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Dissertations / Theses on the topic 'Curves, Plane. Curves, Quartic'
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Jones, Andrew. "Modular elliptic curves over quartic CM fields." Thesis, University of Sheffield, 2015. http://etheses.whiterose.ac.uk/8791/.
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In this thesis I establish the modularity of a number of elliptic curves defined over quartic CM fields, by showing that the Galois representation attached to such curves (arising from the natural Galois action on the l-adic Tate module) is isomorphic to a representation attached to a cuspidal automorphic form for GL(2) over the CM field in question. This is achieved through the study of the Hecke action on the cohomology of certain symmetric spaces, which are known to be isomorphic to spaces of cuspidal automorphic forms by a generalization of the Eichler-Shimura isomorphism.
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Smilovic, Mikhail. "Curves on a plane." Thesis, McGill University, 2012. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=106605.
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In this thesis, we study the space of immersions from the circle to the plane Imm(S¹,R²), modulo the group of diffeomorphisms on S¹. We discuss various Riemannian metrics and find surprisingly that the L²-metric fails to separate points. We show two methods of strengthening this metric, one to obtain a non-vanishing metric, and the other to stabilize the minimizing energy flow. We give the formulas for geodesics, energy and give an example of computed geodesics in the case of concentric circles. We then carry our results over to the larger spaces of immersions from a compact manifold M to a Riemannian manifold (N, g), modulo the group of diffeomorphisms on M.Dans cette thése, nous étudierons l'espace d'immersions d'un cercle au plan Imm(S¹,R²), modulo le groupe de difféomorphisme sur S¹. Nous discuterons de divers métriques riemanniennes et monterons la surprenante impossibilité de séparer des points dans la métrique L². Nous présenterons deux méthodes de renforcer cette métrique, une pour obtenir une métrique non-nulle, et une autre pour stabiliser le flot d'énergie. Nous donnerons les formules pour les géodésiques et l'énergie, et donnerons un exemple de calcul de géodésiques dans le cas des cercles concentriques. Nous étendrons alors nos résultats sur la plus grande espace d'immersion d'une variété M compacte à une variété riemannienne (N,g), modulo le groupe de difféomorphisme sur M.
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Nichols, Margaret E. "Intersection Number of Plane Curves." Oberlin College Honors Theses / OhioLINK, 2013. http://rave.ohiolink.edu/etdc/view?acc_num=oberlin1385137385.
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Debrecht, Johanna M. "Plane Curves, Convex Curves, and Their Deformation Via the Heat Equation." Thesis, University of North Texas, 1998. https://digital.library.unt.edu/ark:/67531/metadc278501/.
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We study the effects of a deformation via the heat equation on closed, plane curves. We begin with an overview of the theory of curves in R3. In particular, we develop the Frenet-Serret equations for any curve parametrized by arc length. This chapter is followed by an examination of curves in R2, and the resultant adjustment of the Frenet-Serret equations. We then prove the rotation index for closed, plane curves is an integer and for simple, closed, plane curves is ±1. We show that a curve is convex if and only if the curvature does not change sign, and we prove the Isoperimetric Inequality, which gives a bound on the area of a closed curve with fixed length. Finally, we study the deformation of plane curves developed by M. Gage and R. S. Hamilton. We observe that convex curves under deformation remain convex, and simple curves remain simple.
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Markwig, Hannah. "The enumeration of plane tropical curves." [S.l.] : [s.n.], 2006. http://deposit.ddb.de/cgi-bin/dokserv?idn=980700736.
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Granholm, Jonas. "Remarkable curves in the Euclidean plane." Thesis, Linköpings universitet, Matematik och tillämpad matematik, 2014. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-112576.
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An important part of mathematics is the construction of good definitions. Some things, like planar graphs, are trivial to define, and other concepts, like compact sets, arise from putting a name on often used requirements (although the notion of compactness has changed over time to be more general). In other cases, such as in set theory, the natural definitions may yield undesired and even contradictory results, and it can be necessary to use a more complicated formalization. The notion of a curve falls in the latter category. While it is intuitively clear what a curve is – line segments, empty geometric shapes, and squiggles like this: – it is not immediately clear how to make a general definition of curves. Their most obvious characteristic is that they have no width, so one idea may be to view curves as what can be drawn with a thin pen. This definition, however, has the weakness that even such a line has the ability to completely fill a square, making it a bad definition of curves. Today curves are generally defined by the condition of having no width, that is, being one-dimensional, together with the conditions of being compact and connected, to avoid strange cases. In this thesis we investigate this definition and a few examples of curves.
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Peternell, Carolin Susanne [Verfasser]. "Birational models for moduli of quartic rational curves / Carolin Susanne Peternell." Mainz : Universitätsbibliothek Mainz, 2018. http://d-nb.info/1164037919/34.
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Radzimski, Vanessa Elena. "Points of small height on plane curves." Thesis, University of British Columbia, 2014. http://hdl.handle.net/2429/46341.
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Let K be an algebraically closed field, and let C be an irreducible plane curve, defined over the algebraic closure of K(t), which is not defined over K. We show that there exists a positive real number c??? such that if P is any point on the curve C whose Weil height is bounded above by c???, then the coordinates of P belong to K.
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Holanda, Felipe D'Angelo. "Introduction to differential geometry of plane curves." Universidade Federal do CearÃ, 2015. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=15052.
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CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel SuperiorA intenÃÃo desse trabalho serà de abordar de forma bÃsica e introdutÃria o estudo da Geometria Diferencial, que por sua vez tem seus estudos iniciados com as Curvas Planas. Serà necessÃrio um conhecimento de CÃlculo Diferencial, Integral e Geometria AnalÃtica para melhor compreensÃo desse trabalho, pois como seu prÃprio nome nos transparece Geometria Diferencial vem de uma junÃÃo do estudo da Geometria envolvendo CÃlculo. Assim abordaremos subtemas como curvas suaves, vetor tangente, comprimento de arco passando por fÃrmulas de Frenet, curvas evolutas e involutas e finalizaremos com alguns teoremas importantes, como o teorema fundamental das curvas planas, teorema de Jordan e o teorema dos quatro vÃrtices. O que, basicamente representa, o capÃtulo 1, 4 e 6 do livro IntroduÃÃo Ãs Curvas Planas de HilÃrio Alencar e Walcy Santos.
The intention of this work is to address in basic form and introductory study of Differential Geometry, which in turn has started his studies with Planas curves. It will require a knowledge of Differential Calculus, Integral and Analytic Geometry for better understanding of this work, because as its name says in Differential Geometry comes from the joint study of geometry involving Calculation. So we discuss sub-themes as smooth curves, tangent vector, arc length through formulas of Frenet, evolutas curves and involute and conclude with some important theorems, as the fundamental theorem of plane curves, Jordan 's theorem and the theorem of four vertices. What basically is, Chapter 1, 4 and 6 of the book Introduction to Plane Curves HilÃrio Alencar and Walcy Santos.
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Al-Shammari, Fahd M. "Jacobians of plane quintic curves of genus one." Diss., The University of Arizona, 2002. http://hdl.handle.net/10150/289840.
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Let K be a number field. By representing genus one curves as plane quintic curves with 5 double points, we construct (up to birational equivalence) the universal elliptic curves defined over the modular curves X₁(5) and X(μ)(5) (X(μ)(5) is the modular curve parameterizing pairs (E, i : (μ)₅ → E) where E is an elliptic curve over Q). We then twist the latter by elements coming from H¹(Gal(K̅/K), (μ)₅) to construct universal families of principal homogeneous spaces for the curves E. Finally we show that every principal homogeneous space arising this way is visible in some abelian variety.
