Open Problems Concerning Michell Trusses

Michell Trusses and Lines of Principal Action

We study the problem of Michell trusses when the system of applied equilibrated forces is a vector measure with compact support. We introduce a class of stress tensors which can be written as a superposition of rank-one tensors carried by curves (lines of principal strains). Optimality conditions are given for such families showing in particular that optimal stress tensors are carried by mutual...

John Michell Clarke

Open problems for equienergetic graphs

The energy of a graph is equal to the sum of the absolute values of its eigenvalues. Two graphs of the same order are said to be equienergetic if their energies are equal. We point out the following two open problems for equienergetic graphs. (1) Although it is known that there are numerous pairs of equienergetic, non-cospectral trees, it is not known how to systematically construct any such pa...

On two problems concerning the Zariski topology of modules

Let $R$ be an associative ring and let $M$ be a left $R$-module.Let $Spec_{R}(M)$ be the collection of all prime submodules of $M$ (equipped with classical Zariski topology). There is a conjecture which says that every irreducible closed subset of $Spec_{R}(M)$ has a generic point. In this article we give an affirmative answer to this conjecture and show that if $M$ has a Noetherian spectrum, t...

Extremal problems concerning Kneser graphs

Let d and S? be two intersecting families of k-subsets of an n-element set. It is proven that l~.JuS?l <(;:i)+(;::) holds for n>f(3+,/?)k, and equality holds only if there exist two points a, b such that {a, b} n F# 0 for all FE d u g, For n=2k+o(Jj;) an example showing that in this case max 1 d u B 1 = (1 o( 1 ))( ;) is given. This disproves an old conjecture of Erdiis [7]. In the second part ...