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Dissertations / Theses on the topic 'Symmetry (Mathematics)'
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1
Breda, d'Azevedo Antonio Joao. "Hypermaps and symmetry." Thesis, University of Southampton, 1991. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.303114.
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Lin, Francesco Ph D. Massachusetts Institute of Technology. "Monopoles and Pin(2)-symmetry." Thesis, Massachusetts Institute of Technology, 2016. http://hdl.handle.net/1721.1/104585.
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Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2016.Cataloged from PDF version of thesis.
Includes bibliographical references (pages 321-326).
In this thesis we generalize the construction of monopole Floer homology due to Kronheimer and Mrowka to the case of a gradient flow with Morse-Bott singularities. Focusing then on the special case of a three-manifold equipped equipped with a spinc structure which is isomorphic to its conjugate, we define the counterpart in this context of Manolescu's recent Pin(2)-equivariant Seiberg-Witten-Floer homology. In particular, we provide an alternative approach to his disproof of the celebrated Triangulation conjecture. Furthermore, we discuss the analogue in this setting of the surgery exact triangle, and perform some sample computations.
by Francesco Lin.
Ph. D.
3
Castro, Sofia Balbina Santos Dias de. "Mode interactions with symmetry." Thesis, University of Warwick, 1993. http://wrap.warwick.ac.uk/4041/.
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This thesis deals with steady-state mode interaction problems with symmetry. We prove several results concerning problems invariant under the action of an arbitrary compact Lie group Γ. These include the existence of mixed-mode solutions and secondary Hopf bifurcations. We also consider the unfolding of the equations characterizing such problems. Where appropriate, we distinguish the case when Γ acts trivially on one of the modes. We then apply the results to the problems of the (1,3)-, (1,5)- and (1,3,5)-mode interactions with spherical symmetry. We also consider the (3,5)- and the (1,3,5)-mode interaction problems with SO(3) symmetry.
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Holtom, Paul Andrew. "Affine-invariant symmetry sets." Thesis, University of Liverpool, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.367704.
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Rostami, M. "Symmetry types of convex polyhedra." Thesis, University of Southampton, 1987. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.375365.
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Nivens, Ryan Andrew. "Fonts and Symmetry." Digital Commons @ East Tennessee State University, 2013. https://dc.etsu.edu/etsu-works/227.
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Using fonts as a context, we will analyze symmetry of fi gures. Diff erent letters and numbers will be measured, and participants will describe items that possess vertical, horizontal, and rotational symmetry. Our discussion and activity will focus on the mathematics of fonts and the presence and absence of symmetry in their design.
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Nivens, Ryan Andrew. "Fonts and Symmetry." Digital Commons @ East Tennessee State University, 2014. https://dc.etsu.edu/etsu-works/224.
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Using fonts as a context, we will analyze symmetry of fi gures. Diff erent letters and numbers will be measured, and participants will describe items that possess vertical, horizontal, and rotational symmetry. Our discussion and activity will focus on the mathematics of fonts and the presence and absence of symmetry in their design.
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Rossi, Paolo. "Symplectic Topology, Mirror Symmetry and Integrable Systems." Doctoral thesis, SISSA, 2008. http://hdl.handle.net/11577/3288900.
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Using Sympelctic Field Theory as a computational tool, we compute Gromov-Witten theory of target curves using gluing formulas and quantum integrable systems. In the smooth case this leads to a relation of the results of Okounkov and Pandharipande with the quantum dispersionless KdV hierarchy, while in the orbifold case we prove triple mirror symmetry between GW theory of target P^1 orbifolds of positive Euler characteristic, singularity theory of a class of polynomials in three variables and extended affine Weyl groups of type ADE.
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Cao, Xiaodong 1972. "Ricci flow on 3-manifolds with symmetry." Thesis, Massachusetts Institute of Technology, 2002. http://hdl.handle.net/1721.1/8398.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2002.Includes bibliographical references (p. 51-53).
In this thesis, we proved that for a 3-manifold S2 x S1 with warped product metric, the isoperimetric ratio on the base manifold S2 has a low positive bound away from zero, if the scalar curvature on the 3-manifold is positive. We also obtained a monotonicity result under the condition that the length of the optimal curve for isoperimetric ratio shrinks to zero under Ricci flow. This result excludes the product of a cigar soliton [Sigma]² with R¹ as the dilation limit of the Ricci flow equation. We also obtained an inequality of the curvature ratio Rmin/Rmax on the dilation limit for compact 3-manifold with positive scalar curvature.
by Xiaodong Cao.
Ph.D.
10
Lari-Lavassani, Ali. "Multiparameter bifurcation with symmetry via singularity theory /." The Ohio State University, 1990. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487683049377079.
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Degenhardt, Sheldon. "Weighted-inversion statistics and their symmetry groups /." The Ohio State University, 1996. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487941504293867.
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Ashwin, Peter Brian. "Applications of dynamical systems with symmetry." Thesis, University of Warwick, 1991. http://wrap.warwick.ac.uk/3985/.