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Hacking, P. "A compactification of the space of plane curves." Thesis, University of Cambridge, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.599821.
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We define a geometrically meaningful compactification of the moduli space of smooth plane curves which can be calculated explicitly. The basic idea is to regard a plane curve DCP2 as a pair (P2, D) of a surface together with a divisor, and allow both the surface and the curve to degenerate. For plane curves of degree d ≥ 4, we obtain a compactification Md which is a moduli space of stable pairs (X, D) using the log minimal model program. A stable pair (X, D) consists of a surface X such that - KX is ample and a divisor D in a given linear system on X with specified singularities. Note that X may be non-normal, and KX is π-Cartier but not Cartier in general. We give a rough classification of stable pairs of arbitrary degree, a complete classification in degrees 4 and 5, and a partial classification in degree 6. The compactification is particularly simple if d is not a multiple of 3- in particular the surface X has at most 2 components. We give a characterisation of these surfaces in terms of the singularities and the Picard numbers of the components. Moreover, we show that Md is smooth in this case.
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Meco, Benjamin. "Expanding flows of curves in the hyperbolic plane." Thesis, Uppsala universitet, Tillämpad matematik och statistik, 2021. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-446016.
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Saloom, Amani Hussain. "Curves in the Minkowski plane and Lorentzian surfaces." Thesis, Durham University, 2012. http://etheses.dur.ac.uk/4451/.
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We investigate in this thesis the generic properties of curves in the Minkowski plane R2 1 and of smooth Lorentzian surfaces. The generic properties of curves in R2 1 are obtained by studying the contacts of curves in R2 1 with lines and pseudo-circles. These contacts are captured by the singularities of the families of height and distancesquared functions on the curves. On the other hand, the generic properties of smooth Lorentzian surfaces are obtained by studying certain Binary Differential Equations defined on the surfaces.
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Blanco, Fernández Guillem. "Bernstein-Sato polynomial of plane curves and Yano's conjecture." Doctoral thesis, Universitat Politècnica de Catalunya, 2020. http://hdl.handle.net/10803/669107.
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The main aim of this thesis is the study of the Bernstein-Sato polynomial of plane curve singularities. In this context, we prove a conjecture posed by Yano about the generic b-exponents of a plane irreducible curve.
In a part of the thesis, we study the Bernstein-Sato polynomial through the analytic continuation of the complex zeta function of a singularity. We obtain several results on the vanishing and non-vanishing of the residues of the complex zeta function. Using these results we obtain a proof of Yano's conjecture under the hypothesis that the eigenvalues of the monodromy are pair-wise different. In another part of the thesis, we study the periods of integrals in the Milnor fiber and their asymptotic expansion. These periods of integrals can be related to the b-exponents and can be constructed in terms of resolution of singularities. Using these techniques, we can present a proof for the general case of Yano's conjecture.
In addition to the Bernstein-Sato polynomial, we also study the minimal Tjurina number of a plane irreducible curve and we answer in the positive a question raised by Dimca and Greuel on the quotient between the Milnor and Tjurina numbers. More precisely, we prove a formula for the minimal Tjurina number of a plane irreducible curve in terms of the multiplicities of the strict transform along its minimal resolution. From this formula, we obtain the positive answer to Dimca and Greuel question.
This thesis also contains computational results for the theory of singularities on smooth complex surfaces. First, we describe an algorithm to compute log-resolutions of ideals on a smooth complex surface. Secondly, we provide an algorithm to compute generators for complete ideals on a smooth complex surface. These algorithms have several applications, for instance, in the computation of the multiplier ideals associated to an ideal on a smooth complex surface.El principal objectiu d'aquesta tesi és l'estudi del polinomi de Bernstein-Sato de singularitats de corbes planes. En aquest context, es demostra una conjectura proposada per Yano el 1982 sobre els \( b \)-exponents genèrics d'una corba plana irreductible. En una part d'aquesta tesi, s'estudia el polinomi de Bernstein-Sato utilitzant la continuació analítica de la funció zeta complexa d'una singularitat. S'obtenen diversos resultat sobre l'anul·lació i no anul·lació del residu de la funció zeta complexa d'una corba plana. Utilitzant aquests resultats, s'obté una demostració de la conjectura de Yano sota la hipòtesi de que els valors propis de la monodromia siguin diferents dos a dos. En un altre part de la tesi, s'estudien els períodes d'integrals en la fibra de Milnor i la seva expansió asimptòtica. Aquesta expansió asimptòtica dels períodes pot ser relacionada amb els b-exponents i pot ser construïda en termes de la resolució de singularitats. Utilitzant aquestes tècniques, es presenta una prova del cas general de la conjectura de Yano. A més a més del polinomi de Bernstein-Sato, també s'estudia el nombre de Tjurina mínim d'una corba plana irreductible i responem positivament a una pregunta formulada per Dimca i Greuel sobre el quocient entre els nombres de Milnor i Tjurina. Concretament, es demostra una fórmula pel nombre de Tjurina mínim en un classe d'equisingularitat de corbes planes irreductibles en termes de la seqüència de multiplicitats de la transformada estricta al llarg de la resolució minimal. A partir d'aquesta fórmula, s'obté la resposta positiva a la pregunta de Dimca i Greuel. Aquesta tesi també conté resultats computacionals per la teoria de singularitats en superfícies complexes llises. Primer, es descriu un algorisme que calcula la log-resolució d'ideals en un superfície complexa llisa. En segon lloc, es dona un algorisme per calcular generadors per ideals complets en una superfície complexa llisa. Aquests algorismes tenen diverses aplicacions, com per exemple, en el càlcul d'ideals multiplicadors associats a un ideal en una superfície complexa llisa.
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Jaramillo, Puentes Andrés. "Rigid isotopy classification of real quintic rational plane curves." Thesis, Paris 6, 2017. http://www.theses.fr/2017PA066116/document.
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Afin d’étudier les classes d'isotopie rigide des courbes rationnelles nodales de degré 5 dans RPP, nous associons à chaque quintique avec un point double réel marque une courbe trigonale dans la surface de Hirzebruch Sigma3 et le dessin reel nodal correspondant dans CP/(z mapsto bar{z}). Les dessins sont des versions réelles, proposées par S. Orevkov dans cite{Orevkov}, des dessins d'enfants de Grothendieck. Un dessin est un graphe contenu dans une surface topologique, muni d'une certaine structure supplémentaire. Dans cette thèse, nous étudions les propriétés combinatoires et les recompositions des dessins correspondants aux courbes trigonales nodales C subset Sigma dans les surfaces réglées réelles Sigma . Les dessins uninodaux sur une surface a bord quelconque et les dessins nodaux sur le disque peuvent être décomposés en blocs correspondant aux dessins cubiques sur le disque D2 , ce qui conduit a une classification des ces dessins. La classification des dessins considérés mène à une classification à isotopie rigide des courbes rationnelles nodales de degré 5 dans RPPIn order to study the rigid isotopy classes of nodal rational curves of degree $5$ in $\RPP$, we associate to every real rational quintic curve with a marked real nodal point a trigonal curve in the Hirzebruch surface $\Sigma_3$ and the corresponding nodal real dessin on~$\CP/(z\mapsto\bar{z})$. The dessins are real versions, proposed by S. Orevkov~\cite{Orevkov}, of Grothendieck's {\it dessins d'enfants}. The {\it dessins} are graphs embedded in a topological surface and endowed with a certain additional structure. We study the combinatorial properties and decompositions of dessins corresponding to real nodal trigonal curves~$C\subset \Sigma$ in real ruled surfaces~$\Sigma$. Uninodal dessins in any surface with non-empty boundary and nodal dessins in the disk can be decomposed in blocks corresponding to cubic dessins in the disk~$\mathbf{D}^2$, which produces a classification of these dessins. The classification of dessins under consideration leads to a rigid isotopy classification of real rational quintics in~$\RPP$
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Eriksson, Östman Albin. "The Total Curvature of Disks Extending Regular Closed Plane Curves." Thesis, Uppsala University, Department of Mathematics, 2008. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-120507.
