QSpace Collection:
http://hdl.handle.net/1974/6158
2015-08-31T23:52:51ZPrediction and Filtering of Stationary Processes: Yaglom’s Method and Minimax Filtering
http://hdl.handle.net/1974/13493
Title: Prediction and Filtering of Stationary Processes: Yaglom’s Method and Minimax Filtering
Authors: Mascher, Philipp
Abstract: The aim of this work is to give a basic introduction to the theory of stationary stochastic processes, particularly to the somewhat specialized problem of prediction and filtering of such processes. Kolmogorov was the first to make
a contribution to its solution using involved mathematical theory. In the years following the publication of Wiener’s famous book, the theory gained considerable popularity from the applied sciences, particularly radio engineering. In this work, we shall present Yaglom’s method to solving the problems considered in Wiener’s book. This alternative approach is entirely based on rather basic facts from Hilbert space theory and the theory of complex variables. As it turns out, the theory of filtering of stationary processes heavily relies on spectral properties of the processes. In particular, Yaglom’s approach assumes complete knowledge of the spectral densities. In this work, however, we shall not be concerned with the problem of estimating such quantities based on a finite sample. Instead, in order to account for uncertainty as frequently encountered in practice, we shall discuss the problem of minimax filtering which has emerged from the practical need of allowing for incomplete knowledge about spectral properties.2015-08-11T04:00:00ZLinearization and Stability of Nonholonomic Mechanical Systems
http://hdl.handle.net/1974/13149
Title: Linearization and Stability of Nonholonomic Mechanical Systems
Authors: Yang, Steven
Abstract: The stability of an equilibrium point of a nonlinear system is typically analyzed in two ways: (1) stability of its linearization, and (2) Lyapunov stability. An unconstrained simple mechanical system is a type of nonlinear system with a special structure, and so the methods for stability analysis can be specialized for this particular class of nonlinear systems. For a simple mechanical system subject to velocity constraints, the situation becomes more complicated. If the constraints are holonomic, then the problem can simply be reduced to that of an unconstrained simple mechanical system by restricting analysis to a certain submanifold of the configuration space. If the constraints are nonholonomic, this approach cannot be taken. In this report we study the differences and additional complexities that arise in these nonholonomic mechanical systems, and derive results with regards to linearization and stability of its equilibria.2015-06-23T04:00:00ZExamining the Probability that the Number of Points on an Elliptic Curve over a Finite Field is Prime
http://hdl.handle.net/1974/13089
Title: Examining the Probability that the Number of Points on an Elliptic Curve over a Finite Field is Prime
Authors: Wheeler, Emilie
Abstract: This project examines the work in the article "The Probability
that the Number of Points on an Elliptic Curve over a Finite
Field is Prime", in which authors Galbraith and McKee ask the
question `What is the probability that a randomly chosen elliptic
curve over Fp has kq points, where k is small and q is prime?' I
performed my own computations and will compare them to their
results.2015-05-27T04:00:00ZAsymptotic Liberation in Free Probability
http://hdl.handle.net/1974/12711
Title: Asymptotic Liberation in Free Probability
Authors: Vazquez Becerra, Josue Daniel
Abstract: Recently G. Anderson and B. Farrel presented the notion of asymptotic liberation on sequences of families of random unitary matrices. They showed that asymptotically liberating sequences of families of random unitary matrices, when used for conjugation, delivers asymptotic freeness, a fundamental concept in free probability theory. Furthermore, applying the Fibonacci-Whitle inequality together with some combinatorial manipulations, they established sufficient conditions on a sequence of families of random unitary matrices in order to be asymptotically liberating.
On the other hand, a theorem by J. Mingo and R. Speicher states that given a graph
$G=(E,V,s,r)$ there exists an optimal rational number $ \mathfrak{r}_{G}$,
depending only on the structure of $G$, such that for any collection of
$n\times n$ complex matrices
$\{ A_{e}=( A_{e} (i_{s(e)},i_{r(e)}) ) \mid e\in E \}$ we have%
\begin{equation*}
\left\vert
\sum_{ i_{v_{1}},\ldots i_{v_{m}} =1 }^{n}
\left( \prod_{e\in E} A_{e} (i_{s(e)},i_{r(e)}) \right)
\right\vert
\leq
n^{\mathfrak{r}_{G}}\prod\limits_{e\in E}\left\Vert A_{e}\right\Vert
\end{equation*}
where $V=\left\{ v_{1},\ldots,v_{m}\right\} $ and $\left\Vert \cdot
\right\Vert $ denotes the operator norm.
In this report we show how to use the latter inequality to prove the same result as G. Anderson and B. Farrell
regarding sufficient conditions for asymptotic liberation.2015-01-26T05:00:00Z