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Dissertations / Theses on the topic 'Polynomial potentials'
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1
Bridle, Ismail Hamzaan. "Non-polynomial scalar field potentials in the local potential approximation." Thesis, University of Southampton, 2017. https://eprints.soton.ac.uk/410270/.
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We present the renormalisation group analysis of O(N) invariant scalar field theory in the local potential approximation. Linearising around the Gaussian fixed point, we find the same eigenoperators solutions exist for both the Wilsonian and the Legendre effective actions, given by solutions to Kummer’s equations. We find the usual polynomial eigenoperators and the Hilbert space they define are a natural subset of these solutions given by a specific set of quantised eigenvalues. Allowing for continuous eigenvalues, we find non-polynomial eigenoperator solutions, the so called Halpern-Huang directions, that exist outside of the polynomial Hilbert space due to the exponential field dependence. Carefully analysing the large field behaviour shows that the exponential dependence implies the Legendre effective action does not have a well defined continuum limit. In comparison, flowing towards the infrared we find that the non-polynomial eigenoperators flow into the polynomial Hilbert space. These conclusions are based off RG flow initiated at an arbitrary scale, implying non-polynomial eigenoperators are dependent upon a scale other than k. Therefore, the asymptotic field behaviour forbids self-similar scaling. These results hold when generalised from the Halpern-Huang directions around the Gaussian fixed point to a general fixed point with a general non-polynomial eigenoperator. Legendre transforming to results of the Polchinski equation, we find the flow of the Wilsonian effective action is much better regulated and always fall into the polynomial Hilbert space. For large Wilsonian effective actions, we find that the non-linear terms of the Polchinski equation forbid any non-polynomial field scaling, regardless of the fixed point. These observations lead to the conclusion that only polynomial eigenoperators show the correct, self-similar, scaling behaviour to construct a non-perturbatively renormalisable scalar QFT.
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Hyder, Asif M. "Green's operator for Hamiltonians with Coulomb plus polynomial potentials." California State University, Long Beach, 2013.
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3
Capraro, Patrick Leonardo [Verfasser]. "Feynman path integrals in configuration space, momentum space and phase space for perturbative and polynomial potentials / Patrick Leonardo Capraro." München : Verlag Dr. Hut, 2018. http://d-nb.info/1155058496/34.
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Capraro, Patrick [Verfasser]. "Feynman path integrals in configuration space, momentum space and phase space for perturbative and polynomial potentials / Patrick Leonardo Capraro." München : Verlag Dr. Hut, 2018. http://d-nb.info/1155058496/34.
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Hoffmann, Jan. "Types with potential: polynomial resource bounds via automatic amortized analysis." Diss., lmu, 2011. http://nbn-resolving.de/urn:nbn:de:bvb:19-139552.
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Zeriahi, Ahmed. "Fonctions plurisousharmoniques extremales, approximation et croissance des fonctions holomorphes sur des ensembles algebriques." Toulouse 3, 1986. http://www.theses.fr/1986TOU30105.
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On etudie les proprietes extremales de la "mesure capacitaire d'equilibre" associee a un "condensateur" d'un espace de stein x de dimension pure. On introduit sur un espace de stein parabolique x la notion de fonction extremale associee a un compact kcx. On generalise la theorie des fonctions extremales de siciak-zaharyuta donnant une nouvelle approche de celle-ci basee sur la theorie du potentiel complexe pour l'operateur de monge ampere complexe. On en deduit des resultats sur la theorie des fonctions a croissance controlee a l'infini. On etudie le cas d'un ensemble algebrique de c**(n). Dans ce cas, on demontre des inegalites polynomiales, une version precise d'un theoreme d'approximation de type bernstein-walsh et on en deduit le comportement asymptotique de certaines suites de polynomes orthogonaux pour la mesure d'equilibre. Pour une variete de c**(n) intersection complete, on construit des operateurs lineaires integraux pour la meilleure approximation polynomiale sur un compact avec un controle precis de l'erreur et on en deduit des resultats sur la croissance des fonctions entieres sur de telles varietes
7
Alexandersson, Per. "On eigenvalues of the Schrödinger operator with a complex-valued polynomial potential." Licentiate thesis, Stockholms universitet, Matematiska institutionen, 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-52064.