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Farag, Eslam Essam Ebrahim. "On the stratification of smooth plane curves by automorphism groups." Doctoral thesis, Universitat Autònoma de Barcelona, 2017. http://hdl.handle.net/10803/457868.
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Curvas proyectivas no singulares sobre un cuerpo con grupo de automorfismo no trivial son de gran interés en Geometría Aritmética. En la tesis, estudiamos la estratificación de las curvas planas no singulares (de género mayor o igual a tres) por sus grupos de automorfismos y tratamos con aspectos de geometría algebraica y aritmética.
En primer lugar, aportamos un estudio general de las clases (modulo -isomorfismo) de curvas planas lisas de género fijo g con un subgrupo de automorfismo fijo , donde denota una clausura algebraica fijada de . En particular, se aporta un estudio de grupos de automorfismos que aparecen y las ecuaciones definitorias asociadas en dichas clases.
En segundo lugar, sea C una curva proyectiva lisa definida sobre , en particular por extensión de escalares obtenemos una curva lisa sobre y suponemos que dicha extension corresponde a una curva plana no singular. Nuestro objetivo es estudiar cuerpos de definición de modelos planos no singulares para y de sus ``twists’’ sobre , usando la inmersión en en lugar de la dada por el modelo canónico en . Más concretamente, preguntamos si es una curva plana lisa sobre o no; y si la respuesta es afirmativa, es entonces cada ``twist’’ de sobre una curva plana lisa sobre ?. Para ambas preguntas la respuesta es no en general. Obtenemos resultados para las cuales las preguntas anteriores siempre tienen una respuesta afirmativa, y mostramos diferentes ejemplos con respecto a la respuesta general negativa. Curiosamente, en la forma de obtener estos ejemplos, tenemos que manejar con superficies no-triviales de Brauer-Severi, y somos capaces de calcular ecuaciones explícitas de una no trivial.
En tercer lugar, obtenemos una denominada clasificación representativa de los estratos por grupo de automorfismos de curvas planas no singulares sobre de género , donde es perfecto de característica o . Curiosamente, en la forma de obtener estas familias, encontramos dos fenómenos notables que no aparecieron anteriormente para género 3. Una, es la existencia de un estrato final no zero dimensional de curvas planas no-singular. Observamos en la tesis que esta es una situación usual cuando el género crece y aportamos una explicación. Describimos explícitamente familias representativas para todos los estratos, excepto para el estrato con grupo de automorfismo cíclico de orden , pero en este caso podemos demostrar la existencia de tal familia aplicando una versión del teorema de Lüroth en dimensión 2. Aquí encontramos la segunda diferencia con el caso de género inferior donde las técnicas conocidas no funcionan completamente.
Por último, sea un cuerpo perfecto de característica distinta de , y sea una curva plana lisa sobre cuyo grupo de automorfismo de C sea -conjugado a un grupo diagonal. Se sabe por el trabajo de B. Huggins en su tesis doctoral (2010, Berkeley) que el cuerpo de moduli de , relativo a la extensión de Galois no necesita ser un cuerpo de definición. Motivados por estos resultados, nos preguntamos sobre las caracterizaciones de tales curvas no definibles sobre su cuerpo de moduli. Distinguimos entre los dos casos dependiendo de si el número de puntos del plano proyectivo fijados por el grupo de automorfismo es finito o infinito. Nuestros resultados pueden ser utilizados como una fuente constructiva de ejemplos para curvas planas lisas con automorfismo cíclico donde el cuerpo de moduli no es un cuerpo de definición.Smooth projective curves over a field with non-trivial automorphism group are always of deep interest in the literature. Following the philosophy of Diophantine equations theory, the simplest case is to consider smooth plane curves over of geometric genus In the thesis, we study the stratification of smooth plane curves by their automorphism groups and we deal with both algebraic and arithmetic geometry aspects. We first give a general study of the stratum, consisting of -isomorphism classes of smooth plane curves of fix genus with a fixed automorphism subgroup , where is a fixed algebraic closure of . In particular, a detailed study of the structure of their automorphism group and the associated defining equations is investigated. Second, let be a smooth projective curve defined over , which is also plane viewed as a smooth curve over . We aim to study fields of definition for non-singular plane models of and also of its twists over k by considering the embedding into instead of the one given by the canonical model into More concretely, we ask whether is a smooth plane curve over or not; and if the answer is yes, is every twist of over also a smooth plane curve over For both questions the answer is no in general, it is not. We obtain results for the curves for which the above questions always have an affirmative answer, and we show different examples concerning the negative general answer. Interestingly, in the way to get these examples, we need to handle with non-trivial Brauer-Severi surfaces, and we are able to compute explicit equations of a non-trivial one. As far as we know, this is the first time that such equations are exhibited. Third, we obtain a so-called representative classification for the strata by automorphism group of smooth -plane curves of genus , where is perfect of characteristic or . Interestingly, in the way to get these families, we find two remarkable phenomena that did not appear before. One is the existence of a non -dimensional final stratum of plane curves. At a first sight it may sound odd, but we will see that this is a normal situation for higher degrees and we will give an explanation for it. We explicitly describe representative families for all strata, except for the stratum with cyclic automorphism group of order (fortunately, we are still able to prove the existence of such family by applying a version of Lüroth’s theorem in dimension 2). Here we find the second difference with the lower genus cases where the known techniques do not fully work. Finally, let be a perfect field of characteristic different from , and be a smooth plane curve over whose automorphism group of is -conjugate to a diagonal group. It is known from the work of B. Huggins in her PhD thesis (2010, Berkeley) that the field of moduli for , relative to the Galois extension does not need to be a field of definition. Motivated by these results, we wonder about characterizations of such curves not definable over their field of moduli. We distinguish between the two cases depending on whether the number of points of the projective plane fixed by the automorphism group is finite or infinite. Our results can be usede as a constructive source of so many examples of smooth plane curves with cyclic automorphism where the field of moduli is not a field of definition.
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Cohen, Camron Alexander Robey. "CURVING TOWARDS BÉZOUT: AN EXAMINATION OF PLANE CURVES AND THEIR INTERSECTION." Oberlin College Honors Theses / OhioLINK, 2020. http://rave.ohiolink.edu/etdc/view?acc_num=oberlin159345184740689.
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Chen, Qun. "Hilbert-Kunz multiplicity of plane curves and a conjecture of K. Pardue." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1998. http://www.collectionscanada.ca/obj/s4/f2/dsk3/ftp04/nq41416.pdf.
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Hilmar, Jan. "Intersection of algebraic plane curves : some results on the (monic) integer transfinite diameter." Thesis, University of Edinburgh, 2008. http://hdl.handle.net/1842/3843.
