Dissertations / Theses on the topic 'Rational degree'

We give a complete characterization of degree two rational maps on P1 with potential good reduction over local fields. We show this happens exactly when the map corresponds to an integral point in the moduli space M2. The proof includes an algorithm by which to conjugate any degree two rational map corresponding to an integral point in M2 into a map with unit resultant. The local fields result is used to solve the same problem for number fields with class number 1. Some additional results are given for degree 2 rational maps over Q. We also give a full description of post-critically finite maps in M2(Q), including the algorithm used to find them.
Thesis (Ph. D.)--University of Hawaii at Manoa, 2012.

In this thesis we lay the foundation for rational degree d as an element of Z[1/p] by using perfectoid analogue of projective space, and consider power series instead of polynomials. We start the groundwork by proving Weierstrass theorems for perfectoid spaces which are analogues of standard Weierstrass theorems in complex analysis. We then move onto defining sheaves for Projective perfectoid analogue and prove perfectoid analogues of Gorthendieck's classication theorem on projective line, Serre's theorem on Cohomology of line bundles. As intermediate results we also compute Picard groups and describe Cartier and Weil divisors for Perfectoid.

Interpolation conditions on rational matrix functions expressed in terms of residues are studied. As a compact way of expressing tangential interpolation conditions of arbitrarily high multiplicity possibly from both sides simultaneously, interpolation conditions are represented in terms of residues. The minimal possible complexity, measured by the McMillan degree, of interpolants is found in terms of the controllability and the observability indices of certain pairs of matrices which are part of given data. An interpolant of such complexity is obtained in realization form. This leads to another approach to the partial realization problem. As a generalization of the well-known Lagrange interpolation problem for scalar polynomials, the problem of seeking for a matrix polynomial interpolant of low complexity is studied. The main tool is state space methods borrowed from systems theory. After adoption of state space methods, problems concerning rational matrix functions are reduced to the realm of linear algebra.
Ph. D.

The general scope of this thesis is aimed at investigating certain classes of discrete equations through the analysis of certain characteristics of the solutions of these equations. We construct new methods of analysis based on the growth of these characteristics that let us single out known integrable discrete equations from certain class of equations. These integrable discrete equations are discrete analogues of the famous Painleve equations.

Досліджені роль і місце інноваційної культури суспільства у системі механізмів формування в Україні середовища сприятливого для інноваційної діяльності. Уточнено склад і функції її окремих підсистем. Виконано оцінку елементів підсистем інноваційної культури суспільства, що дозволяє обґрунтовано впроваджувати систему заходів, спрямованих на її розвиток. Запропоновано систему засадничих принципів формування в Україні інноваційно-сприятливого середовища. Запропоновано рекомендації щодо раціонального вибору ступеня радикалізації розроблюваних конкретною організацією інновацій залежно від співвідношення рівня її інноваційної культури і інноваційної культури суспільства у цілому.
The role and place of society's innovative culture was analyzed through the system of mechanisms of favorable innovative environment formation in Ukraine. The structure and functions of its individual subsystems was specified. The estimation of the subsystems elements of a society's innovative culture was performed, allowing the implementation of reasonable activities aimed at its development. The system of fundamental principles of forming the innovative environment in Ukraine was suggested. The authors advanced recommendations for choosing the rational degree of radicalization of innovations for a particular organization depending on the ratio of the organization’s innovative culture and innovative culture of the society in general.

