A library to create (new) variational integrators to arbitrary order by means of a computer algebra system is presented. The library provides an interface to design as well as analyse variational integrators for dynamical systems, either with non-conservative forces or without.

Our intpakX package extends the computer algebra system Maple. It allows, e.g., verified numerical calculations (computer-assisted proofs) built on arbitrary precision interval operations. Up to now, only the basic operations are supported in a guaranteed way. Concerning higher mathematical functions supported in Maple, there are no data about their accuracies available/published. Thus, it is not possible or at least very hard to build arbitrary precision interval functions using Maple’s intrinsic mathematical functions (nevertheless, intpakX offers such function implementations using some guard digits in an experimental way, which - of course - is not really a reliable mathematical approach).

On the other hand there are software packages supporting reliable multiple precision interval functions like C-XSC, the MPFR and the MPFI libraries, and others. In the talk we discuss the features of some of these libraries in detail. We emphasize the different approaches (arbitrary precision arithmetic, staggered correction arithmetic, functions only for real arguments, functions for complex arguments, ...) and the most important resulting properties of the corresponding implementations. We also compare their performance and we comment on the actual integration of several of these libraries in C-XSC. The missing step is to bring together C-XSC and computer algebra packages like Maple and Mathematica. Combining fast verification methods and symbolic computations deeply extends the range of applications of rigorous mathematical methods.

We propose some cryptographical algorithms based on finite $BN$-pair $G$ defined over the fields $F_q$. We convert the adjacency graph for maximal flags of the geometry of group $G$ into a finite Tits automaton by special colouring of arrows and treat the largest Schubert cell $\mathrm{Sch}=F_q^N$ on this variety as a totality of possible initial states and a totality of accepting states at a time. The computation (encryption map) corresponds to some walk in the graph with the starting and ending points in Sch. To make algorithms fast we will use the embedding of geometry for $G$ into Borel subalgebra of corresponding Lie algebra. We consider the induced subgraph of adjacency graph obtained by deleting all vertices outside of largest Schubert cell and corresponding automaton (Schubert automaton). We consider the following symbolic implementation of Tits and Schubert automata. The symbolic initial state is a string of variables $x_{\alpha }$, where roots $\alpha$ are listed according Bruhat order, choice of label will be governed by linear expression in variables $x_{\alpha }$, where $\alpha$ is a simple root.

Conjugations of such nonlinear map with element of affine group acting on $F_q^N$ can be used in Diffie-Hellman key exchange algorithm based on the complexity of group theoretical discrete logarithm problem in case of Cremona group of this variety. We evaluate the degree of these polynomial maps from above and the maximal order of this transformation from below. For simplicity we assume that $G$ is a simple Lie group of normal type but the algorithm can be easily generalised on wide classes of Tits geometries. In a spirit of algebraic geometry we generalise slightly the algorithm by change of linear governing functions for rational linear maps.

We study the singular locus on the algebraic surface ${\mathfrak{S}}_n$ of genus $2$ curves with a $\left(n,n\right)$-split Jacobian. Such surface was computed by Shaska in [15] for $n=3$, and Shaska at al. in [3] for $n=5$. We show that the singular locus for $n=2$ is exactly th locus of the curves of automorphism group $D_4$ or $D_6$. For $n=3$ we use a birational parametrization of the surface ${\mathfrak{S}}_3$ discovered in [15] to show that the singular locus is a $0$-dimensional subvariety consisting exactly of three genus $2$ curves (up to isomorphism) which have automorphism group $D_4$ or $D_6$. We further show that the birational parametrization used in ${\mathfrak{S}}_3$ would work for all $n\ge 7$ if ${\mathfrak{S}}_n$ is a rational surface.

We say that the sequence $g_n,n\ge 3,n\to \infty$ of polynomial transformation bijective maps of free module $K^n$ over commutative ring $K$ is a sequence of stable degree if the order of $g_n$ is growing with $n$ and the degree of each nonidentical polynomial map of kind ${g_n}^k$ is an independent constant $c$. A transformation $b=\tau {g_n}^k{\tau }^{-1}$, where $\tau$ is affine bijection, $n$ is large and ”$\mathrm{k}$” is relatively small, can be used as a base of group theoretical Diffie-Hellman key exchange algorithm for the Cremona group $C\left(K^n\right)$ of all regular automorphisms of $K^n$. The specific feature of this method is that the order of the base may be unknown for the adversary because of the complexity of its computation. The exchange can be implemented by tools of Computer Algebra (symbolic computations). The adversary can not use the degree of righthand-side in $b^x=d$ to evaluate unknown $x$ in this form for the discrete logarithm problem.

In the paper we introduce the explicit constructions of sequences of elements of stable degree for cases $c=4$ for each commutative ring $K$ containing at least $3$ regular elements and discuss the implementation of related key exchange and public key algorithms.

There is produced a parallel algorithm for symbolic solving systems of partial differential equations by means of multivariate Laplace-Carson transform. There is considered a system of n equations with m as the greatest order of partial derivatives and right hand parts of a special type. Initial and conditions are input. As a result of Laplace-Carson transform of the system according to initial condition we obtain an algebraic system of equations. A method to obtain compatibility conditions is discussed.

We are working with matrices over a ring $K\left[\partial ;\sigma ,\vartheta \right]$ of Ore polynomials over a skew field $K$. Extending a result of [18] for usual polynomials it is shown that in this setting the Hermite and Popov normal forms correspond to Gröbner bases with respect to certain orders. The FGLM algorithm is adapted to this setting and used for converting Popov forms into Hermite forms and vice versa. The approach works for arbitrary, that is, not necessarily square matrices where we establish termination criteria to deal with infinitely dimensional factor spaces.

In the paper we discuss the group theoretical algorithm of Diffie - Hellman key exchange in the cases of symmetrical group $S_{p^n}$ and more general Cremona group of polynomial automorphisms of free module ${\mathbb{K}}^n$ over arbitrary commutative ring $\mathbb{K}$. We show that conjugation of affine map with nonlinear polynomial map $f$ can be element of large order and small degree. Same properties hold for each element of cyclic group generated by such elements. We consider some algorithms for generation of subgroups of large order and small degree of their elements.

In this note we construct special types of rings $A_{\max ,\mathrm{n}}$ which are used in sequel work to define new types of families of continuous Fontaine sheaves. We also study the maps ${\theta }_n$ and $q_n$ providing explicit description of their kernels. Finally, we implement an algorithm which leads to the computation of these kernels.

To improve the teaching of mathematics to students of science and engineering, an holistic approach strongly based on the use of software for symbolic computation with numerical and graphical capabilities is advocated. Within three semesters the quality and quantity of mathematics understood and implementable by those students can be significantly enhanced with this approach, which was tested during an accelerated course that covered all pertinent content within eight weeks.