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Part I discusses the problem of determining the set of intersection points, with corresponding multiplicities, of two algebraic plane curves. We derive an algorithm based on the Euclidean Algorithm for polynomials and show how to use it to find the intersection points of two given curves. We also show that an easy proof of Bézout’s Theorem follows. We then discuss how, for curves with rational coefficients, this algorithm can bemodified to find the intersection points with coordinates expressed in terms of algebraic extensions of the rational numbers. Part II deals with the problem of determining the (monic) integer transfinite diameter of a given real interval. We show how this problem relates to the problem of determining the structure of the spectrum of normalised leading coefficients of polynomials with integer coefficients and all roots in the given interval. We then find dense regions of this spectrum for a number of intervals and discuss algorithms for finding discrete subsets of the spectrum for the interval [0,1]. This leads to an improvement in the known upper bound for the integer transfinite diameter. Finally, we discuss the connection between the infimum of the spectrum and the monic integer transfinite diameter.
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Mokhtarian, Farzin. "A theory of multi-scale, curvature and torsion based shape representation for planar and space curves." Thesis, University of British Columbia, 1990. http://hdl.handle.net/2429/30740.
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This thesis presents a theory of multi-scale, curvature and torsion based shape representation for planar and space curves. The theory presented has been developed to satisfy various criteria considered useful for evaluating shape representation methods in computer vision. The criteria are: invariance, uniqueness, stability, efficiency, ease of implementation and computation of shape properties.
The regular representation for planar curves is referred to as the curvature scale space image and the regular representation for space curves is referred to as the torsion scale space image. Two variants of the regular representations, referred to as the renormalized and resampled curvature and torsion scale space images, have also been proposed. A number of experiments have been carried out on the representations which show that they are very stable under severe noise conditions and very useful for tasks which call for recognition of a noisy curve of arbitrary shape at an arbitrary scale or orientation.
Planar or space curves are described at varying levels of detail by convolving their parametric representations with Gaussian functions of varying standard deviations. The curvature or torsion of each such curve is then computed using mathematical equations which express curvature and torsion in terms of the convolutions of derivatives of Gaussian functions and parametric representations of the input curves. Curvature or torsion zero-crossing points of those curves are then located and combined to form one of the representations mentioned above.
The process of describing a curve at increasing levels of abstraction is referred to as the evolution or arc length evolution of that curve. This thesis contains a number of theorems about evolution and arc length evolution of planar and space curves along with their proofs. Some of these theorems demonstrate that evolution and arc length evolution do not change the physical interpretation of curves as object boundaries and others are in fact statements on the global properties of planar and space curves during evolution and arc length evolution and their representations. Other theoretical results shed light on the local behavior of planar and space curves just before and just after the formation of a cusp point during evolution and arc length evolution.
Together these results provide a sound theoretical foundation for the representation methods proposed in this thesis.Science, Faculty of
Computer Science, Department of
Graduate
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Roos, Joris [Verfasser]. "Singular integrals and maximal operators related to Carleson's theorem and curves in the plane / Joris Roos." Bonn : Universitäts- und Landesbibliothek Bonn, 2017. http://d-nb.info/1139049038/34.
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Killian, Kenneth. "Maxwell’s Problem on Point Charges in the Plane." Scholar Commons, 2008. https://scholarcommons.usf.edu/etd/333.
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This paper deals with approximating an upper bound for the number of equilibrium points of a potential field produced by point charges in the plane. This is a simplified form of a problem posed by Maxwell [4], who considered spatial configurations of the point charges. Using algebraic techniques, we will give an upper bound for planar charges that is sharper than the bound given in [6] for most general configurations of charges. Then we will study an example of a configuration of charges that has exactly the number of equilibrium points that Maxwell's conjecture predicts, and we will look into the nature of the extremal points in this case. We will conclude with a solution to the twin problem for the logarithmic potential, followed by a discussion of the conditions necessary for a degenerate case in the plane.
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Grudsky, Serguey, and Nikolai Tarkhanov. "Conformal reduction of boundary problems for harmonic functions in a plane domain with strong singularities on the boundary." Universität Potsdam, 2012. http://opus.kobv.de/ubp/volltexte/2012/5774/.
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We consider the Dirichlet, Neumann and Zaremba problems for harmonic functions in a bounded plane domain with nonsmooth boundary. The boundary curve belongs to one of the following three classes: sectorial curves, logarithmic spirals and spirals of power type. To study the problem we apply a familiar method of Vekua-Muskhelishvili which consists in using a conformal mapping of the unit disk onto the domain to pull back the problem to a boundary problem for harmonic functions in the disk. This latter is reduced in turn to a Toeplitz operator equation on the unit circle with symbol bearing discontinuities of second kind. We develop a constructive invertibility theory for Toeplitz operators and thus derive solvability conditions as well as explicit formulas for solutions.
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Rimmasch, Gretchen. "Complete Tropical Bezout's Theorem and Intersection Theory in the Tropical Projective Plane." Diss., CLICK HERE for online access, 2008. http://contentdm.lib.byu.edu/ETD/image/etd2507.pdf.
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Fantin, Silas. "Monodromia de curvas algébricas planas." Universidade de São Paulo, 2007. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-10122007-165559/.
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Em 1968, J. Milnor introduziu a monodromia local de Picard-Lefschetz de uma hipersuperfície complexa com singularidade isolada. Em seguida, E. Brieskorn perguntou se esta monodromia é sempre finita. Em 1972, Lê Dúng Trâng provou que a resposta é positiva no caso de germes de curvas planas analíticas irredutíveis. Na época, já eram conhecidos exemplos de curvas planas com dois ramos e monodromia finita. Em 1973, N. A?Campo produziu o primeiro exemplo de germe de curva plana com dois ramos e monodromia infinita. Portanto, a questão mais simples, e ainda em aberto, que se coloca neste contexto, é a determinação da finitude da monodromia para germes de curvas planas com dois ramos. O presente trabalho, consiste em determinar, em várias situações, o polinômio mínimo da monodromia de germes de curvas analíticas planas com dois ramos, cujos gêneros são menores ou iguais a dois, o que permite decidir a sua finitudeIn 1968, J. Milnor introduced the Picard-Lefschetz monodromy of a complex hypersurface with an isolated singularity. Subsequently, E. Brieskorn asked if this monodromy is always finite. In 1972, Lê Dúng Trâng proved that the answer is positive in the case of irreducible analytic germs of plane curves. At this time, examples of plane curves with two branches and finite monodromy were known. In 1973, N. A?Campo produced the first example of a germ of plane curve with two branches and infinite monodromy. Therefore, the simplest and still open problem in this context is to determine whether the monodromy of a plane curve with two branches is finite or infinite. The present work consists in determining, in several situations, the minimal polynomial of the monodromy for germs of plane analytic curves with two branches, whose genera are less or equal than two, wich allows us to decide its finiteness
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Tewari, Ayush Kumar [Verfasser], Michael [Akademischer Betreuer] Joswig, Michael [Gutachter] Joswig, Hannah [Gutachter] Markwig, and Dhruv [Gutachter] Ranganathan. "Realizability of tropical plane curves and tropical incidence geometry / Ayush Kumar Tewari ; Gutachter: Michael Joswig, Hannah Markwig, Dhruv Ranganathan ; Betreuer: Michael Joswig." Berlin : Technische Universität Berlin, 2021. http://d-nb.info/1226217400/34.
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Bartz, Jeremiah. "Multinets in P^2 and P^3." Thesis, University of Oregon, 2013. http://hdl.handle.net/1794/13252.
