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

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

Monomial Dynamical Systems in # P-complete

In this paper, we study boolean monomial dynamical systems. Colón-Reyes, Jarrah, Laubenbacher, and Sturmfels(2006) studied fixed point structure of boolean monomial dynamical systems of f by associating the dynamical systems of f with its dependency graph χf and Jarrah, Laubenbacher, and Veliz-Cuba(2010) extended it and presented lower and upper bound for the number of cycles of a given length for general boolean monomial dynamics. But, it is even difficult to determine the exact number of fixed points of boolean monomial dynamics. We show that the problem of counting fixed points of a boolean monomial dynamical systems is #P-complete, for which no efficient algorithm is known. This is proved by a 1-1 correspondence between fixed points of f sand antichains of the poset of strongly connected components of χf.