13 research outputs found

    Elliptic curves over Galois number fields

    Get PDF
    This thesis is concerned with the statistical behaviour of elliptic curves over extension fields. That is, if K/Q is a finite extension, we study the arithmetic of E/K as E ranges in natural families of elliptic curves defined over Q. We study the statistical properties of the action of the group Aut(K) on E(K) and on the p-Selmer groups Selp(E/K) where p is a prime number. We construct special generalised Selmer groups, and show that these are related to certain representation-theoretic invariants of Selp(E/K). The sizes of these groups are related to the cokernels of the norm maps over the completions of K, which we go on to compute in several cases. In the statistical component of this thesis, we study quadratic twist families of elliptic curves and the family of ‘all elliptic curves’. For quadratic twist families we consider the behaviour over quadratic extensions. Using methods similar to those of Heath-Brown [HB93, HB94] and of Fouvry–Klüners [FK07], we determine the complete distribution of the 2-Selmer groups as Galois modules. This also allows us to determine representation-theoretic properties for the Mordell–Weil groups of 100% of twists. For the family of all elliptic curves over Q, we consider the behaviour with respect to a general finite Galois extension K/F. Writing G = Gal(K/F), our first main result is that the difference in dimension between Selp(E/K) G and Selp(E/F) has bounded average in this family. Using this we are able, with additional assumptions on K/F and p, to bound the average dimension of Selp(E/K) and so the average rank of the Mordell–Weil group E(K). Our methods also allow us to bound how often certain Z[G]-lattices occur as summands of E(K), with additional assumptions on F. We refine our results in the setting where K/Q is multiquadratic and p = 2, and prove strong upper and lower bounds for the average dimension of the 2-Selmer group

    On 2-selmer groups of twists after quadratic extension

    Get PDF
    Let E∕Q be an elliptic curve with full rational 2-torsion. As d varies over squarefree integers, we study the behaviour of the quadratic twists Ed over a fixed quadratic extension K∕Q. We prove that for 100% of twists the dimension of the 2-Selmer group over K is given by an explicit local formula, and use this to show that this dimension follows an Erdos–Kac type distribution. This is in stark contrast to the distribution of the dimension of the corresponding 2-Selmer groups over Q, and this discrepancy allows us to determine the distribution of the 2-torsion in the Shafarevich–Tate groups of the Ed over K also. As a consequence of our methods we prove that, for 100% of twists d, the action of Gal(K∕Q) on the 2-Selmer group of Ed over K is trivial, and the Mordell–Weil group Ed(K) splits integrally as a direct sum of its invariants and anti-invariants. On the other hand, we give examples of thin families of quadratic twists in which a positive proportion of the 2- Selmer groups over K have non-trivial Gal(K∕Q)-action, illustrating that these previous results are genuinely statistical phenomena

    The contemporary visualization and modelling technologies and techniques for the design of the green roofs

    Get PDF
    The contemporary design solutions are merging the boundaries between real and virtual world. The Landscape architecture like the other interdisciplinary field stepped in a contemporary technologies area focused on that, beside the good execution of works, designer solutions has to be more realistic and “touchable”. The opportunities provided by Virtual Reality are certainly not negligible, it is common knowledge that the designs in the world are already presented in this way so the Virtual Reality increasingly used. Following the example of the application of virtual reality in landscape architecture, this paper deals with proposals for the use of virtual reality in landscape architecture so that designers, clients and users would have a virtual sense of scope e.g. rooftop garden, urban areas, parks, roads, etc. It is a programming language that creates a series of images creating a whole, so certain parts can be controlled or even modified in VR. Virtual reality today requires a specific gadget, such as Occulus, HTC Vive, Samsung Gear VR and similar. The aim of this paper is to acquire new theoretical and practical knowledge in the interdisciplinary field of virtual reality, the ability to display using virtual reality methods, and to present through a brief overview the plant species used in the design and construction of an intensive roof garden in a Mediterranean climate, the basic characteristics of roofing gardens as well as the benefits they carry. Virtual and augmented reality as technology is a very powerful tool for landscape architects, when modeling roof gardens, parks, and urban areas. One of the most popular technologies used by landscape architects is Google Tilt Brush, which enables fast modeling. The Google Tilt Brush VR app allows modeling in three-dimensional virtual space using a palette to work with the use of a three-dimensional brush. The terms of two "programmed" realities - virtual reality and augmented reality - are often confused. One thing they have in common, though, is VRML - Virtual Reality Modeling Language. In this paper are shown the ways on which this issue can be solved and by the way, get closer the term of Virtual Reality (VR), also all the opportunities which the Virtual reality offered us. As well, in this paper are shown the conditions of Mediterranean climate, the conceptual solution and the plant species which will be used by execution of intensive green roof on the motel “Marković”

