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COVID19 is the most impactful pandemic of recent times worldwide. It is a highly infectious disease that is caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2 virus), To date there is specific drug nor vaccination against COVID19. Therefor the need for novel and pioneering anti-COVID19 is of paramount importance. In this direction, computer-aided drug design constitutes a very promising antiviral approach for the discovery and analysis of drugs and molecules with biological activity against SARS-CoV2. In silico modelling takes advantage of the massive amounts of biological and chemical data available on the nature of the interactions between the targeted systems and molecules, as well as the rapid progress of computational tools and software. Herein, we describe the potential of the merging of mathematical modelling, artificial intelligence and learning techniques into seamless computational pipelines for the rapid and efficient discovery and design of potent anti- SARS-CoV-2 modulators.

We extend the concept of Grobner bases to relative Grobner bases for ideals in and modules over quotient rings of a polynomial ring over a field. We develop a “relative” variant of both Buchberger’s criteria for avoiding reductions to zero and Schreyer’s theorem for a Grobner basis of the syzygy module. As main contribution, we then introduce the novel notion of relative involutive bases and present an algorithm for their explicit construction. Finally, we define the new notion of relatively quasi-stable ideals and exploit it for the algorithmic determination of coordinates in which finite relative Pommaret bases exist.

Computer algebra methods were used to find the roots of a nonlinear algebraic system that determines the equilibrium orientations for a system of two bodies, connected by a spherical hinge, that moves along a circular orbit under the action of gravitational torque. To determine the equilibrium orientations of two connected bodies the system of 12 algebraic equations was decomposed using algorithms for Grobner basis construction. The number of equilibria was found by analyzing the real roots of the algebraic equations from the calculated Grobner basis. Evolution of the conditions for equilibria existence in the dependence of the parameter of the problem was investigated. The effectiveness of the algorithms for Grobner basis construction was analyzed depending on the number of parameters for the problem under consideration.

In this paper we consider the Voronoi diagram of a finite family of parallel half-lines, with the same orientation, constrained to a compact domain $${\mathscr {D}}_{0} \subset {\mathbb {R}}^3$$ , with respect to the Euclidean distance. We present an efficient approximation algorithm for computing such VD, using a subdivision process, which produces a mesh representing the topology of the VD in $${\mathscr {D}}_{0}$$ . The computed topology may not be correct for degenerate configurations or configurations close to degenerate. In this case, the output is a valid partition, which is close to the exact partition in Voronoi cells if the input data were given with no error. We also present the result of an implementation in Julia language with visualization using Axl software (axl.inria.fr) of the algorithm. Some examples and analysis are shown.

In the twenty-first century, our life will increasingly depend on software-based products and complex interconnected systems. Thus, the quality and security of software-based systems is a world-wide concern. Combinatorial testing is a versatile methodology for finding errors (bugs) and vulnerabilities in software-based systems. This paper offers a review of combinatorial testing. Combinatorial testing (CT) methods evolved from investigations which looked like factorial experiments (FE) with pass/fail outcomes. We will discuss the similarities and differences between FE and CT. Use of CT for detecting errors (bugs) in software-based systems has gained significant interest from the international software testing community. Many successful results have been reported from the use of CT to detect software errors in aerospace, automotive, defense, cybersecurity, electronic medical systems, and financial service industries. Now, combinatorial testing methods are being increasingly used to investigate vulnerabilities in software-based systems. Combinatorial testing could be useful in detecting errors and security vulnerabilities in Internet of Things, Autonomous Systems, and Artificially Intelligent Software.

Implicit locus equations in GeoGebra allow the user to do experiments with generalization of the concept of ellipses, namely with $n$-ellipses. By experimenting we obtain a geometric object that is very similar to a set of two circles., Comment: 13 pages, 14 figures

A lockdown was ordered in Spain on March 2020 due to the COVID-19 pandemic. The first author, advised by the second author, developed a tale and video (in Spanish) with a simplified explanation of virus propagation for their teenager son, justifying the need to stay at home. The tale and video relate the spread of the virus to fractal trees and aim to raise awareness about the transmission of the disease. The video is available from the web page of the Instituto de Matematica Interdisciplinar of the first author’s university. The code was implemented in Scratch 3 and takes advantage of the “Turtle Geometry” (there is an ulterior version using Maple, available from Mapleprimes). This article includes the English version of the original tale, describes the Scratch 3 code, and details possible derived STEM activities. We plan to experiment them in the classroom during the 2020–2021 academic year.

