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Dissertations / Theses on the topic 'Line algebras'

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Published: 4 June 2021
Last updated: 5 February 2022

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1

Ammar, Gregory, Christian Mehl, and Volker Mehrmann. "Schur-Like Forms for Matrix Lie Groups, Lie Algebras and Jordan Algebras." Universitätsbibliothek Chemnitz, 2005. http://nbn-resolving.de/urn:nbn:de:swb:ch1-200501032.

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We describe canonical forms for elements of a classical Lie group of matrices under similarity transformations in the group. Matrices in the associated Lie algebra and Jordan algebra of matrices inherit related forms under these similarity transformations. In general, one cannot achieve diagonal or Schur form, but the form that can be achieved displays the eigenvalues of the matrix. We also discuss matrices in intersections of these classes and their Schur-like forms. Such multistructered matrices arise in applications from quantum physics and quantum chemistry.
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2

Topley, Lewis William. "Centralisers in classical Lie algebras." Thesis, University of Manchester, 2013. https://www.research.manchester.ac.uk/portal/en/theses/centralisers-in-classical-lie-algebras(4138e280-d893-443e-b7f2-c30855dc82ee).html.

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In this thesis we shall discuss some properties of centralisers in classical Lie algebas and related structures. Let K be an algebraically closed field of characteristic p greater than or equal to 0. Let G be a simple algebraic group over K. We shall denote by g = Lie(G) the Lie algebra of G, and for x in g denote by g_x the centraliser. Our results follow three distinct but related themes: the modular representation theory of centralisers, the sheets of simple Lie algebras and the representation theory of finite W-algebras and enveloping algebras. When G is of type A or C and p > 0 is a good prime for G, we show that the invariant algebras S(g_x)^{G_x} and U(g_x)^{G_x} and polynomial algebras on rank g generators, that the algebra S(g_x)^{g_x} is generated by S(g_x)^p and S(g_x)^{G_x}, whilst U(g_x)^{g_x} is generated by U(g_x)^{G_x} and the p-centre, generalising a classical theorem of Veldkamp. We apply the latter result to confirm the first Kac-Weisfeiler conjecture for g_x, giving a precise upper bound for the dimensions of simple U(g_x)-modules. This allows us to characterise the smooth locus of the Zassenhaus variety in algebraic terms. These results correspond to an article, soon to appear in the Journal of Algebra. The results of the next chapter are particular to the case x nilpotent with G connected of type B, C or D in any characteristic good for G. Our discussion is motivated by the theory of finite W-algebras which shall occupy our discussion in the final chapter, although we make several deductions of independent interest. We begin by describing a vector space decomposition for [g_x g_x] which in turn allows us to give a formula for dim g_x^\ab where g_x^\ab := g_x / [g_x g_x]. We then concoct a combinatorial parameterisation of the sheets of g containing x and use it to classify the nilpotent orbits lying in a unique sheet. We call these orbits non-singular. Subsequently we give a formula for the maximal rank of sheets containing x and show that it coincides with dim g_x^\ab if and only if x is non-singular. The latter result is applied to show for any (not necessarily nilpotent) x in g lying in a unique sheet, that the orthogonal complement to [g_x g_x] is the tangent space to the sheet, confirming a recent conjecture. In the final chapter we set p = 0 and consider the finite W-algebra U(g,x), again with G of type B, C or D. The one dimensional representations are parameterised by the maximal spectrum of the maximal abelian quotient E = Specm U(g, x)^\ab and we classify the nilpotent elements in classical types for which E is isomorphic to an affine space A^d_K: they are precisely the non-singular elements of the previous chapter. The component group acts naturally on E and the fixed point space lies in bijective correspondence with the set of primitive ideals of U(g) for which the multiplicity of the correspoding primitive quotient is one. We call them multiplicity free. We show that this fixed point space is always an affine space, and calculate its dimension. Finally we exploit Skryabin's equivalence to study parabolic induction of multiplicity free ideals. In particular we show that every multiplicity free ideals whose associated variety is the closure of an induced orbit is itself induced from a completely prime primitive ideals with nice properties, generalising a theorem of Moeglin. The results of the final two chapters make up a part of a joint work with Alexander Premet.
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3

Dixon, James William Blair. "Rings of semi-algebraic functions on the line." Thesis, University of Manchester, 2013. https://www.research.manchester.ac.uk/portal/en/theses/rings-of-semialgebraic-functions-on-the-line(a5ec78af-26f4-4770-816f-32a6fcfbde0f).html.

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4

Chen, Zhangchi. "Differential invariants of parabolic surfaces and of CR hypersurfaces; Directed harmonic currents near non-hyperbolic linearized singularities; Hartogs’ type extension of holomorphic line bundles; (Non-)invertible circulant matrices On differential invariants of parabolic surfaces A counterexample to Hartogs’ type extension of holomorphic line bundles Directed harmonic currents near non-hyperbolic linearized singularities Affine Homogeneous Surfaces with Hessian rank 2 and Algebras of Differential Invariants On nonsingularity of circulant matrices." Thesis, université Paris-Saclay, 2021. http://www.theses.fr/2021UPASM005.

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La thèse se compose de 6 articles. (1) Nous calculons les générateurs des SA₃(ℝ)-invariants pour les surfaces paraboliques. (2) Nous calculons les invariants rigides relatifs pour les hypersurfaces rigides 2-non-dégénérées de rang de Levi constant 1 dans ℂ³: V₀, I₀, Q₀ ayant 11, 52, 824 monômes au numérateur. (3) Nous organisons tous les modèles affinement homogènes non-dégénérés dans ℂ³ en branches inéquivalentes. (4) Pour un courant harmonique dirigé autour d'une singularité linéarisée non-hyperbolique qui ne charge pas les séparatrices triviales dont l'extension triviale à travers 0 est ddc-fermée, nous démontrons que le nombre de Lelong en 0 est : 4.1) strictement positif si λ>0 ; 4.2) nul si λ est rationnel et négatif ; 4.3) nul si λ est négatif et si T est invariant sous l'action d'un sous-groupe cofini du groupe de monodromie. (5) Nous construisons des fibrés holomorphes en droites en toute dimension n>=2 non-prolongeables au sens de Hartogs. (6) Nous montrons que les matrices circulantes ayant k entrées 1 et k+1 entrées 0 dans leur première rangée sont toujours non singulières lorsque 2k+1 est soit une puissance d'un nombre premier, soit un produit de deux nombres premiers distincts. Pour tout autre entier 2k+1, nous exhibons une matrice circulante singulière
The thesis consists of 6 papers. (1) We calculate the generators of SA₃(ℝ)-invariants for parabolic surfaces. (2) We calculate rigid relative invariants for rigid constant Levi-rank 1 and 2-non-degenerate hypersurfaces in ℂ³: V₀, I₀, Q₀ having 11, 52, 824 monomials in their numerators. (3) We organize all affinely homogeneous nondegenerate surfaces in ℂ³ in inequivalent branches. (4) For a directed harmonic current near a non-hyperbolic linearized singularity which does not give mass to any of the trivial separatrices and whose trivial extension across 0 is ddc-closed, we show that the Lelong number at 0 is: 4.1) strictly positive if the eigenvalue λ>0; 4.2) zero if λ is a negative rational number; 4.3) zero if λ<0 and if T is invariant under the action of some cofinite subgroup of the monodromy group. (5) We construct non-extendable, in the sense of Hartogs, holomorphic line bundles in any dimension n>=2. (6) We show that circulant matrices having k ones and k+1 zeros in the first row are always nonsingular when 2k+1 is either a power of a prime, or a product of two distinct primes. For any other integer 2k+1 we exhibit a singular circulant matrix
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5

Tadanki, Sasidhar. "Multiple resonant multiconductor transmission line resonator design using circulant block matrix algebra." Digital WPI, 2018. https://digitalcommons.wpi.edu/etd-dissertations/249.

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The purpose of this dissertation is to provide a theoretical model to design RF coils using multiconductor transmission line (MTL) structures for MRI applications. In this research, an MTL structure is represented as a multiport network using its port admittance matrix. Resonant conditions and closed-form solutions for different port resonant modes are calculated by solving the eigenvalue problem of port admittance matrix using block matrix algebra. A mathematical proof to show that the solution of the characteristic equation of the port admittance matrix is equivalent to solving the source side input impedance is presented. The proof is derived by writing the transmission chain parameter matrix of an MTL structure, and mathematically manipulating the chain parameter matrix to produce a solution to the characteristic equation of the port admittance matrix. A port admittance matrix can be formulated to take one of the forms depending on the type of MTL structure: a circulant matrix, or a circulant block matrix (CB), or a block circulant circulant block matrix (BCCB). A circulant matrix can be diagonalized by a simple Fourier matrix, and a BCCB matrix can be diagonalized by using matrices formed from Kronecker products of Fourier matrices. For a CB matrix, instead of diagonalizing to compute the eigenvalues, a powerful technique called “reduced dimension method� can be used. In the reduced dimension method, the eigenvalues of a circulant block matrix are computed as a set of the eigenvalues of matrices of reduced dimension. The required reduced dimension matrices are created using a combination of the polynomial representor of a circulant matrix and a permutation matrix. A detailed mathematical formulation of the reduced dimension method is presented in this thesis. With the application of the reduced dimension method for a 2n+1 MTL structure, the computation of eigenvalues for a 4n X 4n port admittance matrix is simplified to the computation of eigenvalues of 2n matrices of size 2 X 2. In addition to reduced computations, the model also facilitates analytical formulations for coil resonant conditions. To demonstrate the effectiveness of the proposed methods (2n port model and reduced dimension method), a two-step approach was adopted. First, a standard published RF coil was analyzed using the proposed models. The obtained resonant conditions are then compared with the published values and are verified by full-wave numerical simulations. Second, two new dual tuned coils, a surface coil design using the 2n port model, and a volume coil design using the reduced dimensions method are proposed, constructed, and bench tested. Their validation was carried out by employing 3D EM simulations as well as undertaking MR imaging on clinical scanners. Imaging experiments were conducted on phantoms, and the investigations indicate that the RF coils achieve good performance characteristics and a high signal-to-noise ratio in the regions of interest.
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6

Stigner, Carl. "Hopf and Frobenius algebras in conformal field theory." Doctoral thesis, Karlstads universitet, Avdelningen för fysik och elektroteknik, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:kau:diva-14456.

