Newton’s method on Bring-Jerrard polynomials

The first and fourth authors were partially supported by P11B2011-30 (Universitat Jaume I) and by the Spanish grant MTM2011-28636-C02-02. The second and third authors were partially supported by the Catalan grant 2009SGR-792, and by the Spanish grant MTM2011-26995-C02-02. The third author also wants to thank the support of the Polish NCN grant decision DEC-2012/06/M/ST1/00168

Newton's method on bring-Jerrard polynomials

In this paper we study the topology of the hyperbolic component of the parameter plane for the Newton's method applied to n-degree BringJerrard polynomials given by $P_{n}(z)=z^{n}-cz +1, c \in \mathbb{C}$. For $n=5$ using the TschirnhausBringJerrard nonlinear transformations, this family controls, at least theoretically, the roots of all quintic polynomials. We also study a bifurcation cascade of the bifurcation locus by considering $c\in\mathbb{R}

The Bring-Jerrard quintic equation, its solutions and a formula for the universal gravitational constant

In this research the Bring-Jerrard quintic polynomial equation is investigated for a formula. Firstly, an explanation given as to why finding a formula and the equation being unsolvable by radicals may appear contradictory when read out of context. Secondly, the reason why some mathematical software programs may fail to render a conclusive test of the formula, and how that can be corrected is explained. As an application, this formula is used to determine another formula that expresses the gravitational constant in terms of other known physical constants. It is also explained why up to now it has been impossible to determine this expression using the current underlying theoretical basis.M. Sc. (Applied Mathematics

Elementos algebraicos y ecuaciones polinomiales de grado pequeño

El objetivo principal de esta memoria es tratar de profundizar en algunos conceptos de la Teor´ıa de Galois. En primer lugar, comenzaremos probando la existencia de n´umeros trascendentes de forma no constructiva utilizando la equipotencia de conjuntos y de forma constructiva, demostrando expl´ıcitamente que el n´umero de Euler es trascendente. Luego usaremos la resultante de dos polinomios para demostrar de forma constructiva que la suma y el producto de elementos algebraicos es tambi´en algebraico. Tambi´en presentamos un m´etodo alternativo de resoluci´on de las ecuaciones de tercer y cuarto grado gracias a las Transformaciones de Tschirnhaus. Finalmente, clasificaremos los grupos de Galois de c´ubicas y cu´articas y caracterizaremos cu´ando un polinomios de grado cinco es resoluble

Stability and Entanglement in an Optomechanical System

Optomechanical systems are currently of great interest as they lie at the boundary between quantum and classical mechanics, promising fundamental insights as well as new technologies. The practical operation of an optomechanical system requires that it satisfy the criteria of mechanical stability. Further, for quantum applications, it is important to characterize the degree of nonclassical correlation present between the mechanical and optical subsystems. In this study, we analyze the stability and entanglement in an optomechanical system where couplings linear as well as quadratic in the mechanical displacement are present simultaneously. Such systems can be realized experimentally. Our analysis of the optomechanical system is accomplished by inspecting the equations of motion that characterize the system. By analyzing the steady state of the system, we find a stability diagram which differs dramatically from the case of pure linear coupling which has been studied earlier. Specifically, we find generally a major loss of stability and a disconnection of the stability diagram when a quadratic coupling is introduced. We derive and state analytically in this thesis the stability criteria for our more generalized system. Further, by linearizing the equations of motion, we characterize the entanglement present in the system, using the logarithmic negativity as a measure. We thereby characterize the changes in the system entanglement that result from the addition of a quadratic coupling to a linearly coupled system

Nestandardni pristupi nizovima Fibonaccijevog tipa

Fibonacci sequence and the limit of the quotient of adjacent Fibonacci numbers, namely the Golden Mean, belong to general knowledge of almost anybody, not only of mathematicians and geometers. There were several attempts to generalize these fundamental concepts which also found applications in art and architecture, as e.g. number series and quadratic equations leading to the so-called ˝Metallic means" by V. DE SPINADEL [8] or the cubic ˝plastic number" by VAN DER LAAN [5] resp. the ˝cubi ratio" by L. ROSENBUSCH [7]. The mentioned generalisations consider series of integers or real numbers. ˝Non-standard aspects" now mean generalisations with respect to a given number field or ring as well as visualisations of the resulting geometric objects. Another aspect concerns Fibonacci type resp. Padovan type combinations of given start objects. Here it turns out that the concept ˝Golden Mean" or ˝van der Laan Mean" also makes sense for vectors, matrices and mappings.Fibonaccijev niz i zlatni rez, limes kvocijenata susjednih Fibonaccijevih brojeva, su pojmovi poznati ne samo matematičarima i geometričarima, već gotovo svima. Oni svoju primjenu nalaze u umjetnosti i arhitekturi. Poznato je nekoliko pokušaja poopćenja ovih pojmova, kao što su nizovi brojeva i kvadratne jednadzbe koje rezultiraju takozvanim ˝metalnim rezovima" V. DE SPINADEL [8], ili kubni ˝plasticni broj" VAN DER LAANA [5], odnosno ˝kubni omjer" L. ROSENBUSCHA [7]. Spomenuta se poopćenja odnose na nizove cijelih ili realnih brojeva. ˝Nestandardnim pristupima" ovdje smatramo poopćenja u odnosu na dano polje ili prsten brojeva, kao i na vizualizaciju dobivenih geometrijskih objekata. Idući se pristup odnosi na Fibonaccijev, odnosno Padovanov tip kombinacija danih početnih objekata. Pokazuje se da pojam zlatnog reza ili van der Laanovog reza ima smisla promatrati i za vektore, matrice i preslikavanja

Resolvent degree, Hilbert's 13th Problem and geometry

We develop the theory of resolvent degree, introduced by Brauer \cite{Br} in order to study the complexity of formulas for roots of polynomials and to give a precise formulation of Hilbert's 13th Problem. We extend the context of this theory to enumerative problems in algebraic geometry, and consider it as an intrinsic invariant of a finite group. As one application of this point of view, we prove that Hilbert's 13th Problem, and his Sextic and Octic Conjectures, are equivalent to various enumerative geometry problems, for example problems of finding lines on a smooth cubic surface or bitangents on a smooth planar quartic.Comment: 65 pages, 2 figures. Minor revisions and correction