In this paper we explain why we decided to abandon the traditional classroom instruction of Linear Algebra and switched to a self-paced, instructor-assisted course format. We also describe our experience with creating a self-paced course in NCLab, and with teaching it at a tier-1 research university for three consecutive semesters. The results of a student survey and student comments are presented.

We find formulas for the generating functions enumerating bargraphs or polyominoes associated with Catalan words according to the number of interior lattice vertices contained within and to the number of vertices of a given degree. As a consequence, one obtains simple explicit formulas for the total values of these statistics over the set of Catalan words of length n.

A symbolic geometry system such as Geometry Expressions can generate symbolic measurements in terms of indeterminate inputs from a geometric figure. It has elements of dynamic geometry system and elements of automated theorem prover. Geometry Expressions is based on the analytical geometry method. We describe the method in the style used by expositions of semi-synthetic theorem provers such as the area method. The analytical geometry method differs in that it considers geometry from a traditional Euclidean/Cartesian perspective. To the extent that theorems are proved, they are only proved for figures sufficiently close to the given figure. This clearly has theoretical disadvantages, however they are balanced by the practical advantage that the geometrical model used is familiar to students and engineers. The method decouples constructions from geometrical measurements, and thus admits a broad variety of measurement types and construction types. An algorithm is presented for automatically deriving simple forms for angle expressions and is shown to be equivalent to a class of traditional proofs. A semi-automated proof system comprises the symbolic geometry system, a CAS and the user. The user’s inclusion in the hybrid system is a key pedagogic advantage. A number of examples are presented to illustrate the breadth of applicability of such a system and the user’s role in proof.

In this paper, the concept of tropical differential Grobner basis is introduced, which is a natural generalization of the tropical Grobner basis to the recently introduced tropical differential algebra. Like the differential Grobner basis, the tropical differential Grobner basis generally contains an infinite number of elements. We give a Buchberger style criterion for the tropical differential Grobner basis. For differential ideals generated by homogeneous linear differential polynomials with constant coefficients, we give a complete algorithm to compute the finite tropical differential Grobner basis.

The aim of this paper is to investigate upper bounds for the maximum degree of the elements of any minimal Janet basis of an ideal generated by a set of homogeneous polynomials. The presented bounds depend on the number of variables and the maximum degree of the generating set of the ideal. For this purpose, by giving a deeper analysis of the method due to Dube (SIAM J Comput 19:750–773, 1990), we improve (and correct) his bound on the degrees of the elements of a reduced Grobner basis. By giving a simple proof, it is shown that this new bound is valid for Pommaret bases, as well. Furthermore, based on Dube’s method, and by introducing two new notions of genericity, so-called J-stable position and prime position, we show that Dube’s (new) bound holds also for the maximum degree of polynomials in any minimal Janet basis of a homogeneous ideal in any of these positions. Finally, we study the introduced generic positions by proposing deterministic algorithms to transform any given homogeneous ideal into these positions.

The torsion module of Kahler differential forms is considered in the context of symbolic computation. Relations between logarithmic differential forms and logarithmic vector fields are investigated. As an application, an effective method is proposed for computing torsion differential forms associated with a hypersurface with an isolated singularity. The main ingredients of the proposed method are logarithmic vector fields and local cohomology.

In this paper, we study the algebraic, rational and formal Puiseux series solutions of certain type of systems of autonomous ordinary differential equations. More precisely, we deal with systems which associated algebraic set is of dimension one. We establish a relationship between the solutions of the system and the solutions of an associated first order autonomous ordinary differential equation, that we call the reduced differential equation. Using results on such equations, we prove the convergence of the formal Puiseux series solutions of the system, expanded around a finite point or at infinity, and we present an algorithm to describe them. In addition, we bound the degree of the possible algebraic and rational solutions, and we provide an algorithm to decide their existence and to compute such solutions if they exist. Moreover, if the reduced differential equation is non trivial, for every given point (x0,y0)∈C2, we prove the existence of a convergent Puiseux series solution y(x) of the original system such that y(x0)=y0., Agencia Estatal de Investigación, Ministerio de Economía y Competitividad, Austrian Science Fund

We consider some perturbation of the Chebyshev polynomials of second kind obtained by modifying one of its recurrence coefficients at an arbitrary order. The goal of this work is to point out that perturbed Chebyshev polynomials of fixed degree and different values of parameters of perturbation have some common points that are zeros of two Chebyshev polynomials of second kind of lower degrees. These common points can be simple or double. We identify the cases in which they are common zeros.

