scholarly journals Tetrahedra on Deformed and Integral Group Cohomology

10.37236/82 ◽  
2009 ◽  
Vol 16 (2) ◽  
Author(s):  
Pavle V. M. Blagojević ◽  
Günter M. Ziegler
Keyword(s):  
Convex Hull ◽  
Index Theory ◽  
Group Cohomology ◽  
Equal Length ◽  
The Other ◽  
Partial Result ◽  
Integral Group ◽  
Integer Coefficients ◽  
Continuous Map ◽  
Borsuk Problem

We show that for every injective continuous map $f:S^2\rightarrow{\Bbb R}^3$ there are four distinct points in the image of $f$ such that the convex hull is a tetrahedron with the property that two opposite edges have the same length and the other four edges are also of equal length. This result represents a partial result for the topological Borsuk problem for ${\Bbb R}^3$. Our proof of the geometrical claim, via Fadell–Husseini index theory, provides an instance where arguments based on group cohomology with integer coefficients yield results that cannot be accessed using only field coefficients.

scholarly journals The maximum surface area polyhedron with five vertices inscribed in the sphere {\bb S}^{2}

2021 ◽  
Vol 77 (1) ◽  
pp. 67-74
Author(s):  
Jessica Donahue ◽  
Steven Hoehner ◽  
Ben Li
Keyword(s):  
Convex Hull ◽  
Surface Area ◽  
Coordination Polyhedron ◽  
Unit Sphere ◽  
The Other ◽  
Optimal Placement ◽  
The North ◽  
Trigonal Bipyramidal ◽  
Bipyramidal Structure ◽  
North And South

This article focuses on the problem of analytically determining the optimal placement of five points on the unit sphere {\bb S}^{2} so that the surface area of the convex hull of the points is maximized. It is shown that the optimal polyhedron has a trigonal bipyramidal structure with two vertices placed at the north and south poles and the other three vertices forming an equilateral triangle inscribed in the equator. This result confirms a conjecture of Akkiraju, who conducted a numerical search for the maximizer. As an application to crystallography, the surface area discrepancy is considered as a measure of distortion between an observed coordination polyhedron and an ideal one. The main result yields a formula for the surface area discrepancy of any coordination polyhedron with five vertices.

A Contribution to the Knowledge of the Bot-flies, Gastrophilus intestinalis, DeG., G. haemorrhoidalis, L., and G. nasalis, L.

1918 ◽  
Vol 9 (2) ◽  
pp. 91-106 ◽  
Author(s):  
S. Hadwen ◽  
A. E. Cameron
Keyword(s):  
The Body ◽  
Entire Length ◽  
Equal Length ◽  
The Other ◽  
Brownish Black

The eggs of the three species of bot-flies discussed in this paper are distinguished by the fact of that of G. haemorrhoidalis being the only one stalked. It is also longer than those of the other two species, which are of about equal length. Further, it is brownish black in colour, that of G. intestinalis being whitish yellow and G. nasalis yellow. The egg of G. intestinalis adheres to the hair by clasping flanges, which run only two-thirds of its length, whilst the flanges of the G. nasalis egg run almost the entire length.The egg of G. haemorrhoidalis is not inserted nor screwed into the skin of the host. The eggs of G. intestinalis are laid indiscriminately on the body of the host, but preferably on the long hairs investing the inside of the foreleg. G. nasalis lays its eggs on the hairs of the intermaxillary space, and G. haemorrhoidalis on the hairs of the lips, preferably the lower.

Assessing Students' Understanding through Conversations

2007 ◽  
Vol 14 (5) ◽  
pp. 260-264
Author(s):  
Cecilia M. Vanderhye ◽  
Cynthia M. Zmijewski Demers
Keyword(s):  
Fourth Grade ◽  
Equal Length ◽  
The Other ◽  
Grade Classroom

In a fourth-grade classroom, the teacher told her students that the area of the rectangle she had drawn on the board was twelve centimeters square. The students knew that one side was three centimeters long, and they were asked to figure out the length of the other sides. One student offered an answer: “The shape looks like a square, so, since all sides of a square have equal length, the other sides must be three inches long.”

scholarly journals Characterization algorithms for shift radix systems with finiteness property

2014 ◽  
Vol 11 (01) ◽  
pp. 211-232 ◽  
Author(s):  
Mario Weitzer
Keyword(s):  
Convex Hull ◽  
Convex Polyhedron ◽  
The Other ◽  
Closed Convex Hull ◽  
Connected Component ◽  
Complicated Structure ◽  
Finiteness Property ◽  
A Chain ◽  
Interior Points

