A Unifying Local Convergence Result for Newton's Method in Riemannian Manifolds

We consider the problem of finding a singularity of a vector field $X$ on a complete Riemannian manifold. In this regard we prove a unified result for local convergence of Newton's method. Inspired by previous work of Zabrejko and Nguen on Kantorovich's majorant method, our approach relies on the introduction of an abstract one-dimensional Newton's method obtained using an adequate Lipschitz-type radial function of the covariant derivative of $X$. The main theorem gives in particular a synthetic view of several famous results, namely the Kantorovich, Smale and Nesterov-Nemirovskii theorems. Concerning real-analytic vector fields an application of the central result leads to improvements of some recent developments in this area

A short survey on Kantorovich-like theorems for Newton's method

We survey influential quantitative results on the convergence of the Newton iterator towards simple roots of continuously differentiable maps defined over Banach spaces. We present a general statement of Kantorovich's theorem, with a concise proof from scratch, dedicated to wide audience. From it, we quickly recover known results, and gather historical notes together with pointers to recent articles

Differential geometric MCMC methods and applications

This thesis presents novel Markov chain Monte Carlo methodology that exploits the natural representation of a statistical model as a Riemannian manifold. The methods developed provide generalisations of the Metropolis-adjusted Langevin algorithm and the Hybrid Monte Carlo algorithm for Bayesian statistical inference, and resolve many shortcomings of existing Monte Carlo algorithms when sampling from target densities that may be high dimensional and exhibit strong correlation structure. The performance of these Riemannian manifold Markov chain Monte Carlo algorithms is rigorously assessed by performing Bayesian inference on logistic regression models, log-Gaussian Cox point process models, stochastic volatility models, and both parameter and model level inference of dynamical systems described by nonlinear differential equations

Higher dimensional theories in physics, following the Kaluza model of unification

This thesis traces the origins and evolution of higher dimensional models in physics, with particular reference to the five-dimensional Kaluza-Klein unification. It includes the motivation needed, and the increasing status and significance of the multidimensional description of reality for the 1990's. The differing conceptualisations are analysed, from the mathematical, via Kasner's embedding dimensions and Schrodinger's waves, to the high status of Kaluza-Klein dimensions in physics today. This includes the use of models, and the metaphysical interpretations needed to translate the mathematics. The main area of original research is the unpublished manuscripts and letters of Theodor Kaiuza, some Einstein letters, further memoirs from his son Theodor Kaiuza Junior and from some of his original students. Unpublished material from Helsinki concerns the Finnish physicist Nordstrom, the real originator of the idea that 'forces' in 4-dimensional spacetime might arise from gravity in higher dimensions. The work of the Swedish physicist Oskar Klein and the reactions of de Broglie and Einstein initiated the Kaluza-Klein connection which is traced through fifty years of neglect to its re-entry into mainstream physics. The cosmological significance and conceptualisation through analogue models is charted by personal correspondence with key scientists across a range of theoretical physics, involving the use of aesthetic criteria where there is no direct physical verification. Qualitative models implicitly indicating multidimensions are identified in the paradoxes and enigmas of existing physics, in Quantum Mechanics and the singularities in General Relativity. The Kaluza-Klein philosophy brings this wide range of models together in the late 1980's via supergravity, superstrings and supermanifolds. This new multidimensional paradigm wave is seen to produce a coherent and consistent metaphysics, a new perspective on reality. It may also have immense potential significance for philosophy and theology. The thesis concludes with the reality question, "Are we a four-dimensional projection of a deeper reality of many, even infinite, dimensions?

Epistemology of Experimental Gravity - Scientific Rationality

The evolution of gravitational tests from an epistemological perspective framed in the concept of rational reconstruction of Imre Lakatos, based on his methodology of research programmes. Unlike other works on the same subject, the evaluated period is very extensive, starting with Newton's natural philosophy and up to the quantum gravity theories of today. In order to explain in a more rational way the complex evolution of the gravity concept of the last century, I propose a natural extension of the methodology of the research programmes of Lakatos that I then use during the paper. I believe that this approach offers a new perspective on how evolved over time the concept of gravity and the methods of testing each theory of gravity, through observations and experiments. I argue, based on the methodology of the research programmes and the studies of scientists and philosophers, that the current theories of quantum gravity are degenerative, due to the lack of experimental evidence over a long period of time and of self-immunization against the possibility of falsification. Moreover, a methodological current is being developed that assigns a secondary, unimportant role to verification through observations and/or experiments. For this reason, it will not be possible to have a complete theory of quantum gravity in its current form, which to include to the limit the general relativity, since physical theories have always been adjusted, during their evolution, based on observational or experimental tests, and verified by the predictions made. Also, contrary to a widespread opinion and current active programs regarding the unification of all the fundamental forces of physics in a single final theory, based on string theory, I argue that this unification is generally unlikely, and it is not possible anyway for a unification to be developed based on current theories of quantum gravity, including string theory. In addition, I support the views of some scientists and philosophers that currently too much resources are being consumed on the idea of developing quantum gravity theories, and in particular string theory, to include general relativity and to unify gravity with other forces, as long as science does not impose such research programs. DOI: 10.13140/RG.2.2.35350.7072

The reasonable effectiveness of Mathematics and its Cognitive roots 1

“At the beginning, Nature set up matters its own way and, later, it constructed human intelligence in such a way that [this intelligence] could understand it” [Galileo Galilei, 1632 (Opere, p. 298)]. “The applicability of our science [mathematics] seems then as a symptom of its rooting, not as a measure of its value. Mathematics, as a tree which freely develops his top, draws its strength by the thousands roots in a ground of intuitions of real representations; it would be disastrous to cut them off, in view of a short-sided utilitarism, or to uproot them from the ground from which they rose ” [H. Weyl, 1910]. Summary. Mathematics stems out from our ways of making the world intelligible by its peculiar conceptual stability and unity; we invented it and used it to single out key regularities of space and language. This is exemplified and summarised below in references to the main foundational approaches to Mathematics, as proposed in the last 150 years. Its unity is also stressed: in this paper, Mathematics is viewed as a "three dimensiona

Mathematical foundations of elasticity

[Preface] This book treats parts of the mathematical foundations of three-dimensional elasticity using modern differential geometry and functional analysis. It is intended for mathematicians, engineers, and physicists who wish to see this classical subject in a modern setting and to see some examples of what newer mathematical tools have to contribute

Network analysis of complex biological systems : boundedness of weakly reversible chemical reaction networks and conditions for synchronisation of coupled oscillators

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