Operator theory has many applications in several main scientific research areas (structural mechan-ics, aeronautics, quantum mechanics, ecology, probability theory, electrical engineering, among others) and theimportance of its study is globally acknowledged. On the study of the operator’s kernel some progress has beenachieved for some specific classes of singular integral operators whose properties allow the use of particular strate-gies. However, the existing algorithms allow, in general, to study the dimension of the kernel of some classes ofsingular integral operators but are not designed to be implemented on a computer. The main goal of this paper is toshow how the symbolic and numeric capabilities of a computer algebra system can be used to study the kernel ofspecial classes of paired singular integral operators with essentially bounded coefficients defined on the unit circle.It is described how some factorization algorithms can be used to compute the dimension of the kernel of specialclasses of singular integral operators. The analytical algorithms [ADimKerPaired-Scalar], [AKerPaired-Scalar],and [ADimKerPaired-Matrix] are presented. The design of these new algorithms was focused on the possibility ofimplementing on a computer all the extensive symbolic and numeric calculations present in the algorithms. Forthe essentially bounded hermitian coefficients case, there exist some relations with Hankel operators. The papercontains some interesting and nontrivial examples obtained with the use of a computer algebra system. Fundação para a Ciência e Tecnologia (Portugal) through the Center for Functional Analysis, Linear Structures and Applications info:eu-repo/semantics/publishedVersion

This paper proposes a new way of testing the effect of labor market reforms in the employment adjustment at the macro level. Following the approach to hysteresis based on the presence of non-convex costs of adjustment, a switching aggregate employment equation is estimated, with an unknown splitting factor, from a computational implementation of the linear play hysteresis operator. The play hysteresis operator describes a dynamics where non-convex adjustment costs create intervals of weak reaction of employment to small variations in forcing variables, while spurts in employment adjustment may occur as a consequence of a large, or cumulative small shocks. The main contribution of this paper is to extend an existing approach for the case where there are disruptive breaks, by accommodating changes in the value of the switching parameter of the employment equation whenever labor market reforms are present. Numerical experiments on the Portuguese economy are performed.

We introduce a fractional order SIRS model with non-linear incidence rate. Existence of a unique positive solution to the model is proved. Stability analysis of the disease free equilibrium and positive fixed points are investigated. Finally, a numerical example is presented., This is a preprint of a paper whose final and definite form is with 'Math. Comput. Sci.', ISSN 1661-8270 (print), ISSN 1661-8289 (electronic), available at [https://www.springer.com/journal/11786]. Submitted 1-Jun-2019; Revised 5-Feb-2020; Accepted 6-Feb-2020

The ability to read, write and manipulate complex mathematical expressions is assumed in most user interfaces for programs designed for authoring, collaboration and reasoning in mathematics. However, little is actually known about how people cognitively process and integrate information when solving complex mathematical problems. In this paper eye-tracking was used to examine how people read and integrate information from mathematical symbols and complex formula, with eye fixations being used as a measure of their current focus of attention. We studied two groups of subjects: mathematically “Expert” and “Non-Expert”. Each participant in the study was presented with a series of stimuli in the form of mathematical problems and their eyes were tracked as they worked through the problems mentally. We discuss the performance differences between Experts and Non-Experts, and between subjects who completed problems Correctly and Incorrectly. From these examinations, we were able to demonstrate differences in both the comprehension and problem-solving approach of the two groups, with the results suggesting that what information is selected, and how, is responsible for a particularly large portion of success in solving such problems. This study has implications in mathematical software usability and accessibility, in particular for educational software, as comprehension of mathematical notation and formalism is often inherent in the implementation of structured mathematical software interfaces.

The Border Basis Algorithm (BBA) still suffers from the lack of analogues of Buchberger’s criteria for avoiding unnecessary reductions. In this paper we develop a signature based technique which provides a first remedial step: signature bounds allow us to recognize multiple reductions of the same ancestor polynomial. The new signature based algorithm is then combined with the Boolean BBA for ideals of Boolean polynomials. Experiments show that it is at least 5 times faster than the standard Boolean BBA.