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In this dissertation, a method for producing multinets from a net in P^3 is presented. Multinets play an important role in the study of resonance varieties of the complement of a complex hyperplane arrangement and very few examples are known. Implementing this method, numerous new and interesting examples of multinets are identified. These examples provide additional evidence supporting the conjecture of Pereira and Yuzvinsky that all multinets are degenerations of nets. Also, a complete description is given of proper weak multinets, a generalization of multinets.
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Alhussain, Mohammed. "Spherical wave AVO response of isotropic and anisotropic media: Laboratory experiment versus numerical simulations." Curtin University of Technology, Department of Exploration Geophysics, 2007. http://espace.library.curtin.edu.au:80/R/?func=dbin-jump-full&object_id=17537.
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A spherical wave AVO response is investigated by measuring ultrasonic reflection amplitudes from a water/Plexiglas interface. The experimental results show substantial deviation from the plane-wave reflection coefficients at large angles. However there is an excellent agreement between experimental data and full-wave numerical simulations performed with the reflectivity algorithm. By comparing the spherical-wave AVO response, modeled with different frequencies, to the plane-wave response, I show that the differences between the two are of such magnitude that three-term AVO inversion based on AVA curvature can be erroneous. I then propose an alternative approach to use critical angle information extracted from AVA curves, and show that this leads to a significant improvement of the estimation of elastic parameters. Azimuthal variation of the AVO response of a vertically fractured model also shows good agreement with anisotropic reflectivity simulations, especially in terms of extracted critical angles which indicated that (1) reflection measurements are consistent with the transmission measurements; (2) the anisotropic numerical simulation algorithm is capable of simulating subtle azimuthal variations with excellent accuracy; (3) the methodology of picking critical angles on seismograms using the inflection point is robust, even in the presence of random and/or systematic noise.
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Salarinoghabi, Mostafa. "Flat and Round Singularity theory." Universidade de São Paulo, 2016. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-09122016-101116/.
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We propose in this thesis a way to study deformations of plane curves that take into consideration the geometry of the curves as well as their singularities. We deal in details with local phenomena that occur generically in two-parameter families of curves. We obtain information on the inflections and vertices appearing on the deformed curves. We also obtain the configurations of the evolutes of the curves and of their deformations, and apply our results to orthogonal projections of space curves. Finally, we consider the profile (outline, apparent contour) of a smooth surface in the Euclidian 3-space. This is the image of the singular set of an orthogonal projection of the surface. The profile is a plane curve and may have singularities. We study the changes in the geometry of the profile as the direction of projection changes locally in the unit sphere.Propomos nesta tese um método para estudar deformações de curvas planas que leva em consideração a geometria delas, bem como as suas singularidades. Consideramos em detalhes os fenômenos locais que ocorrem genericamente em famílias de curvas com dois parâmetros. Obtemos informações sobre as inflexões e vértices que aparecem nas curvas deformadas. Obtemos também as configurações das evolutas das curvas e das suas deformações e aplicamos os nossos resultados nas projeções ortogonais de curvas espaciais. Finalmente, consideramos o perfil de uma superfície regular no espaço Euclidiano R3. O perfil é a imagem do conjunto singular de uma projeção ortogonal da superfície, esta é uma curva plana e pode ter singularidades. Estudamos as alterações na geometria do perfil quando a direção de projeção muda localmente na esfera unitária.
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Sanchez, Luis Florial Espinoza. "Singularidades de curvas na geometria afim." Universidade de São Paulo, 2010. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-07102010-145223/.
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Neste trabalho estudamos a geometria da evoluta afim e da curva normal afim associada à uma curva plana sem inflexões a partir do tipo de singularidade das funções suporte afim. O principal resultado estabelece que se \'\\gamma\' é uma curva plana sem inflexões, satisfazendo certas condições genéricas então dois casos podem ocorrer: 1. se p é um ponto da evoluta afim de \'\\gamma\' em \'s IND. 0\' então temos dois casos: se \'\\gamma\' (\'s IND. 0\') é um ponto sextático então, localmente em p, a evoluta afim é difeomorfa a uma cúspide em \'R POT. 2\' ; se não, localmente em p, a evoluta afim é difeomorfa à uma reta em \'R POT. 2\' , 2. se p = \'\\gamma\' (\'s IND. 0\') é um ponto da normal afim de \'\\gamma\' então temos dois casos: se \'\\gamma\'(\'s IND. 0\') é um ponto parabólico de \'\\gamma\' então, localmente em p, a curva normal afim é difeomorfa a uma cúspide em \'R POT. 2\' ; em outro caso, localmente em p, a curva normal afim é difeomorfa à uma reta em \'R POT. 2\'In this work we study the geometry of the affine evolute and the affine normal curve associated with a plane curve without inflections from the type of singularity of affine support functions. The main result is setting if \'\\gamma\' is a flat curve without inflections, satisfying certain conditions generic then, if p is a point of the affine evolute of \'\\gamma\' at \'s IND. 0\' then two cases: if \'\\gamma\' (\'s IND. 0\') is a sextactic point then locally in p the affine evolute is diffeomorphic to a cusp at \'R POT. 2\', otherwise locally in p the affine evolute is diffeomorphic to a straight in \'R POT. 2\', and second if p = \'\\gamma\' (\'s IND. 0\') is a point of the affine normal curve then two cases: if \'\\gamma\'(\'s IND. 0\') is a parabolic point of \'\\gamma\' then locally in p the affine normal curve is diffeomorphic to a cusp at \'R POT. 2\' , in otherwise locally in p the affine normal curve is diffeomorphic to a line in \'R POT. 2\'
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Costa, Felix Silva 1982. "Áreas e contornos." [s.n.], 2008. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306611.
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Orientador: Sueli Irene Rodrigues CostaDissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica
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Resumo: Neste trabalho são descritos métodos para o cálculo da área de regiões planas delimitadas por curvas simples e algumas propriedades de transformações do plano no plano que preservam áreas. No Capítulo 1, a área de polígonos é introduzida como uma soma de determinantes e utilizada para discutir o cálculo da área de regiões planas contornadas por curvas simples quando estas são aproximadas por polígonos com vértices ajustados por parâmetros geométricos. A fundamentação, baseada no Teorema de Green, de processos mecânicos (planímetros) para o cálculo destas áreas é descrita no Capítulo 2. Propriedades e famílias especiais de aplicações do plano no plano que preservam áreas são apresentadas no Capítulo 3.
Abstract: We describe here methods for the area estimation of plane regions bounded by simple curves and also some properties of plane transformations which preserve area. In Chapter 1 the area of polygons, described as a sum of determinants, is used to discuss the calculus of the area of plane regions bounded by simple curves approached by polygons adjusted through geometric parameters. Mechanical processes ( planimeters) based on the Green's Theorem are described in Chapter 2. Properties and special families of area preserving mappings are presented in Chapter 3.
Mestrado
Geometria
Mestre em Matemática
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Beegle, David J. "Three-dimensional modeling of rigid pavement." Ohio : Ohio University, 1998. http://www.ohiolink.edu/etd/view.cgi?ohiou1176842076.
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Dias, Fabio Scalco. ""Geometria das singularidades de projeções"." Universidade de São Paulo, 2005. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-18122005-190356/.
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Neste trabalho estudamos as singularidades de projeções no plano de curvas genéricas, introduzindo uma nova relação de equivalência para germes e multigermes de curvas planas, denominada A_h-equivalência.In this work singularities of projections to the plane of curves are studied. We introduce a new equivalence relation for germs of plane curves, called A_h-equivalence.
