Bieberbach groups and flat manifolds with finite abelian holonomy from Artin braid groups

Let [Formula: see text]. In this paper, we show that for any abelian subgroup [Formula: see text] of [Formula: see text] the crystallographic group [Formula: see text] has Bieberbach subgroups [Formula: see text] with holonomy group [Formula: see text]. Using this approach, we obtain an explicit description of the holonomy representation of the Bieberbach group [Formula: see text]. As an application, when the holonomy group is cyclic of odd order, we study the holonomy representation of [Formula: see text] and determine the existence of Anosov diffeomorphisms and Kähler geometry of the flat manifold [Formula: see text] with fundamental group the Bieberbach group [Formula: see text].

Heron’s cubic root iteration method with a new approach

Heron’s cubic root iteration formula conjectured by Wertheim is proved and extended for any odd order roots. Some possible proofs are suggested for the roots of even order. An alternative proof of Heron’s general cubic root iterative method is explained. Further, Lagrange’s interpolation formula for nth root of a number is studied and found that Al-Samawal’s and Lagrange’s method are equivalent. Again, counterexamples are discussed to justify the effectiveness of the present investigations.

Structural Properties of Connected Domination Critical Graphs

A graph G is said to be k-γc-critical if the connected domination number γc(G) is equal to k and γc(G+uv)<k for any pair of non-adjacent vertices u and v of G. Let ζ be the number of cut vertices of G and let ζ0 be the maximum number of cut vertices that can be contained in one block. For an integer ℓ≥0, a graph G is ℓ-factor critical if G−S has a perfect matching for any subset S of vertices of size ℓ. It was proved by Ananchuen in 2007 for k=3, Kaemawichanurat and Ananchuen in 2010 for k=4 and by Kaemawichanurat and Ananchuen in 2020 for k≥5 that every k-γc-critical graph has at most k−2 cut vertices and the graphs with maximum number of cut vertices were characterized. In 2020, Kaemawichanurat and Ananchuen proved further that, for k≥4, every k-γc-critical graphs satisfies the inequality ζ0(G)≤mink+23,ζ. In this paper, we characterize all k-γc-critical graphs having k−3 cut vertices. Further, we establish realizability that, for given k≥4, 2≤ζ≤k−2 and 2≤ζ0≤mink+23,ζ, there exists a k-γc-critical graph with ζ cut vertices having a block which contains ζ0 cut vertices. Finally, we proved that every k-γc-critical graph of odd order with minimum degree two is 1-factor critical if and only if 1≤k≤2. Further, we proved that every k-γc-critical K1,3-free graph of even order with minimum degree three is 2-factor critical if and only if 1≤k≤2.

Toy model of harmonic and sum frequency generation in 2D dielectric nanostructures

Optical nonlinearities of matter are often associated with the response of individual atoms. Here, using a toy oscillator model, we show that in the confined geometry of a two-dimensional dielectric nanoparticle a collective nonlinear response of the atomic array can arise from the Coulomb interactions of the bound optical electrons, even if the individual atoms exhibit no nonlinearity. We determine the multipole contributions to the nonlinear response of nanoparticles and demonstrate that the odd order and even order nonlinear electric dipole moments scale with the area and perimeter of the nanoparticle, respectively.

ON SOME EQUIVALENCE RELATION ON NON-ABELIAN $\CA$-GROUPS

A non-abelian group $G$ is called a $\CA$-group ($\CC$-group) if $C_G(x)$ is abelian(cyclic) for all $x\in G\setminus Z(G)$. We say $x\sim y$ if and only if $C_G(x)=C_G(y)$.We denote the equivalence class including $x$ by$[x]_{\sim}$. In this paper, we prove thatif $G$ is a $\CA$-group and $[x]_{\sim}=xZ(G)$, for all $x\in G$, then $2^{r-1}\leq|G'|\leq 2^{r\choose 2}$.where $\frac {|G|}{|Z(G)|}=2^{r}, 2\leq r$ and characterize all groups whose $[x]_{\sim}=xZ(G)$for all $x\in G$ and $|G|\leq 100$. Also, we will show that if $G$ is a $\CC$-group and $[x]_{\sim}=xZ(G)$,for all $x \in G$, then $G\cong C_m\times Q_8$ where $C_m$ is a cyclic group of odd order $m$ andif $G$ is a $\CC$-group and $[x]_{\sim}=x^G$, for all $x\in G\setminus Z(G)$, then $G\cong Q_8$.

From Binary Groups to Terminal Rings

Binary groups are a meaningful step up from non-associative rings and nearrings. It makes sense to study them in terms of their nearrings of zero-fixing polynomial maps. As this involves algebras of a more specialized nature these are looked into in sections three and four. One of the main theorems of this paper occurs in section five where it is shown that a binary group V is a P0(V) ring module if, and only if, it is a rather restricted form of non-associative ring. Properties of these non-associative rings (called terminal rings) are investigated in sections six and seven. The finite case is of special interest since here terminal rings of odd order really are quite restricted. Sections eight to thirteen are taken up with the study of terminal rings of order pn (p an odd prime and n ≥ 1 an integer ≤ 7).

DIAGRAMMATIC CONSTRUCTION OF REPRESENTATIONS OF SMALL QUANTUM $$ \mathfrak{sl} $$2

We provide a combinatorial description of the monoidal category generated by the fundamental representation of the small quantum group of $$ \mathfrak{sl} $$ sl 2 at a root of unity q of odd order. Our approach is diagrammatic, and it relies on an extension of the Temperley–Lieb category specialized at δ = −q − q−1.

Completing Partial Transversals of Cayley Tables of Abelian Groups

In 2003 Grüttmüller proved that if $n\geqslant 3$ is odd, then a partial transversal of the Cayley table of $\mathbb{Z}_n$ with length $2$ is completable to a transversal. Additionally, he conjectured that a partial transversal of the Cayley table of $\mathbb{Z}_n$ with length $k$ is completable to a transversal if and only if $n$ is odd and either $n \in \{k, k + 1\}$ or $n \geqslant 3k - 1$. Cavenagh, Hämäläinen, and Nelson (in 2009) showed the conjecture is true when $k = 3$ and $n$ is prime. In this paper, we prove Grüttmüller’s conjecture for $k = 2$ and $k = 3$ by establishing a more general result for Cayley tables of Abelian groups of odd order.

Polarization Flipping of Even-Order Harmonics in Monolayer Transition-Metal Dichalcogenides

We present a systematic study of the crystal-orientation dependence of high-harmonic generation in monolayer transition-metal dichalcogenides, WS2 and MoSe2, subjected to intense linearly polarized midinfrared laser fields. The measured spectra consist of both odd- and even-order harmonics, with a high-energy cutoff extending beyond the 15th order for a laser-field strength around ~1 V/nm. In WS2, we find that the polarization direction of the odd-order harmonics smoothly follows that of the laser field irrespective of the crystal orientation, whereas the direction of the even-order harmonics is fixed by the crystal mirror planes. Furthermore, the polarization of the even-order harmonics shows a flip in the course of crystal rotation when the laser field lies between two of the crystal mirror planes. By numerically solving the semiconductor Bloch equations for a gapped-graphene model, we qualitatively reproduce these experimental features and find the polarization flipping to be associated with a significant contribution from interband polarization. In contrast, high-harmonic signals from MoSe2 exhibit deviations from the laser-field following of odd-order harmonics and crystal-mirror-plane following of even-order harmonics. We attribute these differences to the competing roles of the intraband and interband contributions, including the deflection of the electron-hole trajectories by nonparabolic crystal bands.