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There are several reasons to be interested in conformal field theories in two dimensions. Apart from arising in various physical applications, ranging from statistical mechanics to string theory, conformal field theory is a class of quantum field theories that is interesting on its own. First of all there is a large amount of symmetries. In addition, many of the interesting theories satisfy a finiteness condition, that together with the symmetries allows for a fully non-perturbative treatment, and even for a complete solution in a mathematically rigorous manner. One of the crucial tools which make such a treatment possible is provided by category theory. This thesis contains results relevant for two different classes of conformal field theory. We partly treat rational conformal field theory, but also derive results that aim at a better understanding of logarithmic conformal field theory. For rational conformal field theory, we generalize the proof that the construction of correlators, via three-dimensional topological field theory, satisfies the consistency conditions to oriented world sheets with defect lines. We also derive a classifying algebra for defects. This is a semisimple commutative associative algebra over the complex numbers whose one-dimensional representations are in bijection with the topological defect lines of the theory. Then we relax the semisimplicity condition of rational conformal field theory and consider a larger class of categories, containing non-semisimple ones, that is relevant for logarithmic conformal field theory. We obtain, for any finite-dimensional factorizable ribbon Hopf algebra H, a family of symmetric commutative Frobenius algebras in the category of bimodules over H. For any such Frobenius algebra, which can be constructed as a coend, we associate to any Riemann surface a morphism in the bimodule category. We prove that this morphism is invariant under a projective action of the mapping class group ofthe Riemann surface. This suggests to regard these morphisms as candidates for correlators of bulk fields of a full conformal field theories whose chiral data are described by the category of left-modules over H.
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7

Sequin, Matthew James. "Comparing Invariants of 3-Manifolds Derived from Hopf Algebras." The Ohio State University, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=osu1338251228.

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8

Purslow, Thomas. "Maximal subalgebras of the exceptional Lie algebras in low characteristic." Thesis, University of Manchester, 2018. https://www.research.manchester.ac.uk/portal/en/theses/maximal-subalgebras-of-the-exceptional-lie-algebras-in-low-characteristic(8ebc7b9a-98fe-4ab0-82a9-ab71ef89fdb9).html.

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9

Benner, P., and R. Byers. "Newtons method with exact line search for solving the algebraic Riccati equation." Universitätsbibliothek Chemnitz, 1998. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-199800775.

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This paper studies Newton's method for solving the algebraic Riccati equation combined with an exact line search. Based on these considerations we present a Newton{like method for solving algebraic Riccati equations. This method can improve the sometimes erratic convergence behavior of Newton's method.
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10

Böhm, Josef. "Linking Geometry, Algebra and Calculus with GeoGebra." Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2012. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-79488.

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GeoGebra is a free, open-source, and multi-platform software that combines dynamic geometry, algebra and calculus in one easy-to-use package. Students from middle-school to university can use it in classrooms and at home. In this workshop, we will introduce the features of GeoGebra with a special focus on not very common applications of a dynamic geometry program. We will inform about plans for developing training and research networks connected to GeoGebra. We can expect that at the time of the conference a spreadsheet will be integrated into GeoGebra which offers new ways teaching mathematics using the interplay between the features of a spreadsheet and the objects of dynamic geometry.
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11

Rajasingam, Prasanthan. "Numerical Solution of the coupled algebraic Riccati equations." OpenSIUC, 2013. https://opensiuc.lib.siu.edu/theses/1323.

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In this paper we develop new and improved results in the numerical solution of the coupled algebraic Riccati equations. First we provide improved matrix upper bounds on the positive semidefinite solution of the unified coupled algebraic Riccati equations. Our approach is largely inspired by recent results established by Liu and Zhang. Our main results tighten the estimates of the relevant dominant eigenvalues. Also by relaxing the key restriction our upper bound applies to a larger number of situations. We also present an iterative algorithm to refine the new upper bounds and the lower bounds and numerically compute the solutions of the unified coupled algebraic Riccati equations. This construction follows the approach of Gao, Xue and Sun but we use different bounds. This leads to different analysis on convergence. Besides, we provide new matrix upper bounds for the positive semidefinite solution of the continuous coupled algebraic Riccati equations. By using an alternative primary assumption we present a new upper bound. We follow the idea of Davies, Shi and Wiltshire for the non-coupled equation and extend their results to the coupled case. We also present an iterative algorithm to improve our upper bounds. Finally we improve the classical Newton's method by the line search technique to compute the solutions of the continuous coupled algebraic Riccati equations. The Newton's method for couple Riccati equations is attributed to Salama and Gourishanar, but we construct the algorithm in a different way using the Fr\'echet derivative and we include line search too. Our algorithm leads to a faster convergence compared with the classical scheme. Numerical evidence is also provided to illustrate the performance of our algorithm.
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12

Emtander, Eric. "Chordal and Complete Structures in Combinatorics and Commutative Algebra." Doctoral thesis, Stockholms universitet, Matematiska institutionen, 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-48241.

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This thesis is divided into two parts. The first part is concerned with the commutative algebra of certain combinatorial structures arising from uniform hypergraphs. The main focus lies on two particular classes of hypergraphs called chordal hypergraphs and complete hypergraphs, respectively. Both these classes arise naturally as generalizations of the corresponding well known classes of simple graphs. The classes of chordal and complete hypergraphs are introduced and studied in Chapter 2 and Chapter 3 respectively. Chapter 4, that is the content of \cite{E5}, answers a question posed at the P.R.A.G.MAT.I.C. summer school held in Catania, Italy, in 2008. In Chapter 5 we study hypergraph analogues of line graphs and cycle graphs. Chapter 6 is concerned with a connectedness notion for hypergraphs and in Chapter 7 we study a weak version of shellability.The second part is concerned with affine monoids and their monoid rings. Chapter 8 provide a combinatorial study of a class of positive affine monoids that behaves in some sense like numerical monoids. Chapter 9 is devoted to the class of numerical monoids of maximal embedding dimension. A combinatorial description of the graded Betti numbers of the corresponding monoid rings in terms of the minimal generators of the monoids is provided. Chapter 10 is concerned with monomial subrings generated by edge sets of complete hypergraphs.
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13

Hazrat, Roozbeh. "On K-theory of classical-like groups." [S.l. : s.n.], 2002. http://deposit.ddb.de/cgi-bin/dokserv?idn=969899742.

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14

Tran, Nguyen Khanh Linh [Verfasser], and Martin [Akademischer Betreuer] Kreuzer. "Kähler Differential Algebras for 0-Dimensional Schemes and Applications / Nguyen Khanh Linh Tran. Betreuer: Martin Kreuzer." Passau : Universität Passau, 2015. http://d-nb.info/1079066950/34.

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15

Larsen, Jeannette M. "Equivalence Classes of Subquotients of Pseudodifferential Operator Modules on the Line." Thesis, University of North Texas, 2012. https://digital.library.unt.edu/ark:/67531/metadc149627/.

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Certain subquotients of Vec(R)-modules of pseudodifferential operators from one tensor density module to another are categorized, giving necessary and sufficient conditions under which two such subquotients are equivalent as Vec(R)-representations. These subquotients split under the projective subalgebra, a copy of ????2, when the members of their composition series have distinct Casimir eigenvalues. Results were obtained using the explicit description of the action of Vec(R) with respect to this splitting. In the length five case, the equivalence classes of the subquotients are determined by two invariants. In an appropriate coordinate system, the level curves of one of these invariants are a pencil of conics, and those of the other are a pencil of cubics.
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16

Borland, Alexander I. "An Invariant of Links on Surfaces via Hopf Algebra Bundles." The Ohio State University, 2017. http://rave.ohiolink.edu/etdc/view?acc_num=osu1503183775028923.

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17

Greiwe, Regina M. Heath Jo W. "A brief exploration of the Sorgenfrey line and the lexicographic order." Auburn, Ala., 2006. http://repo.lib.auburn.edu/2006%20Spring/master's/GREIWE_REGINA_16.pdf.

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18

Veras, Richard Michael. "A Systematic Approach for Obtaining Performance on Matrix-Like Operations." Research Showcase @ CMU, 2017. http://repository.cmu.edu/dissertations/1011.

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Scientific Computation provides a critical role in the scientific process because it allows us ask complex queries and test predictions that would otherwise be unfeasible to perform experimentally. Because of its power, Scientific Computing has helped drive advances in many fields ranging from Engineering and Physics to Biology and Sociology to Economics and Drug Development and even to Machine Learning and Artificial Intelligence. Common among these domains is the desire for timely computational results, thus a considerable amount of human expert effort is spent towards obtaining performance for these scientific codes. However, this is no easy task because each of these domains present their own unique set of challenges to software developers, such as domain specific operations, structurally complex data and ever-growing datasets. Compounding these problems are the myriads of constantly changing, complex and unique hardware platforms that an expert must target. Unfortunately, an expert is typically forced to reproduce their effort across multiple problem domains and hardware platforms. In this thesis, we demonstrate the automatic generation of expert level high-performance scientific codes for Dense Linear Algebra (DLA), Structured Mesh (Stencil), Sparse Linear Algebra and Graph Analytic. In particular, this thesis seeks to address the issue of obtaining performance on many complex platforms for a certain class of matrix-like operations that span across many scientific, engineering and social fields. We do this by automating a method used for obtaining high performance in DLA and extending it to structured, sparse and scale-free domains. We argue that it is through the use of the underlying structure found in the data from these domains that enables this process. Thus, obtaining performance for most operations does not occur in isolation of the data being operated on, but instead depends significantly on the structure of the data.
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19

Coskun, Mustafa Coskun. "ALGEBRAIC METHODS FOR LINK PREDICTIONIN VERY LARGE NETWORKS." Case Western Reserve University School of Graduate Studies / OhioLINK, 2017. http://rave.ohiolink.edu/etdc/view?acc_num=case1499436242956926.

