MacSphere Collection:http://hdl.handle.net/11375/1522024-03-18T16:30:41Z2024-03-18T16:30:41ZThe search for self-contained numbers: k-special 3-smooth representations and the Collatz conjectureStokes, Alunhttp://hdl.handle.net/11375/275432022-05-10T17:56:07Z2021-01-01T00:00:00ZTitle: The search for self-contained numbers: k-special 3-smooth representations and the Collatz conjecture
Authors: Stokes, Alun
Abstract: The Collatz conjecture is a deceptively simply problem that straddles the line between
number theory and dynamical systems. It asks: if we iterate the function that sends
some even n to n+2 and odd n to 3n+1, will this converge to 1 for every natural number?
This problem has long stood unsolved despite attempts in many mathematical disciplines
– in large part due to the difficulty of predicting the multiplicative structure of a number
under addition. In this project, we provide a derivation of the most standard algebraic
reformulation of the non-trivial cycles subproblem. This results in an infinite family of
exponential Diophantine equations which correspond to k-special 3-smooth representations
of integers. By imposing conditions on the exponents in these representations, we
rewrite it in a multiplicative form that admits iterative solving for parameters of the
representation. Doing so while enforcing a maximum value on the largest power of 2
in the representation, we derive a sufficient condition for no non-trivial cycles existing
in this process. We show that a self-contained number, w, is exactly one which has an
odd element of its orbit modularly equivalent to −3^−1 mod w. We then show that non-cyclicity of any self-contained number greater than 5 is sufficient to show that no cycles
exist in the Collatz process. This differs from previous modularity-based results, and
experimental results suggest that self-contained numbers are relatively rare. We show
that exactly 7 such numbers exist less than 10^15 – improving on the previously known
bound of 10^11.2021-01-01T00:00:00ZCOVID-19: Analytics of contagion on inhomogeneous random social networksHurd TRhttp://hdl.handle.net/11375/262812021-04-05T16:15:40Z2021-01-01T00:00:00ZTitle: COVID-19: Analytics of contagion on inhomogeneous random social networks
Authors: Hurd TR
Abstract: Motivated by the need for robust models of the Covid-19 epidemic that adequately reflect the extreme heterogeneity of humans and society, this paper presents a novel framework that treats a population of N individuals as an inhomogeneous random social network (IRSN). The nodes of the network represent individuals of different types and the edges represent significant social relationships. An epidemic is pictured as a contagion process that develops day by day, triggered by a seed infection introduced into the population on day 0. Individuals’ social behaviour and health status are assumed to vary randomly within each type, with probability distributions that vary with their type. A formulation and analysis is given for a SEIR (susceptible-exposed-infective-removed) network contagion model, considered as an agent based model, which focusses on the number of people of each type in each compartment each day. The main result is an analytical formula valid in the large N limit for the stochastic state of the system on day t in terms of the initial conditions. The formula involves only one-dimensional integration. The model can be implemented numerically for any number of types by a deterministic algorithm that efficiently incorporates the discrete Fourier transform. While the paper focusses on fundamental properties rather than far ranging applications, a concluding discussion addresses a number of domains, notably public awareness, infectious disease research and public health policy, where the IRSN framework may provide unique insights.2021-01-01T00:00:00ZA Multiresolution Model for the Simulation of Transient Heat and Mass TransferAlam, Jahrul M.Kevlahan, Nicholas K.-R.Vasilyev, Oleg V.Hossain, Zahangirhttp://hdl.handle.net/11375/168472015-03-19T20:37:12Z2012-05-11T00:00:00ZTitle: A Multiresolution Model for the Simulation of Transient Heat and Mass Transfer
Authors: Alam, Jahrul M.; Kevlahan, Nicholas K.-R.; Vasilyev, Oleg V.; Hossain, Zahangir
Abstract: The development of an efficient computational methodology for transient heat and mass transfer applications is challenging. When a solution is localized on the fraction of a com- putational domain, an appropriate adaptive mesh method could minimize computational work. In this article, we propose a novel adaptive-mesh multiresolution algorithm for the transient momentum and energy equations. The nonlinear dynamics between the velocity and temperature fields are modeled by solving the coupled system of equations simul- taneously, where the rate of convergence has been optimized so that computational cost remains proportional to the number of grid points. Numerical experiments have exhibited good agreements with benchmark simulation data.2012-05-11T00:00:00ZNon-Gaussianity and coherent vortex simulation for two-dimensional turbulence using an adaptive orthogonal wavelet basisFarge, MarieSchneider, KaiKevlahan, Nicholashttp://hdl.handle.net/11375/168452015-03-18T21:12:58Z1999-01-01T00:00:00ZTitle: Non-Gaussianity and coherent vortex simulation for two-dimensional turbulence using an adaptive orthogonal wavelet basis
Authors: Farge, Marie; Schneider, Kai; Kevlahan, Nicholas
Abstract: We decompose turbulent flows into two orthogonal parts: a coherent, inhomogeneous, non-Gaussian component and an incoherent, homogeneous, Gaussian component. The two components have different probability distributions and different correlations, hence different scaling laws. This separation into coherent vortices and incoherent background flow is done for each flow realization before averaging the results and calculating the next time step. To perform this decomposition we have developed a nonlinear scheme based on an objective threshold defined in terms of the wavelet coefficients of the vorticity. Results illustrate the efficiency of this coherent vortex extraction algorithm. As an example we show that in a 2562 computation 0.7% of the modes correspond to the coherent vortices responsible for 99.2% of the energy and 94% of the enstrophy. We also present a detailed analysis of the nonlinear term, split into coherent and incoherent components, and compare it with the classical separation, e.g., used for large eddy simulation, into large scale and small scale components. We then propose a new method, called coherent vortex simulation CVS, designed to compute and model two-dimensional turbulent flows using the previous wavelet decomposition at each time step. This method combines both deterministic and statistical approaches: i Since the coherent vortices are out of statistical equilibrium, they are computed deterministically in a wavelet basis which is remapped at each time step in order to follow their nonlinear motions. ii Since the incoherent background flow is homogeneous and in statistical equilibrium, the classical theory of homogeneous turbulence is valid there and we model statistically the effect of the incoherent background on the coherent vortices. To illustrate the CVS method we apply it to compute a two-dimensional turbulent mixing layer.1999-01-01T00:00:00Z