Related papers: Signed permutations and degree-one dot action repr…
We classify representations of the mapping class group of a surface of genus $g$ (with at most one puncture or boundary component) up to dimension $3g-3$. Any such representation is the direct sum of a representation in dimension $2g$ or…
Recently, Cohen and Wales built a faithful linear representation of the Artin group of type $D_n$, hence showing the linearity of this group. It was later discovered that this representation is reducible for some complex values of its two…
A signed graph is a graph whose edges are labeled either positive or negative. Corresponding to the two signed distance matrices defined for signed graphs, we define two signed distance laplacian matrices. We characterize balance in signed…
In 2005 J.L. Waldspurger proved the following theorem: given a finite real reflection group $W$, the closed positive root cone is tiled by the images of the open weight cone under the action of the linear transformations $id-w$. Shortly…
The paper is devoted to problem of spline approximation. A new method of nodes location for curves and surfaces computer construction by means of B-splines and results of simulink-modeling is presented. The advantages of this paper is that…
For any marked three manifold $(M,\mathcal N)$ and any quantum parameter $q^{\frac{1}{2}}$ (a nonzero complex number), we use $\mathscr{S}_{q^{1/2}}(M,\mathcal{N})$ to denote the stated skein module of $(M,\mathcal{N})$. When…
A signed graph $(G, \sigma)$ is a graph $G$ along with a function $\sigma: E(G) \to \{+,-\}$. A closed walk of a signed graph is positive (resp., negative) if it has an even (resp., odd) number of negative edges, counting repetitions. A…
A total prime labeling of a graph of order $n$ is an extension of a prime labeling in which we distinctly label the vertices and edges. The goal of the labeling is for adjacent vertex labels to be relatively prime, and for each vertex of…
Geometrically continuous splines are piecewise polynomial functions defined on a collection of patches which are stitched together through transition maps. They are called $G^{r}$-splines if, after composition with the transition maps, they…
Multi-degree splines are piecewise polynomial functions having sections of different degrees. They offer significant advantages over the classical uniform-degree framework, as they allow for modeling complex geometries with fewer degrees of…
A permutation may be represented by a collection of paths in the plane. We consider a natural class of such representations, which we call tangles, in which the paths consist of straight segments at 45 degree angles, and the permutation is…
The periodic (ordinal) patterns of a map are the permutations realized by the relative order of the points in its periodic orbits. We give a combinatorial characterization of the periodic patterns of an arbitrary signed shift, in terms of…
Werner Meyer constructed a cocycle in $H^2(Sp(2g, \mathbb{Z}); \mathbb{Z})$ which computes the signature of a closed oriented surface bundle over a surface, with fibre a surface of genus g. By studying properties of this cocycle, he also…
A pseudoline is a homeomorphic image of the real line in the plane so that its complement is disconnected. An arrangement of pseudolines is a set of pseudolines in which every two cross exactly once. A drawing of a graph is pseudolinear if…
In this paper we construct a projective action of certain arithmetic group on the derived category of coherent sheaves on an abelian scheme $A$, which is analogous to Weil representation of the symplectic group. More precisely, the…
For a real matrix $M$, we denote by $sp(M)$ the spectrum of $M$ and by $\left \vert M\right \vert $ its absolute value, that is the matrix obtained from $M$ by replacing each entry of $M$ by its absolute value. Let $A$ be a nonnegative real…
In this dissertation, we explore the structure of inversion graphs of permutations--a class of graphs that naturally arises by representing each permutation as a graph, where vertices correspond to entries and edges encode inversions.…
Given a graph $G$, the graph $[G]$ obtained by adding, for each pair of vertices of $G$, a unique vertex adjacent to both vertices is called the binding graph of $G$. In this work, we show that the class of binding graphs is…
We consider two operations on an edge of an embedded graph (or equivalently a ribbon graph): giving a half-twist to the edge and taking the partial dual with respect to the edge. These two operations give rise to an action of S_3^{|E(G)|},…
We introduce higher strip deformations, which give a way of constructing affine deformations of discrete free groups in the image of the irreducible representation $\operatorname{PSL}_2\mathbb{R}\to \operatorname{SO}(2n,2n-1)$. We use the…