On k-Convex Polygons

We introduce a notion of $k$-convexity and explore polygons in the plane that have this property. Polygons which are \mbox{$k$-convex} can be triangulated with fast yet simple algorithms. However, recognizing them in general is a 3SUM-hard problem. We give a characterization of \mbox{$2$-convex} polygons, a particularly interesting class, and show how to recognize them in \mbox{$O(n \log n)$} time. A description of their shape is given as well, which leads to Erd\H{o}s-Szekeres type results regarding subconfigurations of their vertex sets. Finally, we introduce the concept of generalized geometric permutations, and show that their number can be exponential in the number of \mbox{$2$-convex} objects considered.Comment: 23 pages, 19 figure

SYNTHESIS AND EVALUATION OF ANTIMICROBIAL ACTIVITY OF PHENYL AND FURAN-2-YL[1,2,4] TRIAZOLO[4,3-a]QUINOXALIN-4(5H)-ONE AND THEIR HYDRAZONE PRECURSORS

A variety of 1-(s-phenyl)-[1,2,4]triazolo[4,3-a]quinoxalin-4(5H)-one (3a-3h) and 1-(s-furan-2-yl)-[1,2,4]triazolo[4,3- a]quinoxalin-4(5H)-one (5a-d) were synthesized from thermal annelation of corresponding hydrazones (2a-h) and (4a-d) respectively in the presence of ethylene glycol which is a high boiling solvent. The structures of the compounds prepared were confirmed by analytical and spectral data. Also, the newly synthesized compounds were evaluated for possible antimicrobial activity. 3-(2-(4-hydroxylbenzylidene)hydrazinyl)quinoxalin-2(1H)-one (2e) was the most active antibacterial agent while 1-(5-Chlorofuran-2-yl)-[1,2,4]triazolo[4,3-a]quinoxalin-4(5H)-one (5c) stood out as the most potent antifungal agent

Problems and applications of Discrete and Computational Geometry concerning graphs, polygons, and points in the plane

Esta tesistratasobreproblemasyaplicacionesdelageometríadiscretay computacional enelplano,relacionadosconpolígonos,conjuntosdepuntos y grafos. Después deunprimercapítulointroductorio,enelcapítulo 2 estudiamos una generalizacióndeunfamosoproblemadevisibilidadenelámbitodela O-convexidad. Dadounconjuntodeorientaciones(ángulos) O, decimosque una curvaes O-convexa si suintersecciónconcualquierrectaparalelaauna orientaciónde O es conexa.Cuando O = {0◦, 90◦}, nosencontramosenel caso delaortoconvexidad,consideradodeespecialrelevancia.El O-núcleo de unpolígonoeselconjuntodepuntosdelmismoquepuedenserconectados con cualquierotropuntodelpolígonomedianteunacurva O-convexa.En este trabajoobtenemos,para O = {0◦} y O = {0◦, 90◦}, unalgoritmopara calcular ymantenerel O-núcleodeunpolígonoconformeelconjuntode orientaciones O rota. Dichoalgoritmoproporciona,además,losángulosde rotación paralosqueel O-núcleotieneáreayperímetromáximos. En elcapítulo 3 consideramos unaversiónbicromáticadeunproblema combinatorioplanteadoporNeumann-LarayUrrutia.Enconcreto,de- mostramos quetodoconjuntode n puntosazulesy n puntosrojosenel plano contieneunparbicromáticodepuntostalquetodocírculoquelos tenga ensufronteracontieneensuinterioralmenos n(1− 1 √2 )−o(n) puntos del conjunto.Esteproblemaestáfuertementeligadoalcálculodelosdiagra- mas deVoronoideordensuperiordelconjuntodepuntos,pueslasaristas de estosdiagramascontienenprecisamentetodosloscentrosdeloscírculos que pasanpordospuntosdelconjunto.Porello,nuestralíneadetrabajo actual enesteproblemaconsisteenexplorarestaconexiónrealizandoun estudio detalladodelaspropiedadesdelosdiagramasdeVoronoideorden superior. En loscapítulos 4 y 5, planteamosdosaplicacionesdelateoríadegrafos 6 7 al análisissensorialyalcontroldeltráficoaéreo,respectivamente.Enel primer caso,presentamosunnuevométodoquecombinatécnicasestadísti- cas ygeométricasparaanalizarlasopinionesdelosconsumidores,recogidas a travésdemapeoproyectivo.Estemétodoesunavariacióndelmétodo SensoGraph ypretendecapturarlaesenciadelmapeoproyectivomediante el cálculodelasdistanciaseuclídeasentrelosparesdemuestrasysunor- malización enelintervalo [0, 1]. Acontinuación,aplicamoselmétodoaun ejemplo prácticoycomparamossusresultadosconlosobtenidosmediante métodosclásicosdeanálisissensorialsobreelmismoconjuntodedatos. En elsegundocaso,utilizamoslatécnicadelespectro-coloreadodegrafos para plantearunmodelodecontroldeltráficoaéreoquepretendeoptimizar el consumodecombustibledelosavionesalmismotiempoqueseevitan colisiones entreellos.This thesisdealswithproblemsandapplicationsofdiscreteandcomputa- tional geometryintheplane,concerningpolygons,pointsets,andgraphs. After afirstintroductorychapter,inChapter 2 westudyageneraliza- tion ofafamousvisibilityproblemintheframeworkof O-convexity. Given a setoforientations(angles) O, wesaythatacurveis O-convex if itsin- tersection withanylineparalleltoanorientationin O is connected.When O = {0◦, 90◦}, wefindourselvesinthecaseoforthoconvexity,consideredof specialrelevance.The O-kernel of apolygonisthesubsetofpointsofthe polygonthatcanbeconnectedtoanyotherpointofthepolygonwithan O-convexcurve.Inthisworkweobtain,for O = {0◦} and O = {0◦, 90◦}, an algorithm tocomputeandmaintainthe O-kernelofapolygonasthesetof orientations O rotates. Thisalgorithmalsoprovidestheanglesofrotation that maximizetheareaandperimeterofthe O-kernel. In Chapter 3, weconsiderabichromaticversionofacombinatorialprob- lem posedbyNeumann-LaraandUrrutia.Specifically,weprovethatevery set of n blue and n red pointsintheplanecontainsabichromaticpairof pointssuchthateverycirclehavingthemonitsboundarycontainsatleast n(1 − 1 √2 ) − o(n) pointsofthesetinitsinterior.Thisproblemisclosely related toobtainingthehigherorderVoronoidiagramsofthepointset.The edges ofthesediagramscontain,precisely,allthecentersofthecirclesthat pass throughtwopointsoftheset.Therefore,ourcurrentlineofresearch on thisproblemconsistsonexploringthisconnectionbystudyingindetail the propertiesofhigherorderVoronoidiagrams. In Chapters 4 and 5, weconsidertwoapplicationsofgraphtheoryto sensory analysisandairtrafficmanagement,respectively.Inthefirstcase, weintroduceanewmethodwhichcombinesgeometricandstatisticaltech- niques toanalyzeconsumeropinions,collectedthroughprojectivemapping. This methodisavariationofthemethodSensoGraph.Itaimstocapture 4 5 the essenceofprojectivemappingbycomputingtheEcuclideandistances betweenpairsofsamplesandnormalizingthemtotheinterval [0, 1]. Weap- ply themethodtoareal-lifescenarioandcompareitsperformancewiththe performanceofclassicmethodsofsensoryanalysisoverthesamedataset. In thesecondcase,weusetheSpectrumGraphColoringtechniquetopro- poseamodelforairtrafficmanagementthataimstooptimizetheamount of fuelusedbytheairplanes,whileavoidingcollisionsbetweenthem