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In this thesis, we generalize a recent result of A. Eremenko and A. Gabrielov on irreducibility of the spectral discriminant for the Schroedinger equation with quartic potentials. In the first paper, we consider the eigenvalue problem with a complex-valued polynomial potential of arbitrary degree d and show that the spectral determinant of this problem is connected and irreducible. In other words, every eigenvalue can be reached from any other by analytic continuation. We also prove connectedness of the parameter spaces of the potentials that admit eigenfunctions satisfying k > 2 boundary conditions, except for the case d is even and k = d/2. In the latter case, connected components of the parameter space are distinguished by the number of zeros of the eigenfunctions. In the second paper, we only consider even polynomial potentials, and show that the spectral determinant for the eigenvalue problem consists of two irreducible components. A similar result to that of paper I is proved for k boundary conditions.
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Hackl, Peter. "Optimal Design for Experiments with Potentially Failing Trials." Department of Statistics and Mathematics, WU Vienna University of Economics and Business, 1994. http://epub.wu.ac.at/68/1/document.pdf.
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We discuss the problem of optimal allocation of the design points of an experiment for the case where the trials may fail with non-zero probability. Numerical results for D-optimal designs are given for estimating the coefficients of a polynomial regression. For small sample sizes these designs may deviate substantially from the corresponding designs in the case of certain response. They can be less efficient, but are less affected by failing trials. (author's abstract)Series: Forschungsberichte / Institut für Statistik
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Hoffmann, Jan [Verfasser], and Martin [Akademischer Betreuer] Hofmann. "Types with potential : polynomial resource bounds via automatic amortized analysis / Jan Hoffmann. Betreuer: Martin Hofmann." München : Universitätsbibliothek der Ludwig-Maximilians-Universität, 2011. http://d-nb.info/1020143665/34.
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Haese-Hill, William. "Spectral properties of integrable Schrodinger operators with singular potentials." Thesis, Loughborough University, 2015. https://dspace.lboro.ac.uk/2134/19929.
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The integrable Schrödinger operators often have a singularity on the real line, which creates problems for their spectral analysis. In several particular cases we show that all closed gaps lie on the infinite spectral arc. In the second part we develop a theory of complex exceptional orthogonal polynomials corresponding to integrable rational and trigonometric Schrödinger operators, which may have a singularity on the real line. In particular, we study the properties of the corresponding complex exceptional Hermite polynomials related to Darboux transformations of the harmonic oscillator, and exceptional Laurent orthogonal polynomials related to trigonometric monodromy-free operators.
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Freund, Robert M. "A Potential Reduction Algorithm With User-Specified Phase I - Phase II Balance, for Solving a Linear Program from an Infeasible Warm Start." Massachusetts Institute of Technology, Operations Research Center, 1991. http://hdl.handle.net/1721.1/5409.
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This paper develops a potential reduction algorithm for solving a linear-programming problem directly from a "warm start" initial point that is neither feasible nor optimal. The algorithm is of an "interior point" variety that seeks to reduce a single potential function which simultaneously coerces feasibility improvement (Phase I) and objective value improvement (Phase II). The key feature of the algorithm is the ability to specify beforehand the desired balance between infeasibility and nonoptimality in the following sense. Given a prespecified balancing parameter /3 > 0, the algorithm maintains the following Phase I - Phase II "/3-balancing constraint" throughout: (cTx- Z*) < /3TX, where cTx is the objective function, z* is the (unknown) optimal objective value of the linear program, and Tx measures the infeasibility of the current iterate x. This balancing constraint can be used to either emphasize rapid attainment of feasibility (set large) at the possible expense of good objective function values or to emphasize rapid attainment of good objective values (set /3 small) at the possible expense of a lower infeasibility gap. The algorithm exhibits the following advantageous features: (i) the iterate solutions monotonically decrease the infeasibility measure, (ii) the iterate solutions satisy the /3-balancing constraint, (iii) the iterate solutions achieve constant improvement in both Phase I and Phase II in O(n) iterations, (iv) there is always a possibility of finite termination of the Phase I problem, and (v) the algorithm is amenable to acceleration via linesearch of the potential function.
