I call this Marden’s Theorem because I first read it in M. Marden’s wonderful book [6]. But this material appeared previously in Marden’s earlier paper [5]. In both sources Marden attributes the theorem to Siebeck, citing a paper from 1864 [8]. Indeed, Marden reports appearances of various versions of the theorem in nine papers spanning the period from 1864 to 1928. Of particular interest in wh...

We study a family of Teichmüller curves T (n,m) constructed by Bouw and Möller, and previously by Veech and Ward in the cases n = 2, 3. We simplify the proof that T (n,m) is a Teichmüller curve, avoiding the use Möller’s characterization of Teichmüller curves in terms of maximally Higgs bundles. Our key tool is a description of the period mapping of T (n,m) in terms of Schwarz triangle mappings...

The subject of this article belongs to the general question Under which condition(s) suitably normalized transcendental functions take algebraic values at algebraic arguments? Already the classical examples of Weierstrass’ result concerning the exponential function and Theodor Schneider’s result about the elliptic modular function show that arguments and values in these cases are of particular ...

This paper introduces a new approach to represent logic functions in the form of Sierpinski Gaskets. The structure of the gasket allows to manipulate with the corresponding logic expression using recursive essence of fractals. Thus, the Sierpinski gasket’s pattern has myriad useful properties which can enhance practical features of other graphic representations like decision diagrams. We have c...

Using the Gauss-Manin connection (Picard-Fuchs differential equation) and a result of Malgrange, a special class of algebraic solutions to isomonodromic deformation equations, the geometric isomonodromic deformations, is defined from “families of families” of algebraic varieties. Geometric isomonodromic deformations arise naturally from combinatorial strata in the moduli spaces of elliptic surf...

We present a topologically robust algorithm for Boolean operations on polyhedral boundary models. The algorithm can be proved always to generate a result with valid connectivity if the input shape representations have valid connectivity, irrespective of the type of arithmetic used or the extent of numerical errors in the computations or input data. The main part of the algorithm is based on a s...

The paper begins with a review of screw algebra including dual numbers, unit line vectors, dual vectors, screws, line coordinates, and screw coordinates and summarizes the essential rules of operation for screw algebra. A significant advantage of the algebra is that geometric, kinematic, and dynamic equations of a rigid body can be expressed in a concise and compact form. Several physical quant...

Origami constructions have interesting properties that are not covered by standard euclidean geometry. Such properties have been shown with the help of computer algebra systems. Proofs performed with computer algebra systems can be accompanied by proof documents, still they lack complete mathematical rigorousity, like the one provided by proof assistant checked proofs. Transforming such proofs ...

The asymptotic structure of gauge theories describing fracton interactions is analyzed. Two sets conditions are proposed. Both encompass all known solutions, lead to finite charges and resolve the problem divergent energy coming from monopole contribution. While first set leads expected symmetry algebra, including a dipole charge, second provides soft infinite-dimensional extension it. These pr...

The F-triangle is a refined face count for the generalised cluster complex of Fomin and Reading. We compute the F-triangle explicitly for all irreducible finite root systems. Furthermore, we use these results to partially prove the " M = F Conjecture " of Armstrong which predicts a surprising relation between the F-triangle and the Möbius function of his m-divisible partition poset associated t...