A method for evaluating the results of the educational process:

The article addresses the problem of evaluating the results of specialists as they graduate from their training. There are difficulties, not only with the decreasing number of students in training courses, but also with inadequate methods for educational management. There is a growing need for managers and subject lecturers to obtain information on the results of specialists’ training and graduation. This article sets out a method for evaluating the results of the educational management using the probability multiplication theorem for dependent events. The study shows that the results of the educational process can - and should - be managed if the results of training and graduation are periodically evaluated. At the same time, the evaluation results are of great value to the university and improve the educational management. They show the scale of the specialists’ training and graduation in relation to their training task and create a data base for subjects enabling appropriate managerial decisions aimed at maximizing approaching the results of the specialists’ graduation to their training task. It is of practical and scientific importance for the permanent staff of vocational educational institutions when developing managerial decisions for qualified specialists’ training, as well as for graduate students and researchers involved in the study of the effectiveness of educational management.

An identity of symmetry for the degenerate Frobenius-Euler Polynomials

In this paper, we prove an identity of symmetry for the higher-order degenerate Frobenius-Euler polynomials and derive the recurrence relations and multiplication theorem type result for the degenerate Frobenius-Euler polynomials.

Connection between multiplication theorem for Bernoulli polynomials and first factor hp− $h_p^-$

Based on the multiplication theorem for Bernoulli polynomials we define the matrix

Compatibility of arithmetic and algebraic local constants (the case )

We show that arithmetic local constants attached by Mazur and Rubin to pairs of self-dual Galois representations which are congruent modulo a prime number $p>2$ are compatible with the usual local constants at all primes not dividing $p$ and in two special cases also at primes dividing $p$. We deduce new cases of the $p$-parity conjecture for Selmer groups of abelian varieties with real multiplication (Theorem 4.14) and elliptic curves (Theorem 5.10).

Overcoming roundoff errors in the first-order amplitude for general state-to-state transitions in hydrogen by projectile impact

A computer program to calculate the amplitude for general state-to-state transitions in hydrogen from the analytical result has the potential for significant numerical round-off errors whenever the sum of the angular momenta of the two states is greater than four and the projectile’s impact parameter is less than the ratio of its energy to velocity. This arises from high-order cancellation of terms in the MacDonald functions in that analytical result, whose polynomial portions have been found to obey a new multiplication theorem. The cost for correcting this instability is the replacement of a finite series of MacDonald functions by an infinite series via the multiplication theorem for the full MacDonald function.