'Magic' Configurations of Three-Qubit Observables and Geometric Hyperplanes of the Smallest Split Cayley Hexagon

Recently Waegell and Aravind [J. Phys. A: Math. Theor. 45 (2012), 405301, 13 pages] have given a number of distinct sets of three-qubit observables, each furnishing a proof of the Kochen-Specker theorem. Here it is demonstrated that two of these sets/configurations, namely the $18_{2} - 12_{3}$ and $2_{4}14_{2} - 4_{3}6_{4}$ ones, can uniquely be extended into geometric hyperplanes of the split Cayley hexagon of order two, namely into those of types ${\cal V}_{22}(37; 0, 12, 15, 10)$ and ${\cal V}_{4}(49; 0, 0, 21, 28)$ in the classification of Frohardt and Johnson [Comm. Algebra 22 (1994), 773-797]. Moreover, employing an automorphism of order seven of the hexagon, six more replicas of either of the two configurations are obtained

Distinguished three-qubit 'magicity' via automorphisms of the split Cayley hexagon

Disregarding the identity, the remaining 63 elements of the generalized three-qubit Pauli group are found to contain 12096 distinct copies of Mermin's magic pentagram. Remarkably, 12096 is also the number of automorphisms of the smallest split Cayley hexagon. We give a few solid arguments showing that this may not be a mere coincidence. These arguments are mainly tied to the structure of certain types of geometric hyperplanes of the hexagon. It is further demonstrated that also an (18_{2}, 12_{3})-type of magic configurations, recently proposed by Waegell and Aravind (J. Phys. A: Math. Theor. 45 (2012) 405301), seems to be intricately linked with automorphisms of the hexagon. Finally, the entanglement properties exhibited by edges of both pentagrams and these particular Waegell-Aravind configurations are addressed.Comment: 15 pages, 4 figures, 5 table

Point-primitive generalised hexagons and octagons

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Applications of the amalgam method to the study of locally projective graphs

Since its birth in 1980 with the seminal paper [Gol80] by Goldschmidt, the amalgam method has proved to be one of the most powerful tools in the modern study of groups, with interesting applications to graphs. Consider a connected graph Γ with a family L of complete subgraphs (called lines) with α ∈ {2,3} vertices each, and possessing a vertex- and edge-transitive group G of automorphisms preserving L. It is assumed that for every vertex x of Γ, there is a bijection between the set of lines containing x and the point-set of a projective GF(2)-space. There is a number of important examples of such locally projective graphs, studied and partly classified by Trofimov, Ivanov and Shpectorov, where both classical and sporadic simple groups appear among the automorphism groups. To a locally projective graph one can associate the corresponding locally projective amalgam A = {G(x),G{l}} comprised of the stabilisers in G of a vertex x and of a line l containing it. The renowned Goldschmidt amalgams turn out to belong to this family (α = 3), as well as their densely embedded Djokovic-Miller subamalgams (α = 2). We first determine all the embeddings of the Djokovic-Miller amalgams in the Goldschmidt amalgams, by designing and implementing an algorithm in GAP and MAGMA. This gives, as a by-product, a list of some finite completions for both the Goldschmidt and the Djokovic-Miller amalgams. Next, we consider two examples of locally projective graphs, special for being devoid of densely embedded subgraphs, and we extend their corresponding locally projective amalgams through the notion of a geometric subgraph. In both cases we find a geometric presentation of the amalgams, which we use to prove the simple connectedness of the corresponding geometry. Finally, we use the Goldschmidt’s lemma to classify, up to isomorphism, certain amalgams related to the Mathieu group M24 and the Held group He, as outlined in [Iva18], and we give an explicit construction of the cocycle whose existence and uniqueness is asserted in [Iva18, Lemma 8.5].Open Acces

Tuning the Properties of Isoindigo-Based Organic Semiconductors Through Structural Engineering