Large bichromatic point sets admit empty monochromatic 4-gons

We consider a variation of a problem stated by Erd˝os and Szekeres in 1935 about the existence of a number fES(k) such that any set S of at least fES(k) points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any bichromatic set of n ≥ 5044 points in R2 in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).Postprint (published version

Fundamentals of Restricted-Orientation Convexity

A restricted-orientation convex set, also called an O-convex set, is a set of points whose intersection with lines from some fixed set is empty or connected. The notion of O-convexity generalizes standard convexity and orthogonal convexity. We explore some of the basic properties of O-convex sets in two and higher dimensions. We also study O-connected sets, which are restricted O-convex sets with several special properties. We introduce and investigate restricted-orientation analogs of lines, flats, and hyperplanes, and characterize O-convex and O-connected sets in terms of their intersections with hyperplanes. We then explore properties of O-connected curves; in particular, we show when replacing a segment of an O-connected curve with a new curvilinear segment yields an O-connected curve and when the catenation of several curvilinear segments forms an O- connected segment. We use these results to characterize an O-connected set in terms of O-connected segments that join pairs of it..

Fundamentals of Restricted-Orientation Convexity

A restricted-orientation convex set, also called an O-convex set, is a set of points whose intersection with lines from some fixed set is empty or connected. The notion of O-convexity generalizes standard convexity and orthogonal convexity. We explore some of the basic properties of O-convex sets in two and higher dimensions. We also study O-connected sets, which are a subclass of O-convex sets, with several special properties. We introduce and investigate restricted-orientation analogs of lines, flats, and hyperplanes, and characterize O-convex and O-connected sets in terms of their intersections with hyperplanes. We then explore properties of O- connected curves; in particular, we show when replacing a segment of an O-connected curve with a new curvilinear segment yields an O-connected curve and when the catenation of several curvilinear segments forms an O-connected segment. We use these results to characterize an O-connected set in terms of O-connected segments, joining pairs of..

Fundamentals of Restricted-Orientation Convexity

A restricted-orientation convex set, also called an set, is a set of points whose intersection with lines from some fixed set is empty or connected. The notion generalizes standard convexity and orthogonal convexity