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Moreno, Ávila Carlos Jesús. "Global geometry of surfaces defined by non-positive and negative at infinity valuations." Doctoral thesis, Universitat Jaume I, 2021. http://hdl.handle.net/10803/672247.
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We consider plane divisorial valuations of Hirzebruch surfaces and introduce the concepts of non-positivity and negativity at infinity. We prove that the surfaces given by valuations of the last types have nice global and local geometric properties. Moreover, non-positive at infinity divisorial valuations are those divisorial valuations of Hirzebruch surfaces providing rational surfaces with minimal generated cone of curves. Non-positivity and negativity at infinity are also extended to the class of real valuations of the projective plane and the Hirzebruch surfaces. Finally, we compute the Seshadri-type constants for pairs formed by a big divisor and a divisorial valuation of a Hirzebruch surface and obtain the vertices of the Newton-Okounkov bodies of pairs as above under the non-positivity at infinity property.Introducimos los conceptos de no positividad y negatividad en el infinito para valoraciones planas divisoriales de una superficie de Hirzebruch. Probamos que las superficies dadas por valoraciones con las características anteriores poseen interesantes propiedades globales y locales. Además, las valoraciones divisoriales no positivas en el infinito son aquellas valoraciones divisoriales de superficies de Hirzebruch que dan lugar a superficies racionales tales que su cono de curvas está generado por un número mínimo de generadores. Los conceptos de no positividad y negatividad en el infinito también se extienden a valoraciones reales del plano proyectivo y de superficies de Hirzebruch. Por último, calculamos explícitamente las constantes de tipo Seshadri para pares formados por divisores big y valoraciones divisoriales de superficies de Hirzebruch y obtenemos los vértices de los cuerpos de Newton-Okounkov para pares como los anteriores bajo la condición de no positividad en el infinito.
Programa de Doctorat en Ciències
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Maia, Francisco Everton Pereira. "Curvas planas : clássicas, regulares e de preenchimento." reponame:Repositório Institucional da UFABC, 2016.
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Orientador: Prof. Dr. Vinicius Cifú LopesDissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Mestrado Profissional em Matemática em Rede Nacional, 2016.
Neste trabalho apresentaremos uma visão sobre os princípios das curvas planas. Iniciamos o desenvolvimento dos estudos com as cônicas: parábola, elipse e hipérbole que são aplicadas no Ensino Médio normalmente usando equações cartesianas. Abordaremos o assunto destas e outras curvas usando equações paramétricas, com intuito de mostrar a vantagem de utilizá-las. Abrangeremos em nossos estudos a catenária, a cicloide e a curva de Bézier, curvas as quais não são estudadas no Ensino Básico, mas poderiam ser apresentadas como um desafio motivador ao estudo da Matemática, explorando suas várias aplicações que acontecem de maneira natural em nosso cotidiano. Apresentaremos propriedades gerais das curvas como: continuidade, parametrização, comprimento de arco, curva suave, curvatura e outras, além de realizar a demonstração do teorema fundamental das curvas planas e para finalizar estudaremos uma curva exótica, conhecida como curva de preenchimento de espaço, construída pela primeira vez pelo matemático italiano Giuseppe Peano.
In this work we will present an insight into the principles of flat curves. We start with the conics: parabola, ellipse and hyperbole which are applied in high school usually using Cartesian equations. We will discuss those and other curves using parametric equations, in order to show the advantage of using them. We will cover in our studies the catenary, the cycloid and a Bézier curve, curves which are not studied in basic education, but could be presented as a challenging motivation to the study of Mathematics by exploring their various uses that happen naturally in our everyday lives. We will introduce general properties of curves as: continuity, parameterization, arc length, smooth curve, curvature and others, in addition to the proof of the fundamental theorem of plane curves, and finally we will study an exotic curve, known as space-filling curve, built for the first time by the Italian mathematician Giuseppe Peano.
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Rosa, Lílian Neves Santa. "Invariantes de Arnold de curvas planas." Universidade Federal de Viçosa, 2010. http://locus.ufv.br/handle/123456789/4903.
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This dissertation is devoted to the study of Arnold's invariants of smooth immersed closed curves in the plane. The invariants J± and St were axiomatically defined by Arnold in [Ar1] as numerical characteristic of generic closed curves (immersion of the circle) on IR2: These three Arnold's invariants are associated to the transitions through direct and inverse self-tangencies and triple crossings. In this work, we study and present the Arnold's generic curve invariants and theirs properties. We also introduce and demonstrate the explicit formulas for calculating invariants given by Viro, Shumakovich and Polyak.
Esta dissertação é dedicada ao estudo dos invariantes de Arnold de curvas diferenciáveis fechadas imersas no plano. Os invariantes J± e St foram definidos axiomaticamente por Arnold em [Ar1] como característica numérica de curvas genéricas fechadas (imersões de círculos) no plano. Estes três invariantes estão associados às transições através de auto-tangências diretas e inversas e cruzamentos triplos. Neste trabalho estudamos e introduzimos os invariantes de Arnold de curvas genéricas e suas propriedades. Também introduzimos e demonstramos as fórmulas explícitas para cálculo destes invariantes dadas por Viro, Shumakovich e Polyak.
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Cardim, Breno da Silveira. "Curvas planas: uma visão para o ensino médio." Universidade Federal da Paraíba, 2014. http://tede.biblioteca.ufpb.br:8080/handle/tede/7515.
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In this work, we study the principles of the theory of plane curves, within the context of high school
Neste trabalho estudamos os princípios da teoria das curvas planas, tendo em mente, estudantes do ensino médio. Aqui, é proposta uma introdução ao Cálculo Diferencial e Integral àqueles estudantes, e em seguida um estudo sobre a teoria das curvas, onde alguns exemplos clássicos são apresentados, bem como, conceitos como vetor tangente, área e comprimento de curvas são discutidos.
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Tomasini, Arnaud. "Intersections maximales de quadriques réelles." Thesis, Strasbourg, 2014. http://www.theses.fr/2014STRAD035/document.
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La géométrie algébrique réelle est dans sa définition la plus simple, l'étude des ensembles de solutions d'un système d'équations polynomiales à coefficients réelles. Dans cette vaste thématique, on se concentre sur les intersections de quadriques où déjà le cas de trois quadriques reste largement ouvert. Notre sujet peut être résumé comme l'étude topologique des variétés algébriques réelles et l'interaction entre leur topologie d'une part et leur déformations et dégénérations d'autre part, un problème issu du 16ième problème de Hilbert et enrichi par des développements récents. Au cours de cette thèse, nous allons nous focaliser sur les intersections maximales de quadriques réelles et en particulier démonter l'existence de telles intersections en utilisant des développements issus des recherches effectuées depuis la fin des années 80. Dans le cas d'intersections de trois quadriques, nous allons mettre en évidence le lien très étroits entre ces intersections d'une part et les courbes planes d'autre part, et démontrer que l'étude des M-courbes (une des problématiques du 16ième problème de Hilbert) peut se faire à travers l'étude des intersections maximales. Nous utiliserons ensuite les résultats sur les courbes planes nodales afin de déterminer dans certains cas les classes de déformations d'intersections de trois quadriques réellesReal algebraic geometry is in its simplest definition, the study of sets of solutions of a system of polynomial equations with real coefficients. In this theme, we focus on the intersections of quadrics where already the case of three quadrics remains wide open. Our subject can be summarized as the topological study of real algebraic varieties and interaction between their topology on the one hand and their deformations and degenerations on the other hand, a problem coming from the 16th Hilbert problem and enriched by recent developments. In this thesis, we will focus on maximum intersections of real quadrics and particularly prove the existence of such intersections using research developments made since the late 80. In the case of intersections of three quadrics, we will point the very close link between the intersections on the one hand and on the other plane curves, and show that the study of M-curves (one of the problems of the 16th Hilbert problem) may be done through the study of maximum intersections. Next, we will use the study on nodal plane curves to determine in some cases deformation classes of intersections of three real quadrics
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Barros, Marcelo Miranda. "Identification of Fractal Dimensions from a Dynamical Analogy." Laboratório Nacional de Computação Científica, 2007. http://www.lncc.br/tdmc/tde_busca/arquivo.php?codArquivo=145.
