Design and Complex Dynamics of Potra–Pták-Type Optimal Methods for Solving Nonlinear Equations and Its Applications

In this paper, using the idea of weight functions on the Potra–Pták method, an optimal fourth order method, a non optimal sixth order method, and a family of optimal eighth order methods are proposed. These methods are tested on some numerical examples, and the results are compared with some known methods of the corresponding order. It is proved that the results obtained from the proposed methods are compatible with other methods. The proposed methods are tested on some problems related to engineering and science. Furthermore, applying these methods on quadratic and cubic polynomials, their stability is analyzed by means of their basins of attraction

A sixth-order family of three-point modified Newton-like multiple-root finders and the dynamics behind their extraneous fixed points

The article of record as published may be found at http://dx.doi.org/10.1016/j.amc.2016.02.029A class of three-point sixth-order multiple-root finders and the dynamics behind their extraneous fixed points are investigated by extending modified Newton-like methods with the introduction of the multivariate weight functions in the intermediate steps. The multivariate weight functions dependent on function-to-function ratios play a key role in constructing higher-order iterative methods. Extensive investigation of extraneous fixed points of the proposed iterative methods is carried out for the study of the dynamics associated with corresponding basins of attraction. Numerical experiments applied to a number of test equations strongly support the underlying theory pursued in this paper. Relevant dynamics of the proposed methods is well presented with a variety of illustrative basins of attraction applied to various test polynomials.Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education under the research grant (Project Number: 2015-R1D1A3A-01020808)Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education under the research grant (Project Number: 2015-R1D1A3A-01020808

A class of two-point sixth-order multiple-zero finders of modified double-Newton type and their dynamics

The article of record as published may be found at http://dx.doi.org/10.1016/j.amc.2015.08.039Under the assumption of the known multiplicity of zeros of nonlinear equations, a class of two-point sextic-order multiple-zero finders and their dynamics are investigated in this paper by means of extensive analysis of modified double-Newton type of methods. Wit the introduction of a bivariate weight function dependent on function-to-function and derivative-to-derivative ratios, higher-order convergence is obtained. Additional investigation is carried out for extraneous fixed points of the iterative maps associated with the proposed methods along with a comparison with typically selected cases. Through a variety of test equations, numerical experiments strongly support the theory developed in this paper. In addition, relevant dynamics of the proposed methods is successfully explored for various polynomials with a number of illustrative basins of attraction.National Research Foundation of KoreaMinistry of Education, Science and Technology under the research grant (Project Number: 2015-R1D1A3A-0102080

On the design and analysis of high order Weerakoon-Fernando methods based on weight functions

In this article, using the idea of weight functions on Weerakoon‐Fernando's method, an optimal fourth‐order method and some higher order multipoint methods for solving nonlinear equations are proposed. These methods are tested in some real applications and numerical examples and the results are compared with some existing methods. Their dynamical behavior on complex polynomials is analyzed and basins of attraction of these methods are presented

Cusp Universality for Random Matrices I: Local Law and the Complex Hermitian Case

For complex Wigner-type matrices, i.e. Hermitian random matrices with independent, not necessarily identically distributed entries above the diagonal, we show that at any cusp singularity of the limiting eigenvalue distribution the local eigenvalue statistics are universal and form a Pearcey process. Since the density of states typically exhibits only square root or cubic root cusp singularities, our work complements previous results on the bulk and edge universality and it thus completes the resolution of the Wigner-Dyson-Mehta universality conjecture for the last remaining universality type in the complex Hermitian class. Our analysis holds not only for exact cusps, but approximate cusps as well, where an extended Pearcey process emerges. As a main technical ingredient we prove an optimal local law at the cusp for both symmetry classes. This result is also used in the companion paper [arXiv:1811.04055] where the cusp universality for real symmetric Wigner-type matrices is proven.Comment: 58 pages, 2 figures. Updated introduction and reference

Cusp universality for random matrices I: Local law and the complex Hermitian case

For complex Wigner-type matrices, i.e. Hermitian random matrices with independent, not necessarily identically distributed entries above the diagonal, we show that at any cusp singularity of the limiting eigenvalue distribution the local eigenvalue statistics are universal and form a Pearcey process. Since the density of states typically exhibits only square root or cubic root cusp singularities, our work complements previous results on the bulk and edge universality and it thus completes the resolution of the Wigner–Dyson–Mehta universality conjecture for the last remaining universality type in the complex Hermitian class. Our analysis holds not only for exact cusps, but approximate cusps as well, where an extended Pearcey process emerges. As a main technical ingredient we prove an optimal local law at the cusp for both symmetry classes. This result is also the key input in the companion paper (Cipolloni et al. in Pure Appl Anal, 2018. arXiv:1811.04055) where the cusp universality for real symmetric Wigner-type matrices is proven. The novel cusp fluctuation mechanism is also essential for the recent results on the spectral radius of non-Hermitian random matrices (Alt et al. in Spectral radius of random matrices with independent entries, 2019. arXiv:1907.13631), and the non-Hermitian edge universality (Cipolloni et al. in Edge universality for non-Hermitian random matrices, 2019. arXiv:1908.00969)