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This thesis examines the application of symmetric dynamical systems theory to two areas in applied mathematics: weakly coupled oscillators with symmetry, and bifurcations in flame front equations. After a general introduction in the first chapter, chapter 2 develops a theoretical framework for the study of identical oscillators with arbitrary symmetry group under an assumption of weak coupling. It focusses on networks with 'all to all' Sn coupling. The structure imposed by the symmetry on the phase space for weakly coupled oscillators with Sn, Zn or Dn symmetries is discussed, and the interaction of internal symmetries and network symmetries is shown to cause decoupling under certain conditions. Chapter 3 discusses what this implies for generic dynamical behaviour of coupled oscillator systems, and concentrates on application to small numbers of oscillators (three or four). We find strong restrictions on bifurcations, and structurally stable heteroclinic cycles. Following this, chapter 4 reports on experimental results from electronic oscillator systems and relates it to results in chapter 3. In a forced oscillator system, breakdown of regular motion is observed to occur through break up of tori followed by a symmetric bifurcation of chaotic attractors to fully symmetric chaos. Chapter 5 discusses reduction of a system of identical coupled oscillators to phase equations in a weakly coupled limit, considering them as weakly dissipative Hamiltonian oscillators with very weakly coupling. This provides a derivation of example phase equations discussed in chapter 2. Applications are shown for two van der Pol-Duffing oscillators in the case of a twin-well potential. Finally, we turn our attention to the Kuramoto-Sivashinsky equation. Chapter 6 starts by discussing flame front equations in general, and non-linear models in particular. The Kuramoto-Sivashinsky equation on a rectangular domain with simple boundary conditions is found to be an example of a large class of systems whose linear behaviour gives rise to arbitrarily high order mode interactions. Chapter 7 presents computation of some of these mode interactions using competerised Liapunov-Schmidt reduction onto the kernel of the linearisation, and investigates the bifurcation diagrams in two parameters.
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Hicks, Jesse W. "Classification of Spacetimes with Symmetry." DigitalCommons@USU, 2016. https://digitalcommons.usu.edu/etd/5054.
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Spacetimes with symmetry play a critical role in Einstein's Theory of General Relativity. Missing from the literature is a correct, usable, and computer accessible classification of such spacetimes. This dissertation fills this gap; specifically, we
i) give a new and different approach to the classification of spacetimes with symmetry using modern methods and tools such as the Schmidt method and computer algebra systems, resulting in ninety-two spacetimes;
ii) create digital databases of the classification for easy access and use for researchers;
iii) create software to classify any spacetime metric with symmetry against the new database;
iv) compare results of our classification with those of Petrov and find that Petrov missed six cases and incorrectly normalized a significant number of metrics;
v) classify spacetimes with symmetry in the book Exact Solutions to Einstein’s Field Equations Second Edition by Stephani, Kramer, Macallum, Hoenselaers, and Herlt and in Komrakov’s paper Einstein-Maxwell equation on four-dimensional homogeneous spaces using the new software.
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Whelan, E. A. "Symmetry conditions in ring and module theory." Thesis, University of East Anglia, 1986. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.374275.
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Priestley, Thomas James. "Methods of symmetry reduction and their application." Thesis, University of Kent, 1996. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.319173.
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Henninger, Helen Clare. "The symmetry group of a model of hyperbolic plane geometry and some associated invariant optimal control problems." Thesis, Rhodes University, 2012. http://hdl.handle.net/10962/d1018232.
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In this thesis we study left-invariant control offine systems on the symmetry group of a. model of hyperbolic plane geometry, the matrix Lie group SO(1, 2)₀. We determine that there are 10 distinct classes of such control systems and for typical elements of two of these classes we provide solutions of the left-invariant optimal wntrol problem with quauratic costs. Under the identification of the Lie allgebra .so(l, 2) with Minkowski spacetime R¹̕'², we construct a controllabilility criterion for all left-invariant control affine systems on 50(1. 2)₀ which in the inhomogeneous case depends only on the presence or absence of an element in the image of the system's trace in R¹̕ ²which is identifiable using the inner product. For the solutions of both the optimal control problems, we provide explicit expressions in terms of Jacobi elliptic functions for the solutions of the reduced extremal equations and determine the nonlinear stability of the equilibrium points.
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Drugan, Gregory (Gregory Michael). "Symmetry properties of semilinear elliptic equations with isolated singularities." Thesis, Massachusetts Institute of Technology, 2007. http://hdl.handle.net/1721.1/38999.
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Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2007.Includes bibliographical references (p. 79-80).
In this thesis we use the method of moving planes to establish symmetry properties for positive solutions of semilinear elliptic equations. We give a detailed proof of the result due to Caffarelli, Gidas, and Spruck that a solution in the punctured ball, B\{0}, behaves asymptotically like its spherical average at the origin. We also show that a solution with an isolated singularity in the upper half space Rn+ must be cylindrically symmetric about some axis orthogonal to the boundary aRn+.
by Gregory Drugan.
S.M.
18
Northover, Timothy. "Riemann surfaces with symmetry : algorithms and applications." Thesis, University of Edinburgh, 2011. http://hdl.handle.net/1842/9848.
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Riemann surfaces frequently possess automorphisms which can be exploited to simplify calculations. However, existing computer software (Maple in particular) is designed for maximum generality and has not yet been extended to make use of available symmetries. In many calculations, the symmetries can be most easily used by choosing a specific basis for H₁(Σ,Z) under which the automorphism group acts neatly. This thesis describes a Maple library, designed to be used in conjunction with the existing algcurves, which allows such a choice to be made. In addition we create a visual tool to simplify the presentation of Riemann surfaces as sheeted covers of C and the creation of homology bases suitable for use in the Maple library. Two applications are considered for these techniques, first Klein's curve and then Bring's. Both of these possess maximal symmetry groups for their genus, and this fact is exploited to obtain neat algebraic homology bases. In the Klein case the basis is novel; Bring's is derived from work in the hyperbolic setting by Riera. In both cases previous hyperbolic work and calculations are related to the algebraic results. Vectors of Riemann constants are calculated for both curves, again exploiting the symmetry. Finally this thesis moves back to simpler cases, and presents a general algorithm to convert results from general genus 2 curves into results based on a symmetric basis if one exists. This is applied to algebraic and numeric examples where we discover an elliptic surface covered in each case.