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20

El-Rifai, E. A. "Positive braids and Lorenz links." Thesis, University of Liverpool, 1988. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.384365.

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In this work a new foundation for the study of positive braids in Artin's braid groups is given. The basic braids considered are the set SBn of positive permutation braids, defined as those positive braids where each pair of arcs cross at most once. These are shown to be in 1-1 correspondence with the permutations in S . A canonical n form for positive braids as products of braids in SB is given, ton gether with an explicit algorithm for writing every positive braid in canonical form and a practical test for use in the algorithm. This is a useful approach to braid theory because permutations can be particularly easily handled. Applications of this canonical form are: (1) An easily handled approach to Garside's solution of the word problem in B . n (2) An algorithm to decide whether (/1 ) k is a factor of a positive n braid; this happens if and only if at most k canonical factors have equal to /1 n (where /1 n is the positive braid with each pair of arcs cross exactly once). (3) A proof that a positive braid is a factor of (/1 ) k if and only if n its canonical form has at most k factors. (4) An improvement of Garside's solution of the conjugacy problem, this is by reducing the summit set to a much smaller invariant class under conjugation (super summit set). This includes a necessary and sufficient condition for positive braid to contain /1 n up to conjugation. The linear generators of the Hecke algebras used by Morton. and/ Short are in 1-1 correspondence with the elements of SB. The n canonical forms above give a quick proof that the number of strands in a twist positive braid (one of the form (/1 )2mp for positive braid n P and for positive integer m) is the braid index of the closure of that braid, which was first proved by Franks and Williams. It is also shown that if the 2-variable link invariant P L (v, z) for an oriented link L has width k in the variable v, then it is the same as the polynomial of a closed k-braid, for k = 1, 2. A complete list of 3-braids of width 2, which close to knots, is given. It is also shown that twist positive 3-braids do not admit exchange moves (in the sense of Birman). Consequently the conjugacy class of a twist positive 3-braid representative is a complete link invariant, provided that Birman's conjecture about Markov's moves and exchange moves holds. Lorenz knots and links are studied as an example of positive links. It is proved that a positive permutation braid 1T is a Lorenz braid if and only if all braid words which equal 1T have the same single starting letter. A semicanonical form for a minimal braid representative of a Lorenz link is given. It is proved that every algebraic link of c components is a Lorenz link, for c = 1, 2. (The case for knots was first proved by Birman and Williams). Consequently a necessary and sufficient condition for a knot to be algebraic is given, together with a canonical form for a minimal braid representative for every algebraic knot. To some extent the relation between Lorenz knots and their companions is studied. It is shown that Lorenz knots and links of braid index 3 are determined by conjugacy classes in B 3. A complete list of 3 -braids which close to Lorenz knots and links is given and a complete list of pure 4-braids which close to Lorenz links is also given. It is shown that Lorenz knots and links of braid index 3 are determined by their Alexander polynomials. As a further analogy with the properties of algebraic links it is shown that the linking pattern of a Lorenz link L with pure braid representative and braid index t ~4, determines a unique braid representative for L and so determines L.
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21

Gerhard, Sandra. "Can Early Algebra lead non-proficient students to a better arithmetical understanding?" Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2012. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-79878.

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In mathematics curricula teachers often find the more or less implicit request to link the taught subjects to the previous knowledge of the students, for example using word problems from everyday life. But in today’s multicultural and multisocial society teachers can no longer assume that the children they teach have a more or less equal background and thus everyday live can have a very different meaning for different children. Furthermore there is evidence that good previous knowledge in arithmetic can hinder the approach to other mathematical subjects, like algebra. In this paper I want to provide a brief overview on how previous knowledge in arithmetic can affect student\'s access to algebra and therefore present an early algebra teaching project which introduces elementary school children to algebraic notation by measurement in an action-oriented way. Thereby the chosen approach to algebra explicitly does not come back to the student\'s previous arithmetical knowledge but additionally may support non-proficient students in obtaining more insight in the structure of calculations and hence may help them to have more success in solving calculations and word problems.
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22

Song, Kyle Seregay. "Two-Phase Flow Measurement using Fast X-ray Line Detector System." Diss., Virginia Tech, 2019. http://hdl.handle.net/10919/103371.

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Void fraction is an essential parameter for understanding the interfacial structure, and heat and mass transfer mechanisms in various gas-liquid flow systems. It becomes critically important to accurately measure void fraction as advanced high fidelity two-phase flow models require high-quality validation data. However, void fraction measurement remains a challenging task to date due to the complexity and rapid-changing characteristic of the gas-liquid boundary flow structure. This study aims to develop an advanced void fraction measurement system based on x-ray and fast line detector technologies. The dissertation has covered the major components necessary to develop a complete measurement system. Spectral analysis of x-ray attenuation in two-phase flow has been performed, and a new void fraction model is developed based on the analysis. The newly developed pixel-to-radial conversion algorithm is capable of converting measured void fraction along with the detector array to the radial distribution in a circular pipe for a wide range of void fraction conditions. The x-ray system attains the radial distributions of key measurable factors such as void fraction and gas velocity. The data are compared with the double-sensor conductivity probe and gas flowmeter for various flow conditions. The results show reasonable agreements between the x-ray and the other measurement techniques. Finally, various 2-D tomography algorithms are implemented for the non-axisymmetric two-phase flow reconstruction. A comprehensive summary of classical absorption tomography for the two-phase flow study is provided. An in-depth sensitivity study is carried out using synthetic bubbles, aiming to investigate the effect of various uncertainty factors such as background noise, off-center shift, void profile effect, etc. The sensitivity study provides a general guideline for the performance of existing 2-D reconstruction algorithms.
Doctor of Philosophy
Gas-liquid flow phenomenon exists in an extensive range of natural and engineering systems, for example, hydraulic pipelines in a nuclear reactor, heat exchanger, pump cavitation, and boilers in the gas-fired power stations. Accurate measurement of the void fraction is essential to understand the behaviors of the two-phase flow phenomenon. However, measuring void fraction distribution in two-phase flow is a difficult task due to its complex and fast-changing interfacial structure. This study developed a comprehensive suite of the non-intrusive x-ray measurement techniques, and a pixel-to-radial conversion algorithm to process the line- and time-averaged void fraction information. The newly developed algorithm, called the Area-based Onion-Peeling (ABOP) method, can convert the pixel measurement to the radial void fraction distribution, which is more useful for studying and modeling axisymmetric flows. Various flow conditions are measured and evaluated for the benchmarking of the algorithm. Finally, classical 2-D reconstruction algorithms are investigated for the void fraction measurement in non-axisymmetric flows. A comprehensive summary of the performance of these algorithms for a two-phase flow study is provided. An in-depth sensitivity study using synthetic bubbles has been performed to examine the effect of uncertainty factors and to benchmark the algorithms for the non-axisymmetric flows.
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23

Ramanathan, Krishnan Adithya. "Explicit algebraic subfilter scale modeling for DES-like methods and extension to variable density flows." Thesis, Aix-Marseille, 2019. http://www.theses.fr/2019AIXM0117.

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Dans cette thèse, l’objectif est d’améliorer les capacités prédictives des méthodes hybrides RANS/LES par le développement d’un modèle à l’échelle de sous-filtre qui prend en compte une relation algébrique explicite pour les stresses turbulents de sous-filtre non-isotropes et les flux scalaires turbulents. Tout d'abord, un modèle explicite algébrique (EARSM) est développé et calibré avec le modèle BSL de Menter pour les écoulements incompressibles dans un contexte RANS. Deuxièmement, le modèle EARSM est étendu dans le cadre hybrides RANS/LES. Méthodes hybrides RANS/LES spécifiquement dans le cadre de l'Equivalent-Detached Eddy Simulation, aboutissant au modèle hybride explicite algébrique EAHSM. Le travail de validation est réalisé pour l'écoulement entièrement développé du canal à un nombre de Reynolds de frottement à 550 et l'écoulement dans un tuyau carré à un nombre de Reynolds de frottement de 600. Enfin, en supposant que l'approximation de Boussinesq soit valide, les modèles EARSM et EAHSM sont étendus à des écoulements à densité légèrement variable. Suite à la solution directe des relations algébriques implicites, le modèle algébrique explicite pour les contraintes de Reynolds et les flux scalaires est obtenu dans un cadre RANS amené au modèle explicite algébrique de flux scalaire (EASFM). Une méthode itérative est utilisée pour traiter la non-linéarité des expressions couplées pour les relations algébriques. Ensuite, l’EASFM est étendu au cadre des méthodes hybrides RANS/LES. Le comportement des modèles est évalué sur l'écoulement homogène, en stratification stable
The aim of this work is to improve the predictive capabilities of hybrid RANS/LES methods HRLM through the development of a subfilter scale model which considers an explicit algebraic relation for the non-isotropic turbulent subfilter stress and turbulent scalar fluxes, contributing to the improvement of the safety analysis concerning hydrogen hazards. Firstly, an Explicit Algebraic Reynolds Stress Model EARSM is developed using the direct solution method and calibrated with Menter's BSL model for incompressible flows in a RANS framework. Secondly, the EARSM model is extended to seamless HRLM specifically in the framework of Equivalent-Detached Eddy Simulation, arriving at the Explicit Algebraic Hybrid Stress Model EAHSM. The calibration of the model constant is performed on the decay of isotropic turbulence. The validation is performed against the DNS data available in the literature for the fully developed Channel flow at a moderate friction Reynolds number of 550 and flow in a square pipe at a friction Reynolds number of 600. Finally, assuming the Boussinesq approximation to be valid, the developed EARSM and the EAHSM models are extended to slightly variable density flows. Following the direct solution of the implicit algebraic relationships, the explicit algebraic model for both the Reynolds stresses and the scalar flux is obtained in a RANS framework which leads to the Explicit Algebraic Scalar Flux Model(EASFM). An effective iterative solution method is used to treat the nonlinearity of the coupled expressions for the algebraic relations. The EASFM is extended to the framework of seamless HRLM. The behaviour of the models is assessed for stably stratified flows
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24

Hill, Thomas. "Dispersive Estimates of Schrodinger and Schrodinger-Like Equations in One Dimension." University of Cincinnati / OhioLINK, 2020. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1595850253465442.