Kehrein and Kreuzer (J Pure Appl Algebra 205(2):279–295, 2006) presented an algorithm, referred to as the BorderBasis algorithm, for computing border bases. The main objective of this paper is to improve this computation by applying new structures from the theory of Grobner bases. In doing so, based on the method developed by Gao et al. (Math Comput 85(297):449–465, 2016), a new variant of the BorderBasis algorithm is proposed to compute simultaneously a border basis and also a Grobner basis for the syzygy module of the input polynomials (this pair of bases will be called a coupled border basis). The new algorithm, and also the original form of the BorderBasis algorithm, have been implemented in the computer algebra systems Maple and ApCoCoA. We compare the performance of our implementation of these algorithms via a set of benchmark polynomials. Our experiments show that our algorithm performs more efficiently than the original BorderBasis algorithm.

Hyperbolic polynomials are real multivariate polynomials with only real roots along a fixed pencil of lines. Testing whether a given polynomial is hyperbolic is a difficult task in general. We examine different ways of translating hyperbolicity into nonnegativity conditions, which can then be tested via sum-of-squares relaxations., 12 pages, 1 figure; minor changes in Section 2 and a few corrections

The aim of this paper is to present the research book “The Grobner Cover” Montes (The Grobner Cover, ACM-series, Springer, Berlin, 2019) recently published. This book is divided into two parts, one theoretical and one focusing on applications, and offers a complete description of the Canonical Grobner Cover, to the author’s best knowledge, the most accurate algebraic method for discussing parametric polynomial systems. It also includes applications to the automatic deduction of geometric theorems, loci computation and envelopes. The theoretical part is a self-contained exposition on the theory of Parametric Grobner Systems and Bases. It begins with Weispfenning introduction of Comprehensive Grobner Systems (CGS), a fundamental contribution made in 1992, and provides a complete description of the Canonical Grobner Cover (GC), which includes a canonical discussion of a set of parametric polynomial equations developed in Montes, Wibmer (J Symb Comput 45:1391–1425, 2010). In turn, the application part selects three problems for which the Grobner Cover offers valuable new perspectives. The automatic deduction of geometric theorems (ADGT) becomes fully automatic and straightforward using GC, representing a major improvement on all previous methods. In terms of loci and envelope computation, GC makes it possible to introduce a taxonomy of the components and automatically compute it. The book also generalizes the definition of the envelope of a family of hyper-surfaces, and provides algorithms for its computation, as well as for discussing how to determine the real envelope. All the algorithms described in the book have also been included in the Singular software library grobcov.lib implemented by the author and H. Schonemann, the book serving also as User Manual for the library.

Grothendieck local residue is considered in the context of symbolic computation. An effective method based on the theory of holonomic D-modules is proposed for computing Grothendieck local residues. The key is the notion of Noether operator associated to a local cohomology class. The resulting algorithm and an implementation are described with illustrations.

This article is related to the problem of the construction of planetary motion theory. We have expanded the Hamiltonian of the four-planetary problem into the Poisson series in osculating elements of the second Poincare system. The series expansion is constructed up to the third degree of the small parameter. The averaging procedure of the Hamiltonian is performed by the Hori–Deprit method. It allows to eliminate short-periodic perturbations and sufficiently increase time step of the integration of the equations of motion. This method is based on Lie transformation theory. The equations of motion in averaged elements are constructed as the Poisson brackets of the averaged Hamiltonian and corresponding orbital element. The transformation between averaged and osculating elements is given by the change-variable functions, which are obtained in the second approximation of the Hori–Deprit method. We used computer algebra system Piranha for the implementation of the Hori–Deprit method. Piranha is an echeloned Poisson series processor authored by F. Biscani. The properties of the obtained series are discussed. The numerical integration of the equations of motion is performed by Everhart method for the Solar system’s giant planets.

The object of this research is the process of small oscillations of a three-dimensional elastic pendulum, tuned to a 1:1:2 resonance to vertical and horizontal oscillations. The purpose is to develop a symbolic algorithm for calculating small oscillations of the pendulum. The primary efforts are aimed at building a software package that generates formulas that approximate the movements of the pendulum with sufficient accuracy. The algorithms of this work are developed based on the resonant normal form method. The importance of the study is due to the wide range of applicability of the method of normal forms for constructing approximations of periodic and conditionally periodic local families of solutions of ODEs. For high-dimensional resonance systems, the technique is a generalization of the Poincare–Linstedt method, and for coarse systems, the Carleman linearization method. The reliability of the results is confirmed by comparisons with the results of numerical solutions. Results can be useful for specialists working at the interface of computational mathematics and continuum mechanics. Approaches and methodologies developed in this paper can be applied to solve various modeling problems.