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Elmhirst, Toby. "Symmetry and emergence in polymorphism and sympatric speciation." Thesis, University of Warwick, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.275232.
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Barany, Ernest J. "Algebraic aspects of broken symmetry : irreducible representations of SO(3) /." The Ohio State University, 1988. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487591658174108.
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Pokhrel, Dipendra. "Local PT-Symmetry preserves the no-signaling principle." DigitalCommons@Robert W. Woodruff Library, Atlanta University Center, 2016. http://digitalcommons.auctr.edu/dissertations/3901.
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Bender and Boettcher explored a quantum theory based on a non-Hermitian PT symmetric Hamiltonian , where PT is the operator of the space-time reflection and demonstrated that the PT-symmetric Hamiltonian can possess entirely real spectra. In this thesis, we point out that in the framework of PT-symmetric quantum mechanics; the calculation of matrix elements in the Hilbert space is ill-defined. We point out the importance of using CPT inner product in PT-symmetric systems. We manifested our assessment using the CPT inner product prescription for the entangled wave function of the composite system. We show that for a composite system with a local PT-symmetry, it preserves the no-signaling condition and the orthogonality of the states.The reduced density matrix is diagonal and independent of the non-Hermitian parameter . We reaffirm the consistency of PT-symmetric quantum mechanics as a candidate for a fundamental theory .
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Coles, Robert Anthony. "The effects of symmetry in non-linear sigma models." Thesis, King's College London (University of London), 1997. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.265148.
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Bott, Christopher James. "Mirror Symmetry for K3 Surfaces with Non-symplectic Automorphism." BYU ScholarsArchive, 2018. https://scholarsarchive.byu.edu/etd/7456.
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Mirror symmetry is the phenomenon, originally discovered by physicists, that Calabi-Yau manifolds come in dual pairs, with each member of the pair producing the same physics. Mathematicians studying enumerative geometry became interested in mirror symmetry around 1990, and since then, mirror symmetry has become a major research topic in pure mathematics. One important problem in mirror symmetry is that there may be several ways to construct a mirror dual for a Calabi-Yau manifold. Hence it is a natural question to ask: when two different mirror symmetry constructions apply, do they agree?We specifically consider two mirror symmetry constructions for K3 surfaces known as BHK and LPK3 mirror symmetry. BHK mirror symmetry was inspired by the LandauGinzburg/Calabi-Yau correspondence, while LPK3 mirror symmetry is more classical. In particular, for algebraic K3 surfaces with a purely non-symplectic automorphism of order n, we ask if these two constructions agree. Results of Artebani Boissi`ere-Sarti originally showed that they agree when n = 2, and more recently Comparin-Lyon-Priddis-Suggs showed that they agree when n is prime. However, the n being composite case required more sophisticated methods. Whenever n is not divisible by four (or n = 16), this problem was solved by Comparin and Priddis by studying the associated lattice theory more carefully. In this thesis, we complete the remaining case of the problem when n is divisible by four by finding new isomorphisms and deformations of the K3 surfaces in question, develop new computational methods, and use these results to complete the investigation, thereby showing that the BHK and LPK3 mirror symmetry constructions also agree when n is composite.
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Lu, Wenxuan. "Instanton correction, wall crossing and mirror symmetry of Hitchin's moduli spaces." Thesis, Massachusetts Institute of Technology, 2011. http://hdl.handle.net/1721.1/67809.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011.Cataloged from PDF version of thesis.
Includes bibliographical references (p. 205-210).
We study two instanton correction problems of Hitchin's moduli spaces along with their wall crossing formulas. The hyperkahler metric of a Hitchin's moduli space can be put into an instanton-corrected form according to physicists Gaiotto, Moore and Neitzke. The problem boils down to the construction of a set of special coordinates which can be constructed as Fock-Goncharov coordinates associated with foliations of quadratic differentials on a Riemann surface. A wall crossing formula of Kontsevich and Soibelman arises both as a crucial consistency condition and an effective computational tool. On the other hand Gross and Siebert have succeeded in determining instanton corrections of complex structures of Calabi-Yau varieties in the context of mirror symmetry from a singular affine structure with additional data. We will show that the two instanton correction problems are equivalent in an appropriate sense via the identification of the wall crossing formulas in the metric problem with consistency conditions in the complex structure problem. This is a nontrivial statement of mirror symmetry of Hitchin's moduli spaces which till now has been mostly studied in the framework of geometric Langlands duality. This result provides examples of Calabi-Yau varieties where the instanton correction (in the sense of mirror symmetry) of metrics and complex structures can be determined. This equivalence also relates certain enumerative problems in foliations to some gluing constructions of affine varieties.
by Wenxuan Lu.
Ph.D.
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Vaintrob, Dmitry. "Mirror symmetry and the K theory of a p-adic group." Thesis, Massachusetts Institute of Technology, 2016. http://hdl.handle.net/1721.1/104578.
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Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2016.Cataloged from PDF version of thesis.
