Mathematics

Momentum Trading is a common trading strategy extensively applied in technical and quantitative analysis. One of the most basic indicator is moving average crossover. The general implementation is, provided a short-term moving average (short for STMA) and long-term moving average(short for LTMA), once the STMA is higher than LTMA, shown as the STMA curve cross LTMA curve from below, we take a long position in the securities generating such MAs. While the STMA is lower than LTMA, shown as the STMA curve cross LTMA curve from above, we take a short position in the securities of such MAs. We hereinafter will call the former signal as crossover, or gold cross as the term in our nation, and the latter signal as crossdown, or death cross in our nation. In the following content, we will analyze the practicability and efficiency of momentum trading by constructing 4 portfolio and tracking their return streams.

This thesis is a detailed examination on Schopenhauer’s metaphysics of will. I first present the inconsistencies in Schopenhauer’s philosophy and point to the problems related to his metaphysical claims. I then demonstrate an alternative reading of Schopenhauer’s philosophy, focusing on its two characteristics: its contraction of reason and its phenomenological emphasis. I concede that Schopenhauer’s philosophy is burdened with the tension between his adaptation of Kant’s epistemology and his ambitious metaphysical goal to break away from the Kantian tradition. However, these inconsistencies and problems do not render Schopenhauer’s philosophy valueless. In contrary, they open up possibilities for Schopenhauer to address another arena of philosophy that Kant does not pay attention to, that is, human embodiment and human agency. Understanding Schopenhauer’s theories on this topic sheds great light upon his metaphysical project as a whole.

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The open book decompositions of the 3-sphere S3 whose pages are pairs of pants have been fully understood for some time, through the lens of contact geometry. The purpose of this note is to exhibit a purely topological derivation of the classification of such open books, in terms of the links that form their bindings and the corresponding monodromies. We construct all of the links and their pair-of-pants fiber surfaces from the simplest example, a connected sum of two Hopf links, through performing (generalized) Stallings twists. Then, by applying the now-classical theory of genus two Heegaard diagrams in S3, we verify that the monodromies of the links in this family are the only ones corresponding to pair-of-pants open book decompositions of S3.

The two dimensional incompressible Navier-Stokes equation on D δ := [0, 2πδ] × [0, 2π] with δ ≈ 1, periodic boundary conditions, and viscosity 0 < ν 1 is considered. Bars and dipoles, two explicitly given quasi-stationary states of the system, evolve on the time scale O(e −νt) and have been shown to play a key role in its long-time evolution. Of particular interest is the role that δ plays in selecting which of these two states is observed. Recent numerical studies suggest that, after a transient period of rapid decay of the high Fourier modes, the bar state will be selected if δ = 1, while the dipole will be selected if δ = 1. Our results support this claim and seek to mathematically formalize it. We consider the system in Fourier space, project it onto a center manifold consisting of the lowest eight Fourier modes, and use this as a model to study the selection of bars and dipoles. It is shown for this ODE model that the value of δ controls the behavior of the asymptotic ratio of the low modes, thus determining the likelihood of observing a bar state or dipole after an initial transient period. Moreover, in our model, for all δ ≈ 1, there is an initial time period in which the high modes decay at the rapid rate O(e −t/ν), while the low modes evolve at the slower O(e −νt) rate. The results for the ODE model are proven using energy estimates and invariant manifolds and further supported by formal asymptotic expansions and numerics.