In this paper we present some examples of calculation the Lebesgue outer measure of some subsets of $$\mathbb {R}^2$$R2 directly from definition 1. We will consider the following subsets of $$\mathbb {R}^2$$R2: $$\displaystyle \{(x, y)\in \mathbb {R}^2: 0\le y \le x^2, x\in [0, 1]\}$${(x,y)∈R2:0≤y≤x2,x∈[0,1]}, $$\displaystyle \{(x, y)\in \mathbb {R}^2: 0\le y \le \exp (-x), x\ge 0\}$${(x,y)∈R2:0≤y≤exp(-x),x≥0}, $$\big \{ (x,y) \in \mathbb {R}^2: \ln x \le y \le 0, x\in (0, 1]\big \}$${(x,y)∈R2:lnx≤y≤0,x∈(0,1]}, $$\big \{ (x,y) \in \mathbb {R}^2: 0\le y \le 1/x, x\ge 1\big \}$${(x,y)∈R2:0≤y≤1/x,x≥1}, $$\displaystyle \{(x, y)\in \mathbb {R}^2: 0\le y \le \sin x, x\in [0, \pi /2]\}$${(x,y)∈R2:0≤y≤sinx,x∈[0,π/2]}, $$\displaystyle \{(x, y)\in \mathbb {R}^2: 0\le y \le \exp (x), x\in [0, 1]\}$${(x,y)∈R2:0≤y≤exp(x),x∈[0,1]}, $$\displaystyle \{(x, y)\in \mathbb {R}^2: 0\le y \le \ln (1-2r \cos x+r^2), x \in [0, \pi ]\}$${(x,y)∈R2:0≤y≤ln(1-2rcosx+r2),x∈[0,π]}, $$r>1$$r>1 and some others. We could not find any analogical examples in available literature (except for rectangle and countable sets), so this paper is an attempt to fill this gap. We calculate sums, limits and plot graphs and dynamic plots of needed sets and unions of rectangles sums of which volumes approximate Lebesgue outer measure of the sets, using Mathematica. We also show how to calculate the needed sums and limits by hand (without CAS). The title of this paper is very similar to the title of author’s article (Wojas and Krupa in Math Comput Sci 11:363–381, 2017) which deals with definition of Lebesgue integral but this paper deals with definition of Lebesgue outer measure instead.

The classical problem of three bodies of variable masses is considered in the case when two of the bodies are protoplanets and all the masses vary non-isotropically at different rates. The problem is analyzed in the framework of the planetary perturbation theory in terms of the osculating elements of aperiodic motion on quasi-conic sections. An algorithm for symbolic computation of the disturbing function and its expansion into power series in terms of the eccentricities and inclinations is discussed in detail. Differential equations describing the long-term evolution of the orbital parameters are derived in the form of Lagrange’s planetary equations. All the relevant calculations are done with the computer algebra system Wolfram Mathematica.

This submission presents a particular example of the use of the free dynamic mathematics software GeoGebra to determine the dual curve to the given curve and to inspect its properties. The example is aimed at students of mathematics teaching. The combination of dynamic geometry tools with computer algebra functions, namely the functions for the computation of Groebner bases for polynomial ideals and a function for eliminating variables from the system of polynomial equations, based also on the method of Groebner bases, allows them to apply naturally the knowledge they acquire during their study of mathematics.

