We provide a formal proof of equivalence between the class of fractals created by Recursive Turtle Programs (RTP) and Iterated Affine Transformations (IAT). We begin by reviewing RTP ( a geometric interpretation of non-bracketed L-systems with a single production rule) and IAT (Iterated Function Systems restricted to affine transformations). Next, we provide a simple extension to RTP that gener...

this article presents a unified approach to the abstract notions of partial convolution and involution in $l^p$-function spaces over semi-direct product of locally compact groups. let $h$ and $k$ be locally compact groups and $tau:hto aut(k)$ be a continuous homomorphism.  let $g_tau=hltimes_tau k$ be the semi-direct product of $h$ and $k$ with respect to $tau$. we define left and right $tau$-c...

In this paper, we establish the expected order of magnitude $k$th-moment quadratic Dirichlet $L$-functions associated to hyperelliptic curves genus $g$ as well prime moduli over a fixed finite field $\mathbb{F}_{q}$ for all real $k \geq 0$.

Almost everywhere convergence on the solution of Schr\"odinger equation is an important problem raised by Carleson in harmonic analysis. In recent years, this was essentially solved building sharp $L^p$-estimate maximal function. Du-Guth-Li \cite{DGL} proved $L^p$-estimates for all $p \geq 2$ $\mathbb{R}^{2+1}$. Du-Zhang \cite{DZ} $L^2$-estimate $\mathbb{R}^{n+1}$ with $n 3$, but $p>2$ function...