For d ∈ ℕ and r ∈ ℝd, let τr : ℤd → ℤd, where τr(a) = (a2, …, ad, -⌊ra⌋) for a = (a1, …, ad), denote the (d-dimensional) shift radix system associated with r. τr is said to have the finiteness property if and only if all orbits of τr end up in (0, …, 0); the set of all corresponding r ∈ ℝd is denoted by [Formula: see text], whereas 𝒟d consists of those r ∈ ℝd for which all orbits are eventually periodic. [Formula: see text] has a very complicated structure even for d = 2. In the present paper, two algorithms are presented which allow the characterization of the intersection of [Formula: see text] and any closed convex hull of finitely many interior points of 𝒟d which is completely contained in the interior of 𝒟d. One of the algorithms is used to determine the structure of [Formula: see text] in a region considerably larger than previously possible, and to settle two questions on its topology: It is shown that [Formula: see text] is disconnected and that the largest connected component has non-trivial fundamental group. The other is the first algorithm characterizing [Formula: see text] in a given convex polyhedron which terminates for all inputs. Furthermore, several infinite families of "cutout polygons" are deduced settling the finiteness property for a chain of regions touching the boundary of 𝒟2.

scholarly journals A pair of non-homeomorphic product measures on the Cantor set

2007 ◽  
Vol 142 (1) ◽  
pp. 103-110 ◽  
Author(s):  
TIM D. AUSTIN
Keyword(s):  
Cantor Set ◽  
The Other ◽  
Continuous Image ◽  
Continuous Map ◽  
Bernoulli Measure ◽  
Product Measures ◽  
Infinite Power

AbstractFor r ∈ [0, 1] let μr be the Bernoulli measure on the Cantor set given as the infinite power of the measure on {0, 1} with weights r and 1 − r. For r, s ∈ [0, 1] it is known that the measure μr is continuously reducible to μs (that is, there is a continuous map sending μr to μs) if and only if s can be written as a certain kind of polynomial in r; in this case s is said to be binomially reducible to r. In this paper we answer in the negative the following question posed by Mauldin:Is it true that the product measures μrand μsare homeomorphic if and only if each is a continuous image of the other, or, equivalently, each of the numbersrandsis binomially reducible to the other?

scholarly journals THE GEODESIC RULE FOR HIGHER CODIMENSIONAL GLOBAL DEFECTS

2008 ◽  
Vol 23 (25) ◽  
pp. 2053-2066
Author(s):  
ANTHONY J. CREACO ◽  
NIKOS KALOGEROPOULOS
Keyword(s):  
Liquid Crystals ◽  
Diffusion Equation ◽  
Convex Hull ◽  
Geometric Structure ◽  
Order Parameter ◽  
The Other ◽  
Equation Approach ◽  
Totally Geodesic ◽  
Geodesic Submanifolds ◽  
Totally Geodesic Submanifolds

We generalize the geodesic rule to the case of formation of higher codimensional global defects. Relying on energetic arguments, we argue that, for such defects, the geometric structures of interest are the totally geodesic submanifolds. On the other hand, stochastic arguments lead to a diffusion equation approach, from which the geodesic rule is deduced. It turns out that the most appropriate geometric structure that one should consider is the convex hull of the values of the order parameter on the causal volumes whose collision gives rise to the defect. We explain why these two approaches lead to similar results when calculating the density of global defects by using a theorem of Cheeger and Gromoll. We present a computation of the probability of formation of strings/vortices in the case of a system, such as nematic liquid crystals, whose vacuum is ℝP2.

Chain recurrence and attraction in non-compact spaces

1991 ◽  
Vol 11 (4) ◽  
pp. 709-729 ◽  
Author(s):  
Mike Hurley
Keyword(s):  
Metric Space ◽  
Basins Of Attraction ◽  
The Other ◽  
Compact Metric Space ◽  
Chain Recurrence ◽  
Fundamental Interest ◽  
Compact Spaces ◽  
Continuous Map ◽  
Chain Recurrent Set ◽  
Recurrent Set

AbstractIn the study of a dynamical systemf:X→Xgenerated by a continuous mapfon a compact metric spaceX, thechain recurrent setis an object of fundamental interest. This set was defined by C. Conley, who showed that it has two rather different looking, but equivalent, definitions: one given in terms of ‘approximate orbits’ through individual points (pseudo-orbits, or ε-chains), and the other given in terms of the global structure of the class of ‘attractors’ and ‘basins of attraction’ off. The first of these definitions generalizes directly to dynamical systems on any metric space, compact or not. The main purpose of this paper is to extend the second definition to non-compact spaces in such a way that it remains equivalent to the first.

