We demonstrate the role of ${D}_{(s)}^{*}$ and their contributions in quasi-two-body decays ${B}_{(s)}\ensuremath{\rightarrow}{D}_{(s)}h{h}^{\ensuremath{'}}$ ($h,{h}^{\ensuremath{'}}={\ensuremath{\pi},K}$) perturbative QCD approach, stemming from quark flavor changing $\overline{b}\ensuremath{\rightarrow}\overline{c}{q}_{2}{\overline{q}}_{1}$ $\overline{b}\ensuremath{\rightarrow}c\overline{{q}_...

In this paper, we resolve a conjecture of Green and Liebeck (2019) [3] on codes in PGL(2,q). To be specific, show that: if D is dihedral subgroup order 2(q+1) G=PGL(2,q), A={g∈G:gq+1=1,g2≠1}, then λG=A⋅D, where λ=q or q−1 according as q even odd.

We give four new extension theorems for linear codes over Fq: (a) For q = 2, h ≥ 3, every [n, k, d]q code with d odd whose weights are congruent to 0 or d (mod q/2) is extendable. (b) For q = 2, h ≥ 3, every [n, k, d]q code with gcd(d, q) = 2 whose weights are congruent to 0 or d (mod q) is doubly extendable. (c) For integers h,m, t with 0 ≤ m < t ≤ h and prime p, every [n, k, d]q code with gcd...

In this paper, we study the number of $\mathbb F_{q^n}$-rational points on affine curve $\mathcal{X}_{d,a,b}$ given by equation $$ y^d=ax\text{Tr}(x)+b,$$ where $\text{Tr}$ denote trace function from F_{q^n}$ to F_{q}$ and $d$ is a positive integer. particular, present bounds for F_{q}$-rational and, cases satisfies natural condition, explicit formulas rational are obtained. Particularly, compl...

There are a number of results concerning the generation of a collineation group by two of its elements. A. A. Albert and J. Thompson [1] were the first to exhibit two elements generating the little projective group PSL(d, q) of PG(d − 1, q) (for each d and q). According to a theorem of W. M. Kantor and A. Lubotzky [8], “almost every” pair of its elements generates PSL(d, q) as qd→∞ (asymptotica...

Proof of Lemma 5: Because both D(q||u) = D(1− q||1− u) and GF(q||u) = GF(1− q||1− u), we can assume u ≥ q without loss of generality. Keeping q fixed, we want to show that the result holds for all u. Note that D(q||q) = GF(q||q) = 0. Thus, differentiating with respect to u, it is sufficient to prove that GF ′(q||u) ≥ D′(q||u)/2 for all u ≥ q,u ≤ 1. We change variables by setting y = u−q. We use...

We prove that | ∑{ d≤x, (d,q)=1 μ(d)/d| ≤ 2.4 (q/φ(q))/ log(x/q) for every x > q ≥ 1 and similar estimates for the Liouville functions. We give also better constants when x/q is larger.