We discuss the problem of fitting a smooth regular curve $$\gamma {:}[0,T]{\rightarrow }\mathbb {E}^n$$ based on reduced data$$Q_m = \{q_i\}_{i = 0}^m$$ in arbitrary Euclidean space $$\mathbb {E}^n$$. The respective interpolation knots $${\mathcal T} = \{t_i\}_{i = 0}^m$$ satisfying $$q_i = \gamma (t_i)$$ are assumed to be unknown. In our setting the substitutes $${\mathcal T}_{\lambda }=\{{\hat{t}}_i\}_{i = 0}^m$$ of $${{\mathcal {T}}}$$ are selected according to the so-called exponential parameterization governed by $$Q_m$$ and $$\lambda \in [0,1]$$. A modified Hermite interpolant $$\hat{\gamma }^H$$ introduced in Kozera and Noakes (Fundam Inf 61(3–4):285–301, 2004) is used here to fit $$(\hat{{\mathcal {T}}}_{\lambda },Q_m)$$. The case of $$\lambda = 1$$ (i.e. for cumulative chords) for general class of admissible samplings yields a sharp quartic convergence order in estimating $$\gamma {\in } C^4$$ by $${\hat{\gamma }}^H$$ [see Kozera (Stud Inf 25(4B–61):1–140, 2004) and Kozera and Noakes (Fundam Inf 61(3–4):285–301, 2004)]. As recently shown in Kozera and Wilkolazka (Math Comput Sci, 2018. https://doi.org/10.1007/s11786-018-0362-4) the remaining $$\lambda \in [0,1)$$ render a linear convergence order in $${\hat{\gamma }}^H\approx \gamma $$ for any $$Q_m$$ sampled more-or-less uniformly. The related analysis relies on comparing the difference $$\gamma -{\hat{\gamma }}^H\circ \phi ^H$$ in which $$\phi ^H$$ forms a special mapping between [0, T] and $$[0,{\hat{T}}]$$ with $${\hat{T}} = {\hat{t}}_m$$. In this paper: (a) several sufficient conditions enforcing $$\phi ^H$$ to yield a genuine reparameterization are first formulated and then analytically and symbolically simplified. The latter covers also the asymptotic case expressed in a simple form. Ultimately, the reformulated conditions can be algebraically verified and/or geometrically visualized, (b) additionally in Sect. 3, the sharpness of the asymptotics of $$\gamma -{\hat{\gamma }}^H\circ \phi ^H$$ [from Kozera and Wilkolazka (Math Comput Sci, 2018. https://doi.org/10.1007/s11786-018-0362-4)] is proved upon applying symbolic and analytic calculations in Mathematica.

Isoptic curves of plane curves are a live domain of study, mostly for closed, smooth, strictly convex curves. A technology-rich environment allows for a two-fold development: dynamical geometry systems enable us to perform experiments and to derive conjectures, and computer algebra systems (CAS) are the appropriate environments for an algebraic approach for determining isoptics, with its Grobner bases packages, amongst others. Closed Fermat curves in the affine plane present a specific problem: the variables of the involved polynomials represent the coordinates of points on the Fermat curve, which avoid the dense set of points with two rational coordinates (excepted 4 “trivial” points). Therefore, automated computation has to be performed over the field $$\mathbb {R}$$ of the real numbers, with which the Grobner bases packages do not work. Instead, other packages implemented in CASs have to be used. First, we present a study of the orthoptics of closed Fermat curves of even order. Then we proceed to an algebraic study of these curves using a CAS. The generalization to angles other than $$90^\circ $$ is performed afterwards, with an algebraic approach using support functions, then using numerical methods.

In this note, we apply some techniques developed in Bernal et al. (IEEE Trans Inf Theory 62(2):655–668, 2016; Adv Math Commun 10:459–474, 2016; IEEE Trans Inf Theory 2018. https://doi.org/10.1109/TIT.2018.2868446) to give a particular construction of bivariate Abelian Codes from cyclic codes, multiplying their dimension and preserving their apparent distance. We show that, in the case of cyclic codes whose maximum BCH bound equals its minimum distance the obtained abelian code verifies the same property; that is, the strong apparent distance and the minimum distance coincide. We finally use this construction to multiply Reed–Solomon codes to abelian codes.

The second author was one of the authors of a computer package written in Maple that could automatically generate railway maps of a network for any date. This package was presented at ACA’2008 and its design and implementation is described in a previous article. Each section of the network was colored accordingly to its characteristics. The position of the nodes (stations, junctions, $$\ldots $$) was obtained from a list of geographical coordinates. The work presented here deals with a similar although not identical case: now subway networks are treated as graphs with the help of a computer algebra system in order to obtain the diagrammatic map for any date. Most subway network maps follow more or less closely the ideas introduced by Harry Beck in his diagrammatic design of London subway map (the distances between stations and geographic orientation of the lines don’t have to be respected, as the clarity and the number of stations between two stations is the key information to be visualized). Therefore allocating nodes is far simpler, and we have decided to manually allocate the stations on a predefined grid and to automatize the drawing of the network. The situation is also simpler because all lines are double track and electrified. For instance in Madrid subway there are minor differences between lines, such as the kind of catenary (classic or rigid), the loading gauge (narrow/broad), $$\ldots $$ that will not be considered here. Each node and edge of the graph has dates associated: inauguration date/closure date–the latter if applies. The package takes advantage of the simplifications w.r.t. the previous article mentioned above and the features of Maple’sGraphTheory package. This way the approach, although general, can be implemented in relatively few lines of code. We know of no other similar works. The work is illustrated with the case of Madrid subway network, one of the biggest ones in the world.