Includes bibliographical references (pages 59-61).
Let G be a split, semisimple p-adic group. We construct a derived localization functor Loc : ... from the compactified category of [BK2] associated to G to the category of equivariant sheaves on the Bruhat-Tits building whose stalks have finite-multiplicity isotypic components as representations of the stabilizer. Our construction is motivated by the "coherent-constructible correspondence" functor in toric mirror symmetry and a construction of [CCC]. We show that Loc has a number of useful properties, including the fact that the sections ... compactifying the finitely-generated representation V. We also construct a depth by Dmitry A. Vaintrob.
Ph. D.
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Sheridan, Nicholas (Nicholas James). "Homological mirror symmetry for a Calabi-Yau hypersurface in projective space." Thesis, Massachusetts Institute of Technology, 2012. http://hdl.handle.net/1721.1/73374.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2012.Cataloged from PDF version of thesis.
Includes bibliographical references (p. 365-369).
This thesis is concerned with Kontsevich's Homological Mirror Symmetry conjecture. In Chapter 1, which is based on [1], we consider the n-dimensional pair of pants, which is defined to be the complement of n + 2 generic hyperplanes in CPn. The pair of pants is conjectured to be mirror to the Landau-Ginzburg model (Cn+2 , W), where W = z1...zn+2 We construct an immersed Lagrangian sphere in the pair of pants, and show that its endomorphism A.. algebra in the Fukaya category is quasi-isomorphic to the endomorphism dg algebra of the structure sheaf of the origin in the mirror,.giving some evidence for the Homological Mirror Symmetry conjecture in this case. In Chapter 2, which is based on [2], we build on these results to prove Homological Mirror Symmetry for a smooth d-dimensional Calabi-Yau hypersurface in projective space, for any d =/> 3.
by Nicholas Sheridan.
Ph.D.
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Gillespie, Maria Monks. "A combinatorial approach to the q; t-symmetry in Macdonald polynomials." Thesis, University of California, Berkeley, 2016. http://pqdtopen.proquest.com/#viewpdf?dispub=10150833.
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Using the combinatorial formula for the transformed Macdonald polynomials of Haglund, Haiman, and Loehr, we investigate the combinatorics of the symmetry relation H˜μ*(x; q,t) = H˜ μ(x; t,q). We provide a purely combinatorial proof of the relation in the case of Hall-Littlewood polynomials (q = 0) when mu is a partition with at most three rows, and for the coefficients of the square-free monomials in X={x_1,x_2,...} for all shapes mu. We also provide a proof for the full relation in the case when mu is a hook shape, and for all shapes at the specialization t = 1. Our work in the Hall-Littlewood case reveals a new recursive structure for the cocharge statistic on words.
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Gavhi, Mpfareleni Rejoyce. "Interpolatory refinement pairs with properties of symmetry and polynomial filling." Thesis, Link to the online version, 2008. http://hdl.handle.net/10019/840.
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Jonsson, Denice. "Matematikläromedels erbjudande av symmetri : En kvalitativ och kvantitativ studie anpassad för årskurs 1-3." Thesis, Högskolan i Jönköping, Högskolan för lärande och kommunikation, 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:hj:diva-39998.
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I den svenska skolan styr matematikläromedel till stor del matematikundervisningen i klassrummet. Det har visat sig att lärare använder matematikläromedlet som en ersättning av kursplanen. Förlagen är däremot inte skyldiga att inkludera alla delar av kursplanen för matematik i läromedlen. Ett grundläggande arbetsområde i geometrin är symmetri. Symmetri beskrivs som ett eget avsnitt under det centrala innehållet i kursplanen för matematik. Symmetri har däremot inte alltid behandlats lika tydligt som i den nuvarande kursplanen. Därför handlar den här studien om matematikläromedels erbjudande av symmetri. Syftet med den här studien är att undersöka hur symmetribegreppet erbjuds i matematikläromedel för årskurs 1-3 samt hur de stämmer överens med kursplanen i matematik. Till studien har tre forskningsfrågor tagits fram. Den första forskningsfrågan är: Vilka symmetrier erbjuds av matematikläromedlet för årskurs 1 – 3? Den andra frågan: I hur stor omfattning förekommer symmetri uppgifter i läromedel? Den tredje frågan: På vilket sätt erbjuds eleven att se, upptäcka, konstruera och beskriva symmetrier i bilder och natur? Den här studien bygger på en flermetodsforskning eftersom den innehåller både en kvantitativ innehållsanalys och en kvalitativ dokumentationsanalys. Undersökningen har genomförts som en litteraturstudie med inslag av en handlingserbjudande teori. Totalt har 9 serier av matematikläromedel anpassade för årskurs 1-3 analyserats. Läromedel som har analyserats är elevläromedel och inte lärarhandledning. Resultatet visar att eleven får erbjudande att arbeta med spegelsymmetri, translation, tesselering, symmetriska respektive asymmetriska uppgifter i matematikläromedel. Resultatet visar att omfattningen av uppgifter som behandlar symmetri varierar i erbjudande. Resultatet visar även att eleven vid flera tillfällen får erbjudande att se, upptäcka och konstruera symmetrier. Däremot är det inte lika vanligt att eleven får beskriva symmetrier.The mathematics teaching material is an important part of the Swedish education. It has been proven that mathematics teaching material are been used as an replacement of the curriculum in school. But the publisher does not need to include every part of the curriculum for mathematics. A basic part in geometry is symmetry. Symmetry is described as an own section in the curriculum, but it has not always been so. Therefore, this study is about mathematics teaching materials offers of symmetry. The aim of this study is to examine mathematics teaching materials offer in symmetry for pupils in grade 1-3 and how they match with the curriculum in mathematics. To this study, three research questions have been produced. The first one: Witch symmetry offers in mathematics teaching materials for grade 1-3? The second one: In how big extent presents symmetry in mathematics teaching materials? The third one: In witch way does mathematics teaching material offers pupils to, see, discover, construct and describe symmetry in pictures and in nature? The method that has been used in the study is a quantitative content analysis and a qualitative document analysis, with an affordance theory. With a total of nine series of mathematics teaching materials grade in 1-3 in Swedish school. The analysis shows that pupils are appointed with the possibility to work with mirrorsymmetry, translation, tesselering, symmetric and asymmetric tasks. The analysis also shows that the extent of symmetry task varies in different offers in teaching materials. The analysis also shows that pupils offers to see, discover and construct symmetries. However, it is uncommon that pupils get the opportunity to describe different types of symmetry.