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Several areas of knowledge use fractal geometry to help to understand natural objects and phenomena. Irregular self-similar - in which parts resemble the whole - objects may be better understood through fractal dimensions which provide how a property varies with resolution or scale. We present a new approach to calculate fractal dimensions that, instead of the frequently used methods based on covering, seeks geometry information from physical characteristics. Here, we treat the element of a fractal sequence as structures. Imposing constraints on the structures, we build simple harmonic oscillators. The variation of the period of these oscillators with respect to a determined measure of length provides a fractal dimension. This techinique was tested for a family of continuous self-similar plane curves, including the classical Koch triadic. We show that this dynamical dimension may be related to Hausdorff-Besicovitch dimension. With random geometry, the techinique besides providing a fractal dimension, identifies randomness. A new kind of fractal is also presented. The ideia is to use more than one generator in the generation process of a fractal to obtain mixed fractals.Diversas áreas do conhecimento têm utilizado a geometria fractal para melhor entender muitos objetos e fenômenos naturais. Objetos irregulares com padrão auto-similar onde as partes se assemelham ao todo podem ser melhor compreendidos através de dimensões fractais que fornecem como o valor de uma propriedade varia dependendo da resolução, ou escala, em que o objeto é observado ou medido. Apresentamos uma nova abordagem para calcular dimensões fractais através de características físicas. Neste trabalho busca-se uma caracterização da dinâmica de estruturas lineares com geometria fractal. Trata-se os elementos de uma sequência geradora de um fractal como estruturas. Osciladores harmônicos simples são construídos com tais estruturas. A variação do período de vibração desses osciladores com uma determinada medida de comprimento nos fornece uma dimensão fractal. A técnica foi testada para a família de curvas contínuas e auto-similares no plano, onde está incluída a clássica triádica de Koch. Mostramos que essa dimensão dinâmica pode ser relacionada à dimensão de Hausdorff-Besicovitch. Com geometria aleatória, a técnica além de fornecer a dimensão fractal, identifica a aleatoriedade. Um novo tipo de fractal é apresentado. A idéia é usar mais de um gerador no processo de geração de um fractal para obter os fractais mistos.
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Saleur, Benoît. "Trois problèmes géométriques d'hyperbolicité complexe et presque complexe." Thesis, Paris 11, 2011. http://www.theses.fr/2011PA112256/document.
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Cette thèse est consacrée à l'étude de trois problèmes d'hyperbolicité complexe et presque complexe. La première partie est dédiée à la recherche d'une conséquence quantitative de l'hyperbolicité au sens de Kobayashi, qui est une propriété qualitative. Le résultat obtenu prend la forme d'une inégalité isopérimétrique qui évoque l'inégalité d'Ahlfors relative aux recouvrements des surfaces de surfaces. Sa démonstration est purement riemannienne.La deuxième partie de la thèse est consacrée à la démonstration d'une version presque complexe du théorème de Borel, qui affirme que les courbes entières dans le plan projectif complexe évitant quatre droites en position générale sont linéairement dégénérées. Dans un plan projectif presque complexe, les J-droites substituent aux droites projectives et nous disposons d'un énoncé analogue pour les J-courbes entières. La démonstration de ce résultat repose sur l'utilisation de projections centrales et sur la théorie de recouvrement des surfaces d'Ahlfors.La dernière partie est consacrée à la démonstration d'une version presque complexe du théorème de Bloch, qui affirme qu'une suite non normale de disques holomorphes du plan projectif évitant quatre droites en position générale converge, en un certain sens, vers une réunion de trois droites. Notre résultat implique en particulier l'hyperbolicité du complémentaire dans le plan projectif presque complexe de quatre J-droites modulo trois J-droitesThis thesis is dedicated to the study of three problems of complex and almost complex hyperbolicity. Its first part is dedicated to the research of a quantitative consequence to Kobayashi hyperbolicity, which is a qualitative property. The result we obtain has the form of an isoperimetric inequality that suggests Ahlfors' inequality, the central result of the theory of covering surfaces. Its proof uses only riemannian tools.The second part of the thesis is dedicated to the proof of an almost complex version of Borel's theorem, which says that an entire curve in the compex preojective plane missing four lines in general position is degenerate. In an almost compex context, we can obtain a similar result for entire J-curves just by replacing projective lines by J-lines. The proof of this result uses central projections and Ahlfors' theory of covering surfaces.The last part is dedicated to the proof of an almost complex version of Bloch's theorem, which says that given a sequence of holomorphic discs in the projective plane, either it is normal, either it converges in some sens to a reunion of three lines. Our result will show in particular that the complementary set of four J-lines in general position is hyperbolic modulo three J-lines
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Oliveira, Marina Mariano de. "Curvas pedais e Teorema dos Quatro Vértices : uma introdução à geometria diferencial." reponame:Repositório Institucional da UFABC, 2018.
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Orientador: Prof. Dr. Marcus Antônio Mendonça MarrocosDissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Mestrado Profissional em Matemática em Rede Nacional - PROFMAT, Santo André, 2018.
Neste trabalho, apresentamos a geometria diferencial das curvas planas de um modo mais acessível para um leitor não especialista no assunto, mas de forma a despertar seu interesse. A Teoria Local das Curvas Planas é desenvolvida por meio de exemplos e, em particular, exibimos a família das curvas pedais. Ilustramos a Teoria Global por meio do Teorema dos Quatro Vértices e apresentamos, também, formas de explorar os conceitos de geometria diferencial na Educação Básica, com resultados geométricos interessantes e visualmente atraentes. Para isso, contamos com o auxílio do GeoGebra, um software de matemática dinâmica, e da string art, um estilo de arte caracterizado por um arranjo de cordas que formam padrões geométricos. Com isso, buscamos proporcionar ao leitor uma forma diferente de experimentar a geometria diferencial das curvas planas, bem como proporcionar aos alunos do Ensino Médio um aprendizado interessante de geometria analítica.
In this work, we present the differential geometry of the plane curves in an accessible way for not specialized readers in the subject, but in order to arouse their interest. The Local Theory of Plane Curves is developed by means of several examples and, in particular, we bring out the class of pedal curves. In order to ilustrate the Global Theory we present the Four-Vertex Theorem and we also present a way to introduce differential geometry concepts to secondary school students with interesting and visually attractive geometric results. To do this, we use the software GeoGebra, a interactive geometry and algebra application, and string art, a sort of art characterized by an arrangement of strings that form geometric patterns. We hope to provide to the readers a pratical experience of differential geometry of plane curves, as well as providing them the students of High School with an interesting learning of analytical geometry.
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Abdallah, Nancy. "Cohomologie des courbes planes algébriques." Phd thesis, Université Nice Sophia Antipolis, 2014. http://tel.archives-ouvertes.fr/tel-01064511.