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25

Viu, Sos Juan. "Periods and line arrangements : contributions to the Kontsevich-Zagier period conjecture and to the Terao conjecture." Thesis, Pau, 2015. http://www.theses.fr/2015PAUU3022/document.

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La première partie concerne un problème de théorie des nombres, pour laquel nous développons une approche géométrique basé sur des outils provenant de la géométrie algébrique et de la géométrique combinatoire. Introduites par M. Kontsevich et D. Zagier en 2001, les périodes sont des nombres complexes obtenus comme valeurs des intégrales d'une forme particulier, où le domaine et l'intégrande s'expriment par des polynômes avec coefficients rationnels. La conjecture de périodes de Kontsevich-Zagier affirme que n'importe quelle relation polynomiale entre périodes peut s'obtenir par des relations linéaires entre différentes représentations intégrales, exprimées par des règles classiques du calcul intégrale. En utilisant des résolutions de singularités, on introduit une réduction semi-canonique de périodes en se concentrant sur le fait d'obtenir une méthode algorithmique et constructive respectant les règles classiques de transformation intégrale: nous prouvons que n'importe quelle période non nulle, représentée par une certaine intégrale, peut être exprimée sauf signe comme le volume d'un ensemble semi-algébrique compact. La réduction semi-canonique permet une reformulation de la conjecture de périodes de Kontsevich-Zagier en termes de changement de variables préservant le volume entre ensembles semi-algébriques compacts. Via des triangulations et méthodes de la géométrie-PL, nous étudions les obstructions de cette approche comme la généralisation du 3ème Problème de Hilbert. Nous complétons les travaux de J. Wan dans le développement d'une théorie du degré pour les périodes, basée sur la dimension minimale de l'espace ambiance nécessaire pour obtenir une telle réduction compacte, en donnant une première notion géométrique sur la transcendance de périodes. Nous étendons cet étude en introduisant des notions de complexité géométrique et arithmétique pour le périodes basées sur la complexité polynomiale minimale parmi les réductions semi-canoniques d'une période. La seconde partie s'occupe de la compréhension d'objets provenant de la géométrie algébrique avec une forte connexion avec la géométrique combinatoire, pour lesquels nous avons développé une approche dynamique. Les champs de vecteurs logarithmiques sont un outils algébro-analytique utilisés dans l'étude des sous-variétés et des germes dans des variétés analytiques. Nous nous sommes concentré sur le cas des arrangements de droites dans des espaces affines ou projectifs. On s'est plus particulièrement intéressé à comprendre comment la combinatoire d'un arrangement détermine les relations entre les champs de vecteurs logarithmiques associés: ce problème est connu sous le nom de conjecture de Terao. Nous étudions le module des champs de vecteurs logarithmiques d'un arrangement de droites affin en utilisant la filtration induite par le degré des composantes polynomiales. Nous déterminons qu'il n'existent que deux types de champs de vecteurs polynomiaux qui fixent une infinité de droites. Ensuite, nous décrivons l'influence de la combinatoire de l'arrangement de droites sur le degré minimal attendu pour ce type de champs de vecteurs. Nous prouvons que la combinatoire ne détermine pas le degré minimal des champs de vecteurs logarithmiques d'un arrangement de droites affin, en présentant deux pairs de contre-exemples, chaque qu'un d'eux correspondant à une notion différente de combinatoire. Nous déterminons que la dimension des espaces de filtration suit une croissance quadratique à partir d'un certain degré, en dépendant uniquement de la combinatoire de l'arrangement. A fin d'étudier de façon calculatoire une telle filtration, nous développons une librairie de fonctions sur le software de calcul formel Sage
The first part concerns a problem of number theory, for which we develop a geometrical approach based on tools coming from algebraic geometry and combinatorial geometry. Introduced by M. Kontsevich and D. Zagier in 2001, periods are complex numbers expressed as values of integrals of a special form, where both the domain and the integrand are expressed using polynomials with rational coefficients. The Kontsevich-Zagier period conjecture affirms that any polynomial relation between periods can be obtained by linear relations between their integral representations, expressed by classical rules of integral calculus. Using resolution of singularities, we introduce a semi-canonical reduction for periods focusing on give constructible and algorithmic methods respecting the classical rules of integral transformations: we prove that any non-zero real period, represented by an integral, can be expressed up to sign as the volume of a compact semi-algebraic set. The semi-canonical reduction permit a reformulation of the Kontsevich-Zagier conjecture in terms of volume-preserving change of variables between compact semi-algebraic sets. Via triangulations and methods of PL–geometry, we study the obstructions of this approach as a generalization of the Third Hilbert Problem. We complete the works of J. Wan to develop a degree theory for periods based on the minimality of the ambient space needed to obtain such a compact reduction, this gives a first geometric notion of transcendence of periods. We extend this study introducing notions of geometric and arithmetic complexities for periods based in the minimal polynomial complexity among the semi-canonical reductions of a period. The second part deals with the understanding of particular objects coming from algebraic geometry with a strong background in combinatorial geometry, for which we develop a dynamical approach. The logarithmic vector fields are an algebraic-analytic tool used to study sub-varieties and germs of analytic manifolds. We are concerned with the case of line arrangements in the affine or projective space. One is interested to study how the combinatorial data of the arrangement determines relations between its associated logarithmic vector fields: this problem is known as the Terao conjecture. We study the module of logarithmic vector fields of an affine line arrangement by the filtration induced by the degree of the polynomial components. We determine that there exist only two types of non-trivial polynomial vector fields fixing an infinitely many lines. Then, we describe the influence of the combinatorics of the arrangement on the expected minimal degree for these kind of vector fields. We prove that the combinatorics do not determine the minimal degree of the logarithmic vector fields of an affine line arrangement, giving two pair of counter-examples, each pair corresponding to a different notion of combinatorics. We determine that the dimension of the filtered spaces follows a quadratic growth from a certain degree, depending only on the combinatorics of the arrangements. We illustrate these formula by computations over some examples. In order to study computationally these filtration, we develop a library of functions in the mathematical software Sage
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26

Mullins, Larry Andrew. "An upperbound on the ropelength of arborescent links." CSUSB ScholarWorks, 2007. https://scholarworks.lib.csusb.edu/etd-project/3133.

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27

Mansuroglu, Nil. "The structure of the second derived ideal of free centre-by-metabelian Lie rings." Thesis, University of Manchester, 2014. https://www.research.manchester.ac.uk/portal/en/theses/the-structure-of-the-second-derived-ideal-of-free-centrebymetabelian-lie-rings(1193df12-e488-4993-8e6f-ad8014fdadb3).html.

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We study the free centre-by-metabelian Lie ring, that is, the free Lie ring with the property that the second derived ideal is contained in the centre. We exhibit explicit generating sets for the homogeneous components and the fine homogeneous components of the second derived ideal. Each of these components is a direct sum of a free abelian group and a (possibly trivial) elementary abelian $2$-group. Our generating sets are such that some of their elements generate the torsion subgroup while the remaining ones freely generate a free abelian group. A key ingredient of our approach is the determination of the dimensions of the corresponding homogeneous components of the free centre-by-metabelian Lie algebra over fields of characteristic other than $2$. For this we exploit a $6$-term exact sequence of modules over a polynomial ring that is originally defined over the integers, but turns into a sequence whose terms are projective modules after tensoring with a suitable field. Our results correct a partly erroneous theorem in the literature. Moreover, we study the product of three homogeneous components of a free Lie algebra. Let $L$ be a free Lie algebra of finite rank over a field and let $L_n$ denote the degree $n$ homogeneous component of $L$. Formulae for the dimension of the subspaces $[L_n,L_m]$ for all $n$ and $m$ were obtained by Ralph St\"{o}hr and Michael Vaughan-Lee. Formulae for the dimension of the subspaces of the form $[L_n,L_m,L_k]$ under certain conditions on $n,m$ and $k$ were obtained by Nil Mansuro\u{g}lu and Ralph St\"{o}hr. Surprisingly, in contrast to the case of a product of two homogeneous components, the dimension of such products may depend on the characteristic of the field. For example, the dimension of $[L_2,L_2,L_1]$ over fields of characteristic $2$ is different from the dimension over fields of characteristic other than $2$.
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28

Gunturkun, Mustafa Hakan. "Using Tropical Degenerations For Proving The Nonexistence Of Certain Nets." Phd thesis, METU, 2010. http://etd.lib.metu.edu.tr/upload/12612076/index.pdf.

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A net is a special configuration of lines and points in the projective plane. There are certain restrictions on the number of its lines and points. We proved that there cannot be any (4,4) nets in CP^2. In order to show this, we use tropical algebraic geometry. We tropicalize the hypothetical net and show that there cannot be such a configuration in CP^2.
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29

Unger, Benjamin [Verfasser], Volker [Akademischer Betreuer] Mehrmann, Vu Hoang [Gutachter] Linh, Volker [Gutachter] Mehrmann, and Wim [Gutachter] Michiels. "Well-posedness and realization theory for delay differential-algebraic equations / Benjamin Unger ; Gutachter: Vu Hoang Linh, Volker Mehrmann, Wim Michiels ; Betreuer: Volker Mehrmann." Berlin : Technische Universität Berlin, 2020. http://d-nb.info/1221668277/34.