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Boily, Patrick. "Spiral wave dynamics under full Euclidean symmetry-breaking: A dynamical system approach." Thesis, University of Ottawa (Canada), 2006. http://hdl.handle.net/10393/29341.
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Spirals are common in Nature: the snail's shell and the ordering of seeds in the sunflower are amongst the most widely-known occurrences. While these are static, dynamic spirals can also be observed in excitable systems such as heart tissue, retina, certain chemical reactions, slime mold aggregates, flame fronts, etc. The images associated with these spirals are often breathtaking, but spirals have also been linked to cardiac arrhythmias, a potentially fatal heart ailment.
In the literature, very specific models depending on the excitable system of interest are used to explain the observed behaviour of spirals (such as anchoring or drifting). Barkley [5] first noticed that the Euclidean symmetry of these models, and not the model itself, is responsible for the observed behaviour. But in experiments, the physical domain is never Euclidean. The heart, for instance, is finite, anisotropic and littered with inhomogeneities. To capture this loss of symmetry, LeBlanc and Wulff [48,51] introduced forced Euclidean symmetry-breaking (FESB) in the analysis.
To accurately model the physical situation, two basic types of symmetry-breaking perturbations are used: translational symmetry-breaking (TSB) and rotational symmetry-breaking (RSB) terms. LeBlanc and Wulff, [51] and LeBlanc [48] have studied the effects of these individual perturbations, and they have shown that phenomena such as anchoring and quasi-periodic meandering can be explained by FESB. However, these specific perturbations only tell part of the story.
In this thesis, the effects of multiple TSB perturbations, as well as those of combined TSB-RSB perturbations are studied and provide a more complete explanation for two aspects of spiral dynamics: anchoring and boundary drifting. Higher co-dimension phenomena are also considered.
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Johnson, Jared Drew. "An Algebra Isomorphism for the Landau-Ginzburg Mirror Symmetry Conjecture." BYU ScholarsArchive, 2011. https://scholarsarchive.byu.edu/etd/2793.
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Landau-Ginzburg mirror symmetry takes place in the context of affine singularities in CN. Given such a singularity defined by a quasihomogeneous polynomial W and an appropriate group of symmetries G, one can construct the FJRW theory (see [3]). This construction fills the role of the A-model in a mirror symmetry proposal of Berglund and H ubsch [1]. The conjecture is that the A-model of W and G should match the B-model of a dual singularity and dual group (which we denote by WT and GT). The B-model construction is based on the Milnor ring, or local algebra, of the singularity. We verify this conjecture for a wide class of singularities on the level of Frobenius algebras, generalizing work of Krawitz [10]. We also review the relevant parts of the constructions.
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George, Timothy Edward. "Symmetric representation of elements of finite groups." CSUSB ScholarWorks, 2006. https://scholarworks.lib.csusb.edu/etd-project/3105.
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The purpose of the thesis is to give an alternative and more efficient method for working with finite groups by constructing finite groups as homomorphic images of progenitors. The method introduced can be applied to all finite groups that possess symmetric generating sets of involutions. Such groups include all finite non-abelian simple groups, which can then be constructed by the technique of manual double coset enumeration.
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Nass, Aminu Ma'aruf. "Point symmetry methods for Itô Stochastic Differential Equations (SDE) with a finite jump process." Doctoral thesis, University of Cape Town, 2017. http://hdl.handle.net/11427/25387.
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The mixture of Wiener and a Poisson processes are the primary tools used in creating jump-diffusion process which is very popular in mathematical modeling. In financial mathematics, they are used to describe the change of stock rates and bonanzas, and they are often used in mathematical biology modeling and population dynamics. In this thesis, we extended the Lie point symmetry theory of deterministic differential equations to the class of jump-diffusion stochastic differential equations, i.e., a stochastic process driven by both Wiener and Poisson processes. The Poisson process generates the jumps whereas the Brownian motion path is continuous. The determining equations for a stochastic differential equation with finite jump are successfully derived in an Itô calculus context and are found to be deterministic, even though they represent a stochastic process. This work leads to an understanding of the random time change formulae for Poisson driven process in the context of Lie point symmetries without having to consult much of the intense Itô calculus theory needed to formally derive it. We apply the invariance methodology of Lie point transformation together with the more generalized Itô formulae, without enforcing any conditions to the moments of the stochastic processes to derive the determining equations and apply it to few models. In the first part of the thesis, point symmetry of Poisson-driven stochastic differential equations is discussed, by considering the infinitesimals of not only spatial and temporal variables but also infinitesimals of the Poisson process variable. This was later extended, in the second part, to define the symmetry of jumpdiffusion stochastic differential equations (i.e., stochastic differential equations driven by both Wiener and Poisson processes).