On s'intéresse, dans cette thèse, à des questions concernant le nombre maximum de points rationnels d'une courbe singulière définie sur un corps fini, sujet qui, depuis Weil, a été amplement abordé dans le cas lisse. Cette étude se déroule en deux temps. Tout d'abord on présente une construction de courbes singulières de genres et corps de base donnés, possédant un grand nombre de points rationnels : cette construction, qui repose sur des notions et outils de géométrie algébrique et d'algèbre commutative, permet de construire, en partant d'une courbe lisse X, une courbe à singularités X', de telle sorte que X soit la normalisée de X', et que les singularités ajoutées soient rationnelles sur le corps de base et de degré de singularité prescrit. Ensuite, en utilisant une approche euclidienne, on prouve une nouvelle borne sur le nombre de points fermés de degré deux d'une courbe lisse définie sur un corps fini.La combinaison de ces résultats, à priori indépendants, permet notamment d'étudier le problème de savoir quand la borne d'Aubry-Perret, analogue de la borne de Weil dans le cas singulier, est atteinte. Cela nous amène de façon naturelle à l'étude des propriétés des courbes maximales et, lorsque la cardinalité du corps de base est un carré, à l'analyse du spectre des genres de ces dernières
In this PhD thesis, we focus on some issues about the maximum number of rational points on a singular curve defined over a finite field. This topic has been extensively discussed in the smooth case since Weil's works. We have split our study into two stages. First, we provide a construction of singular curves of prescribed genera and base field and with many rational points: such a construction, based on some notions and tools from algebraic geometry and commutative algebra, yields a method for constructing, given a smooth curve X, another curve X' with singularities, such that X is the normalization of X', and the added singularities are rational on the base field and with the prescribed singularity degree. Then, using a Euclidian approach, we prove a new bound for the number of closed points of degree two on a smooth curve defined over a finite field.Combining these two a priori independent results, we can study the following question: when is the Aubry-Perret bound (the analogue of the Weil bound in the singular case) reached? This leads naturally to the study of the properties of maximal curves and, when the cardinality of the base field is a square, to the analysis of the spectrum of their genera

This thesis explores the bilateral relationship between the European Union (EU) and Japan as interregional partners and as united global actors. The principal aim is to navigate the nature of relations between these two diverse entities asserting that ideology, culture and identity formation contribute significantly to the joint policies undertaken by these two groups. The theoretical assumptions of this thesis are discussed in parallel to observable phenomena relative to the nature of relations between the EU and Japan, 1990-2005. This research analyses EU and Japan interactions from the Joint Declaration 1991 and including the Action Plan 2001 and aims to use the theoretical concepts of Identity. The frameworks of the Four Point Plan and the Analytic Narrative are appropriate to gain insight to the nature of these relations. Both the Four Point Plan and the Analytic Narrative are broad in scope but defined in nature, thus they are most suitable for the purpose of this analysis. This multidisciplinary approach enables the deconstruction of the influence held by 'identity' in domestic and international policy, outlining that policy influences such as preference and 'stories'! affect the relationship between the EU and Japan. This modem research angle serves to emphasise the uniqueness of the EU and Japan as international actors. The United States and rise of China act as intervening variables, and their secondary role highlights the changing global environment over this period.

Une variété horosphérique est une variété algébrique normale dans laquelle un groupe algébrique réductif opère avec une orbite ouverte fibrée en tores sur une variété de drapeaux. La dimension de ces tores est appelée le rang de la variété horosphérique. En particulier, les variétés toriques et les variétés de drapeaux sont horosphériques. Dans cette thèse, on classifie les variétés horosphériques de Fano en termes de certains polytopes rationnels qui généralisent les polytopes réflexifs considérés par V.Batyrev. Puis on obtient une majoration du degré des variétés horosphériques lisses de Fano, analogue à celle donnée par O.Debarre dans le cas torique. On étend un résultat récent de C.Casagrande : les variétés horosphériques Q-factorielles de Fano ont leur nombre de Picard majoré par deux fois la dimension. On donne aussi de nombreux exemples en rang 2.

