More maximal arcs in Desarguesian projective planes and their geometric structure

In a previous paper R. Mathon gave a new construction method for maximal arcs in finite Desarguesian projective planes via closed sets of conics, as well as giving many new examples of maximal arcs. In the current paper, new classes of maximal arcs are constructed, and it is shown that every maximal arc so constructed gives rise to an infinite class of maximal arcs. Apart from when they are of Denniston type or dual hyperovals, closed sets of conics are shown to give maximal arcs that are not isomorphic to the known constructions. An easy characterisation of when a closed set of conics is of Denniston type is given. Results on the geometric structure of the maximal arcs and their duals are proved, as well as on elements of their collineation stabilisers.

Algorithms for Finding Unitals and Maximal Arcs in Projective Planes of Order 16

The paper has been presented at the International Conference Pioneers of Bulgarian Mathematics, Dedicated to Nikola Obreshkoff and Lubomir Tschakalo ff , Sofia, July, 2006.

A Relation between the Weyl Group W(e8) and Eight-Line Arrangements on a Real Projective Plane

The paper has been presented at the 12th International Conference on Applications of Computer Algebra, Varna, Bulgaria, June, 2006.

On Projective Plane of Order 13 with a Frobenius Group of Order 39 as a Collineation Group

One of the most outstanding problems in combinatorial mathematics and geometry is the problem of existence of finite projective planes whose order is not a prime power.

When does a Bounded Domain Cover a Projective Manifold? (Survey)

* The research has been partially supported by Bulgarian Funding Organizations, sponsoring the Algebra Section of the Mathematics Institute, Bulgarian Academy of Sciences, a Contract between the Humboldt Univestit¨at and the University of Sofia, and Grant MM 412 / 94 from the Bulgarian Board of Education and Technology

Projective Methods of Image Recognition

We propose a method for image recognition on the base of projections. Radon transform gives an opportunity to map image into space of its projections. Projection properties allow constructing informative features on the base of moments that can be successfully used for invariant recognition. Offered approach gives about 91-97% of correct recognition.

Predegree Polynomials of Plane Configurations in Projective Space

2000 Mathematics Subject Classification: 14N10, 14C17.

Le jeune adulte et la maturité vocationnelle

Plusieurs collèges compilent à chaque session les taux d'abandon, d'échec et de changement de programme de leurs étudiants. Bien que certains facteurs et causes d'échec, d'abandon ou de changement de programme aient pu être relevés, ces compilations statistiques ne s'avèrent guère utiles aux professionnels des services de consultation. Ces données permettent de constater les faits. Une analyse plus en détail nous permettant d'identifier les sources du problème serait beaucoup plus utile. En plus des étudiants qui échouent, abandonnent des cours ou changent de programme, que dire de ceux qui ont fixé trop rapidement ou de façon trop rigide leur choix vocationnel et de ceux qui se maintiennent dans un état d'indécision chronique au niveau de leur choix vocationnel ? Comment entreprendre des actions préventives à l'égard de ces phénomènes du cheminement scolaire des étudiants ? Nous identifions comme étant une des sources de ces phénomènes la maturité vocationnelle du jeune adulte, puisque c'est celle-ci qui influencera la qualité de ses choix et son adaptation professionnelle (Super; voir Laflamme et Petit, 1973). En effet, le jeune adulte doit être en mesure d'évaluer les motifs et les conséquences des gestes qu'il pose en fonction de la réalisation de son projet vocationnel, pour contrôler son cheminement scolaire. Or nous constatons, que même si les jeunes adultes semblent préoccupés par la question de leur orientation, bon nombre d'entre eux ne manifestent pas l'intérêt de prendre en charge leur démarche d'orientation. Plusieurs semblent s'occuper de cette question à la dernière minute, par exemple, juste avant les dates limites des demandes d'admission pour l'entrée au collège ou à l'université. D'autres semblent procéder par essais et erreurs sans nécessairement être conscients des valeurs, buts, ambitions qui les poussent à poser ces choix successifs (nouveaux programmes) parfois peu satisfaisants. Certains refuseront même de poser des choix reléguant au hasard ce qui arrivera d'eux. Tous ces faits et impressions tirés de notre expérience nous font penser que le jeune adulte ne semble pas s'impliquer personnellement dans les actes qu'il pose. Il ne semble pas croire qu'il a du pouvoir sur sa vie et qu'il peut prendre en charge les évènements qui l'entourent, particulièrement ici, son cheminement scolaire et sa démarche d'orientation. Il laisse plutôt l'environnement exercer un contrôle sur lui, ce qui a pour effet, croyons-nous, de le rendre passif quant à son cheminement scolaire et à sa démarche d'orientation. Il semble y avoir une contradiction: le jeune adulte est préoccupé par la question de son orientation mais ne semble pas motivé à investir beaucoup d'énergie pour son développement vocationnel. Il ne se prend pas en charge, il ne passe pas à l'action claire et manifeste. A l'observation de ces comportements, il nous apparaît alors que beaucoup de jeunes adultes ne sont pas réellement en possession de leur projet vocationnel. C'est là justement qu'il nous faut vérifier si le jeune adulte de niveau collégial a atteint une certaine maturité vocationnelle. Nous tenterons donc de voir comment se situe le jeune adulte du niveau collégial dans son développement vocationnel et ce, par rapport aux théories existantes et comment mesurer son niveau de maturité vocationnelle. Pour ce faire nous effectuerons, dans un premier temps, une revue de la littérature scientifique en ce qui a trait à la notion de maturité vocationnelle; puis nous effectuerons une revue et une critique des travaux antérieurs sur la mesure et l'évaluation de la maturité vocationnelle. Cette dernière étape vous permettra de déterminer quel instrument serait le plus adéquat pour mesurer la maturité vocationnelle du jeune adulte de niveau collégial.