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Harris, Elena Yavorska. "Symmetric representation of elements of sporadic groups." CSUSB ScholarWorks, 2005. https://scholarworks.lib.csusb.edu/etd-project/2844.
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Uses the techniques of symmetric presentations to manipulate elements of large sporadic groups and to represent elements of these groups in much shorter forms than their corresponding permutation or matrix representation. Undertakes to develop a nested algorithm and a computer program to manipulate elements of large sporadic groups.
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Léonard, Christian, Sylvie Roelly, and Jean-Claude Zambrini. "Temporal symmetry of some classes of stochastic processes." Universität Potsdam, 2013. http://opus.kobv.de/ubp/volltexte/2013/6459/.
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In this article we analyse the structure of Markov processes and reciprocal processes to underline their time symmetrical properties, and to compare them. Our originality consists in adopting a unifying approach of reciprocal processes, independently of special frameworks in which the theory was developped till now (diffusions, or pure jump processes). This leads to some new results, too.
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Hood, Simon. "Nonclassical symmetry reductions and exact solutions of nonlinear partial differential equations." Thesis, University of Exeter, 1993. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.357042.
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Amdjadi, F. "Numerical methods for bifurcations and mode interactions in problems with symmetry." Thesis, University of Surrey, 1994. http://epubs.surrey.ac.uk/843720/.
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The thesis studies two problems. The first is a problem with 0(2) symmetry which gives rise to a branch with Z2 symmetry. The Jacobian along the Z2-symmetric branch is always singular, due to the group orbit of solutions. The situation when a pair of complex eigenvalues cross the imaginary axis is considered and canonical coordinates are used to remove the degeneracy from the system so that the standard theory can be applied. This method is used in order to understand the type of solutions that occur. The method cannot be applied to partial differential equations and so a method involving a phase condition is employed which has also been used by Aston, Spence and Wu. Two examples have been considered. The first is a simple example on C3 and the second example is the Kuramoto Sivashinsky equation for which a multiple Hopf bifurcation on a D3 symmetric branch has been studied. A Hopf bifurcation on a so called strange fixed point has also been considered and results compared with the numerical results of Hyman and Nicolaenko. The second problem involves mode interactions between a steady state bifurcation and a Hopf bifurcation in problems with Z2 symmetry. Two different extended systems have been considered and it is shown that the mode interaction corresponds to a bifurcation of these systems. Finally a method for predicting the existence of a tertiary Hopf bifurcation arising from a mode interaction, using only the bifurcation diagrams, has also been considered. Initially a problem with Z2 ⊗ Z2 symmetry, which involves a steady state/steady state mode interaction, has been considered and then similar results are obtained for Hopf/steady state mode interactions.
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Grayland, Andrews. "Automated static symmetry breaking in constraint satisfaction problems." Thesis, University of St Andrews, 2011. http://hdl.handle.net/10023/1718.
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Variable symmetries in constraint satisfaction problems can be broken by adding lexicographic ordering constraints. Existing general methods of generating such sets of ordering constraints can produce a huge number of additional constraints. This adds an unacceptable overhead to the solving process. Methods exist by which this large set of constraints can be reduced to a much smaller set automatically, but their application is also prohibitively costly. In contrast, this thesis takes a bottom up approach to generating symmetry breaking constraints. This will involve examining some commonly-occurring families of mathematical groups and deriving a general formula to produce a minimal set of ordering constraints which are sufficient to break all of the symmetry that each group describes. In some cases it is known that there exists no manageable sized sets of constraints to break all symmetries. One example of this occurs with matrix row and column symmetries. In such cases, incomplete symmetry breaking has been used to great effect. Double lex is a commonly used incomplete symmetry breaking technique for row and column symmetries. This thesis also describes another similar method which compares favourably to double lex. The general formulae investigated are used as building blocks to generate small sets of ordering constraints for more complex groups, constructed by combining smaller groups. Through the utilisation of graph automorphism tools and the groups and permutations software GAP we provide a method of defining variable symmetries in a problem as a group. Where this group can be described as the product of smaller groups, with known general formulae, we can construct a minimal set of ordering constraints for that problem automatically. In summary, this thesis provides the theoretical background necessary to apply efficient static symmetry breaking to constraint satisfaction problems. It also goes further, describing how this process can be automated to remove the necessity of having an expert CP practitioner, thus opening the field to a larger number of potential users.
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Mehraban, Arash. "Non-Classical Symmetry Solutions to the Fitzhugh Nagumo Equation." Digital Commons @ East Tennessee State University, 2010. https://dc.etsu.edu/etd/1736.
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In Reaction-Diffusion systems, some parameters can influence the behavior of other parameters in that system. Thus reaction diffusion equations are often used to model the behavior of biological phenomena. The Fitzhugh Nagumo partial differential equation is a reaction diffusion equation that arises both in population genetics and in modeling the transmission of action potentials in the nervous system. In this paper we are interested in finding solutions to this equation. Using Lie groups in particular, we would like to find symmetries of the Fitzhugh Nagumo equation that reduce this non-linear PDE to an Ordinary Differential Equation. In order to accomplish this task, the non-classical method is utilized to find the infinitesimal generator and the invariant surface condition for the subgroup where the solutions for the desired PDE exist. Using the infinitesimal generator and the invariant surface condition, we reduce the PDE to a mildly nonlinear ordinary differential equation that could be explored numerically or perhaps solved in closed form.