The larval development of three species ofMagelona(Polychaeta) from localities near Plymouth

1982 ◽  
Vol 62 (2) ◽  
pp. 385-401 ◽  
Author(s):  
Douglas P. Wilson
Keyword(s):  
Larval Development ◽  
Equal Length ◽  
The Other ◽  
The Right ◽  
Early Larvae ◽  
Made In

Fertilizations ofMagelona filiformis, M. alleni, andM. mirabiliswere made in the laboratory and the resulting larvae reared through the earliest stages. Later stages of all three species were obtained from the plankton and induced to metamorphose in the laboratory. The long larval tentacles derive from the prototroch and associated tissues. Adult tentacles appear during later pelagic life as thickenings at the bases of the larval tentacles; the latter, together with the long provisional bristles, are discarded at metamorphosis. The early larvae ofM. alleniare considerably larger than equivalent stages of the other two species. The early larvae ofM. mirabilisare peculiar in that at first the larval tentacle of the left side is a mere stump while that of the right side is relatively large; in the other two species both tentacles are of equal length. Larvae of the three species are readily identifiable by characters described in the text.

scholarly journals Comparison of maxillary first molar occlusal outlines of Neandertals from the Meuse River Basin of Belgium using elliptical Fourier analysis

2017 ◽  
Vol 80 (3) ◽  
pp. 273-286
Author(s):  
Frank L’Engle Williams ◽  
Katherine M. Lane ◽  
William G. Anderson
Keyword(s):  
Convex Hull ◽  
Fourier Analysis ◽  
River Basin ◽  
Principal Components ◽  
The Other ◽  
Outline Shape ◽  
Maxillary Molars ◽  
Elliptical Fourier Analysis ◽  
Mis 5 ◽  
Meuse River

AbstractSeveral Neandertals derive from the karstic caves of the Meuse river tributaries of Belgium, including Engis 2, Scladina 4A-4 and Spy 1. These may form a group that is distinct in maxillary first molar occlusal outlines compared to La Quina 5 from Southwest France. Alternatively, chronological differences may separate individuals given that Scladina 4A-4 from MIS 5 is older than the others from MIS 3. Neolithic samples (n = 42) from Belgium (Maurenne Caverne de la Cave, Hastière Caverne M, Hastière Trou Garçon, Sclaigneaux and Bois Madame) dated to 4.6–3.9 kyr provide a context for the Neandertals. Dental casts were prepared from dental impressions of the original maxillary molars. Crown and occlusal areas as well as mesiodistal lengths were measured by calibrated Motic 3.0 microscope cameras. Occlusal outlines of the casts were captured through photostereomicroscopy and non-landmark smooth tracing methods. Occlusal outlines were processed using elliptical Fourier analysis within SHAPE v1.3 which reduced amplitudes of the harmonics into principal components (PC) axes. The first two PC axes group the Neandertals, although Scladina 4A-4 falls nearly outside the convex hull for the Neolithic sample. Neandertals are imperfectly separated from the Neolithic sample on PC3 and PC4, and completely distinct on PC5 and PC6. Scladina 4A-4 differs from the other Neandertals on most PC axes. Chronology may best explain the separation of Scladina 4A-4 from the more recent fossils, and particularly Spy 1 and La Quina 5 which are the most similar in maxillary first molar occlusal outline shape.

scholarly journals INVARIANT HYPERSURFACES

2020 ◽  
pp. 1-27
Author(s):  
Jason Bell ◽  
Rahim Moosa ◽  
Adam Topaz
Keyword(s):  
Rational Function ◽  
Finite Type ◽  
Characteristic Zero ◽  
Positive Characteristic ◽  
Inverse Image ◽  
The Other ◽  
Partial Result ◽  
Rational Maps ◽  
Special Cases ◽  
The One

The following theorem, which includes as very special cases results of Jouanolou and Hrushovski on algebraic $D$ -varieties on the one hand, and of Cantat on rational dynamics on the other, is established: Working over a field of characteristic zero, suppose $\unicode[STIX]{x1D719}_{1},\unicode[STIX]{x1D719}_{2}:Z\rightarrow X$ are dominant rational maps from an (possibly nonreduced) irreducible scheme $Z$ of finite type to an algebraic variety $X$ , with the property that there are infinitely many hypersurfaces on  $X$ whose scheme-theoretic inverse images under $\unicode[STIX]{x1D719}_{1}$ and $\unicode[STIX]{x1D719}_{2}$ agree. Then there is a nonconstant rational function $g$ on $X$ such that $g\unicode[STIX]{x1D719}_{1}=g\unicode[STIX]{x1D719}_{2}$ . In the case where $Z$ is also reduced, the scheme-theoretic inverse image can be replaced by the proper transform. A partial result is obtained in positive characteristic. Applications include an extension of the Jouanolou–Hrushovski theorem to generalised algebraic ${\mathcal{D}}$ -varieties and of Cantat’s theorem to self-correspondences.

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