On a necessary and sufficient cyclicity condition for a quadrilateral

A convex quadrilateral with sides a,b,c,d, and diagonals p,q is cyclic iff abp-bcq+cdp-daq=0. This condition, in spite of its simplicity, appears to be unnoted and unexpectedly proof-resilient. We employ advanced methods of computer algebra and nonlinear analysis.Comment: 28 pp, 2 figures, A4 paper, in Russia

Towards effective two-level supercompilation

The paper presents a number of improvements to the method of two-level supercompilation: a fast technique of lemma discovering by analyzing the expressions in the partial process tree, an enhancement to the algorithm of checking improvement lemmas based on the normalization of tick annotations, and a few techniques of finding simplified versions of lemmas discovered in the process of two-level supercompilation

Supercompiler HOSC 1.1: proof of termination

The paper contributes the proof of termination of an experimental supercompiler HOSC dealing with higher-order functions

Integration of ODEs on Riemann surfaces with an arbitrary precision

Abstract: We consider analytical systems of ODEs with a real or complex time. Integration of such ODEs is equivalent to an analytical continuation of a solution along some path, which usually belongs to the real axis. The problems that may appear along this path are often caused by singularities of the solution that lie outside the real axis. It is possible to circumvent problematic parts of the path (including singularities) by going on the Riemann surface of the solution (i.e., in the complex domain). A natural way to realize this program is to use the method of Taylor expansions, which does not require a formal complexification of the system (i.e., a change of variables). We use two classical problems, i.e., the Restricted Three-Body problem, and Van der Pol equation, to demonstrate how Taylor expansions can be used for integration of ODEs with an arbitrary precision. We obtained some new results in these problems.Note: Research direction:Mathematical modelling in actual problems of science and technic

Supercompiler HOSC 1.0: under the hood

The paper describes the internal structure of HOSC, an experimental supercompiler dealing with programs written in a higher-order functional language. A detailed and formal account is given of the concepts and algorithms the supercompiler is based upon

MRSC: a toolkit for building multi-result supercompilers

The paper explains the principles of multi-result supercompilation. We introduce a formalism for representing supercompilation algorithms as rewriting rules for graphs of congurations. Some low-level technical details related to the implementation of multi-result supercompilation in MRSC are discussed. In particular, we consider the advantages of using spaghetti stacks for representing graphs of configurations

Supercompiler HOSC 1.5: homeomorphic embedding and generalization in a higher-order setting

The paper describes the algorithm of the supercompiler HOSC 1.5, an experimental specializer dealing with programs written in a higher-order functional language. The design decisions behind the algorithm are illustrated through a series of examples. Of particular interest are the decisions related to generalization and homeomorphic embedding of expressions with bound variables

Supercompiler HOSC: proof of correctness

The paper presents the proof of correctness of an experimental supercompiler HOSC dealing with higher-order functions

Study of the accuracy of active magnetic damping algorithm

Abstract: Attitude motion of a satellite equipped with an active magnetic attitude control system is considered. Control system implements «-Bdot» damping algorithm. Satellite behavior is analyzed in a steady-state motion. Slow spinning around the principal axis of maximum inertia is proven, angular velocity is found. Attitude accuracy in dipole geomagnetic field model is studied. Numerical analysis is carried out.Note: Research direction:Theoretical and applied problems of mechanicsRussia

Simulation of anode effect in aluminum electrolytic cell

Abstract: The paper describes three-phase three-dimensional mathematical model of aluminum electrolytic cell, which takes into account the relationship of the four main groups of processes: the hydrodynamic, electromagnetic, thermal and electrochemical. The process of anode effect in electrolytic cell was researched with presented mathematical model. It was found that anode effect in some cases leads to MHD instability, which increases loss of metal.Note: Research direction:Mathematical modelling in actual problems of science and technic