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On décrit dans cette thèse les dimensions des groupes quotients gradués associés à la cohomologie du complémentaire d'une courbe plane par rapport à la filtration de Hodge en fonction de certains invariants géométriques. Le cas des courbes à singularités ordinaires est détaillé. En particulier, on trouve le polynôme de Hodge-Deligne d'une courbe C quelconque à singularités isolées et celui de son complémentaire duquel on déduit les nombres de Hodge mixtes ainsi que les nombres de Betti correspondants. Dans le cas des courbes dont les singularités sont des nœuds et des points triples ordinaires, on donne des relations importantes avec l'algèbre de Milnor du polynôme homogène f qui définit C, les syzygies de l'idéal Jacobien de f et la filtration par l'ordre de pôle du groupe cohomologique d'ordre 2 du complémentaire de la courbe.
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Barros, Marcelo Miranda. "Identificação de dimensões fractais a partir de uma analogia dinâmica." Laboratório Nacional de Computação Científica, 2007. https://tede.lncc.br/handle/tede/74.
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Previous issue date: 2007-03-23Several areas of knowledge use fractal geometry to help to understand natural objects and phenomena. Irregular self-similar - in which parts resemble the whole - objects may be better understood through fractal dimensions which provide how a property varies with resolution or scale. We present a new approach to calculate fractal dimensions that, instead of the frequently used methods based on covering, seeks geometry information from physical characteristics. Here, we treat the element of a fractal sequence as structures. Imposing constraints on the structures, we build simple harmonic oscillators. The variation of the period of these oscillators with respect to a determined measure of length provides a fractal dimension. This techinique was tested for a family of continuous self-similar plane curves, including the classical Koch triadic. We show that this dynamical dimension may be related to Hausdorff-Besicovitch dimension. With random geometry, the techinique besides providing a fractal dimension, identifies randomness. A new kind of fractal is also presented. The ideia is to use more than one generator in the generation process of a fractal to obtain mixed fractals.
Diversas áreas do conhecimento têm utilizado a geometria fractal para melhor entender muitos objetos e fenômenos naturais. Objetos irregulares com padrão auto-similar onde as partes se assemelham ao todo podem ser melhor compreendidos através de dimensões fractais que fornecem como o valor de uma propriedade varia dependendo da resolução, ou escala, em que o objeto é observado ou medido. Apresentamos uma nova abordagem para calcular dimensões fractais através de características físicas. Neste trabalho busca-se uma caracterização da dinâmica de estruturas lineares com geometria fractal. Trata-se os elementos de uma sequência geradora de um fractal como estruturas. Osciladores harmônicos simples são construídos com tais estruturas. A variação do período de vibração desses osciladores com uma determinada medida de comprimento nos fornece uma dimensão fractal. A técnica foi testada para a família de curvas contínuas e auto-similares no plano, onde está incluída a clássica triádica de Koch. Mostramos que essa dimensão dinâmica pode ser relacionada à dimensão de Hausdorff-Besicovitch. Com geometria aleatória, a técnica além de fornecer a dimensão fractal, identifica a aleatoriedade. Um novo tipo de fractal é apresentado. A idéia é usar mais de um gerador no processo de geração de um fractal para obter os fractais mistos.
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Morán, Cañón Mario. "Étude schématique du schéma des arcs." Thesis, Rennes 1, 2020. http://www.theses.fr/2020REN1S079.
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Le schéma des arcs associé à une variété algébrique définie sur un corps paramètre les germes formels de courbes que l'on peut tracer sur la variété considérée. Nous étudions certaines propriétés schématiques locales du schéma des arcs d’une variété. Étant donnée une courbe affine plane singulière définie par un polynôme réduit homogène ou homogène à poids, nous calculons, principalement par des arguments d'algèbre différentielle, des présentations de l'idéal définissant l'adhérence du lieu lisse de l'espace tangent qui est toujours une composante irréductible de cet espace. En particulier, nous obtenons une base de Gröbner de cet idéal, ce qui nous permet de décrire les fonctions de l'espace tangent de la variété qui sont nilpotentes dans le schéma des arcs. Par ailleurs, nous étudions le voisinage formel dans le schéma des arcs d’une variété torique normale de certains arcs appartenant à l’ensemble de Nash associé à une valuation divisorielle torique. Nous établissons un théorème de comparaison, dans le schéma des arcs, entre le voisinage formel du point générique de l’ensemble de Nash et celui d'un arc rationnel suffisamment général dans ce même ensemble de NashThe arc scheme associated with an algebraic variety defined over a field parameterizes the formal germs of curves lying on the considered variety. We study some local schematic properties of the arc scheme of a variety. Given an affine plane curve singularity defined by a reduced homogeneous or weighted homogeneous polynomial, we compute, mainly using arguments from differential algebra, presentations of the ideal defining the Zariski closure of the smooth locus of the tangent space, which is always an irreducible component of this space. In particular, we obtain a Groebner basis of such ideal, which gives a complete description of the functions of the tangent space of the variety which are nilpotent in the arc scheme. On the other hand, we study the formal neighbourhood in the arc scheme of a normal toric variety of certain arcs belonging to the Nash set associated with a divisorial toric valuation. We establish a comparison theorem, in the arc scheme, between the formal neighbourhood of the generic point of the Nash set and that of a rational arc sufficiently generic in the same Nash set
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Chen, Shin-Ling, and 陳時霖. "Geometry of Convex Closed Plane Curves." Thesis, 2014. http://ndltd.ncl.edu.tw/handle/skmzds.
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碩士國立清華大學
數學系
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In the first five sections we introduce some important geometric quantities of a curve, such as curvature, torsion, the support function and the width. In the rest part of this thesis we derive some famous geometric inequalities such as the isoperimetric inequality, Gage’s isoperimetric inequality and Wirtinger inequality. Also we introduce an important idea for a curve – the parallel curve. Using the idea of the parallel curve we can obtain our final goal of this thesis – the entropy estimate.
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Markwig, Hannah [Verfasser]. "The enumeration of plane tropical curves / Hannah Markwig." 2006. http://d-nb.info/980700736/34.
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Lin, Yu-Chu, and 林育竹. "Evolving Convex Closed Plane Curves and Related Topics." Thesis, 2009. http://ndltd.ncl.edu.tw/handle/84131208125293366252.
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Borges, Filho Herivelto Martins. "Characterization of multi-Frobenius non-classical plane curves and construction of complete plane (N, d)-arcs." 2009. http://hdl.handle.net/2152/6522.
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This work is composed of two independent parts, both addressing problems related
to algebraic curves over finite fields.
In the first part, we characterize all irreducible plane curves defined over Fq which are Frobenius non-classical for different powers of q. Such characterization
gives rise to many previously unknown curves which turn out to have some interesting properties. For instance, for n [greater-than or equal to] 3 a curve which is both q- and qn-Frobenius
non-classical will have its number of Fqn-rational points attaining the Stöhr-Voloch bound. In the second part, we study the arc property of several plane curves and
present new complete (N, d)-arcs in PG(2, q). Some of these arcs (viewed as linear (N, 3,N - d)-codes) are just a small constant away from the Griesmer bound and for some small values of q the bound is achieved. In addition, this part also answers
a question of Voloch about the arc property of a certain family of curves with many
rational points, and another question of Giulietti et al about the arc property of
q-Frobenius non-classical plane curves.text
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Wright, James Robert. "Lp estimates for operators associated to oscillating plane curves." 1990. http://catalog.hathitrust.org/api/volumes/oclc/23438635.html.
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