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Parain, Dominique. "Analyse des potentiels évoqués somesthésiques à l'aide de la double transformation de Karhunen-Loeve." Rouen, 1990. http://www.theses.fr/1990ROUES045.
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Nous présentons un nouveau traitement des composantes de fréquences rapides des potentiels évoqués somesthésiques (P. E. S. ). Cette méthode comprend plusieurs parties : 1) un filtrage par ajustement polynomial ; 2) un traitement du signal (analyse en séries de Fourier) ; 3) une reconnaissance de forme (double transformation de Karhunen-Loeve). La nouvelle méthode de filtrage polynomial par segment sans changement de phase est décrite. L'intérêt du coefficient polynomial de courbure des principales composantes du P. E. S. Est étudié. La transformation de Karhunen-Loeve entraîne une importante réduction des données. La méthode d'analyse compare une série de trois P. E. S. à trois fonctions créneaux (un mobile et deux fixes). Ce traitement est un essai de quantification des latences interpics des P. E. S.
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Reinhold, Küstner. "Asymptotic zero distribution of orthogonal polynomials with respect to complex measures having argument of bounded variation." Nice, 2003. http://www.theses.fr/2003NICE4054.
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On détermine la distribution asymptotique des pôles pour trois types de meilleurs approximants (Padé à l’infini, rationnel en L2 sur le cercle unité, méromorphe dans le disque unité en Lp sur le cercle unité, p>2) de la transformée de Cauchy d’une mesure complexe sous l’hypothèse que le support S de la mesure soit de capacité positive et inclus dans (-1, 1), que la mesure satisfasse une condition de densité et que l’argument de la mesure soit la restriction d’une fonction à variation bornée. Les polynômes dénominateurs des approximants satisfont des relations d’orthogonalité. Au moyen d’un théorème de Kestelman, on obtient des contraintes géométriques pour les zéros qui impliquent que chaque mesure limite faible des mesures de comptage associées à son support inclus dans S. Puis, à l’aide de résultats de la théorie du potentiel dans le plan, on montre que les mesures de comptage convergent faiblement vers la distribution d’équilibre logarithmique respectivement hyperbolique de SWe determine the asymptotic pole distribution for three types of best approximants (Padé at infinity, rational in L2 on the unit circle, meromorphic in the unit disk in Lp on the unit circle, p>2) of the Cauchy transform of a complex measure under the hypothesis that the support S of the measure is of positive capacity and included in (-1 1), that the measure satisfies a density condition and that the argument of the measure is the restriction of a function of bounded variation ? The denominator polynomials of the approximants satisfay orthogonality relations ? By means of a theorem of Kestelman we obtain geometric constraints for the zeros which imply that every weak limit measure of the associated counting measures has support included in S. Then, with the help of results from potential theory in the plane, we show that the counting measures converge weakly to the logarithmic respectively hyperbolic equilibrium distribution of S
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Findley, Elliot M. "Christoffel Function Asymptotics and Universality for Szegő Weights in the Complex Plane." Scholar Commons, 2009. https://scholarcommons.usf.edu/etd/1965.
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In 1991, A. Máté precisely calculated the first-order asymptotic behavior of the sequence of Christoffel functions associated with Szego measures on the unit circle. Our principal goal is the abstraction of his result in two directions: We compute the translated asymptotics, limn λn(µ, x + a/n), and obtain, as a corollary, a universality limit for the fairly broad class of Szego weights. Finally, we prove Máté’s result for measures supported on smooth curves in the plane. Our proof of the latter derives, in part, from a precise estimate of certain weighted means of the Faber polynomials associated with the support of the measure. Finally, we investigate a variety of applications, including two novel applications to ill-posed problems in Hilbert space and the mean ergodic theorem.
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Cruz, Neto Francisco Alves da. "O oscilador de Klein-Gordon (2+1)-D sujeito a interações externas." Universidade Federal do Maranhão, 2016. http://tedebc.ufma.br:8080/jspui/handle/tede/1557.