    The benefits of an additional practice in descriptive geomerty course: non obligatory workshop at the Faculty of Civil Engineering in Belgrade

    Get PDF
    At the Faculty of Civil Engineering in Belgrade, in the Descriptive geometry (DG) course, non-obligatory workshops named “facultative task” are held for the three generations of freshman students with the aim to give students the opportunity to get higher final grade on the exam. The content of this workshop was a creative task, performed by a group of three students, offering free choice of a topic, i.e. the geometric structure associated with some real or imagery architectural/art-work object. After the workshops a questionnaire (composed by the professors at the course) is given to the students, in order to get their response on teaching/learning materials for the DG course and the workshop. During the workshop students performed one of the common tests for testing spatial abilities, named “paper folding". Based on the results of the questionnairethe investigation of the linkages between:students’ final achievements and spatial abilities, as well as students’ expectations of their performance on the exam, and how the students’ capacity to correctly estimate their grades were associated with expected and final grades, is provided. The goal was to give an evidence that a creative work, performed by a small group of students and self-assessment of their performances are a good way of helping students to maintain motivation and to accomplish their achievement. The final conclusion is addressed to the benefits of additional workshops employment in the course, which confirmhigherfinal scores-grades, achievement of creative results (facultative tasks) and confirmation of DG knowledge adaption

    Тhe benefits of an additional practice in descriptive geometry course: non obligatory workshop at the Faculty of civil engineering in Belgrade

    Get PDF
    At the Faculty of Civil Engineering in Belgrade, in the Descriptive geometry (DG) course, non-obligatory workshops named “facultative task” are held for the three generations of freshman students with the aim to give students the opportunity to get higher final grade on the exam. The content of this workshop was a creative task, performed by a group of three students, offering free choice of a topic, i.e. the geometric structure associated with some real or imagery architectural/art-work object. After the workshops a questionnaire (composed by the professors at the course) is given to the students, in order to get their response on teaching/learning materials for the DG course and the workshop. During the workshop students performed one of the common tests for testing spatial abilities, named “paper folding". Based on the results of the questionnairethe investigation of the linkages between:students’ final achievements and spatial abilities, as well as students’ expectations of their performance on the exam, and how the students’ capacity to correctly estimate their grades were associated with expected and final grades, is provided. The goal was to give an evidence that a creative work, performed by a small group of students and self-assessment of their performances are a good way of helping students to maintain motivation and to accomplish their achievement. The final conclusion is addressed to the benefits of additional workshops employment in the course, which confirmhigherfinal scores-grades, achievement of creative results (facultative tasks) and confirmation of DG knowledge adaption

    The contemporary visualization and modelling technologies and the techniques for the design of the green roofs