We study a new (large) class of algebras (that was introduced in Bavula in Math Comput Sci 11(3–4):253–268, 2017)—the skew category algebras. Any such an algebra $$ \mathcal{C}(\sigma )$$C(σ) is constructed from a category $$ \mathcal{C}$$C and a functor $$\sigma $$σ from the category $$ \mathcal{C}$$C to the category of algebras. Criteria are given for the algebra $$ \mathcal{C}(\sigma )$$C(σ) to be simple or left Noetherian or right Noetherian or semiprime or have 1.

The aim of this work is to describe the structure and properties of the binary isodual codes having automorphisms of odd prime order and to present a method for their construction. As an example, we construct binary isodual codes of length 30 and minimum weight $$\ge \,7$$ having an automorphism of order 5.

We discuss how the geometric theory of differential equations can be used for the numerical integration and visualisation of implicit ordinary differential equations, in particular around singularities of the equation. The Vessiot theory automatically transforms an implicit differential equation into a vector field distribution on a manifold and thus reduces its analysis to standard problems in dynamical systems theory like the integration of a vector field and the determination of invariant manifolds. For the visualisation of low-dimensional situations we adapt the streamlines algorithm of Jobard and Lefer to 2.5 and 3 dimensions. A concrete implementation in Matlab is discussed and some concrete examples are presented., Comment: 14 pages, 3 figures. Math.Comput.Sci. (2019)

In this paper asymptotic behavior of the service process for the multi-channel network with input flow controlled by semi-Markov process is studied. Conditions in which a stationary regime exists are pointed out. Integral representation for generating function of the stationary distribution is constructed. In case of exponentially distributed service times limit Gaussian and diffusion Ornstein–Uhlenbeck processes for normalized service process of network processing under heavy traffic conditions are obtained.

In this paper, we present a new subclass $${\mathcal {T}}_{\varSigma }(\mu )$$ of bi univalent functions belong to $$\varSigma $$ in the open unit disc $${\mathcal {U}} =\left\{ z\, :\,\,z\in {\mathcal {C}}\,\,and \,\, |z

In this paper, we explore the application of attractive ellipsoid methodology (Poznyak et al. in Attractive ellipsoids in robust control. Springer, Berlin, 2014) on a new demand and supply model. The balance between the demand and supply is expressed by the Lin and Yang model (Li and Yang in Int J Bifurc Chaos 27(01):1750016, 2017), described by differential equations, even if in the such a demand-supply model, other factors change and have significant effect on demand-supply dynamics. Analysis is compared with the situation when the prices of most products do not stay close to their equilibrium values and the Li and Yang demand and supply model is described by both ordinary differential equation and differential algebraic equation system. To achieve a specific economic goal, we will be able to design a management strategy based on minimum size of the invariant attractive ellipsoid, associated with the dynamic system, with a good performance in the rejection of external disturbances. We can consider a new transformed problem instead of the original problem with respect to solvability and related questions. Theoretical results are illustrated by an example.

Grothendieck local residue is considered in the context of symbolic computation. Basic ideas of our approach are the use of local cohomology, holonomic D-modules, and Noether operators. An effective method is introduced for computing Grothendieck local residues of rational n-forms under shape basis condition. Resulting algorithms that avoid the use of Grobner bases on the Weyl algebra and an implementation are described. Some examples are also given for illustration.

Symmetry breaking is a crucial technique to solve many graph problems. However, current state-of-the-art techniques break graph symmetries only partially, causing search algorithms to unnecessarily explore many isomorphic parts of the search space. We study properties of perfect symmetry breaking for graph problems. One promising and surprising result on small-sized graphs—up to order five—is that perfect symmetry breaking can be achieved using a compact propositional formula in which each literal occurs at most twice. At least for small graphs, perfect symmetry breaking can be expressed more compactly than the existing (partial) symmetry-breaking methods. We present several techniques to compute and analyze perfect symmetry-breaking formulas.

