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Introduction to Formal Mathematics: Home

Introduction

Welcome to the Pre-Calculus Subject Guide. This guide will help you research topics related to Pre-Calculus. Start with the tabs above to find books, articles, websites, and other relevant resources. If you have any questions or need help researching a topic, drop me a line, and I'll try to help you out.

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A World Without Quantum Computers - John Conway Spirited Seminar Series

The lecture will briefly discuss my theory explaining why quantum computers are not possible. I will address the question if Google's Sycamore and other recent experiments falsify my theory, and discuss consequences of my theory to classical philosophical questions of predictability, free will, and reductionism. On the mathematical side, the connection between noise and Fourier expansion will play a role.

Spectral Statistics Of Quantum Graphs

The spectral geometry concerns the relation between the geometric structure of a mathematical object and the spectra (eigenvalues) of differential operators acting on that object. The idea that interactions of systems of entities can be represented in the form of a network has been long-established and researched. One of the most conventional practices in graph theory is to deduce the characteristics and structure from graph’s spectrum. In discrete networks, the spectrum plays a central role in understanding numerous physical and social phenomena. Although these discrete network models can successfully describe complex systems, the geometry of the connections between the nodes is neglected. This results in one fundamental limitation: each edge in the network is identified as a pair of vertices it is connecting, and the length of each edge plays no role. It is, therefore, more realistic to use metric graphs with edges having meaningful and precise lengths. The corresponding, continuous as opposed to discrete, dynamics is described by differential operators. A metric graph together with a differential operator is generally called a quantum graph. In this talk I will discuss the spectral dependence of quantum graphs on its structural parameters. I will also discuss some basic techniques based on variational principle for estimating the discrete eigenvalues.

Numerical Solutions of Some Nonlinear Fractional Flow Problems

Physical models of viscoelastic flow in terms of fractional order derivatives is a fascinating subject, in various fields of engineering, fluid dynamics, quantum computing, etc. The viscoelastic fluid models represent more realistic behavior as compared to the integer order derivatives in fluid dynamics. This is mainly because of the hereditary effects are taken into account by the fractional order models while formulating the flow problems. Mathematical models for such phenomenon usually result in the form of fractional partial differential equations (PDEs). In order to solve the fractional order PDEs, there are several analytical methods such as Laplace transform and Fourier transform methods in literature which had been usually employed to solve linear fractional order PDEs. However, it is difficult to solve non-linear fractional order PDEs using these analytical techniques. Hence, the researchers use numerical techniques to find approximate solutions for such fractional PDEs.

 

 

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