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30

Lê, Francois. "Vingt-sept droites sur une surface cubique : rencontres entre groupes, équations et géométrie dans la deuxième moitié du XIXe siècle." Thesis, Paris 6, 2015. http://www.theses.fr/2015PA066135/document.

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En 1849, Arthur Cayley et George Salmon démontrent que toute surface cubique contient exactement vingt-sept droites. Résultat célèbre de la deuxième moitié du 19ème siècle, ce théorème a notamment donné lieu à des recherches sur une équation algébrique particulière appelée "équation aux vingt-sept droites". Dans notre thèse, nous étudions les rapprochements entre groupes, équations et géométrie opérés dans ces recherches. Après un travail préparatoire mettant en place certains points mathématiques et chronologiques associés aux vingt-sept droites, nous nous intéressons au Traité des substitutions et des équations algébriques de Camille Jordan, publié en 1870. Cet ouvrage contient une section consacrée à l'équation aux vingt-sept droites dont nous analysons en détail les mathématiques. Pour mettre en contexte certains points, un corpus plus large est ensuite construit autour des "équations de la géométrie", famille d'équations associées à des configurations géométriques dont les vingt-sept droites ne sont qu'un exemple. Ce corpus s'étend de 1847 à 1896, et ses principaux auteurs sont Jordan, Alfred Clebsch et Felix Klein. Dans le but de rendre compte de l'organisation particulière du savoir partagé dans le corpus, nous discutons et utilisons alors la notion de "culture". Enfin, nous étudions précisément deux textes du corpus proposant de géométriser certaines parties de l'algèbre et nous montrons en quoi les équations de la géométrie ont participé à une compréhension géométrique de la théorie des substitutions ainsi qu'à l'élaboration des idées du Programme d'Erlangen de Klein (1872)
In 1849, Arthur Cayley and George Salmon proved that every cubic surface contains exactly twenty-seven lines. A famous result in the second half of the 19th century, this theorem gave rise to research about a particular algebraic equation called the "twenty-seven lines equation." In our thesis, we study how groups, equations, and geometry interact throughout this research. After a preparatory work presenting some mathematical and chronological points about the twenty-seven lines, we look into Camille Jordan's Traité des substitutions et des équations algébriques, published in 1870. This book contained a section devoted to the twenty-seven lines equation, the mathematics of which we thoroughly study. In order to contextualize some elements, a larger corpus is then built around "geometrical equations," a family of equations linked to geometrical configurations among which the twenty-seven lines are just one example. The corpus extends from 1847 to 1896 and its main authors are Jordan, Alfred Clebsch, and Felix Klein. Aiming at describing the particular organization of the knowledge shared in the corpus, we then discuss and use the notion of "culture." Finally, we closely study two texts of the corpus, each of them presenting a geometrization of a part of algebra, and we ascertain that geometrical equations participated to a geometrical understanding of substitution theory as well as the elaboration of the ideas of Klein's Erlanger Programm (1872)
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31

Cokelaer, François. "Amélioration des ouvertures par chemins pour l'analyse d'images à N dimensions et implémentations optimisées." Phd thesis, Université de Grenoble, 2013. http://tel.archives-ouvertes.fr/tel-00952306.

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La détection de structures fines et orientées dans une image peut mener à un très large champ d'applications en particulier dans le domaine de l'imagerie médicale, des sciences des matériaux ou de la télédétection. Les ouvertures et fermetures par chemins sont des opérateurs morphologiques utilisant des chemins orientés et flexibles en guise d'éléments structurants. Ils sont utilisés de la même manière que les opérateurs morphologiques utilisant des segments orientés comme éléments structurants mais sont plus efficaces lorsqu'il s'agit de détecter des structures pouvant être localement non rigides. Récemment, une nouvelle implémentation des opérateurs par chemins a été proposée leur permettant d'être appliqués à des images 2D et 3D de manière très efficace. Cependant, cette implémentation est limitée par le fait qu'elle n'est pas robuste au bruit affectant les structures fines. En effet, pour être efficaces, les opérateurs par chemins doivent être suffisamment longs pour pouvoir correspondre à la longueur des structures à détecter et deviennent de ce fait beaucoup plus sensibles au bruit de l'image. La première partie de ces travaux est dédiée à répondre à ce problème en proposant un algorithme robuste permettant de traiter des images 2D et 3D. Nous avons proposé les opérateurs par chemins robustes, utilisant une famille plus grande d'éléments structurants et qui, donnant une longueur L et un paramètre de robustesse G, vont permettre la propagation du chemin à travers des déconnexions plus petites ou égales à G, rendant le paramètre G indépendant de L. Cette simple proposition mènera à une implémentation plus efficace en terme de complexité de calculs et d'utilisation mémoire que l'état de l'art. Les opérateurs développés ont été comparés avec succès avec d'autres méthodes classiques de la détection des structures curvilinéaires de manière qualitative et quantitative. Ces nouveaux opérateurs ont été par la suite intégrés dans une chaîne complète de traitement d'images et de modélisation pour la caractérisation des matériaux composite renforcés avec des fibres de verres. Notre étude nous a ensuite amenés à nous intéresser à des filtres morphologiques récents basés sur la mesure de caractéristiques géodésiques. Ces filtres sont une bonne alternative aux ouvertures par chemins car ils sont très efficaces lorsqu'il s'agit de détecter des structures présentant de fortes tortuosités ce qui est précisément la limitation majeure des ouvertures par chemins. La combinaison de la robustesse locale des ouvertures par chemins robustes et la capacité des filtres par attributs géodésiques à recouvrer les structures tortueuses nous ont permis de proposer un nouvel algorithme, les ouvertures par chemins robustes et sélectives.
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32

Souto, Antonio Marcos da Silva. "A reta de Euler e a circunferência dos nove pontos: um olhar algébrico." Universidade Federal da Paraíba, 2013. http://tede.biblioteca.ufpb.br:8080/handle/tede/7478.

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This work is the result of a research on the Euler line and the circumference of the nine points. The software geogebra was used to illustrate geometric constructions and present some practical activities for the study of notable points of the triangle, the Euler line and the circumference of the nine points to high school students. However, the work was based on the proof, with the use of Modern Algebra and Linear Algebra, the existence and properties of the object of this research, especially the universal property of points in the plane, critical in these demonstrations.
Este trabalho é o resultado de uma pesquisa sobre a reta de Euler e a circunferência dos nove pontos. Foi utilizado o software geogebra para ilustrar as construções geométricas e apresentar algumas atividades práticas para o estudo dos pontos notá- veis do triângulo, da reta de Euler e da circunferência dos nove pontos aos estudantes do Ensino Médio. Todavia, o trabalho se baseou nas demonstrações, com o uso da Álgebra Moderna e da Álgebra Linear, da existência e das propriedades do objeto desta pesquisa, sobretudo da propriedade universal dos pontos no plano, fundamental nestas demonstrações.
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33

Kohli, Ben-Michael. "Les invariants de Links-Gould comme généralisations du polynôme d’Alexander." Thesis, Dijon, 2016. http://www.theses.fr/2016DIJOS062/document.

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On s’intéresse dans cette thèse aux rapports qui existent entre deux invariants d’entrelacs. D’une part l’invariant d’Alexander ∆ qui est l’invariant de nœuds le plus classique, et le plus étudié avec le polynôme de Jones, et d’autre part la famille des invariants de Links-Gould LGn,m qui sont des invariants quantiques dérivés des super algèbres de Hopf Uqgl(n|m). On démontre en particulier un cas de la conjecture de De Wit-Ishii-Links : certaines spécialisa- tions des polynômes de Links-Gould fournissent des puissances du polynôme d’Alexander. Les polynômes LG sont donc des généralisations du polynôme d’Alexander. On conjecture de plus que ces invariants conservent certaines propriétés homologiques bien connues de ∆ permettant d’évaluer le genre des entrelacs et de tester le caractère fibré des nœuds
In this thesis we focus on the connections that exist between two link invariants: first the Alexander-Conway invariant ∆ that was the first polynomial link invariant to be discovered, and one of the most thoroughly studied since alongside with the Jones polynomial, and on the other hand the family of Links-Gould invariants LGn,m that are quantum link invariants derived from super Hopf algebras Uqgl(n|m). We prove a case of the De Wit-Ishii-Links conjecture: in some cases we can recover powers of the Alexander polynomial as evaluations of the Links-Gould invariants. So the LG polynomials are generalizations of the Alexander invariant. Moreover we give evidence that these invariants should still have some of the most remarkable properties of the Alexander polynomial: they seem to offer a lower bound for the genus of links and a criterion for fiberedness of knots
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34

Will, Etienne. "Constructions tropicales de noeuds algébriques dans IRP3." Phd thesis, Université de Strasbourg, 2012. http://tel.archives-ouvertes.fr/tel-00733721.

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Cette thèse présente la construction de courbes tropicales réelles dans R^3 dont la projectivisation, qui est un entrelacs projectif dans IRP^3, est constituée de 2 composantes, I'une étant isotope à un noeud donné au départ. Dans le cas de certains noeuds toriques, il est possible de modifier cette construction pour que I'entrelacs projectif correspondant ait une seule composante isotope au noeud torique considéré. Pour chacune de ces courbes tropicales réelles, nous faisons appel au théorème récent de G. Mikhalkin, qui affirme l'existence d'une algébrique réelle non singulière dans IRP^3, de même genre et degré que la courbe tropicale réelle considérée, et qui est isotope à l'entrelacs projectif correspondant.
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35

Haubold, Niko. "Compressed Decision Problems in Groups." Doctoral thesis, Universitätsbibliothek Leipzig, 2012. http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-85413.