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Constantinescu, Radu 1968. "Cicular symmetry in topological quantum field theory and the topology of the index bundle." Thesis, Massachusetts Institute of Technology, 1998. http://hdl.handle.net/1721.1/10131.
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Redmond, Brian F. "Bifurcation analysis of a class of delay-differential equations with reflectional symmetry: Applications to ENSO." Thesis, University of Ottawa (Canada), 2002. http://hdl.handle.net/10393/6242.
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We consider a general class of first-order nonlinear delay-differential equations (DDEs) with reflectional symmetry, and study completely the bifurcations of the trivial equilibrium under some generic conditions on the Taylor coefficients of the DDE. Our analysis reveals a Hopf bifurcation curve terminating on a pitchfork bifurcation line at a codimension two Takens-Bogdanov point in parameter space. We compute the normal form coefficients of the reduced vector field on the centre manifold in terms of the Taylor coefficients of the original DDE, and in contrast to many previous bifurcation analyses of DDEs, we also compute the unfolding parameters in terms of these coefficients. For application purposes, this is important since one can now identify the possible asymptotic dynamics of the DDE near the bifurcation points by computing quantities which depend explicitly on the Taylor coefficients of the original DDE. We illustrate these results using simple model systems relevant to the areas of neural networks and atmospheric physics, and show that the results agree with numerical simulations. Finally, note that most of the results of this thesis have already been refereed and published elsewhere (see [26]).
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Bright, Chelsea. "Radial symmetry and mass-independent boundedness of stationary states of aggregation-diffusion models." Diss., University of Pretoria, 2020. http://hdl.handle.net/2263/75051.
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General aggregation diffusion equations have been used in a variety of different settings, including the modelling of chemotaxis and the biological aggregation of insects and herding of animals. We consider a non-local aggregation diffusion equation, where the repulsion is modelled by nonlinear diffusion (Laplace operator applied to $ m $th power of the spatial density) and attraction modelled by non-local interaction. The competition between these forces gives rise to characteristic time-independent morphologies. When the attractive interaction kernel is radially symmetric and strictly increasing with respect to the norm in the $ n $-dimensional linear space of the space variable, it is previously known that all stationary solutions are radially symmetric and decreasing up to a translation. We extend this result to attractive kernels with compact support, where a wider variety of time-independent patterns occur. We prove that for compactly supported attractive kernels and for power in the diffusion term $ m>1 $, all stationary states are radially symmetric and decreasing up to a translation on each connected component of their support. Furthermore, for $ m>2 $, we prove analytically that stationary states have an upper-bound independent of the initial data, confirming previous numerical results given in the literature. This result is valid for both attractive kernels with compact support and unbounded support. Finally, we investigate a model that incorporates both non-local attraction and non-local repulsion. We show that this model may be considered as a generalization of the aggregation diffusion equation and we present numerical results showing that $ m=2 $ is a threshold value such that, for $ m>2 $, stationary states of the fully non-local model possess a mass-independent upper-bound.Dissertation (MSc)--University of Pretoria, 2020.
Masters Research Bursary UP Mast Research Renewal
Mathematics and Applied Mathematics
MSc
Unrestricted
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Parker, Martyn. "Forced symmetry breaking of Euclidean equivariant partial differential equations, pattern formation and Turing instabilities." Thesis, University of Warwick, 2003. http://wrap.warwick.ac.uk/74320/.
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Many natural phenomena may be modelled using systems of differential equations that possess symmetry. Often the modelling process introduces additional symmetries that are only approximately present in the real physical system. This thesis investigates how the inclusion of small symmetry breaking effects changes the behaviour of the original solutions, such a process is called forced symmetry breaking. Part I introduces the general equivariant bifurcation theory required for the rest of this work. In particular, we generalise previous techniques used to study forced symmetry breaking to a certain class of Euclidean invariant problems. This allows the study of the effects of forced symmetry breaking on spatially periodic solutions to differential equations. Part II considers spatially periodic solutions in two dimensions that are supported by the square or hexagonal lattices. The methods of Part I are applied to investigate how the translation free solutions, supported by these lattices, are altered when the perturbation term possesses certain symmetries. This leads to a partial classification theorem, describing the behaviour of these solutions. This classification is extended in Part III to three-dimensional solutions. In particular, the cubic lattices: simple, face centred, and body centred cubic, are considered. The analysis follows the same lines as Part II, but is necessarily more complex. This complexity is also present in the results, there are much richer dynamical possibilities. Parts II and III lead to a partial classification of the behaviour of spatially periodic solutions to differential equations in two and three dimensions. Finally in Part IV the results of Part III, concerning the body centred cubic lattice, are applied to the black-eye Turing instability. In particular, the model of Gomes [39] is cast in a new light where forced symmetry breaking is present, leading to several qualitative predictions. Nonlinear optical systems and the Polyacrylamide-Methylene Blue-Oxygen reaction are also discussed.
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Cummins, C. J. "Applications of S-function techniques to the representation theory of Lie superalgebras and symmetry breaking." Thesis, University of Southampton, 1986. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.374751.
