COMBINATORIAL INSCRIBABILITY OBSTRUCTIONS FOR HIGHER DIMENSIONAL POLYTOPES

For 3-dimensional convex polytopes, inscribability is a classical property that is relatively well-understood due to its relation with Delaunay subdivisions of the plane and hyperbolic geometry. In particular, inscribability can be tested in polynomial time, and for every f-vector of 3-polytopes, there exists an inscribable polytope with that f-vector. For higher dimensional polytopes, much less is known. Of course, for any inscribable polytope, all of its lower dimensional faces need to be inscribable, but this condition does not appear to be very strong. We observe non-trivial new obstructions to the inscribability of polytopes that arise when imposing that a certain inscribable face be inscribed. Using this obstruction, we show that the duals of the 4-dimensional cyclic polytopes with at least eight vertices - all of whose faces are inscribable - are not inscribable. This result is optimal in the following sense: We prove that the duals of the cyclic 4-polytopes with up to seven vertices are, in fact, inscribable. Moreover, we interpret this obstruction combinatorially as a forbidden subposet of the face lattice of a polytope, show that d-dimensional cyclic polytopes with at least d+4 vertices are not circumscribable, and that no dual of a neighborly 4-polytope with eight vertices, that is, no polytope with f-vector (20,40,28,8), is inscribable

Spherical CR uniformization of Dehn surgeries of the Whitehead link complement

We apply a spherical CR Dehn surgery theorem in order to obtain infinitely many Dehn surgeries of the Whitehead link complement that carry spherical CR structures. We consider as starting point the spherical CR uniformization of the Whitehead link complement constructed by Parker and Will, using a Ford domain in the complex hyperbolic plane $\mathbb{H}^2_{\mathbb{C}}$. We deform the Ford domain of Parker and Will in $\mathbb{H}^2_{\mathbb{C}}$ in a one parameter family. On the one side, we obtain infinitely many spherical CR uniformizations on a particular Dehn surgery on one of the cusps of the Whitehead link complement. On the other side, we obtain spherical CR uniformizations for infinitely many Dehn surgeries on the same cusp of the Whitehead link complement. These manifolds are parametrized by an integer $n \geq 4$, and the spherical CR structure obtained for $n = 4$ is the Deraux-Falbel spherical CR uniformization of the Figure Eight knot complement.Comment: 70 pages, 15 figures, comments are welcom

An introduction to multifractal geometry of wave sea states on the west and south-east coasts of South Africa