Renormalization in complex dynamics

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2018, Director: Núria Fagella Rabionet[en] Renormalization theory is a powerful technique both in mathematics and physics. In particular, it is essential in the study of the MLC conjecture, which wonders whether or not the Mandelbrot set is locally connected. The main purpose of this project is to comprehend the Straightening Theorem with the utmost rigour, that is an essential result behind the aforementioned technique. In order to approach our aim, we proceed to concentrate on quasiconformal geometry and give some basic background concerning dynamical systems. Besides, we explore some applications of the cornerstone of this thesis within the framework of complex dynamics. Lastly, we outline the notion of renormalization when dealing with percolation theory in physics

Entropie‐dominierte Selbstorganisationsprozesse birnenförmiger Teilchensysteme

The ambition to recreate highly complex and functional nanostructures found in living organisms marks one of the pillars of today‘s research in bio- and soft matter physics. Here, self-assembly has evolved into a prominent strategy in nanostructure formation and has proven to be a useful tool for many complex structures. However, it is still a challenge to design and realise particle properties such that they self-organise into a desired target configuration. One of the key design parameters is the shape of the constituent particles. This thesis focuses in particular on the shape sensitivity of liquid crystal phases by addressing the entropically driven colloidal self-assembly of tapered ellipsoids, reminiscent of „pear-shaped“ particles. Therefore, we analyse the formation of the gyroid and of the accompanying bilayer architecture, reported earlier in the so-called pear hard Gaussian overlap (PHGO) approximation, by applying various geometrical tools like Set-Voronoi tessellation and clustering algorithms. Using computational simulations, we also indicate a method to stabilise other bicontinuous structures like the diamond phase. Moreover, we investigate both computationally and theoretically(density functional theory) the influence of minor variations in shape on different pearshaped particle systems, including the stability of the PHGO gyroid phase. We show that the formation of the gyroid is due to small non-additive properties of the PHGO potential. This phase does not form in pears with a „true“ hard pear-shaped potential. Overall our results allow for a better general understanding of necessity and sufficiency of particle shape in regards to colloidal self-assembly processes. Furthermore, the pear-shaped particle system sheds light on a unique collective mechanism to generate bicontinuous phases. It suggests a new alternative pathway which might help us to solve still unknown characteristics and properties of naturally occurring gyroid-like nano- and microstructures.Ein wichtiger Bestandteil der heutigen Forschung in Bio- und Soft Matter Physik besteht daraus, Technologien zu entwickeln, um hoch komplexe und funktionelle Strukturen, die uns aus der Natur bekannt sind, nachzubilden. Hinsichtlich dessen ist vor allem die Methode der Selbstorganisation von Mikro- und Nanoteilchen hervorzuheben, durch die eine Vielzahl verschiedener Strukturen erzeugt werden konnten. Jedoch stehen wir bei diesem Verfahren noch immer vor der Herausforderung, Teilchen mit bestimmten Eigenschaften zu entwerfen, welche die spontane Anordnung der Teilchen in eine gewünschte Struktur bewirken. Einer der wichtigsten Designparameter ist dabei die Form der Bausteinteilchen. In dieser Dissertation konzentrieren wir uns besonders auf die Anfälligkeit von Flüssigkristallphasen bezüglich kleiner Änderungen der Teilchenform und nutzen dabei das Beispiel der Selbstorganisation von Entropie-dominierter Kolloide, die dem Umriss nach verjüngten Ellipsoiden oder "Birnen" ähneln. Mit Hilfe von geometrischen Werkzeugen wie z.B. Set-Voronoi Tessellation oder Cluster-Algorithmen analysieren wir insbesondere die Entstehung der Gyroidphase und der dazugehörigen Bilagenformation, welche bereits in Systemen von harten Birnen, die durch das pear hard Gaussian overlap (PHGO) Potential angenähert werden, entdeckt wurden. Des Weiteren zeigen wir durch Computersimulationen eine Strategie auf, um andere bikontinuierliche Strukturen, wie die Diamentenphase, zu stabilisieren. Schlussendlich betrachten wir sowohl rechnerisch (durch Simulationen) als auch theoretisch (durch Dichtefunktionaltheorie) die Auswirkungen kleiner Abweichungen der Teilchenform auf das Verhalten des kolloiden, birnenförmigen Teilchensystems, inklusive der Stabilität der PHGO Gyroidphase. Wir zeigen, dass die Entstehung des Gyroids auf kleinen nicht-additiven Eigenschaften des PHGO Birnenmodells beruhen. In ''echten'' harten Teilchensystemen entwickelt sich diese Struktur nicht. Insgesamt ermöglichen unsere Ergebnisse einen besseren Einblick auf das Konzept von notwendiger und hinreichender Teilchenform in Selbstorganistationsprozessen. Die birnenförmigen Teilchensysteme geben außerdem Aufschluss über einen ungewöhnlichen, kollektiven Mechanismus, um bikontinuierliche Phasen zu erzeugen. Dies deutet auf einen neuen, alternativen Konstruktionsweg hin, der uns möglicherweise hilft, noch unbekannte Eigenschaften natürlich vorkommender, gyroidähnlicher Nano- und Mikrostrukturen zu erklären