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Wir beschäftigen uns mit Problemen der algorithmischen Gruppentheorie und untersuchen dabei die Komplexität von komprimierten Versionen des Wortproblems und des Konjugationsproblems für endlich erzeugte Gruppen. Das Wortproblem fragt für eine feste, endlich erzeugte Gruppe ob ein gegebenes Wort über der Erzeugermenge das neutrale Element der Gruppe repräsentiert. Wir betrachten das gegebene Wort jedoch in einer komprimierten Form, als Straight-line Program (SLP) und untersuchen die Komplexität dieses Problems, das wir \'komprimiertes Wortproblem\' nennen. SLPs sind kontextfreie Grammatiken, die genau einen String erzeugen. Die Eingabegröße ist dabei stets die Größe des gegebenen SLPs. Eine Hauptmotivation ist dabei, dass für eine feste endlich erzeugte Gruppe das Wortproblem ihrer Automorphismengruppe durch eine Turingmaschine in Polynomialzeit auf das komprimierte Wortproblem der Gruppe selbst reduzierbar ist. Wir untersuchen das komprimierte Wortproblem für die verbreiteten Gruppenerweiterungen HNN-Erweiterungen (amalgamierte Produkte und Graphprodukte) und können zeigen, dass sich Instanzen des komprimierten Wortproblems von einer Turingmaschine in Polynomialzeit auf Instanzen des komprimierten Wortproblems der Basisgruppe (respektive Basisgruppen und Knotengruppen) reduzieren lassen. Weiterhin zeigen wir, dass das komprimierte Wortproblem für endlich erzeugte nilpotente Gruppen von einer Turingmaschine in Polynomialzeit entscheidbar ist. Wir betrachten außerdem eine komprimierte Variante des Konjugationsproblems. Das unkomprimierte Konjugationsproblem fragt für zwei gegebene Wörter über den Erzeugern einer festen endlich erzeugten Gruppe, ob sie in dieser Gruppe konjugiert sind. Beim komprimierten Konjugationsproblem besteht die Eingabe aus zwei SLPs und es wird gefragt, ob die beiden Wörter die von den SLPs erzeugt werden in der Gruppe konjugierte Elemente präsentieren. Wir konnten zeigen, dass sich das komprimierte Konjugationsproblem für Graphgruppen in Polynomialzeit entscheiden lässt. Weiterhin haben wir das Wortproblem der äußeren Automorphismengruppen von Graphprodukten endlich erzeugter Gruppen untersucht. Durch den engen Zusammenhang des komprimierten Konjugationsproblems einer Gruppe mit dem Wortproblem der äußeren Automorphismengruppe konnten wir zeigen, dass sich das Wortproblem der äußeren Automorphismengruppe eines Graphprodukts von endlich erzeugten Gruppen durch eine Turingmaschine in Polynomialzeit auf Instanzen von simultanen komprimierten Konjugationsproblemen der Knotengruppen und Instanzen von komprimierten Wortproblemen der Knotengruppen reduzieren lässt. Als Anwendung gelten obige Resultate auch für right-angled Coxetergruppen und Graphgruppen, da beide spezielle Graphprodukte sind. So folgt beispielsweise, dass das komprimierte Wortproblem einer right-angled Coxetergruppe in Polynomialzeit entscheidbar ist.
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36

Najman, Laurent. "Morphologie Mathématique: de la Segmentation d'Images à l'Analyse Multivoque." Phd thesis, Université Paris Dauphine - Paris IX, 1994. http://tel.archives-ouvertes.fr/tel-00742889.

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La première partie de cette thèse étudie la ligne de partage des eaux, un des outils fondamentaux développés par la morphologie mathématique dans le but de segmenter des images. Une caractérisation de cet objet pour des fonctions régulières est donnée, et un théorème de convergence de l'algorithme associe est démontré. Les liens entre la ligne de partage des eaux et le squelette par zones d'influence euclidien (ou diagramme de voronoï), ainsi qu'avec l'équation eikonale utilisée en shape from shading sont ensuite mis en valeur. Des algorithmes pour la reconstruction géodésique et pour la segmentation avec points d'ancrage sont construits sur le principe de celui de la ligne de partage des eaux. Enfin, un algorithme de segmentation hiérarchique fonde sur un nouveau principe de dynamique des contours, est développé. Il permet d'obtenir dans une seule image toute l'information du gradient utilisable pour la segmentation. La deuxième partie de cette thèse applique des outils de l'analyse multivoque et mutationnelle a la morphologie mathématique. La dérivée mutationnelle du tube de dilatation est calculée, justifiant de manière rigoureuse l'intuition selon laquelle un objet se dilate suivant ses normales en chacun de ses points. Les propriétés algébriques et de continuité d'applications induites par des inclusions différentielles et agissant sur des ensembles fermés sont caractérisées. Enfin, un algorithme d'optimisation (l'algorithme des montagnes russes), de nature non probabiliste, garantissant la convergence vers un minimum global, est proposé.
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37

Nyman, Peter. "On relations between classical and quantum theories of information and probability." Doctoral thesis, Linnéuniversitetet, Institutionen för datavetenskap, fysik och matematik, DFM, 2011. http://urn.kb.se/resolve?urn=urn:nbn:se:lnu:diva-13830.

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In this thesis we study quantum-like representation and simulation of quantum algorithms by using classical computers.The quantum--like representation algorithm (QLRA) was  introduced by A. Khrennikov (1997) to solve the ``inverse Born's rule problem'', i.e. to construct a representation of probabilistic data-- measured in any context of science-- and represent this data by a complex or more general probability amplitude which matches a generalization of Born's rule.The outcome from QLRA matches the formula of total probability with an additional trigonometric, hyperbolic or hyper-trigonometric interference term and this is in fact a generalization of the familiar formula of interference of probabilities. We study representation of statistical data (of any origin) by a probability amplitude in a complex algebra and a Clifford algebra (algebra of hyperbolic numbers). The statistical data is collected from measurements of two dichotomous and trichotomous observables respectively. We see that only special statistical data (satisfying a number of nonlinear constraints) have a quantum--like representation. We also study simulations of quantum computers on classical computers.Although it can not be denied that great progress have been made in quantum technologies, it is clear that there is still a huge gap between the creation of experimental quantum computers and realization of a quantum computer that can be used in applications. Therefore the simulation of quantum computations on classical computers became an important part in the attempt to cover this gap between the theoretical mathematical formulation of quantum mechanics and the realization of quantum computers. Of course, it can not be expected that quantum algorithms would help to solve NP problems for polynomial time on classical computers. However, this is not at all the aim of classical simulation.  The second part of this thesis is devoted to adaptation of the Mathematica symbolic language to known quantum algorithms and corresponding simulations on classical computers. Concretely we represent Simon's algorithm, Deutsch-Josza algorithm, Shor's algorithm, Grover's algorithm and quantum error-correcting codes in the Mathematica symbolic language. We see that the same framework can be used for all these algorithms. This framework will contain the characteristic property of the symbolic language representation of quantum computing and it will be a straightforward matter to include future algorithms in this framework.
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Liu, Linyuan. "Cohomologie des fibrés en droites sur SL3/B en caractéristique positive : deux filtrations et conséquences." Thesis, Sorbonne université, 2019. http://www.theses.fr/2019SORUS229.

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Soit G un groupe algébrique semi-simple sur un corps k algébriquement clos de caractéristique positive et soit B un sous-groupe de Borel. La cohomologie des fibrés en droites G-équivariants sur G/B induits par des caractères de B sont des objets importants dans la théorie des représentations de G. Dans cette thèse, on se concentre sur G = SL3. Dans le premier chapitre,on montre l’existence d’une filtration à deux étages de H1(μ) et H2(μ) pour μ dans l’adhérence de la région de Griffith. Dans le deuxième chapitre, on montre l’existence d’une p-Hi-D-filtration de Hi(μ) pour tout i et μ, qui généralise la p filtration de H0(μ) introduite par Jantzen. Dans le troisième chapitre, on étudie et détermine la structure des modules apparaissants dans la p-Hi-D-filtration.Dans le dernier chapitre, on donne une description explicite et combinatoire de H2(μ) pour μ dans la région de Griffith et on généralise cette description à Hd(G/B, μ) pour G = SLd+1 et certains poids μ
Let G be a semi-simple algebraic group over an algebraically closed field of positive characteristic. The cohomology of G-equivariant line bundles over G/B induced by a character of B are important objects in the representation theory of G. In this thesis, we concentrate on G = SL3. In the first chapter,we prove the existence of a two-step filtration of H1(μ) and H2(μ) when μ is in the closure of the Griffith region. In the second chapter, we prove the existence ofa p-Hi-D-filtration of Hi(μ) for all i and μ, which generalizes the p-filtration ofH0(μ) introduced by Jantzen. In the third chapter, we study and determine the structure of the modules appearing in the p-Hi-D-filtration. In the last chapter,we give an explicit and combinatorial description of H2(μ) for μ in the Griffith region and we generalize this description to Hd(G/B, μ) for G = SLd+1 and certain weights μ
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39

Lewis, Elizabeth Faith. "Peter Guthrie Tait : new insights into aspects of his life and work : and associated topics in the history of mathematics." Thesis, University of St Andrews, 2015. http://hdl.handle.net/10023/6330.