Solar power is one of the most prominent renewable energy technologies vying to replace fossil fuels. Traditionally, this field has been dominated by silicon solar cells; recent innovations, such as organic photovoltaic devices (OPVs), offer the possibility of lightweight, flexible solar power generation with a wide range of applications, from building façades to textiles. Unfortunately, organic solar cells suffer from low power conversion efficiencies. To overcome this problem, the design of new organic semiconductors has become an area of intense research. In engineering these materials, it is essential to develop strong links between molecular structure and the optoelectronic properties of the resulting semiconductors, such as optical band gap, extinction coefficient, frontier orbital energies, and charge carrier mobility. This thesis explores the relationship between structural differences in isoindigo derivatives, such as increasing electron deficiency, or increased molecular planarity, and differences in the resulting material’s optoelectronic properties. The first two sections of this thesis investigate the effects of heteroatom substitution on isoindigo-based semiconductors. Four target compounds were synthesized, each containing either electron-withdrawing nitrogen atoms, or electron rich alkoxy groups. The semiconductors were incorporated into the active layer of organic solar cells. Alkoxy substitution was shown to improve device efficiency; conversely, nitrogen substitution led to lowered device efficiency. In a follow-up study, it was shown that the azaisoindigo groups were capable of coordinating to a Lewis acid; this coordination caused a red-shift in the molecule’s S0S1 transition. The Lewis adduct was identified using UV/vis spectroscopy, NMR spectroscopy, and (TD)DFT calculations. It was then demonstrated that the coordination reaction could be performed with vapor phase Lewis acids. The third project in this thesis focuses on the synthesis of two isoindigo dimers. The first, bisisoindigo, is a ring-fused dimer of isoindigo. This was chosen to study the effects of increased planarity and conjugation length on the optoelectronic properties of isoindigo. Initially, both bisisoindigo and a donor-acceptor molecular semiconductor based on bisisoindigo were synthesized, characterized, and used in OPVs. Poor active layer morphology, due to aggregation of the bisisoindigo, led to low efficiencies in the OPVs. Following work on the ring-fused isoindigo structure, a second dimer was synthesized in which two isoindigo units were joined by a single bond; this design provides free rotation between isoindigo units. Both dimers, as well as isoindigo, were used as electron acceptors in a study of the effects of acceptor number and planarity in donor-acceptor copolymers. The acceptors were copolymerized with thiophene and terthiophene to yield a total of six polymers. Both the optoelectronic properties of these polymers, and their performance in OPVs, were compared to discover trends in donor-acceptor semiconductor properties with increasing acceptor content. Over the course of four major projects it has been demonstrated that altering the structure of the conjugated building block isoindigo has major effects on the optical band gap, orbital energies, and charge transport characteristics of the resulting organic semiconductors. The final chapter of this thesis will serve to link these projects in a general discussion of how the design of isoindigo-based organic semiconductors can be used to produce desired optoelectronic properties in the resulting materials. The thesis will conclude with a brief look towards the prospects of this area of research, including establishing the general applicability of these design strategies to organic semiconductors beyond those based on isoindigo

Las magníficas ironías de Dios, análisis metatextual de "La Biblioteca de Babel" de Jorge Luis Borges

"Lo común para interpretar una obra narrativa es leerla, analizarla y generar alguna hipótesis de lectura en función de lo que presenta el narrador, cómo narra, cómo describe. "La Biblioteca de Babel" es un cuento sui generis porque el propio narrador plantea muchas hipótesis para interpretar algunos de los enigmas centrales del cuento, como ¿quién creó la Biblioteca?, ¿quién creó a esos hombres?, ¿quién escribió los libros?, ¿qué dicen esos libros?, etc. El narrador consigna una serie de conjeturas, ajenas o propias, para responder a estas preguntas; no hay una perspectiva única. Hay suficientes elementos en el relato para leerlo en clave literaria, filosófica, matemática, astronómica, religiosa, etc. Si bien, uno de los rasgos comunes a casi toda la literatura es su carácter polisémico, este rasgo se acentúa en "La Biblioteca de Babel" debido a todo ese laberinto de perspectivas, a veces complementarias; a menudo, contradictorias. Para nuestra investigación hemos elegido enfocarnos en el ángulo religioso, pues, de acuerdo con nuestra revisión de la bibliografía crítica, ha sido menos explorado que otros, así que hemos visto en ese enfoque una oportunidad para aportar algo al estudio de este cuento"

An interrelated approach to teaching mathematics in secondary schools

This thesis is primarily concerned with the production and evaluation of ideas and materials, based upon an interrelated approach to teaching, which is aimed at arousing curiosity and interest in pupils in secondary schools from the age of fourteen upwards. A case is presented for the consideration of such an approach and a brief account given of how early ideas were formulated. These ideas resulted in the establishment of positive guidelines and strategies upon which the research was to be based. Much emphasis is placed on the significance of effective and attractive written materials for pupils with one chapter being specifically aimed at outlining important aspects of general module preparation. The thesis presents in some detail evaluations of trials carried out with groups of students studying a variety of topics involving mathematical principles. It attempts to describe the successes and failures of various modules of study devised during the research programme and takes special account of comments made by pupils and staff who participated in trials. With the recommendation for a new approach to teaching, effective in-service training of teachers is an essential exercise. Various in-service training programmes organised for teachers and ideas produced during these sessions by enthusiastic, stimulated participants are reported. In addition, the thesis contains proposals for the establishment of a professional centre for mathematical education in schools and colleges within Leicestershire where ideas produced from research projects such as this can be extended, developed fully and subsequently disseminated in an effective manner. In conclusion, the achievements of the research programme are discussed and recommendations and suggestions made for wider use of the interrelated approach to teaching in secondary schools

How To Draw A Hexagon

. Pictures of the G2(2) hexagon and its dual are presented. A way to obtain these pictures is discussed. 1. Introduction This paper presents pictures of the two classical generalised hexagons over the field with two elements. Information on these generalised hexagons is employed so that the pictures make some properties of these hexagons obvious. Familiarity with the theory of generalised polygons is not assumed. Perhaps this paper will convince the uninitiated reader that even new and seemingly abstract geometrical structures sometimes have nice visual presentations. Generalised polygons were introduced by Tits in [7]. They serve, among other things, as a geometric realisation of certain groups. Generalised triangles are essentially projective planes, generalised quadrangles are also known as polar spaces of rank 2. Classical (thick) generalised n-gons, n ? 2, exist only for n 2 f3; 4; 6; 8g. By [3] also thick finite generalised n-gons, n ? 2, exist only for n 2 f3; 4; 6; 8g. If the ..