Motivic Decomposition of Projective Pseudo-Homogeneous Varieties

Let G be a semi-simple algebraic group over a field k. Projective G-homogeneous varieties are projective varieties over which G acts transitively. The stabilizer or the isotropy subgroup at a point on such a variety is a parabolic subgroup which is always smooth when the characteristic of k is zero. However, when k has positive characteristic, we encounter projective varieties with transitive G-action where the isotropy subgroup need not be smooth. We call these varieties projective pseudo-homogeneous varieties. To every such variety, we can associate a corresponding projective homogeneous variety. In this thesis, we extensively study the Chow motives (with coefficients from a finite connected ring) of projective pseudo-homogeneous varieties for G inner type over k and compare them to the Chow motives of the corresponding projective homogeneous varieties. This is done by proving a generic criterion for the motive of a variety to be isomorphic to the motive of a projective homogeneous variety which works for any characteristic of k. As a corollary, we give some applications and examples of Chow motives that exhibit an interesting phenomenon. We also show that the motives of projective pseudo-homogeneous varieties satisfy properties such as Rost Nilpotence and Krull-Schmidt.

Truncated projective spaces, Brown–Gitler spectra and indecomposable A(1)-modules

MSC 19L41; 55S10.

Étude des propriétés psychométriques de l’échelle affective du Questionnaire de maturité vocationnelle

Les conseillers d’orientation (c.o.) en milieu scolaire ont plusieurs rôles à remplir auprès des élèves, entre autres la réponse aux demandes des élèves, le maintien de l’accès à une documentation pertinente et l’accompagnement dans une meilleure connaissance de soi et du monde du travail (Dupont, Gingras et Marceau, 2002a). Pour ce faire, plusieurs moyens sont à leur disposition, dont l’évaluation psychométrique (OCCOQ, 2013). Un concept qui pourrait utile de mesurer auprès des adolescents est la maturité vocationnelle (Jordaan et Heyde, 1979). Des instruments de mesure évaluant la maturité vocationnelle existent déjà, mais aucun d’entre eux ne semble convenir parfaitement à la réalité du Québec, emmenant la création d’un nouvel outil, le Questionnaire de maturité vocationnelle (QMV). L’objectif de ce mémoire est de vérifier les propriétés psychométriques de l’échelle affective du QMV. Les participants de l’étude sont 1 164 adolescents québécois âgés entre 14 et 18 ans (moyenne : 15,30 ans), provenant de 14 écoles secondaires. Pour ce qui est du niveau scolaire des participants, 511 (43,9 %) sont en 3e secondaire, 303 (26 %) sont en 4e secondaire et 350 (30,1 %) sont en 5e secondaire. Les résultats obtenus appuient la fidélité et la validité de l’instrument de mesure.