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The dynamics of scalar particle spin-zero in a plane has drawn attention recently due to new phenomena such as quantum Hall effect and topological insulators for bosonic systems. We study the dynamics of a particle spin-zero scalar Klein-Gordon an oscillator coupled to a potential mixture of potential nature scalar and vector Cornell type in the (2 + 1) dimensions. Applying the method of separation of variables, the radial equation may be expressed as a Schr¨odinger equation with an effective candidate compound the three-dimensional harmonic oscillator potential Cornell another. Using an appropriate change of variable radial equation can be expressed in terms of the differential equation of second order called biconfluente of Heun. Following proper procedure, that is, correctly applying the boundary conditions, the radial equation solution can be expressed in terms of polynomials Heun. From the boundary conditions the quantization condition is also obtained and show that for this fundamental state problem is defined by the quantum number n = 0 under restrictions of the values of potential parameters. We also analyze the solutions to some particular cases already discussed in the literature. In this context, when we consider the scalar potential of the linear type and vector Coulomb type, the ground state is also defined by the number n = 0 as opposed to what was reported in the literature. We also observed that when we consider only the vector Coulomb interaction type, in this case the ground state is defined by quantum number n = 1, in agreement with other studies reported in the literature.
A dinâmica de partículas escalares de spin-zero num plano tem chamado a atenção recentemente devido a novos fenômenos como por exemplo o efeito Hall quântico e isolantes topológicos para sistemas bosônicos. Neste trabalho estudamos a dinâmica de uma partícula escalar de spin-zero num potencial oscilador de Klein-Gordon acoplado a uma mistura de potenciais de natureza escalar e vetorial do tipo Cornell em (2+1) dimensões. Aplicando o método de separação de variáveis, a equação radial pode ser expressa como uma equação de Schrördinger com um potencial efetivo composto do oscilador harmônico tridimensional mais um potencial Cornell. Usando uma apropriada mudança de variável a equação radial pode ser expressa em termos da equação diferencial de segunda ordem chamada biconfluente de Heun. Seguindo o procedimento adequado, é dizer, aplicando corretamente as condições de contorno, a solução da equação radial pode ser expressa em termos dos polinômios de Heun. A partir das condições de contorno a condição de quantização também é obtida e mostramos que para este problema o estado fundamental é definido pelo número quântico n=0 mediante restrições dos valores dos parâmetros do potencial. Também analisamos as soluções para alguns casos particulares já discutidos na literatura. Neste contexto, quando consideramos o potencial escalar do tipo linear e vetor do tipo Coulomb, o estado fundamental também é definido pelo número n=0 em oposição ao que foi divulgado na literatura. Observamos ainda que quando consideramos apenas a interação vetorial do tipo Coulomb, neste caso o estado fundamental é definido pelo número quântico n=1, em concordância com outros trabalhos divulgados na literatura.
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Singh, Pranav. "High accuracy computational methods for the semiclassical Schrödinger equation." Thesis, University of Cambridge, 2018. https://www.repository.cam.ac.uk/handle/1810/274913.
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The computation of Schrödinger equations in the semiclassical regime presents several enduring challenges due to the presence of the small semiclassical parameter. Standard approaches for solving these equations commence with spatial discretisation followed by exponentiation of the discretised Hamiltonian via exponential splittings. In this thesis we follow an alternative strategy${-}$we develop a new technique, called the symmetric Zassenhaus splitting procedure, which involves directly splitting the exponential of the undiscretised Hamiltonian. This technique allows us to design methods that are highly efficient in the semiclassical regime. Our analysis takes place in the Lie algebra generated by multiplicative operators and polynomials of the differential operator. This Lie algebra is completely characterised by Jordan polynomials in the differential operator, which constitute naturally symmetrised differential operators. Combined with the $\mathbb{Z}_2$-graded structure of this Lie algebra, the symmetry results in skew-Hermiticity of the exponents for Zassenhaus-style splittings, resulting in unitary evolution and numerical stability. The properties of commutator simplification and height reduction in these Lie algebras result in a highly effective form of $\textit{asymptotic splitting:} $exponential splittings where consecutive terms are scaled by increasing powers of the small semiclassical parameter. This leads to high accuracy methods whose costs grow quadratically with higher orders of accuracy. Time-dependent potentials are tackled by developing commutator-free Magnus expansions in our Lie algebra, which are subsequently split using the Zassenhaus algorithm. We present two approaches for developing arbitrarily high-order Magnus--Zassenhaus schemes${-}$one where the integrals are discretised using Gauss--Legendre quadrature at the outset and another where integrals are preserved throughout. These schemes feature high accuracy, allow large time steps, and the quadratic growth of their costs is found to be superior to traditional approaches such as Magnus--Lanczos methods and Yoshida splittings based on traditional Magnus expansions that feature nested commutators of matrices. An analysis of these operatorial splittings and expansions is carried out by characterising the highly oscillatory behaviour of the solution.