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Strazzullo, Francesco. "Symmetry Analysis of General Rank-3 Pfaffian Systems in Five Variables." DigitalCommons@USU, 2009. https://digitalcommons.usu.edu/etd/449.
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In this dissertation we applied geometric methods to study underdetermined second order scalar ordinary differential equations (called general Monge equations), nonlinear involutive systems of two scalar partial differential equations in two independent variables and one unknown and non-Monge-Ampere Goursat parabolic scalar PDE in the plane. These particular kinds of differential equations are related to general rank-3 Pfaffian systems in five variables. Cartan studied these objects in his 1910 paper. In this work Cartan provided normal forms only for some general rank-3 Pfaffian systems with 14-, 7-, and 6-dimensional symmetry algebra. We applied our normal forms to
[i] sharpen Cartan's integration method of nonlinear involutive systems,
[ii] classify all general Monge equations with a freely acting transverse 3-dimensional symmetry algebra, of which many new examples are presented, and
[iii] provide a broad classification of non-Monge-Ampere Darboux integrable hyperbolic PDE in the plane.
We developed a computer software, called FiveVariables, that classifies general rank-3 Pfaffian systems. FiveVariables runs in the environment DifferentialGeometry of Maple, version 11 and later.
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Dinolov, Georgi. "Swarm Control Through Symmetry and Distribution Characterization." Scholarship @ Claremont, 2011. http://scholarship.claremont.edu/hmc_theses/2.
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Two methods for control of swarms are described. The first of these methods, the Virtual Attractive-Repulsive (VARP) method, is based on potentials defined between swarm elements. The second control method, or the abstraction method, is based on controlling the macroscopic characteristics of a swarm. The derivation of a new control law based on the second method is described. Numerical simulation and analytical interpretation of the result is also presented.
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Neves, Paulo Roberto Vargas [UNESP]. "O uso do caleidoscópio no ensino de grupos de simetria e transformações geométricas." Universidade Estadual Paulista (UNESP), 2011. http://hdl.handle.net/11449/91059.
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neves_prv_me_rcla.pdf: 1296931 bytes, checksum: b45ea16895bf9e20db063325be68a349 (MD5)Este trabalho teve o objetivo de produzir um conjunto de atividades para analisar como o uso do caleidoscópio associado ao estudo dos ornamentos planos pode contribuir no ensino de grupos de simetria e transformações geométricas em um curso de graduação em Matemática. Esta pesquisa tem caráter qualitativo e foi desenvolvida segundo a proposta metodológica de Romberg. Elaborou-se uma proposta de ensino baseada na metodologia de Resolução de Problemas que foi aplicada a um grupo de professores (alguns em fase de formação) de matemática. As atividades tiveram a finalidade de fazer com que os alunos usassem o caleidoscópio para reproduzir ornamentos planos e, a partir de então, discutissem, com base em argumentos geométricos e algébricos, quais as possibilidades (e impossibilidades) que esse instrumento oferece para obtenção desses ornamentos e suas respectivas justificativas. A coleta de dados ocorreu, essencialmente, por observação participante em sala de aula por meio do uso de questionários, anotações e registros fotográficos. Após a coleta de dados, foi feita uma análise das possibilidades e limitações do material desenvolvido para o ensino de grupos de simetria e transformações geométricas, bem como o uso do caleidoscópio enquanto recurso didático
The purpose of this work was to develop a set of activities to analyze how the use of kaleidoscope associated to the study of ornaments can contribute to the teaching of symmetry groups and geometric transformations on a undergraduate course in Mathematics. This is a qualitative research and it was developed according to the methodological proposal of Romberg. A teaching proposal was drafted and was applied to a group of mathematics teachers. Activities were designed following the methodology of problem-solving and intended to make students to use the kaleidoscope to reproduce some ornaments and thereafter, discuss, based on geometric and algebraic arguments, the possibilities and impossibilities that this tool provides to obtain ornaments and their respective justifications. Data collection occurred primarily by participant observation in the classroom through the use of questionnaires, notes and photographic records. After the end of the course a viability analysis of the activities was done (possibilities and limitations) for teaching symmetry groups and geometric transformations as well as the use of Kaleidoscope as a didactic tool
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Morris, Richard James. "Symmetry of curves and the geometry of surfaces : two explorations with the aid of computer graphics." Thesis, University of Liverpool, 1990. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.293136.
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Berry, Robert D. "A New Approach to Lie Symmetry Groups of Minimal Surfaces." BYU ScholarsArchive, 2004. https://scholarsarchive.byu.edu/etd/321.
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The Lie symmetry groups of minimal surfaces by way of planar harmonic functions are determined. It is shown that a symmetry group acting on the minimal surfaces is isomorphic with H × H^2 — the analytic functions and the harmonic functions. A subgroup of this gives a generalization of the associated family which is examined.
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Alsufyani, Nada. "The Iterative Method for Quantum Double-well and Symmetry-breaking Potentials." DigitalCommons@Robert W. Woodruff Library, Atlanta University Center, 2017. http://digitalcommons.auctr.edu/cauetds/62.
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Numerical solutions of quantum mechanical problems have witnessed tremendous advances over the past years. In this thesis, we develop an iterative approach to problems of double-well potentials and their variants with parity-time-reversal symmetry- breaking perturbations. We show that the method provides an efficient scheme for obtaining accurate energies and wave functions. We discuss in this thesis potential applications to a variety of related topics such as phase transitions, symmetry breaking, and external field-induced effects.