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In this thesis I present new insights into aspects of Peter Guthrie Tait's life and work, derived principally from largely-unexplored primary source material: Tait's scrapbook, the Tait–Maxwell school-book and Tait's pocket notebook. By way of associated historical insights, I also come to discuss the innovative and far-reaching mathematics of the elusive Frenchman, C.-V. Mourey. P. G. Tait (1831–1901) F.R.S.E., Professor of Mathematics at the Queen's College, Belfast (1854–1860) and of Natural Philosophy at the University of Edinburgh (1860–1901), was one of the leading physicists and mathematicians in Europe in the nineteenth century. His expertise encompassed the breadth of physical science and mathematics. However, since the nineteenth century he has been unfortunately overlooked—overshadowed, perhaps, by the brilliance of his personal friends, James Clerk Maxwell (1831–1879), Sir William Rowan Hamilton (1805–1865) and William Thomson (1824–1907), later Lord Kelvin. Here I present the results of extensive research into the Tait family history. I explore the spiritual aspect of Tait's life in connection with The Unseen Universe (1875) which Tait co-authored with Balfour Stewart (1828–1887). I also reveal Tait's surprising involvement in statistics and give an account of his introduction to complex numbers, as a schoolboy at the Edinburgh Academy. A highlight of the thesis is a re-evaluation of C.-V. Mourey's 1828 work, La Vraie Théorie des quantités négatives et des quantités prétendues imaginaires, which I consider from the perspective of algebraic reform. The thesis also contains: (i) a transcription of an unpublished paper by Hamilton on the fundamental theorem of algebra which was inspired by Mourey and (ii) new biographical information on Mourey.
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Maldonado, Juan Carlos Nuñez. "Conectividade de variedades semi-algébricas." Universidade de São Paulo, 2017. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-27072017-103944/.

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Neste projeto apresentamos os teoremas de estrutura, decomposição celular, e o teorema da existência da triangulação para conjuntos semi-algébricos compactos. Como aplicações destes teoremas mostramos o lema de seleção da curva local e global. Além disso, apresentamos uma breve descrição da topologia da fibra de Milnor local e global, bem como alguns resultados sobre o grau de conexidade da fibra genérica global de uma função polinomial complexa, que mostram a íntima relação entre o grau de conexidade com a dimensão do conjunto singular.
In this project we present some structure theorems, cell decomposition, and the theorem on the existence of triangulation for compact semi-algebraic sets. As applications we prove the curve selection lemma in the local and global cases. Moreover, we present a brief description about the topology of local and global Milnor´s fibers, as well as, some results about the connectivity degree of the generic fibers of a complex polynomial function, that show the close relation between the connectivity degree and the dimension of the singular locus.
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41

Yang, Dapeng. "Approche algébrique pour l’analyse de systèmes modélisés par bond graph." Thesis, Ecole centrale de Lille, 2012. http://www.theses.fr/2012ECLI0007/document.

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La commande de systèmes physiques s’avère être une tâche difficile en général. En fonction du modèle choisi, les outils mathématiques pour l’analyse et la conception de lois de commande peuvent changés. Pour les systèmes décrits par une représentation entrée-sortie, type transfert, ou par une équation de type état, les principales informations exploitées lors de la phase d’analyse concerne la structure interne du modèle (structure finie) et la structure externe (structure à l’infini) qui permettent avant la phase de synthèse de connaître, sur le modèle en boucle ouverte, les propriétés des lois de commande envisagées ainsi que les propriétés du système piloté (stabilité…).Le travail porte principalement sur l’étude des zéros invariants des systèmes physiques représentés par bond graph, en particulier dans un contexte de modèle type LTV. L’approche algébrique est essentielle dans ce contexte car même si les aspects graphiques restent très proches du cas linéaire classique, l’extension aux modèles LTV reste très complexe d’un point de vue mathématique, en particulier pour le calcul de racines de polynômes. De nouvelles techniques d’analyse des zéros invariants utilisant conjointement l’approche bond graph (exploitation de la causalité) et l’approche algébriques ont permis de mettre en perspective certains modules associés à ces zéros invariants et de clarifier le problème d’annulation des grandeurs de sortie. L’application aux problèmes d’observateurs à entrées inconnues a permis d’illustrer nos propos sur des exemples physiques, avec certaines extensions, problèmes pour lesquels les zéros invariants apparaissent aussi comme éléments essentiels
The control synthesis of physical systems is a complex task because it requires the knowledge of a "good model" and according to the choice of a model some specific tools must be developed. These tools, mainly developed from a mathematical and theoretical point of view, must be used from the analysis step (analysis of model properties) to the control synthesis step. It is well-known that in many approaches, the properties of the controlled systems can be analyzed from the initial model. If the system is described with an input-output representation or with a state space representation, two kinds of information are often pointed out: the external structure (infinite structure) and the internal structure (finite structure). The first one is often related to the existence of some control strategies (input-output decoupling, disturbance decoupling...) and the second one gives some focus on the stability property of the controlled system.In this report, the focus has been on the study of invariant zeros of bond graph models in the context of LTV models. The algebraic approach was essential because, even if the problem is already solved for LTI bond graph models, the extension to LTV models is not so easy. The simultaneous use of algebraic and graphical approaches has been proven to be effective and convenient to solve this problem. First, some tools from the algebraic approach have been recalled in chapter one and results for the study of invariant zeros of LTI bond graph models recalled in chapter two. Some new developments are proposed in chapter three and some applications for the unknown input observer problem with some physical applications conclude this work
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42

Ibn, Taarit Kaouther. "Contribution à l'identification des systèmes à retards et d'une classe de systèmes hybrides." Phd thesis, Ecole Centrale de Lille, 2010. http://tel.archives-ouvertes.fr/tel-00587336.

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Les travaux présentés dans cette thèse concernent le problème d'identification des systèmes à retards et d'une certaine classe de systèmes hybrides appelés systèmes "impulsifs".Dans la première partie, un algorithme d'identification rapide a été proposé pour les systèmes à entrée retardée. Il est basé sur une méthode d'estimation distributionnelle non asymptotique initiée pour les systèmes sans retard. Une telle technique mène à des schémas de réalisation simples, impliquant des intégrateurs, des multiplicateurs et des fonctions continues par morceaux polynomiales ou exponentielles. Dans le but de généraliser cette approche pour les systèmes à retard, trois exemples d'applications ont été étudiées. La deuxième partie a été consacrée à l'identification des systèmes impulsifs. En se basant sur le formalisme des distributions, une procédure d'identification a été élaborée afin d'annihiler les termes singuliers des équations différentielles représentant ces systèmes. Par conséquent, une estimation en ligne des instants de commutations et des paramètres inconnus est prévue indépendamment des lois de commutations. Des simulations numériques d'un pendule simple soumis à des frottements secs illustrent notre méthodologie
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43

Robertson, Ian. "The Euler class group of a line bundle on an affine algebraic variety over a real closed field /." 2000. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&res_dat=xri:pqdiss&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&rft_dat=xri:pqdiss:9965149.

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44

Linde, Klaus-Jürgen [Verfasser]. "Global vertex algebras on Riemann surfaces / vorgelegt von Klaus-Jürgen Linde." 2004. http://d-nb.info/974026107/34.

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45

Heydeman, Matthew Thomas Edwin. "Supersymmetric Scattering Amplitudes and Algebraic Aspects of Holography from the Projective Line." Thesis, 2019. https://thesis.library.caltech.edu/11735/24/Heydeman_Matthew_Thesis_2019.pdf.

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In this thesis, we consider two topics in string theory and quantum field theory which are related by the common appearance of one-dimensional projective geometry. In the first half of the thesis, we study six-dimensional (6D) supersymmetric quantum field theories and supergravity at the leading (tree) approximation and compute the complete S-matrix for these theories as world-sheet integrals over the punctured Riemann sphere. This exploits the analytic structure of tree amplitudes which are rational and holomorphic in the kinematics and naturally related to the geometry of points on the complex projective line. The 6D n-particle S-matrix makes many symmetries and hidden properties manifest and generalizes the well-studied formulas for four-dimensional amplitudes in the form of twistor string theory and the rational curves formalism. While the systems we study are all field theories, they are in essence low-energy effective field theory limits of string theory and M-theory backgrounds. This includes theories such as those with 6D (2,0) supersymmetry which contain U(1) self-dual tensor fields which are difficult to treat from a Lagrangian point of view. Our formulas circumvent this difficulty and allow a generalization and unification of a large class of 6D scattering amplitudes which permit a sensible classical limit, including the abelian world-volume of the M-theory Five-brane. Dimensional reduction to four dimensions is also possible, leading to new formulas for 4D physics from 6D.

In the second half of the thesis, we discuss the projective algebraic and geometric structure of the AdS3/CFT2 correspondence. In the usual statement of this correspondence, two-dimensional conformal field theory (CFT) on the Riemann sphere or a higher-genus surface is holographically dual to features of topological gravity in three dimensions with negative curvature. Since every compact Riemann surface is a projective algebraic curve, many constructions of interest in physics (which a priori depend on the analytic structure of the spacetime) can be formulated in purely algebraic language. We generalize the AdS (anti-de Sitter space)/CFT correspondence according to this principle using projective geometry over the p-adic numbers, Qp. The result is a formulation of holography in which the bulk geometry is discrete---the Bruhat--Tits tree for PGL(2,Qp)---but the group of bulk isometries nonetheless agrees with that of boundary conformal transformations and is not broken by discretization. Parallel to the usual holographic correspondence, semi-classical dynamics of fields in the bulk compute the correlation functions of local operators on the boundary. Beyond correlators on the p-adic line, we propose a tensor network model in which the patterns of entanglement on the boundary are computed by discrete geometries in the bulk. We suggest that this forms the natural geometric setting for tensor networks that have been proposed as models of bulk reconstruction via quantum error correcting codes. The model is built from tensors based on projective geometry over finite fields, Fp, and correctly computes the Ryu-Takayanagi formula, holographic entanglement and black hole entropy, and multiple interval entanglement inequalities.

