arXiv subjects
Publications and source records attributed to Nasser Heydari.
In this article, we study similarity of triangles in the Susa Mathematical Texts (\textbf{SMT}). We also suggest that the Susa scribes were aware of intercept theory because they used this theorem in solving a complicated system of equations.
This article studies the systems of equations appearing in the Susa Mathematical Texts (\textbf{SMT}) and the different approaches used by the Susa scribes to solve them.
In this article, we study some of quadratic equations and their solutions found in the Susa Mathematical Texts (\textbf{SMT}). We show that the Susa scribes used this group of equations in different problems and took a standard approach, known as completing the square, to find solutions
This article studies the application of the Pythagorean theorem in the Susa Mathematical Texts (\textbf{SMT}) and we discuss those texts whose problems and related calculations demonstrate its use. Among these texts, \textbf{SMT No.\,1} might be the most important as it contains a geometric application of the Pythagorean theorem.
In this article, we study the problems found in the Susa Mathematical Texts No.\,24 and No.\,25 (\textbf{SMT No.\,24} and \textbf{SMT No.\,25}) which concern excavation projects such as canals and holes. We also examine certain Elamite structures, such as the canal systems serving Susa and a reservoir at the ziggurat of Chogha Zanbil, in whose construction geometry might well have played an important role.
This article studies three-dimensional objects and their volumes in Elamite mathematics, particularly those found in the Susa Mathematical Tablet No.\,14 (\textbf{SMT No.\,14}). In our discussion, we identify some basic solids whose volumes have been correctly computed in Babylonian and Elamite mathematics. We also show that the Elamite scribes knew the right formula for calculating the volume of a certain pyramid which is a rare phenomenon occurring in the Babylonian mathematical tablets.
In this article, we study a particular group of plane figures whose constants are listed in the Susa Mathematical Tablet No.\,3 (\textbf{SMT No.\,3}). We explain possible ways to define these figures and seek to demonstrate that the Susa scribes used complicated calculations to obtain such numbers. We also give examples of circular figures used for Elamite artifacts through which one can appreciate the significance of these figures in Elamite art.
The bisection of trapezoids by transversal lines has many examples in Babylonian mathematics. In this article, we study a similar problem in Elamite mathematics, inscribed on a clay tablet held in the collection of the Louvre Museum and thought to date from between 1894--1595 BC. We seek to demonstrate that this problem is different from typical Babylonian problems about bisecting trapezoids by transversal lines. We also identify some of the possible mathematical ideas underlying this problem and the innovative approach that might have motivated its design.
In this article, we study the inscription on the reverse of Susa Mathematical Text No.\,2, a clay tablet held in the collection of the Louvre Museum and thought to date from between 1894--1595 BC. We focus on the formula given in this text for the approximate area of a regular heptagon. We give a geometric explanation for the formula and show that this approximation is more accurate than other contemporaneous formulas in Babylonian mathematics and even that of Greek mathematician Heron who proved it almost 1800 years later. We also consider the possible ways the Susa scribes might have applied this formula to construct the regular heptagon for inscription on a clay tablet.
Given a Hamiltonian system $ (M,ω, G,μ) $ where $(M,ω)$ is a symplectic manifold, $G$ is a compact connected Lie group acting on $(M,ω)$ with moment map $ μ:M \rightarrow\mathfrak{g}^{*}$, then one may construct the symplectic quotient $(M//G, ω_{red})$ where $M//G := μ^{-1}(0)/G$. Kirwan used the norm-square of the moment map, $|μ|^2$, as a G-equivariant Morse function on $M$ to derive formulas for the rational Betti numbers of $M//G$. A real Hamiltonian system $(M,ω, G,μ, σ, ϕ) $ is a Hamiltonian system along with a pair of involutions $(σ:M \rightarrow M, ϕ:G \rightarrow G) $ satisfying certain compatibility conditions. These imply that the fixed point set $M^σ$ is a Lagrangian submanifold of $(M,ω)$ and that $M^σ//G^ϕ := (μ^{-1}(0) \cap M^σ)/G^ϕ$ is a Lagrangian submanifold of $(M//G, ω_{red})$. In this paper we prove analogues of Kirwan's Theorems that can be used to calculate the $\mathbb{Z}_2$-Betti numbers of $M^σ//G^ϕ $. In particular, we prove (under appropriate hypotheses) that $|μ|^2$ restricts to a $G^ϕ$-equivariantly perfect Morse-Kirwan function on $M^σ$ over $\mathbb{Z}_2$ coefficients, describe its critical set using explicit real Hamiltonian subsystems, prove equivariant formality for $G^ϕ$ acting on $M^σ$, and combine these results to produce formulas for the $\mathbb{Z}_2$-Betti numbers of $M^σ//G^ϕ$.