Access to computers

In one of Duncan Dowson’s talks on the early attempts to solve the elastohydrodynamic lubrication problem, he commented that progress only became possible when he and Gordon Higginson were granted private access to the University computer at weekends. And how these words brought back memories! For John Tripp and I could never have solved our numerical problem without the courtesy of the Norwalk State Technical College allowing us access to its computer out of term. That was in 1964. In this century, such a story is hard to believe, when every researcher has on his own desk, for his own use, a computer more powerful than either of the two referred to above. It seems worth recalling how computing has changed in my scientific lifetime.

Integrality of Seshadri constants and irreducibility of principal polarizations on products of two isogenous elliptic curves

In this paper we consider the question of when all Seshadri constants on a product of two isogenous elliptic curves $$E_1\times E_2$$ E 1 × E 2 without complex multiplication are integers. By studying elliptic curves on $$E_1\times E_2$$ E 1 × E 2 we translate this question into a purely numerical problem expressed by quadratic forms. By solving that problem, we show that all Seshadri constants on $$E_1\times E_2$$ E 1 × E 2 are integers if and only if the minimal degree of an isogeny $$E_1\rightarrow E_2$$ E 1 → E 2 equals 1 or 2. Furthermore, this method enables a characterization of irreducible principal polarizations on $$E_1\times E_2$$ E 1 × E 2 .

Ramsey-type numbers involving graphs and hypergraphs with large girth

Erdős asked if, for every pair of positive integers g and k, there exists a graph H having girth (H) = k and the property that every r-colouring of the edges of H yields a monochromatic cycle C k . The existence of such graphs H was confirmed by the third author and Ruciński. We consider the related numerical problem of estimating the order of the smallest graph H with this property for given integers r and k. We show that there exists a graph H on R10k2; k15k3 vertices (where R = R(C k ; r) is the r-colour Ramsey number for the cycle C k ) having girth (H) = k and the Ramsey property that every r-colouring of the edges of H yields a monochromatic C k Two related numerical problems regarding arithmetic progressions in subsets of the integers and cliques in graphs are also considered.

Unknown Values Estimation in Incomplete Fuzzy Soft and Interval Valued Soft Matrices

In day-to-day problems, incomplete information due to unknown values in data is the cause of the loss of information which leads to uncertainly, ambiguity and vagueness. There are many reasons for unknown values in data, like errors in data collection, lack of data information, inappropriate technique, and illegibility of data which cause incompleteness in data. Thus, estimating the unknown values in data of various information systems is an important area of research. In this communication, the definitions of incomplete Fuzzy soft and interval-valued fuzzy soft matrices are given with application in numerical problems. An algorithm is proposed for unknown values estimation in an incomplete fuzzy soft matrix and applied in solving a numerical problem. An application of an incomplete fuzzy soft matrix after inserting the unknown values in dimension reduction is studied. The unknown values in the incomplete interval-valued fuzzy soft matrix are also estimated and applied in the multi-criteria decision-making method.

Simulation of Nonlinear Magnetic Systems by the Finite Element Method Using BLR-Factorization

The possibility of practical application of BLR-factorization (low-rank approximation of the matrix of un-knowns of a system of linear equations) for finite element modeling of the electromagnetic field topology of nonlinear magnetic systems is considered. A method for estimating the accuracy of the computed solution of the SLAE and the nature of the influence of the given accuracy of the low-rank approximation of the matrix of un-knowns on the upper limit of the relative forward error of the computed solution of the SLAE are shown. Using a model problem as an example, the dependence of the accuracy of calculating the integral characteristics of an electromechanical apparatus on the tolerance of the low-rank approximation of the matrix of unknowns is shown, as well as its effect on the convergence of the process of solving a nonlinear numerical problem. A quantitative assessment of the reduction in the computational complexity of the process of solving a numerical problem and the required amount of computer memory for solving the SLAE is carried out. The applicability of BLR-factorization for finite element modeling of the topology of the electromagnetic field without the use of numerical methods of the Krylov subspace is estimated.

A parallel-GPU code for asteroid aggregation problems with angular particles

ABSTRACT The paper presents a numerical implementation of the gravitational N-body problem with contact interactions between non-spherically shaped bodies. The work builds up on a previous implementation of the code and extends its capabilities. The number of bodies handled is significantly increased through the use of a CUDA/GPU-parallel octree structure. The implementation of the code is discussed and its performance is compared against direct N2 integration. The code features both smooth (force-based) and non-smooth (impulse-based) methods, as well as a visco-elastic non-smooth method, to handle contact interaction between bodies. The numerical problem of simulating ‘rubble-pile’ asteroid gravitational aggregation processes is addressed. We discuss the features of the problem and derive criteria to set up the numerical simulation from the dynamical constraints of the combined gravitational–collisional problem. Examples of asteroid aggregation scenarios that could benefit from such implementation are finally presented.

Particle-Based Modeling of Electron–Phonon Interactions

An important challenge in particle-based modeling of electron–phonon interactions is the large difference in the statistical weight of the particles in the two simulated populations. Each change in the state of a simulated phonon during scattering is statistically representative of an interaction with multiple simulated electrons, which results in a large numerical burden accurately represent both populations. We developed two stochastic approaches to mitigate this numerical problem. The first approach is based on Poisson modeling of the scattering processes coupled with a thinning algorithm, which works effectively at steady-state, but it is prone to statistical errors in the energy during the transient regime. The second approach is based on point process (PP) modeling of the scattering, allowing stochastical book-keeping, which corrects the energy error. Here, we present a mathematical description of the problem and the two stochastic approaches along with the numerical results we obtained for the synchronous transient simulation of the electron and phonon populations.