In Chapter 2, we present tree-level n-particle on-shell scattering amplitudes of various brane theories with 16 conserved supercharges which are generalizations of Dirac--Born--Infeld theory. These include the world-volume theory of a probe D3-brane or D5-brane in 10D Minkowski spacetime as well as a probe M5-brane in 11D Minkowski spacetime, which describes self interactions of an abelian tensor supermultiplet with 6D (2,0) supersymmetry. We propose twistor-string-like formulas for tree-level scattering amplitudes of all multiplicities for each of these theories, and the amplitudes are written as integrals over the moduli space of certain rational maps localized on the (n-3)! solutions of the scattering equations. The R symmetry of the D3-brane theory is shown to be SU(4) x U(1), and the U(1) factor implies that its amplitudes are helicity conserving. Each of 6D theories (D5-brane and M5-brane) reduces to the D3-brane theory by dimensional reduction. As special cases of the general M5-brane amplitudes, we present compact formulas for examples involving only the self-dual B field with n=4,6,8.

In Chapter 3, we extend this formalism to n-particle tree-level scattering amplitudes of six-dimensional N=(1,1) super Yang--Mills (SYM) and N=(2,2) supergravity (SUGRA). The SYM theory arises on the world volume of coincident D5-branes, and the supergravity is the result of toroidal compactification of string theory. These theories have non-abelian interactions which allow for both even and odd-point amplitudes, unlike the branes of Chapter 2. Due to the properties of spinor-helicity variables in six dimensions, the even-n and odd-n formulas are quite different and have to be treated separately. We first propose a manifestly supersymmetric expression for the even-n amplitudes of N=(1,1) SYM theory and perform various consistency checks. By considering soft-gluon limits of the even-n amplitudes, we deduce the form of the rational maps and the integrand for n odd. The odd-n formulas obtained in this way have a new redundancy that is intertwined with the usual SL(2,C) invariance on the Riemann sphere. We also propose an alternative form of the formulas, analogous to the Witten--RSV (Roiban, Spradlin, and Volovich) formulation, and explore its relationship with the symplectic (or Lagrangian) Grassmannian. Since the amplitudes are formulated in a way that manifests double-copy properties, formulas for the six-dimensional N=(2,2) SUGRA amplitudes follow. These six-dimensional results allow us to deduce new formulas for five-dimensional SYM and SUGRA amplitudes, as well as massive amplitudes of four-dimensional N=4 SYM on the Coulomb branch.

In Chapter 4, we consider half-maximal supergravity and present a twistor-like formula for the complete tree-level S matrix of chiral 6D (2,0) supergravity coupled to 21 abelian tensor multiplets. This is the low-energy effective theory that corresponds to Type IIB superstring theory compactified on a K3 surface. As in previous chapters, the formula is expressed as an integral over the moduli space of certain rational maps of the punctured Riemann sphere; the new ingredient is an integrand which successfully incorporates both gravitons and multiple flavors of tensors. By studying soft limits of the formula, we are able to explore the local moduli space of this theory, SO(5,21)/(SO(5) x SO(21)). Finally, by dimensional reduction, we also obtain a new formula for the tree-level S-matrix of 4D N=4 Einstein--Maxwell theory.

In Chapter 5, we introduce p-adic AdS/CFT and discuss several physical and mathematical features of the holographic correspondence between conformal field theories on P1(Qp) and lattice models on the Bruhat--Tits tree of PGL(2,Qp), an infinite tree of p+1 valence which has the p-adic projective line as its boundary. We review the p-adic numbers, the Bruhat--Tits tree, and some of their applications to physics including p-adic CFT. A key feature of these constructions is the discrete and hierarchical nature of the tree and the corresponding field theories, which serve as a toy model of holography in which there are no gravitons and no conformal descendants. Standard holographic results for massive free scalar fields in a fixed background carry over to the tree; semi-classical dynamics in the bulk compute correlation functions in the dual field theory and we obtain a precise relationship between the bulk mass and the scaling dimensions of local operators. It is also possible to interpret the vertical direction in the tree a renormalization-group scale for modes in the boundary CFT. Higher-genus bulk geometries (the BTZ black hole and its generalizations) can be understood straightforwardly in our setting and their construction parallels the story in AdS_3 topological gravity.

In Chapter 6, we consider a class of holographic quantum error-correcting codes, built from perfect tensors in network configurations dual to Bruhat--Tits trees and their quotients by Schottky groups corresponding to BTZ black holes. The resulting holographic states can be constructed in the limit of infinite network size. We obtain a p-adic version of entropy which obeys a Ryu--Takayanagi like formula for bipartite entanglement of connected or disconnected regions, in both genus-zero and genus-one p-adic backgrounds, along with a Bekenstein--Hawking-type formula for black hole entropy. We prove entropy inequalities obeyed by such tensor networks, such as subadditivity, strong subadditivity, and monogamy of mutual information (which is always saturated). In addition, we construct infinite classes of perfect tensors directly from semi-classical states in phase spaces over finite fields, generalizing the CRSS algorithm. These codes and the resulting networks provide a natural bulk geometric interpretation of non-Archimedean notions of entanglement in holographic boundary states.

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46

Lin, Shihchun, and 林施君. "The Study of Line-Diagram Representation on Algebraic Problem Solving Performance for Elementary School Students." Thesis, 2013. http://ndltd.ncl.edu.tw/handle/70845027182030620507.

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碩士
國立屏東教育大學
數理教育研究所
101
The aim of this research was to find out the fifth grade students’ solving performance of the algebra problems by the teaching strategy of using “line-diagram” representation. The quasi-experimental methodology was adopted and 60 fifth-graders from two classes were chosen as subjects. The researcher taught the experimental group by five steps of “line-diagram” strategy. On the other side, the control group was given the normal teaching strategy eight lessons were implemented in two weeks. This research using the post-test and delay test which researchers designed as assessment tools. Data were analyzed by Analysis of Covariance (ANCOVA) and nonparametric statistic test. According to the students' problem-solving performance, we could find out their strategies using and errors made on problem solving. The result shows that the experimental group students’ performances were significantly higher than the control group on post and delay test. The participants in experimental group have nice learning effects and learning retention effects. Besides, for the middle-achievement students shows more obvious progress on learning. On the problem solving, the control group students were easy to use key-word and functional reciprocal strategy, and it causes the failures on problem solving. However, the experimental group students tried to draw the line segments, and it clearly showed the relations between the numbers. They understood the reciprocal concepts from line segments, so that they can solve the problem successfully. Finally, according to the result, this study brings out some suggestions for teaching and the further research.
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47

Fishback, Paul Edward. "Holomorphic functions that map continuous, nonanalytic functions into the disc algebra, and nicely placed subsets of the real line." 1992. http://catalog.hathitrust.org/api/volumes/oclc/27399439.html.

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Thesis (Ph. D.)--University of Wisconsin--Madison, 1992.
Typescript. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 67-68).
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48

Mirjalalieh, Shirazi Mirhamed. "Equiangular Lines and Antipodal Covers." Thesis, 2010. http://hdl.handle.net/10012/5493.

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It is not hard to see that the number of equiangular lines in a complex space of dimension $d$ is at most $d^{2}$. A set of $d^{2}$ equiangular lines in a $d$-dimensional complex space is of significant importance in Quantum Computing as it corresponds to a measurement for which its statistics determine completely the quantum state on which the measurement is carried out. The existence of $d^{2}$ equiangular lines in a $d$-dimensional complex space is only known for a few values of $d$, although physicists conjecture that they do exist for any value of $d$. The main results in this thesis are: \begin{enumerate} \item Abelian covers of complete graphs that have certain parameters can be used to construct sets of $d^2$ equiangular lines in $d$-dimen\-sion\-al space; \item we exhibit infinitely many parameter sets that satisfy all the known necessary conditions for the existence of such a cover; and \item we find the decompose of the space into irreducible modules over the Terwilliger algebra of covers of complete graphs. \end{enumerate} A few techniques are known for constructing covers of complete graphs, none of which can be used to construct covers that lead to sets of $d^{2}$ equiangular lines in $d$-dimensional complex spaces. The third main result is developed in the hope of assisting such construction.
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49

Cheng, Ching-Lin, and 程景麟. "Effects of Using On-Line Cooperative Problem-Posing for the Sixth Grade in Elementary School on the Performance of Algebra Concept." Thesis, 2013. http://ndltd.ncl.edu.tw/handle/43996852991187183700.

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碩士
國立臺灣師範大學
資訊教育學系
101
The purpose of this studywasto investigate whether using an on-linecooperativeenvironmentaccompany with problem posing activity could affect elementary school student’s ability and algebra concept. The study used Moodle to construct theon-line problem posing system. A quasi experimental non –control group designed was used in this study. The participants were 60 sixth grade studentsin Taipei City. There were two classes assigned to experimental group and control group. The experimental group received cooperative problem posing materials while control group received individual problem posing materials. After eight weeks practice, we conducted posttest. After the experiment, results of this studywere concluded as following: 1.The experimental group demonstrated better performance in the posttest than the control group. 2.The experimental group demonstrated better problem posing ability than the control group. 3.The experimental group demonstrated better comprehensiveability ofalgebra conceptthan the control group. Keywords: algebra concept, online cooperativelearning,problem posing, Moodle
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50

Haque, Mohammad Moinul. "Realizability of tropical lines in the fan tropical plane." 2013. http://hdl.handle.net/2152/21209.

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In this thesis we construct an analogue in tropical geometry for a class of Schubert varieties from classical geometry. In particular, we look at the collection of tropical lines contained in the fan tropical plane. We call these tropical spaces "tropical Schubert prevarieties", and develop them after creating a tropical analogue for flag varieties that we call the "flag Dressian". Having constructed this tropical analogue of Schubert varieties we then determine that the 2-skeleton of these tropical Schubert prevarieties is realizable. In fact, as long as the lift of the fan tropical plane is in general position, only the 2-skeleton of the tropical Schubert prevariety is realizable.
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