Related papers: The Dehn function of SL(n;Z)
A set $X\subseteq\mathbb N$ is S-recognizable for an abstract numeration system S if the set $\rep_S(X)$ of its representations is accepted by a finite automaton. We show that the growth function of an S-recognizable set is always either…
If a (cusped) surface S admits an ideal triangulation T with no shears, we show an efficient algorithm to give S as a quotient of hypebolic plane by a subgroup of PSL(2, Z). The algorithm runs in time O(n log n), where n is the number of…
In the classical theory of cubic interpolation splines there exists an algorithm which works with only $O\left( n\right)$ arithmetic operations. Also, the smoothing cubic splines may be computed via the algorithm of Reinsch which reduces…
Iterating Newton's method symbolically for the general quadratic yields a rational function, the numerator and denominator of which are polynomials with highly composite coefficients.
This is the first of two papers devoted to connections between asymptotic functions of groups and computational complexity. One of the main results of this paper states that if for every $m$ the first $m$ digits of a real number $\alpha\ge…
The decomposition of the polynomials on the quaternionic unit sphere in $\Hd$ into irreducible modules under the action of the quaternionic unitary (symplectic) group and quaternionic scalar multiplication has been studied by several…
We decompose the $\hat{\mathfrak{sl}}(n)$-module $V(\Lambda_0) \otimes V(\Lambda_0)$ and give generating function identities for the outer multiplicities. In the process we discover some seemingly new partition identities in the cases…
Viewing Dehn's algorithm as a rewriting system, we generalise to allow an alphabet containing letters which do not necessarily represent group elements. This extends the class of groups for which the algorithm solves the word problem to…
We give a new algorithm to simplify a given triangulation with respect to a given curve. The simplification uses flips together with powers of Dehn twists in order to complete in polynomial time in the bit-size of the curve.
We prove that all rational slopes are characterizing for the knot $5_2$, except possibly for positive integers. Along the way, we classify the Dehn surgeries on knots in $S^3$ that produce the Brieskorn sphere $\Sigma(2,3,11)$, and we study…
The Dinitz conjecture states that, for each $n$ and for every collection of $n$-element sets $S_{ij}$, an $n\times n$ partial latin square can be found with the $(i,j)$\<th entry taken from $S_{ij}$. The analogous statement for $(n-1)\times…
In this paper, we derive the quadratic formula as a consequence of constructively proving the existence of standard and factored forms for general form real quadratic functions. Emphasis is put on connections to graphing of corresponding…
In the March 2025 issue of Pour la Science, Jean-Paul Delahaye described a wonderful solution to the following problem: How many ways can you divide a 3 by 2n rectangle into two connected, congruent pieces? We show that this problem can be…
For odd $n$ we construct a path $\rho_t\colon \pi_1(S) \to SL(n,\mathbb{R})$ of discrete, faithful and Zariski dense representations of a surface group such that $\rho_t(\pi_1(S)) \subset SL(n,\mathbb{Q})$ for every $t\in \mathbb{Q}$.
This article provides a simple proof of the quadratic formula, which also produces an efficient and natural method for solving general quadratic equations. The derivation is computationally light and conceptually natural, and has the…
In Euclidean space, one can use the dot product to give a formula for the area of a triangle in terms of the coordinates of each vertex. Since this formula involves only addition, subtraction, and multiplication, it can be used as a…
A sum-of-squares is a polynomial that can be expressed as a sum of squares of other polynomials. Determining if a sum-of-squares decomposition exists for a given polynomial is equivalent to a linear matrix inequality feasibility problem.…
We define zeta functions for the adjoint action of GL(n) on its Lie algebra and study their analytic properties. For n<4 we are able to fully analyse these functions, and recover the Shintani zeta function for the prehomogeneous vector…
Let $\sigma_n=\lfloor1+n\cdot\log_23\rfloor$. For the Collatz 3x + 1 function exists for each $n\in\mathbb{N}$ a set of different residue classes $(\text{mod}\ 2^{\sigma_n})$ of starting numbers $s$ with finite stopping time…
The Dedekind zeta function of a quadratic number field factors as a product of the Riemann zeta function and the $L$-function of a quadratic Dirichlet character. We categorify this formula using objective linear algebra in the abstract…