Thesis (PhD)--Stellenbosch University, 2015.ENGLISH ABSTRACT: All of the Ports along the South African coastline are subject to bound infragravity wave action to a greater or lesser degree, for example, at the Ports of Saldanha Bay on the west coast and Ngqura on the south-east coast. Saldanha Bay harbour principally services loose- and liquid-bulk carriers and the Ngqura harbour mainly services container ships. The long wave actions when severe cause moorings to be broken, ships to leave the quay and loading to stop. This research has confirmed that the sea surface is a multifractal structure characterised by many singularities ranging from highly irregular or rough features to smooth or calm features. Any wave train is comprised of the full range of these features to various degrees and in varying percentages of occupancy. Notwithstanding this problem, relatively little is known about them in the South African context due to the fact that they cannot be visually detected and specialised, sophisticated equipment is required to physically measure them. The country is currently planning the development of a new port and the expansion of others for larger ships. Under these circumstances this research is seen to be appropriate from the point of view of obtaining a new method for the characterisation of these hazardous wave conditions. The objective of the research was achieved. This was to identify a set of fractal dimensions that describe the surface geometry of a hazardous bound infragravity wave sea state. In order to achieve the objective, a set of fractal dimensions was firstly determined from video imagery of an open water wave field, by analysing a set of single point time series data derived from the imagery. This has been done in order to be able to visually compare the derived set of fractal dimensions with video imagery of the sea surface that they represent. It also has the advantage of proving that fractal methods of analysis are applicable for the study of sea surface single point time series data. Secondly, periods when long wave action occurs at both Saldanha Bay and Ngqura harbours were identified by the presence of their actions in the harbours. Thirdly, single point time series data recorded by the Council for Scientific and Industrial Research (CSIR) were obtained during the identified periods as well as two days before these times and fractal sets of dimensions for the periods were determined. This was achieved by means of the following methods of analysis: ● The rescaled range (R/S) method, ● The Multifractal Detrended Fluctuation Analysis (MDFA) method, ● The Power Spectral Density (PSD) method in both the Fourier and the wavelet domains, and ● The Wavelet Transform method. Fourthly, the fractal data sets from each harbour were compared to confirm that the sets of dimensions for the hazardous sea state are clearly different from those of the non-hazardous sea state and can be used to describe the condition. Finally, the fractal sets of dimensions for hazardous sea states at both harbours were compared to identify any variances between them. During the research it was found that a hazardous sea state could be profiled for identification purposes and for complementing the currently determined significant wave height and peak period details by means of fractal indices. These indices were identified by comparison with a similar set of indices for nonhazardous sea states at the same location, as part of a ‘calibration’ process and clearly identified shifts in the Holder exponents of the sea states enabled the unambiguous identification of the hazardous condition. Having completed the research and analysis work, the author has identified other areas of coastal engineering, besides the identification of hazardous bound infragravity wave sea states, where a study of multifractal geometry could be applied advantageously.AFRIKAANSE OPSOMMING: Alle hawens langs Suid Afrika se kuslyn is tot ‘n meerdere of mindere mate onderworpe aan gebonde infragravitasie golf aksie. Hierdie probleem is egter veral straf by die hawens van Saldanhabaai aan die weskus en Ngqura, aan die suidooskus. Saldanhabaai voorsien dienste hoofsaaklik aan massa draers van los stowwe en vloeistof, terwyl Ngqura hoofsaaklik houerskepe bedien. Die lang golf aksies veroorsaak dat ankertoue breek en die skepe die kaai verlaat, sodat laai van die skepe tot stilstand kom. In die loop van hierdie navorsing is gevind dat die seevlak ‘n multifraktale struktuur is, met singulariteitseienskappe wat wissel van hoogs onreëlmatig of rowwe eienskappe tot gladde, reëlmatige eienskappe. Enige golfreeks behels die volle omvang van hierdie eienskappe in verskillende grade en wisselende teenwoordigheids persentasies. Die navorsing is gefokus op die geometrie van gebonde infragravitasie golfaksie seetoestande, wat oral langs die Suid Afrikaanse kuslyn voorkom, en in twee hawens ‘n beduidende bedreiging vir vasgemaakte skepe is. Ondanks die probleem, is min bekend oor hierdie toestande in Suid Afrikaanse konteks, omdat hulle nie visueel bespeur kan word nie en spesiale gesofistikeerde gereedskap nodig is om hulle fisies te kan meet. Daar word tans beplan om ‘n nuwe hawe te ontwikkel, wat hierdie navorsing veral gepas maak, met die doel om ‘n meer volledige beskrywing van hierdie bedreigende golftoestande te weeg te bring. Die doel van die navorsing is om ‘n stel fraktale dimensies te identifiseer wat die oppervlakgeometrie van ‘n bedreigende gebonde infragravitasie golf-seetoestand omskryf. Hierdie dimensies kan dan gebruik word om ‘n indentifiserende profiel van die seetoestand te teken om die inligting tans beskikbaar oor beduidende golfhoogte en piektye, aan te vul. Om hierdie doel te bereik is ‘n stel fraktale dimensies eerstens bepaal deur middel van videobeelding van ‘n oopwater golfveld. ‘n Stel enkelpunt tydserie data, afgelei van die beelding, word dan ge-analiseer. Dit het visuele vergelyking tussen die afgeleide stel fraktale dimensies en die videobeelding van die seevlak wat dit verteenwoordig het, moontlik gemaak. ‘n Verdere voordeel is dat dit bewys het dat fraktale analisemetodes toepaslik is vir die bestudering van seevlak enkelpunt tydreeks data. Tweedens is die tye wanneer lang golfaksie teenwoordig was in die hawens by Saldanha en Ngqura, vasgestel deur die uitwerking daarvan in die hawens. Derdens is enkelpunt tydreeks data wat deur die WNNR aangeteken is vir die vasgestelde tydperke, sowel as twee dae voor elke tydperk, verkry en is fraktale dimensiestelle vir elke tydperk vasgestel. Vierdens is die fraktale datastelle van albei die hawens vergelyk om te bevestig dat die stelle dimensies vir bedreigende seetoestande duidelik verskil van die vir niebedreigende toestande, en dus geskik is om die seetoestand te beskryf. Ten slotte is die fraktale dimensiestelle vir bedreigende seetoestande in die twee hawens vergelyk om enige verskille tussen hulle te bepaal. Na voltooiing van die navorsing en analise is ander gebiede van kusingenieurswese behalwe die bepaling van bedreigende gebonde infragravitasie golf seetoestande, identifiseer waar multifraktale geometrie ook tot voordeel aangewend kan word

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

Proof and Proving in Mathematics Education

undefine