Narrow Results

Let $A_2(𝔐)$ and $L_2({𝔄}^2)$ be, respectively, the 2-generated free metabelian associative and Lie algebra over the field of complex numbers. In the associative case we find a finite set of generators of the algebra $A_2{(𝔐)}^{D_{2n}}$ of the dihedral group of order $2n$, $n\ge 3$. In the Lie case we find a minimal system of generators as a $ℂ{[x,y]}^{D_{2n}}$-module of the $D_{2n}$-invariants $L_2^{\prime }{({𝔄}^2)}^{D_{2n}}$ in the commutator ideal $L_2^{\prime }({𝔄}^2)$ of $L_2({𝔄}^2)$. In both cases we compute the Hilbert (or Poincaré) series of the algebras $A_2{(𝔐)}^{D_{2n}}$ and $L_2{({𝔄}^2)}^{D_{2n}}$.

In this paper, we introduce a new topic on geometric patterns issued from Truchet tile, that we agree to call combinatorial arabesque. The originally Truchet tile, by reference to the French scientist of 17th century Sébastien Truchet, is a square split along the diagonal into two triangles of contrasting colors. We define an equivalence relation on the set of all square tiling of same size, leading naturally to investigate the equivalence classes and their cardinality. Thanks to this class notion, it will be possible to measure the beauty degree of a Truchet square tiling by means of an appropriate algebraic group. Also, we define many specific arabesques such as entirely symmetric, magic and hyper-maximal arabesques. Mathematical characterizations of such arabesques are established facilitating thereby their enumeration and their algorithmic generating. Finally the notion of irreducibility is introduced on arabesques.

In this paper, we extend the results for connectivity and convex hulls of Sierpiński relatives whose convex hulls have right isosceles triangle boundaries to the relatives corresponding to equilateral triangle version. Furthermore, we use a particular subclass of the relatives called trapezoid-shaped relatives to build new hexagonal fractals, which helps to visualize and distinguish the subgroups of symmetries of the hexagon. These results contribute to identify the topologies of Sierpiński relatives.

Let G be a group and let w = w(x₁, x₂,…, x_n) be a word in the absolutely free group F_n on free variables x₁, x₂,…, x_n. The set S⁽ⁿ⁾(G) of all words w such that the equality w(g_σ1, g_σ2,…, g_σn) = w(g₁, g₂,…, g_n) holds for all g₁, g₂,…, g_n∈G and all permutations σ ∈ S_n is a subgroup of F_n, called the subgroup of n-symmetric words for G. In this paper, the groups S⁽²⁾(D_p) and S⁽³⁾(D_p) for dihedral groups D_p are determined, where p > 3 is a prime. In particular, it turns out that the groups S⁽³⁾(D_p) are not abelian.

The main purpose of this paper is to give an intrinsic interpretation of the space Θ(D) generated by the binary theta series ϑ_f attached to the positive binary quadratic forms f whose discriminant has the form D(f) = D/t², for some integer t. It turns out that , the space of modular forms of weight 1 and of level |D| which have complex multiplication (CM) by their Nebentypus character . As an application, we obtain a structure theorem of the space . The proof of this theorem rests on the results of [The space of binary theta series, Ann. Sci. Math. Québec36 (2012) 501–534] together with a characterization of the newforms f which have CM by their Nebentypus character in terms of properties of the associated Deligne–Serre Galois representationρ_f.

In this paper, we establish the structure of $U(F{D}_7)$, $U(F{D}_{12})$, $U(F{D}_{14})$ and $U(F{D}_{24})$ where $F$ is a finite field and $D_n$ is the dihedral group of order $2n$. For an odd prime $p$, we also establish the structure of $U(F{D}_{2^k\cdot p})$, when characteristic of $F$ is $2$ or $p$.