The scheduling problem, which is the core of all approaches related to real-time systems, has received proper attention from the research community. However, while preemptive scheduling has benefited from most of the results to date, the more difficult case of non-preemptive scheduling is still lacking similar achievements. This paper is approaching non-preemptive scheduling from two different angles. First, the number of processors that would allow a feasible schedule for a given task set is analyzed, yielding both lower and upper limits which can be determined in polynomial time. Second, a hybrid scheduling algorithm, combining two widely known techniques, namely EDF and LLF, is proposed and tested. A common feature of both objectives is the transition from a single-instance task to a periodic task. The relationships between these two cases are investigated, resulting in a better understanding of periodic behavior.

Hamiltonian dynamical systems can be studied from a variety of viewpoints. Our intention in this paper is to show some examples of usage of two Maxima packages for symbolical and numerical analysis (pdynamics and poincare, respectively), along with the set of scripts for obtaining the code corresponding to graphical representations of Poincare sections, including animation movies.

One way to compute a GCD of a pair of multivariate polynomials is by finding a certain syzygy. We can weaken this to create an “approximate syzygy”, for the purpose of computing an approximate GCD. The primary tools are Grobner bases and optimization. Depending on specifics of the formulation, one might use quadratic programming, linear programming, unconstrained with quadratic main term and quartic penalty, or a penalty-free sum-of-squares optimization. There are relative strengths and weaknesses to all four approaches, trade-offs in terms of speed vs. quality of result, size of problem that can be handled, and the like. Once a syzygy is found, there is a polynomial quotient to form, in order to get an approximation to an exact quotient. This step too can be tricky and requires careful handling. We will show what seem to be reasonable formulations for the optimization and quotient steps. We illustrate with several examples from the literature.

A survey on the speed of real-time animation of loci and envelopes in the dynamic geometry software GeoGebra is presented.

We introduce the meandric colliers as planar self-avoiding closed continuous curves crossing the 2n arcs of a meander 2r times. We describe their construction from meandering curves obtained by replacing the horizontal line of an open meander with a concave or convex line and we study their properties and deal with their enumeration. We also consider the problem of refolding a meander of great order into a vector of components of a meandric collier.

Fractal dimension is a powerful tool employed as a measurement of geometric aspects. In this work we propose a method of topological fractal analysis for 2D binary digital images by using a graph-based topological model of them, called Homological Spanning Forest (HSF, for short). Defined at interpixel level, this set of two trees allows to topologically describe the (black and white) connected component distribution within the image with regards to the relationship “to be surrounded by”. This distribution is condensed into a rooted tree, such that its nodes are connected components determined by some special sub-trees of the previous HSF and the levels of the tree specify the degree of nesting of each connected component. We ask for topological auto-similarity by comparing this topological description of the whole image with a regular rooted tree pattern. Such an analysis can be used to directly quantify some characteristics of biomedical images (e.g. cells samples or clinical images) that are not so noticeable when using geometrical approaches. Ministerio de Economía y Competitividad TEC2016-77785-P Ministerio de Economía y Competitividad MTM2016-81030-P

Multivariate Hermitian quadratic forms play an important role in the real quantifier elimination algorithm based on the computation of comprehensive Grobner systems introduced by V. Weispfenning and further improved by us. Our algorithm needs the computation of a certain type of saturation ideal in a parametric polynomial ring. In this paper, we study multivariate Hermitian quadratic forms in more detail and show several facts which have special importance in a parametric polynomial ring. Our results enable us to have an efficient method to compute the saturation ideal, which brings us a drastic improvement of our real quantifier elimination software.

We give a new construction of the outer automorphism of the symmetric group on six points. Our construction features a complex Hadamard matrix of order six containing third roots of unity and the algebra of split quaternions over the real numbers.

Suppose there exists a Hadamard 2- $$(m,\frac{m-1}{2},\frac{m-3}{4})$$ design with skew incidence matrix, and a conference graph with v vertices, where $$v = 2m-1$$ . Under this assumption we prove that there exists a Siamese twin Menon design with parameters $$(4m^{2},2m^{2}-m,m^{2}-m)$$ intersecting in a balanced incomplete block design $$\mathrm {BIBD}(2m^{2} - m, m^{2} - m, m^{2} - m - 1)$$ and a pairwise balanced design $$\mathrm {PBD}(2m^{2} - m, \{m^{2}, m^{2} - m\}, m^{2} - m - 1)$$ . These Menon designs lead to regular amicable Hadamard matrices of orders not previously constructed. Further we construct complex orthogonal designs of order $$4m^2$$ and Butson Hadamard matrices $$\mathrm {BH}(4m^{2},2k)$$ for all k. Some results regarding automorphisms of the constructed Menon designs are proven.