Elliptic curves, group law, and efficient computation

This thesis is about the derivation of the addition law on an arbitrary elliptic curve and efficiently adding points on this elliptic curve using the derived addition law. The outcomes of this research guarantee practical speedups in higher level operations which depend on point additions. In particular, the contributions immediately find applications in cryptology. Mastered by the 19th century mathematicians, the study of the theory of elliptic curves has been active for decades. Elliptic curves over finite fields made their way into public key cryptography in late 1980’s with independent proposals by Miller [Mil86] and Koblitz [Kob87]. Elliptic Curve Cryptography (ECC), following Miller’s and Koblitz’s proposals, employs the group of rational points on an elliptic curve in building discrete logarithm based public key cryptosystems. Starting from late 1990’s, the emergence of the ECC market has boosted the research in computational aspects of elliptic curves. This thesis falls into this same area of research where the main aim is to speed up the additions of rational points on an arbitrary elliptic curve (over a field of large characteristic). The outcomes of this work can be used to speed up applications which are based on elliptic curves, including cryptographic applications in ECC. The aforementioned goals of this thesis are achieved in five main steps. As the first step, this thesis brings together several algebraic tools in order to derive the unique group law of an elliptic curve. This step also includes an investigation of recent computer algebra packages relating to their capabilities. Although the group law is unique, its evaluation can be performed using abundant (in fact infinitely many) formulae. As the second step, this thesis progresses the finding of the best formulae for efficient addition of points. In the third step, the group law is stated explicitly by handling all possible summands. The fourth step presents the algorithms to be used for efficient point additions. In the fifth and final step, optimized software implementations of the proposed algorithms are presented in order to show that theoretical speedups of step four can be practically obtained. In each of the five steps, this thesis focuses on five forms of elliptic curves over finite fields of large characteristic. A list of these forms and their defining equations are given as follows: (a) Short Weierstrass form, y2 = x3 + ax + b, (b) Extended Jacobi quartic form, y2 = dx4 + 2ax2 + 1, (c) Twisted Hessian form, ax3 + y3 + 1 = dxy, (d) Twisted Edwards form, ax2 + y2 = 1 + dx2y2, (e) Twisted Jacobi intersection form, bs2 + c2 = 1, as2 + d2 = 1, These forms are the most promising candidates for efficient computations and thus considered in this work. Nevertheless, the methods employed in this thesis are capable of handling arbitrary elliptic curves. From a high level point of view, the following outcomes are achieved in this thesis. - Related literature results are brought together and further revisited. For most of the cases several missed formulae, algorithms, and efficient point representations are discovered. - Analogies are made among all studied forms. For instance, it is shown that two sets of affine addition formulae are sufficient to cover all possible affine inputs as long as the output is also an affine point in any of these forms. In the literature, many special cases, especially interactions with points at infinity were omitted from discussion. This thesis handles all of the possibilities. - Several new point doubling/addition formulae and algorithms are introduced, which are more efficient than the existing alternatives in the literature. Most notably, the speed of extended Jacobi quartic, twisted Edwards, and Jacobi intersection forms are improved. New unified addition formulae are proposed for short Weierstrass form. New coordinate systems are studied for the first time. - An optimized implementation is developed using a combination of generic x86-64 assembly instructions and the plain C language. The practical advantages of the proposed algorithms are supported by computer experiments. - All formulae, presented in the body of this thesis, are checked for correctness using computer algebra scripts together with details on register allocations.

Obstacle-free range determination for rail track maintenance vehicles

Maintenance trains travel in convoy. In Australia, only the first train of the convoy pays attention to the track sig- nalization (the other convoy vehicles simply follow the preceding vehicle). Because of human errors, collisions can happen between the maintenance vehicles. Although an anti-collision system based on a laser distance meter is already in operation, the existing system has a limited range due to the curvature of the tracks. In this paper, we introduce an anti-collision system based on vision. The two main ideas are, (1) to warp the camera image into an image where the rails are parallel through a projective transform, and (2) to track the two rail curves simultaneously by evaluating small parallel segments. The performance of the system is demonstrated on an image dataset.

Multiple camera management using wide baseline matching

Camera calibration information is required in order for multiple camera networks to deliver more than the sum of many single camera systems. Methods exist for manually calibrating cameras with high accuracy. Manually calibrating networks with many cameras is, however, time consuming, expensive and impractical for networks that undergo frequent change. For this reason, automatic calibration techniques have been vigorously researched in recent years. Fully automatic calibration methods depend on the ability to automatically find point correspondences between overlapping views. In typical camera networks, cameras are placed far apart to maximise coverage. This is referred to as a wide base-line scenario. Finding sufficient correspondences for camera calibration in wide base-line scenarios presents a significant challenge. This thesis focuses on developing more effective and efficient techniques for finding correspondences in uncalibrated, wide baseline, multiple-camera scenarios. The project consists of two major areas of work. The first is the development of more effective and efficient view covariant local feature extractors. The second area involves finding methods to extract scene information using the information contained in a limited set of matched affine features. Several novel affine adaptation techniques for salient features have been developed. A method is presented for efficiently computing the discrete scale space primal sketch of local image features. A scale selection method was implemented that makes use of the primal sketch. The primal sketch-based scale selection method has several advantages over the existing methods. It allows greater freedom in how the scale space is sampled, enables more accurate scale selection, is more effective at combining different functions for spatial position and scale selection, and leads to greater computational efficiency. Existing affine adaptation methods make use of the second moment matrix to estimate the local affine shape of local image features. In this thesis, it is shown that the Hessian matrix can be used in a similar way to estimate local feature shape. The Hessian matrix is effective for estimating the shape of blob-like structures, but is less effective for corner structures. It is simpler to compute than the second moment matrix, leading to a significant reduction in computational cost. A wide baseline dense correspondence extraction system, called WiDense, is presented in this thesis. It allows the extraction of large numbers of additional accurate correspondences, given only a few initial putative correspondences. It consists of the following algorithms: An affine region alignment algorithm that ensures accurate alignment between matched features; A method for extracting more matches in the vicinity of a matched pair of affine features, using the alignment information contained in the match; An algorithm for extracting large numbers of highly accurate point correspondences from an aligned pair of feature regions. Experiments show that the correspondences generated by the WiDense system improves the success rate of computing the epipolar geometry of very widely separated views. This new method is successful in many cases where the features produced by the best wide baseline matching algorithms are insufficient for computing the scene geometry.