Discrete Dynamics over Finite Fields

A dynamical system consists of a set V and a map f : V → V . The primary goal is to characterize points in V according to their limiting behaviors under iteration of the map f . Especially understanding dynamics of nonlinear maps is an important but difficult problem, and there are not many methods available. This work concentrates on dynamics of certain nonlinear maps over finite fields. First we study monomial dynamics over finite fields. We show that determining the number of fixed points of a boolean monomial dynamics is #P–complete problem and consider various cases in which the dynamics can be explained efficiently. We also extend the result to the monomial dynamics over general finite fields. Then we study the dynamics of a simple nonlinear map, f(x) = x + x-1, on fields of characteristic two. The main idea is to lift the map f to a proper finite covering map whose dynamics is easier to understand. We lift the map of f to an isogeny g on an elliptic curve where the dynamics of g can be further reduced to that of a linear map on Z –module. As an application of finite covering, we construct a new family of permutation maps over finite fields from the known permutation maps

Global Classical Solutions to the Relativistic Boltzmann Equation Without Angular Cut-off

We prove the unique existence and exponential decay of global in time classical solutions to the special relativistic Boltzmann equation without any angular cut-off assumptions with initial perturbations in some weighted Sobolev spaces. We consider perturbations of the relativistic Maxwellian equilibrium states. We work in the case of a spatially periodic box. We consider the general conditions on the collision kernel from Dudynski and Ekiel-Jezewska (Commun. Math. Phys. 115(4):607-629,1985). Additionally, we prove sharp constructive upper and coercive lower bounds for the linearized relativistic Boltzmann collision operator in terms of a geometric fractional Sobolev norm; this shows that a spectral gap exists and that this behavior is similar to that of the non-relativistic case as shown by Gressman and Strain (Journal of AMS 24(3), 771-847, 2011). We also derive the relativistic analogue of Carleman dual representation of Boltzmann collision operator. Lastly, we explicitly compute the Jacobian of a collision map (p, q) to (cp\u27 + (1-c)p, q) for a fixed c in (0, 1), and it is shown that the Jacobian is bounded above in p and q. This is the first global existence and stability result for relativistic Boltzmann equation without angular cutoff and this resolves the open question of perturbative global existence for the relativistic kinetic theory without the Grad\u27s angular cut-off assumption

Three Essays on the Mobility of Human Capital and Knowledge Transfer

Both economics and strategic management literature illuminate the impacts of human capital and knowledge on the growth of regional economies and firms. Despite their different characteristics as factors of production, human capital and knowledge often move (or stay) together, as knowledge is embedded in the human brain. Also, human capital and knowledge have shared characteristics; a region or firm cannot be free from the risk of unintended leakage because human capital can move deliberately and knowledge can spill over through various channels. Therefore, it is in the best interests of firms and regions to foster (or retain) human capital and knowledge. The three papers constituting this thesis all address what affects mobility decision of human capital across regions or firms, address what are the antecedents of firms and regions attracting (or poaching) human capital, and also address how knowledge transfer is affected by such human capital mobility

Data compressive paradigm for spectral sensing and classification using electrically tunable detectors

This dissertation contains three major parts: (1) demonstration of the algorithmic spectrometry in the mid-IR sensing regime using spectrally tunable quantum dots-in-a-well (DWELL) IR detector without employing any spectral filters; (2) further demonstration of the spectral-classification capability of tunable DWELL IR focal-plane array (FPA), again without using any spectral filters; and (3) development of a generalized filter-free data-compressive spectral sensing paradigm using the DWELL detector that enables arbitrarily specified MS sensing (e.g., spectral matched filtering, slope sensing, multicolor sensing, etc.) without using any spectral filters and possibly under constrained acquisition times

Proportional signs in the works of Heinrich Schutz

Some time signatures used in the Neue Schütz Ausgabe (Bärenreither, 1955-2008) differ from both modern signatures and contemporary mensuration signs, obscuring Schütz\u27s original intentions. A review of the history of proportion signs from the late 14th century to the 17th century shows that the four basic mensuration signs of the late 14th century were the foundation of the proportion system throughout the period, and that the proportion signs of the 16th and 17th century were adaptations of modus cum tempore signs and fractions. Although confusion was created through misunderstandings of the meanings of the signs and by attempts to reform the system, the original meanings of the mensuration-proportion signs were retained throughout the period. A study of the proportion signs used in the Psalmen Davids (1619) and in the Symphoniae Sacrae III (1650), as well as several signs found in a few of his other works, shows that Schütz\u27s notation is within the conventional practice of mensurationiv proportion notation. Some of Schütz\u27s signs are open to more than one interpretation, requiring an explanation of possible interpretations of the signs and some suggestions for modern performance