Isogeometric Design, Analysis and Optimisation of Lattice-Skin Structures

The advancements in additive manufacturing techniques enable novel designs using lattice structures in mechanical parts, lightweight materials, biomaterials and so forth. Lattice-skin structures are a class of structures that couple thin-shells with lattices, which potentially combine the advantages of the thin-shell and the lattice structure. A new and systematic isogeometric analysis approach that integrates the geometric design, structural analysis and optimisation of lattice-skin structures is proposed in the dissertation. In the geometric design of lattice-skin structures, a novel shape interrogation scheme for splines, specifically subdivision surfaces, is proposed, which is able to compute the line/surface intersection efficiently and robustly without resorting to successive refinements or iterations as in Newton-Raphson method. The line/surface intersection algorithm involves two steps: intersection detection and intersection computation. In the intersection detection process, a bounding volume tree of k-dops (discrete oriented polytopes) for the subdivision surface is first created in order to accelerate the intersection detection between the line and the surface. The spline patches which are detected to be possibly intersected by the line are converted to Bézier representations. For the intersection computation, a matrix-based algorithm is applied, which converts the nonlinear intersection computation into solving a sequence of linear algebra problems using the singular value decomposition (SVD). Finally, the lattice-skin geometry is generated by projecting selected lattice nodes to the nearest intersection points intersected by the lattice edges. The Stanford bunny example demonstrates the efficiency and accuracy of the developed algorithm. The structural analysis of lattice-skin structures follows the isogeometric approach, in which the thin-shell is discretised with spline basis functions and the lattice structure is modelled with pin-jointed truss elements. In order to consider the lattice-skin coupling, a Lagrange multiplier approach is implemented to enforce the displacement compatibility between the coupled lattice nodes and the thin-shell. More importantly, the parametric coordinates of the coupled lattice nodes on the thin-shell surface are obtained directly from the lattice-skin geometry generation, which integrates the design and analysis process of lattice-skin structures. A sandwich plate example is analysed to verify the implementation and the accuracy of the lattice-skin coupling computation. In addition, a SIMP-like lattice topology optimisation method is proposed. The topology optimisation results of lattice structures are analysed and compared with several examples adapted from the benchmark examples commonly used in continuum topology optimisation. The SIMP-like lattice topology optimisation proposed is further applied to optimise the lattice in lattice-skin structures. The lattice-skin topology optimisation is fully integrated with the lattice-skin geometry design since the sensitivity analysis in the proposed method is based on lattice unit cells which are inherited from the geometry design stage. Finally, shape optimisation of lattice-skin structures using the free-form deformation (FFD) technique is studied. The corresponding shape sensitivity of lattice-skin structures is derived. The geometry update of the lattice-skin structure is determined by the deformation of the FFD control volume, and in this process the coupling between lattice nodes and the thin-shell is guaranteed by keeping the parametric coordinates of coupled lattice nodes which are obtained in the lattice-skin geometry design stage. A pentagon roof example is used to explore the combination of lattice topology optimisation and shape optimisation of lattice-skin structures

Symmetry in Applied Mathematics

Applied mathematics and symmetry work together as a powerful tool for problem reduction and solving. We are communicating applications in probability theory and statistics (A Test Detecting the Outliers for Continuous Distributions Based on the Cumulative Distribution Function of the Data Being Tested, The Asymmetric Alpha-Power Skew-t Distribution), fractals - geometry and alike (Khovanov Homology of Three-Strand Braid Links, Volume Preserving Maps Between p-Balls, Generation of Julia and Mandelbrot Sets via Fixed Points), supersymmetry - physics, nanostructures -chemistry, taxonomy - biology and alike (A Continuous Coordinate System for the Plane by Triangular Symmetry, One-Dimensional Optimal System for 2D Rotating Ideal Gas, Minimal Energy Configurations of Finite Molecular Arrays, Noether-Like Operators and First Integrals for Generalized Systems of Lane-Emden Equations), algorithms, programs and software analysis (Algorithm for Neutrosophic Soft Sets in Stochastic Multi-Criteria Group Decision Making Based on Prospect Theory, On a Reduced Cost Higher Order Traub-Steffensen-Like Method for Nonlinear Systems, On a Class of Optimal Fourth Order Multiple Root Solvers without Using Derivatives) to specific subjects (Facility Location Problem Approach for Distributed Drones, Parametric Jensen-Shannon Statistical Complexity and Its Applications on Full-Scale Compartment Fire Data). Diverse topics are thus combined to map out the mathematical core of practical problems