(1992). Matrices: Methods and Applications. Journal of the Operational Research Society: Vol. 43, No. 12, pp. 1185-1185.

/pdf/matrices-methods-and-applications-3mhi2vv26h.pdf

Matrices: Methods and Applications

Graphs and matrices

Cellular automata (CAs) are dynamical systems which exhibit complex global behavior from simple local interaction and computation. Since the inception of cellular automaton (CA) by von Neumann in 1950s, it has attracted the attention of several researchers over various backgrounds and fields for modeling different physical, natural as well as real-life phenomena. Classically, CAs are uniform. However, non-uniformity has also been introduced in update pattern, lattice structure, neighborhood dependency and local rule. In this survey, we tour to the various types of CAs introduced till date, the different characterization tools, the global behavior of CAs, like universality, reversibility, dynamics etc. Special attention is given to non-uniformity in CAs and especially to non-uniform elementary CAs, which have been very useful in solving several real-life problems.

A survey of cellular automata: types, dynamics, non-uniformity and applications

This paper has two goals. The first is to present the construction, due to the author, of measures on non-archimedean analytic varieties associated to metrized line bundles and some of its applications. We take this opportunity to add remarks, examples and mention related results.

Heights and measures on analytic spaces. A survey of recent results, and some remarks

The normalized Laplacian, degree-Kirchhoff index and the spanning tree numbers of generalized phenylenes

We prove the Effective Bogomolov Conjecture, and so the Bogomolov Conjecture, over a function field of characteristic 0 by proving Zhang’s Conjecture about certain invariants of metrized graphs. In the function field case, these conjectures were previously known to be true only for curves of good reduction, for curves of genus at most 4 and a few other special cases. We also either verify or improve the previous results. We relate the invariants involved in Zhang’s Conjecture to the tau constant of metrized graphs. Then we use and extend our previous results on the tau constant. By proving another Conjecture of Zhang, we obtain a new proof of the slope inequality for Faltings heights on moduli space of curves.

Zhang's Conjecture and the Effective Bogomolov Conjecture over function fields

We prove the Effective Bogomolov Conjecture, and so the Bogomolov Conjecture, over a function field of characteristic 0 by proving Zhang's Conjecture about certain invariants of metrized graphs. In the function field case, these conjectures were previously known to be true only for curves of genus at most 4 and a few other special cases. We also either verify or improve the previous results. We relate the invariants involved in Zhang's Conjecture to the tau constant of metrized graphs. Then we use and extend our previous results on the tau constant. By proving another Conjecture of Zhang, we obtain a new proof of the slope inequality for Faltings heights on moduli space of curves.

The reversibility problem for linear cellular automata with null boundary defined by a rule matrix in the form of a pentadiagonal matrix was studied recently over the binary field ℤ2 (del Rey and Rodriguez Sanchez in Appl Math Comput, 2011, doi:101016/jamc201103033) In this paper, we study one-dimensional linear cellular automata with periodic boundary conditions over any finite field ℤp For any given p≥2, we show that the reversibility problem can be reduced to solving a recurrence relation depending on the number of cells and the coefficients of the local rules defining the one-dimensional linear cellular automata More specifically, for any given values (from any fixed field ℤp) of the coefficients of the local rules, we outline a computer algorithm determining the recurrence relation which can be solved by testing reversibility of the cellular automaton for some finite number of cells As an example, we give the full criteria for the reversibility of the one-dimensional linear cellular automata over the fields ℤ2 and ℤ3

Reversibility of 1D Cellular Automata with Periodic Boundary over Finite Fields \pmb{ {\mathbb{Z}}}_{p}

An elementary algorithm for computing the determinant of pentadiagonal Toeplitz matrices

We explicitly compute the effective resistances between any two vertices of a ladder graph by using circuit reductions. Using our findings, we obtain explicit formulas for Kirchhoff index of a ladder graph. Comparing our formula for Kirchhoff index and previous results in the literature, we obtain an explicit sum formula involving trigonometric functions. We also expressed our formulas in terms of certain generalized Fibonacci numbers that are the values of the Chebyshev polynomials of the second kind at 2.

Zubeyir Cinkir

Papers

Zhang's Conjecture and the Effective Bogomolov Conjecture over function fields

Zhang's Conjecture and the Effective Bogomolov Conjecture over function fields

Reversibility of 1D Cellular Automata with Periodic Boundary over Finite Fields \pmb{ {\mathbb{Z}}}_{p}

An elementary algorithm for computing the determinant of pentadiagonal Toeplitz matrices

Effective resistances and Kirchhoff index of ladder graphs