In virtue of the Belyi Theorem a complex algebraic curve can be defined over the algebraic numbers if and only if the corresponding Riemann surface can be uniformized by a subgroup of a Fuchsian triangle group. Such surfaces are known as Belyi surfaces. Here we study certain natural actions of the alternating groups ${\mathrm{A}}_n$ on them. We show that they are symmetric and calculate the number of connected components, called ovals, of the corresponding real forms. We show that all symmetries with ovals are conjugate and we calculate the number of purely imaginary real forms both in case of ${\mathrm{A}}_n$ considered here and ${\mathrm{S}}_n$ considered in an earlier paper [2].

Let $\varphi$ be an analytic self-map of the open unit disk $\mathbb{D}$ in the complex plane. Such a map induces through composition the linear composition operator $C_{\varphi }:H\left(\mathbb{D}\right)\to H\left(\mathbb{D}\right)$. The eigenvalues and the spectrum of such an operator acting on different spaces of analytic functions have been investigated in several articles, see e.g. [1], [8], [16], [28] and [29]. In this article we continue this line of research by combining the composition operator with the differentiation $D:H\left(\mathbb{D}\right)\to H\left(\mathbb{D}\right),f\to f\prime$. Then we obtain two linear operators $D{C}_{\varphi }:H\left(\mathbb{D}\right)\to H\left(\mathbb{D}\right),f\mapsto \varphi \prime \left(f\prime \circ \varphi \right)$ and $C_{\varphi }D:H\left(\mathbb{D}\right)\to H\left(\mathbb{D}\right),f\mapsto f\prime \circ \varphi$. Now, we calculate the eigenvalues of the operators $D{C}_{\varphi }$ and $C_{\varphi }D$.

In this present work, we give an characterization of quasi-symmetric Laguerre-Hahn orthogonal polynomial sequences of class one through the study of the differential functional equation fulfilled by its corresponding regular linear form.

In this article we present a construction of error correcting codes, that have representation as very sparse matrices and belong to the class of Low Density Parity Check Codes. LDPC codes are in the classical Hamming metric. They are very close to well known Shannon bound. The ability to use graphs for code construction was first discussed by Tanner in 1981 and has been used in a number of very effective implementations. We describe how to construct such codes by using special a family of graphs introduced by Ustimenko and Woldar. Graphs that we used are bipartite, bi-regular, very sparse and do not have short cycles $C_4$. Due to the very low density of such graphs, the obtained codes are fast decodable. We describe how to choose parameters to obtain a desired code rate. We also show results of computer simulations of BER (bit error rate) of the obtained codes in order to compare them with other known LDPC codes.

Në këtë artikull ne trajtojmë se si mësimdhënia e matematikës në shkollat 9-vjeçare dhe të mesme mund të përmirësohet në mënyrë të ndjeshme kur motivimi i koncepteve dhe ideve bëhet nëpërmjet problemeve klasike dhe historisë së matematikës. Kjo përmirëson intuitën e studentëve, zgjon kuriozitetin e tyre, shmang mësimin përmendësh të formulave të panevojshme, dhe i vendos konceptet në këndvështrimin e duhur historik. Ne ilustrojmë me disa probleme si diagonalizimi i formave kuadratike, integralet eliptike, dallorin e polinomeve me gradë të lartë, dhe disa ndërtime gjeometrike.

In this paper we discuss how teaching of mathematics for middle school and high school students can be improved dramatically when motivation of concepts and ideas is done through the classical problems and the history of mathematics. This method improves intuition of students, awakens their curiosity, avoids memorizing useless formulas, and put concepts in a historical prospective. To illustrate we show how diagonalizing quadratic forms, elliptic integrals, discriminants of high degree polynomials, and geometric constructions can be introduced successfully in high school level.