Research and Development engineering is at the corner stone of humanity’s evolution. It is perceived to be a systematic creative process which ultimately improves the living standard of a society through the creation of new applications and products. The commercial paradigm that governs project selection, resource allocation and market penetration prevails when the focus shifts from pure research to applied research. Furthermore, the road to success through commercialisation is difficult for most inventors, especially in a vast and isolated country such as Australia which is located a long way from wealthy and developed economies. While market leading products are considered unique, the actual process to achieve these products is essentially the same; progressing from an idea, through development to an outcome (if successful). Unfortunately, statistics indicate that only 3% of ‘ideas’ are significantly successful, 4% are moderately successful, and the remainder ‘evaporate’ in that form (Michael Quinn, Chairman, Innovation Capital Associates Pty Ltd). This study demonstrates and analyses two techniques developed by the author which reduce uncertainty in the engineering design and development phase of new product development and therefore increase the probability of a successful outcome. This study expands the existing knowledge of the engineering design and development stage in the new product development process and is couched in the identification of practical methods, which have been successfully used to develop new products by Australian Small Medium Enterprise (SME) Excel Technology Group Pty Ltd (ETG). Process theory is the term most commonly used to describe scientific study that identifies occurrences that result from a specified input state to an output state, thus detailing the process used to achieve an outcome. The thesis identifies relevant material and analyses recognised and established engineering processes utilised in developing new products. The literature identified that case studies are a particularly useful method for supporting problem-solving processes in settings where there are no clear answers or where problems are unstructured, as in New Product Development (NPD). This study describes, defines, and demonstrates the process of new product development within the context of historical product development and a ‘live’ case study associated with an Australian Government START grant awarded to Excel Technology Group in 2004 to assist in the development of an image-based vehicle detection product. This study proposes two techniques which reduce uncertainty and thereby improve the probability of a successful outcome. The first technique provides a predicted project development path or forward engineering plan which transforms the initial ‘fuzzy idea’ into a potential and achievable outcome. This process qualifies the ‘fuzzy idea’ as a potential, rationale or tangible outcome which is within the capability of the organisation. Additionally, this process proposes that a tangible or rationale idea can be deconstructed in reverse engineering process in order to create a forward engineering development plan. A detailed structured forward engineering plan reduces the uncertainty associated with new product development unknowns and therefore contributes to a successful outcome. This is described as the RETRO technique. The study recognises however that this claim requires qualification and proposes a second technique. The second technique proposes that a two dimensional spatial representation which has productivity and consumed resources as its axes, provides an effective means to qualify progress and expediently identify variation from the predicted plan. This spatial representation technique allows a quick response which in itself has a prediction attribute associated with directing the project back onto its predicted path. This process involves a coterminous comparison between the predicted development path and the evolving actual project development path. A consequence of this process is verification of progress or the application of informed, timely and quantified corrective action. This process also identifies the degree of success achieved in the engineering design and development phase of new product development where success is defined as achieving a predicted outcome. This spatial representation technique is referred to as NPD Mapping. The study demonstrates that these are useful techniques which aid SMEs in achieving successful new product outcomes because the technique are easily administered, measure and represent relevant development process related elements and functions, and enable expedient quantified responsive action when the evolving path varies from the predicted path. These techniques go beyond time line representations as represented in GANTT charts and PERT analysis, and represent the base variables of consumed resource and productivity/technical achievement in a manner that facilitates higher level interpretation of time, effort, degree of difficulty, and product complexity in order to facilitate informed decision making. This study presents, describes, analyses and demonstrates an SME focused engineering development technique, developed by the author, that produces a successful new product outcome which begins with a ‘fuzzy idea’ in the mind of the inventor and concludes with a successful new product outcome that is delivered on time and within budget. Further research on a wider range of SME organisations undertaking new product development is recommended.

Dans cette thèse, nous nous intéressons à l'étude de l'arithmétique (le principe de Hasse, l'approximation faible, et l'obstruction de Brauer-Manin) des zéro-cycles sur les variétés algébriques définies sur des corps de nombres. Nous introduisons la notion de sous-ensemble hilbertien généralisé. En utilisant la méthode de fibration, nous démontrons que l'obstruction de Brauer-Manin est la seule au principe de Hasse et à l'approximation faible pour les zéro-cycles de degré 1; et établissons l'exactitude d'une suite de type global-local concernant les groupes de Chow des zéro-cycles, pour certaines variétés qui admettent une structure de fibration au-dessus d'une courbe lisse ou au-dessus de l'espace projectif, où les hypothèses arithmétiques sont posées seulement sur les fibres au-dessus d'un sous-ensemble hilbertien généralisé.De plus, nous relions l'arithmétique des points rationnels et l'arithmétique des zérocycles de degré 1 sur les variétés géométriquement rationnellement connexes. Comme application, nous trouvons que l'obstruction de Brauer-Manin est la seule au principe de Hasse et à l'approximation faible pour les zéro-cycles de degré 1 sur- les espaces homogènes d'un groupe algébrique linéaire à stabilisateur connexe,- certains fibrés en surfaces de Châtelet au-dessus d'une courbe lisse ou au-dessus de l'espace projectif (en particulier, les solides de Poonen).

One of the fundamental problems in Algebraic Geometry is to study solutions to certain systems of polynomial equations in several variables, or in other words, find rational points on a given variety which is defined by equations. In this paper, we discuss the existence of del Pezzo surface of degree 1 and 2 with a unique rational point over any finite field [Special characters omitted.] , and we will give a lower bound on the number of rational points to each q. Furthermore, we will give explicit equations of del Pezzo surfaces with a unique rational point. Also we will discuss the rationality property of the del Pezzo surfaces especially in lower degrees.