Related papers: Sarrus rules and dihedral groups
In this article, we shall derive by elementary calculations the Gauss map, spherical image, Weingarten map and the curvatures at identity of the special linear group, that is, the matrices of determinant 1. We could not find any reference…
In this paper, we investigate sums of four squares of integers whose prime factorizations are restricted, making progress towards a conjecture of Sun that states that two of the integers may be restricted to the forms $2^a3^b$ and $2^c5^d$.…
We prove a structural result for sets of integers with doubling at most $4 + \delta$, with $\delta>0$ sufficiently small. This generalises earlier work of Eberhard--Green--Manners which dealt with sets of integers with doubling strictly…
The formula for the dihedral angle of the simplex of n dimensions, arccos(1/n), is derived using classical geometry.
Math is widely considered as a powerful tool and its strong appeal depends on the high level of abstraction it allows in modelling a huge number of heterogeneous phenomena and problems, spanning from the static of buildings to the flight of…
We determine the minimal non-trivial integer group determinant for the dicyclic group of order $4n$ when $n$ is odd. We also discuss the set of all integer group determinants for the dicyclic groups of order $4p$.
A superposition rule is a particular type of map that enables one to express the general solution of certain systems of first-order ordinary differential equations, the so-called Lie systems, out of generic families of particular solutions…
In this article we aim to develop from first principles a theory of sum sets and partial sum sets, which are defined analogously to difference sets and partial difference sets. We obtain non-existence results and characterisations. In…
Quaternionic modular forms on $\mathsf{G}_2$ carry a surprisingly rich arithmetic structure. For example, they have a theory of Fourier expansions where the Fourier coefficients are indexed by totally real cubic rings. For quaternionic…
In this paper we will relate hyperstructures and the general $\mathscr{H}$-principle to known mathematical structures, and also discuss how they may give rise to new mathematical structures. The main purpose is to point out new ideas and…
In this pedagogical text aimed at those wanting to start thinking about or brush up on probabilistic inference, I review the rules by which probability distribution functions can (and cannot) be combined. I connect these rules to the…
We give here a general, best-possible, and smoothly-derived form of the Master Theorem for divide-and-conquer recurrences.
These course notes are about computing modular forms and some of their arithmetic properties. Their aim is to explain and prove the modular symbols algorithm in as elementary and as explicit terms as possible, and to enable the devoted…
The Bayesian Learning Rule provides a framework for generic algorithm design but can be difficult to use for three reasons. First, it requires a specific parameterization of exponential family. Second, it uses gradients which can be…
In this paper we intend to give some calculus rules for tangent sets in the sense of Bouligand and Ursescu, as well as for corresponding derivatives of set-valued maps. Both first and second order objects are envisaged and the assumptions…
A formula is presented for the determinant of the second additive compound of a square matrix in terms of coefficients of its characteristic polynomial. This formula can be used to make claims about the eigenvalues of polynomial matrices,…
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 this short note, we prove a formula for the group inverse of a block matrix and consider the pseudo principal pivot transform expressed in terms of group inverses. Extensions of the usual principal pivot transform, where the usual…
We offer two comments on the beautiful papers of Giles Gardam and Alan Murray that yield counterexamples to the Kaplansky unit conjecture. First we discuss the determinants of these units in a certain $4\times 4$ matrix representation of…
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…