    Get PDF
    The contemporary design solutions are merging the boundaries between real and virtual world. The Landscape architecture like the other interdisciplinary field stepped in a contemporary technologies area focused on that, beside the good execution of works, designer solutions has to be more realistic and “touchable”. The opportunities provided by Virtual Reality are certainly not negligible, it is common knowledge that the designs in the world are already presented in this way so the Virtual Reality increasingly used. Following the example of the application of virtual reality in landscape architecture, this paper deals with proposals for the use of virtual reality in landscape architecture so that designers, clients and users would have a virtual sense of scope e.g. rooftop garden, urban areas, parks, roads, etc. It is a programming language that creates a series of images creating a whole, so certain parts can be controlled or even modified in VR. Virtual reality today requires a specific gadget, such as Occulus, HTC Vive, Samsung Gear VR and similar. The aim of this paper is to acquire new theoretical and practical knowledge in the interdisciplinary field of virtual reality, the ability to display using virtual reality methods, and to present through a brief overview the plant species used in the design and construction of an intensive roof garden in a Mediterranean climate, the basic characteristics of roofing gardens as well as the benefits they carry. Virtual and augmented reality as technology is a very powerful tool for landscape architects, when modeling roof gardens, parks, and urban areas. One of the most popular technologies used by landscape architects is Google Tilt Brush, which enables fast modeling. The Google Tilt Brush VR app allows modeling in three-dimensional virtual space using a palette to work with the use of a three dimensional brush. The terms of two "programmed" realities - virtual reality and augmented reality - are often confused. One thing they have in common, though, is VRML - Virtual Reality Modeling Language. In this paper are shown the ways on which this issue can be solved and by the way, get closer the term of Virtual Reality (VR), also all the opportunities which the Virtual reality offered us. As well, in this paper are shown the conditions of Mediterranean climate, the conceptual solution and the plant species which will be used by execution of intensive green roof on the motel “Marković”

    Modular forms and lattice point counting problems

    Full text link
    Tesis Doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de lectura: 14-12-201

    Generating the focal-directorial geometric forms as a designing pattern of the architectural-urban space

    Get PDF
    Истраживањем различитих типова равних кривих и површи које су генерисане као геометријско место тачака (ГМТ) са константном сумом/производом растојања до фокуса, показано је да постоје и такве геометријске форме које настају као ГМТ са константном сумом/производом растојања до директриса. Захваљујући оваквој генези, добијена је веза фокалних и директрисних геометријских форми са Паладијевим пропорционалним системом, али и са Ферма-Веберовим скупом тачака, тј. са Веберовим локацијским проблемом. Такође, анализирањем ових елемената уочила се њихова повезаност са неким геометријским неједнакостима многоугла, и на основу тога, формиран је прелазни тип. Прелазни тип је назван фокално-директрисно генерисани елемент (фокално-директрисна крива и фокално-директрисна површ)...Through explorations of various types of planar curves and surfaces generated as locus of points with constant sum/product of distances to the focus, it was shown that there also exist geometric forms that are locus of points with constant sum/product of distances to the directrix. Due to such genesis, one could obtain a connection between focal and directorial geometric forms with Palladio’s proportional system, as well as with Fermat-Weber’s set of points, i.e. with Weber’s location problem. In addition, analysis of these elements led to observations of their connections with some geometric polygonal inequalities, which then led to introduction of a transitory element type. This transitory type of an element is called focal-directorial generated element (focal-directorial curve and focal-directorial surface)..

    On the extension of the Erdös-Mordell type inequalities

    Get PDF
    We discuss the extension of inequality R_A >= c/a * r_b + b/a * r_c to the plane of triangle ABC. Based on the obtained extension, in regard to all three vertices of the triangle, we get the extension of Erdos-Mordell inequality, and some inequalities of Erdos-Mordell type.This is the peer-reviewed version of the article: B. J. Malešević, M. Petrović, M. Obradović, and B. Popkonstantinović, ‘On the extension of the Erdös-Mordell type inequalities’, Mathematical Inequalities & Applications, vol. 17, no. 1, pp. 269–281, 2014, [http://dx.doi.org/10.7153/mia-17-22
    corecore