, especially in Chapter 3, and we have added some new diagrams and computer plots

In this note, we will explain the connection between the Seven Circles Theorem and hyperbolic geometry, then prove a stronger result about hyperbolic geometry hexagons which implies the Seven Circles Theorem as a special case., Comment: 5 figures

Meyerson's Theorem says that all but at most 2 points of any Jordan loop are vertices of inscribed equilateral triangles. We show that for any Jordan loop there are uncountable many other triangle shapes for which this same result is true. Our result comes from taking the limit of a structural result about spaces of triangles inscribed in polygons., Comment: The proof of the result in Chapter 4 (the Component Lemma) has a gap which I cannot seem to fix. I will return the paper to the arXiv if I figure out how to fix it

Image saturation has been an issue for several instruments in solar astronomy, mainly at EUV wavelengths. However, with the launch of the Atmospheric Imaging Assembly (AIA) as part of the payload of the Solar Dynamic Observatory (SDO) image saturation has become a big data issue, involving around 10^$ frames of the impressive dataset this beautiful telescope has been providing every year since February 2010. This paper introduces a novel desaturation method, which is able to recover the signal in the saturated region of any AIA image by exploiting no other information but the one contained in the image itself. This peculiar methodological property, jointly with the unprecedented statistical reliability of the desaturated images, could make this algorithm the perfect tool for the realization of a reconstruction pipeline for AIA data, able to work properly even in the case of long-lasting, very energetic flaring events.

A Steiner chain of length k consists of k circles, tangent to two given non-intersecting circles (the parent circles) and tangent to each other in a cyclic pattern. The Steiner porism states that once a chain of k circles exists, there exists a 1-parameter family of such chains with the same parent circles that can be constructed starting with any initial circle, tangent to the parent circles. What do the circles in these 1-parameter family of Steiner chains of length k have in common? We prove that the first k-1 moments of their curvatures remain constant within a 1-parameter family. For k=3, this follows from the Descartes Circle Theorem. We extend our result to Steiner chains in the spherical and hyperbolic geometries and present a related more general theorem involving spherical designs.

We prove an integral formula for continuous paths of rectangles inscribed in a piecewise smooth loop. We then use this integral formula to show that (with a very mild genericity hypothesis) the number of rectangle coincidences, informally described as the number of inscribed rectangles minus the number of isometry classes of inscribed rectangles, grows linearly with the number of positively oriented extremal chords -- a.k.a. diameters -- of the polygon, Comment: 19 pages, traditional proof. This update corrects a glitch in which I left off the statement that the extremal chords need to be positively oriented, in a sense that is explained in the paper

Let g be an arbitrary Jordan loop and let G denote the space of rectangles R which are inscribed in g in such a way that the cyclic order of the vertices of R is the same whether it is induced by R or by g. We prove that G contains a connected set S satisfying one of three properties: 1. S consists of rectangles of uniformly large area, including a square, and every point of g is the vertex of a rectangle in S. 2. S consists of rectangles having all possible aspect ratios, and all but at most 4 points of g are vertices of rectangles in S. 3. S contains rectangles of every sufficiently small diameter, and all but at most 2 points of g are vertices of rectangles in S., Comment: 32 pages. This paper is a revision of the first version, inspired in part by comments from an anonymous referee. The current version of the paper will probably appear in Geometriae Dedicata

The Miniature X-ray Solar Spectrometer (MinXSS) CubeSat is the first solar science oriented CubeSat mission flown for the NASA Science Mission Directorate, with the main objective of measuring the solar soft X-ray (SXR) flux and a science goal of determining its influence on Earth's ionosphere and thermosphere. These observations can also be used to investigate solar quiescent, active region, and flare properties. The MinXSS X-ray instruments consist of a spectrometer, called X123, with a nominal 0.15 keV full-width-half-maximum (FWHM) resolution at 5.9 keV and a broadband X-ray photometer, called XP. Both instruments are designed to obtain measurements from 0.5 - 30 keV at a nominal time cadence of 10 seconds. A description of the MinXSS instruments, performance capabilities, and relation to the Geostationary Operational Environmental Satellite (GOES) 0.1 - 0.8 nm flux are discussed in this article. Early MinXSS results demonstrate the capability to measure variations of the solar spectral SXR flux between 0.8 - 12 keV from at least GOES A5 - M5 (5 $\times$ 10$^{-8}$ - 5 $\times$ 10$^{-5}$ W m$^{-2}$) levels and infer physical properties (temperature and emission measure) from the MinXSS data alone. Moreover, coronal elemental abundances can be inferred, specifically Fe, Ca, Si, Mg, S, Ar, and Ni, when there is sufficiently high count rate at each elemental spectral feature. Additionally, temperature response curves and emission measure loci demonstrate the MinXSS sensitivity to plasma emission at different temperatures. MinXSS observations coupled with those from other solar observatories can help address some of the most compelling questions in solar coronal physics. Finally, simultaneous observations by MinXSS and Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI) can provide the most spectrally complete soft X-ray solar flare photon flux measurements to date., Comment: 45 pages, 16 figures, accepted for publication in Solar Physics

We describe a new technique to measure the height of the X-ray limb with observations from occulted X-ray flare sources as observed by the RHESSI (the Reuven Ramaty High-Energy Spectroscopic Imager) satellite. This method has model dependencies different from those present in traditional observations at optical wavelengths, which depend upon detailed modeling involving radiative transfer in a medium with complicated geometry and flows. It thus provides an independent and more rigorous measurement of the "true" solar radius, meaning that of the mass distribution. RHESSI's measurement makes use of the flare X-ray source's spatial Fourier components (the visibilities), which are sensitive to the presence of the sharp edge at the lower boundary of the occulted source. We have found a suitable flare event for analysis, SOL2011-10-20T03:25 (M1.7), and report a first result from this novel technique here. Using a 4-minute integration over the 3-25 keV photon energy range, we find $R_{\mathrm{X-ray}} = 960.11\ \pm\ 0.15 \pm 0.29$ arcsec, at 1 AU, where the uncertainties include statistical uncertainties from the method and a systematic error. The standard VAL-C model predicts a value of 959.94 arcsec, about 1$\sigma$ below our value., Comment: 12 pages, 5 figures, accepted for publication in ApJ

Double coronal hard X-ray (HXR) sources are believed to be critical observational evidence of bi-directional energy release through magnetic reconnection in a large-scale current sheet in solar ares. Here we present a study on double coronal sources observed in both HXR and microwave regimes, revealing new characteristics distinct from earlier reports. This event is associated with a footpoint-occulted X1.3-class flare (25 April 2014, starting at 00:17 UT) and a coronal mass ejection that are likely triggered by the magnetic breakout process, with the lower source extending upward from the top of the partially-occulted flare loops and the upper source co-incident with rapidly squeezing-in side lobes (at a speed of ~250 km/s on both sides). The upper source can be identified at energies as high as 70-100 keV. The X-ray upper source is characterized by flux curves different from the lower source, a weak energy dependence of projected centroid altitude above 20 keV, a shorter duration and a HXR photon spectrum slightly-harder than those of the lower source. In addition, the microwave emission at 34 GHz also exhibits a similar double source structure and the microwave spectra at both sources are in line with gyro-synchrotron emission given by non- thermal energetic electrons. These observations, especially the co-incidence of the very-fast squeezing-in motion of side lobes and the upper source, indicate that the upper source is associated with (possibly caused by) this fast motion of arcades. This sheds new lights on the origin of the corona double-source structure observed in both HXRs and microwaves., Comment: 7 figures