The problem of Euler/Tarry revisited

The problem of Euler/Tarry concerning 36 officers can be formulated in mathematical terms: Can a latin square of order 6 have an orthogonal square, or equivalently, are there 6 pairwise disjoint transversals? This was first answered (in the negative) by Tarry (1900/01). We prove the following Theorem: If a latin square of order 6 admits a reflection, i. e. an automorphism of order two which fixes the main diagonal elementwise, then it has no orthogonal square. We list the 12 isomorphism types of latin squares of order 6 and see: they all admit such a reflection. So we get a solution of the Euler problem without the tedious task of tracing the transversals.

Improving operation efficiency of transportation vehicles equipped with a flexible tractive element under conditions of mining enterprises

Usually, basic traffic flows of mining enterprises are effected with the help of transportation vehicles with a flexible tractive element, i.e. belt conveyor, ground ropeway etc. Parameters of transport and operation procedures depend upon mining and geological conditions of mineral mining, technical characteristics, and operation parameters of the transportation vehicles. Objective is to substantiate a new equation of friction of flexible bodies which corresponds to classical concepts of friction of bodies. The methods are based upon the adjustment of common idea of friction of bodies; factors, working upon friction; and solutions of Euler problem on flexible body slip along the fixed block. As a result, common factors of friction of inflexible bodies, i.e. friction force and normal reaction between linearly connected bodies, have been determined; new equation of friction of flexible bodies has been substantiated. Scientific novelty has been represented by means of a new equation of friction of flexible bodies (in the indirect form) which: involves friction force and normal reaction between linearly connected bodies; corresponds to equilibrium conditions of a mechanical system; and coincides with friction law of Coulomb bodies being common one. Practical implication is in the proposed new scientific knowledge giving corrected idea of friction of flexible bodies. The knowledge improves both education level and research level stipulating technological expansion as well as upgrade of transportation vehicles with flexible body. The findings can be applied to improve both efficiency and safety of transportation vehicles with a flexible tractive element under complicated operation conditions; among other things, it concerns mining enterprises.

PROCEEDURE FOR SOLVING GOLDBACH-EULER PROBLEM USING THE NUMBERS OF A SPECIAL TYPE

Since the elements are closed, the set Θ = {6λ ± 1; λ ∈ N }, is a semigroup with respect to the operation of multiplication. The paper focuses on presenting even numbers ζ > 8 in the form of sums of two elements: θ1 = 6λ1 ± 1 and θ2 = 6λ2 ± 1 from the set Θ. Any even number ζ > 8 is comparable with one of the numbers m = (0, 2, -2), according to (mod 6). According to the remnants listed m , even numbers ζ > 8 are divided into 3 types. Each type has its own way of presenting sums in the form of two elements from the set Θ. For any even number ζ > 8 on the segment [1 ÷ ν] there is always at least a pair of numbers (λ1, λ2) ∈ [1 ÷ ν], that both are elements from the union of sets: the parameters of the prime numbers (twins) and the parameters (composite and prime) of numbers Θ. A variant of the solution of Goldbach - Euler conjecture for even numbers ζ > 8 is given on the set of primes P. Goldbach - Euler conjecture is also solvable in the set of prime numbers (twins), if the parameters of numbers θ1 and θ2, i.e. λ1 and λ2 belong to the set N \ FN , where FN is the set of parameters of the composite numbers Θ on the segment [1 ÷ ν].

Dynamical convexity of the Euler problem of two fixed centers

We give thorough analysis for the rotation functions of the critical orbits from which one can understand bifurcations of periodic orbits. Moreover, we give explicit formulas of the Conley–Zehnder indices of the interior and exterior collision orbits and show that the universal cover of the regularised energy hypersurface of the Euler problem is dynamically convex for energies below the critical Jacobi energy.