Welcome to this first special issue of the Tbilisi Mathematical Journal (TMJ). Special in that it is devoted to a single topic; and also because it deals with one of the most important developments of todays mathematics: Symbolic Computation.

In 2010, Snolb [9] studied the structure of nilpotent Lie algebras admitting a Levi extension. As a corollary of the results therein, it is shown that the classes of characteristically nilpotent or filiform Lie algebras do not admit Levi extensions. The paper ends by asking for the possibility of finding series of nilpotent Lie algebras in arbitrary dimension not being abelian or Heisenberg and allowing such extensions. Our goal in this work is to present computational examples of this type of algebras by using Sage software. In the case of nilpotent Lie algebras admitting $\mathfrak{sl}_2(k)$ as Levi factor special constructions will be given by means of Sage routines based on transvections over $\mathfrak{sl}_2(k)$-irreducible modules.

Let $f:GF(2)^n \rightarrow GF(2)$ be a monotone Boolean function. Associated to $f$ is the Cayley graph $X$ whose vertices correspond to points of $GF(2)^n$ and whose edges correspond to pairs of vectors $(v,w)$ whose sum is in the support of $f$. The spectrum of $X$ (the set of eigenvalues of its adjacency matrix) can be computed in terms of the Walsh-Hadamard transform of $f$. We show that if $f$ is atomic, the adjacency matrix of $X$ is singular if and only if the support of $f$ has an even number of elements. We ask whether it is true that for every even monotone function the adjacency matrix of the Cayley graph must be singular. We give an example in dimension $n=6$ to show that the answer to this question is no. We use Sage to compute some examples of monotone Boolean functions, their Cayley graphs, and the graph spectra. We include some interesting characterizations of monotone functions. We give some conditions on a monotone function that imply that the function is not bent. Finally, we ask whether it is true that no even monotone function is bent, for $n>2$.

As the literature points out, students can benefit from the study of a detailed answer of an exercise after trying to solve it in a traditional pen-and-paper way. This encouraged the creation of a new software package for the growing community of Sage users. Thus, this package aims at enabling the author to add, change and delete parameterized exercise templates, with detailed answer, from one or more personal databases. Moreover, it allows the production of new exercises in a booklet style or exam form.

This paper describes the package, presenting some examples of its functionalities.

In this paper, we discuss informally some practical experiences and results of getting students to use Sage while studying and solving problems in engineering mathematics within an existing curriculum and with minimum effort on our part. We had no lofty goals. We were simply trying to see if technology now existed for our students to effectively and efficiently check their by-hand solutions to problems in single and multivariable calculus in order to help them pass written exams. With Sage, our students could quickly and freely start by just checking calculations and/or create interactive 3D plots of all types to help them visualize and explore. This did seem to help them with their exams. And we found that encouraging students to use this method of study also helped them to think about the process of problem solving in an environment particularly well adapted for students of engineering. We explain by detailing an example.

We describe a computational approach to the verification of Maeda's conjecture for the Hecke operator $T_2$ on the space of cusp forms of level one. We provide experimental evidence for all weights less than 14000, as well as some applications of these results. The algorithm was implemented using the mathematical software Sage, and the code and resulting data were made freely available.

The aim of this work is to present a new program written in Sage which allows us to visualize the attraction basins associated to the end points of a discrete semi-flow induced by a rational function on the Riemann sphere by using its geometry and complex structure. One interesting novelty brought by the developed program is that it is able to plot fractals not only in a determined rectangular area of the complex plane, but also on the whole surface of the Riemann sphere. Another advantage of the program is that it permits us to visualize not only the attraction basins of fixed points, but also the basins of end points associated with periodic points. In addition, some applications of the described algorithms to Numerical Analysis and Fractal Geometry are suggested.

Numpy and SciPy are program libraries for the Python scripting language, which apply to a large spectrum of numerical and scientific computing tasks. The Sage project provides a multiplatform software environment which enables one to use, in a unified way, a large number of software components, including NumPy and SciPy, and which has Python as its command language. We review several examples, typical for scientific computation courses, and their solution using these tools in the Sage environment.

The aim of this paper is to review the Mathematics Grammar Library (MGL) approach to multilingual mathematics and to present $\texttt{gfsage}$, a prototype of a multilingual interface to $\textsf{Sage}$.