The result of [6] is the existence of an infinite family of Zariski dense surface subgroups of fixed genus inside SL(3,Z); here we exhibit such subgroups inside SL(4,Z) and symplectic groups. In this setting the power of such a result comes in large part from the conclusion that the groups are Zariski dense the existence of surface groups inside SL(4,Z) can be proved fairly easily, since it’s n...

Let X = C. In this paper we present an algorithm that computes the de Rham cohomology groups H dR(U, C) where U is the complement of an arbitrary Zariski-closed set Y in X . Our algorithm is a merger of the algorithm given by T. Oaku and N. Takayama ([7]), who considered the case where Y is a hypersurface, and our methods from [9] for the computation of local cohomology. We further extend the a...

Modern model theory can be viewed as a subject obsessed with notions of dimension, with the key examples furnished by linear dimension on the one hand, and the dimension of an algebraic variety (or, from another point of view, transcendences degree) on the other. There are several rigorous, and not always equivalent, notions of abstract dimension in use. For historical reasons the one we use is...

This study has been inspired by the paper "An efficient 3D topology optimization code written in MATLAB” written by Liu and Tovar (2014) demonstrating that SIMP-based three-dimensional (3D) topology optimization of continuum structures can be implemented in 169 lines of MATLAB code. Based on the above paper, we show here that, by simple and easy-to-understand modificati...

In his Ph.D. thesis, Cadegan-Schlieper constructs an invariant of the embedded topology a line arrangement which generalizes $$\mathcal {I}$$ -invariant introduced by Artal, Florens and author. This new is called loop-linking number in present paper. We refine result proving that homeomorphism type complement. give two effective methods to compute this invariant, both are based on braid monodro...

We study Zariski pairs of sextics which are distinguished by the Alexander polynomials. For this purpose, we present two constructive methods to produce explicit sextics of non-torus type with given configuration of simple singularities.