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Hryniewicki, Maciej Konrad. "Accurate and Efficient Evaluation of the Second Virial Coefficient Using Practical Intermolecular Potentials for Gases." Thesis, 2011. http://hdl.handle.net/1807/29559.
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The virial equation of state p = ρRT[
1 + B(T) ρ + C(T) ρ2 + · · ·] for high pressure and density gases is used for computing chemical equilibrium properties and mixture compositions of strong shock and detonation waves. The second and third temperature-dependent virial coefficients B(T) and C(T) are included in tabular form in computer codes, and they are evaluated by polynomial interpolation. A very accurate numerical integration method is presented for computing B(T) and its derivatives for tables, and a sophisticated method is introduced for interpolating B(T) more accurately and efficiently than previously possible. Tabulated B(T) values are non-uniformly distributed using an adaptive grid, to minimize the size and storage of the tables and to control the maximum relative error of interpolated values. The methods introduced for evaluating B(T) apply equally well to the intermolecular potentials of Lennard-Jones in 1924, Buckingham and Corner in 1947, and Rice and Hirschfelder in 1954.
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Moreira, Sara Barros. "Resource Analysis for Lazy Evaluation with Polynomial Potential." Master's thesis, 2020. https://hdl.handle.net/10216/131436.
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Moreira, Sara Barros. "Resource Analysis for Lazy Evaluation with Polynomial Potential." Dissertação, 2020. https://hdl.handle.net/10216/131436.
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"Polynomial-time algorithms for linear programming based only on primal scaling and projected gradients of a potential function." Sloan School of Management, Massachusetts Institute of Technology, 1988. http://hdl.handle.net/1721.1/2207.
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Bandyopadhyay, Choiti. "The Role Of Potential Theory In Complex Dynamics." Thesis, 2012. http://etd.iisc.ernet.in/handle/2005/2291.
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Potential theory is the name given to the broad field of analysis encompassing such topics as harmonic and subharmonic functions, the Dirichlet problem, Green’s functions, potentials and capacity. In this text, our main goal will be to gain a deeper understanding towards complex dynamics, the study of dynamical systems defined by the iteration of analytic functions, using the tools and techniques of potential theory. We will restrict ourselves to holomorphic polynomials in C.
At first, we will discuss briefly about harmonic and subharmonic functions. In course, potential theory will repay its debt to complex analysis in the form of some beautiful applications regarding the Julia sets (defined in Chapter 8) of a certain family of polynomials, or a single one.
We will be able to provide an explicit formula for computing the capacity of a Julia set, which in some sense, gives us a finer measurement of the set. In turn, this provides us with a sharp estimate for the diameter of the Julia set. Further if we pick any point w from the Julia set, then the inverse images q−n(w) span the whole Julia set. In fact, the point-mass measures with support at the discrete set consisting of roots of the polynomial, (qn-w) will eventually converge to the equilibrium measure of the Julia set, in the weak*-sense. This provides us with a very effective insight into the analytic structure of the set.
Hausdorff dimension is one of the most effective notions of fractal dimension in use. With the help of potential theory and some ergodic theory, we can show that for a certain holomorphic family of polynomials varying over a simply connected domain D, one can gain nice control over how the Hausdorff dimensions of the respective Julia sets change with the parameter λ in D.