Pointing out that Λ-fractional analysis is the unique fractional calculus theory including mathematically acceptable fractional derivatives, variational calculus for Λ-fractional analysis is established. Since Λ-fractional analysis is a non-local procedure, global extremals are only accepted. That means the extremals should satisfy not only the Euler–Lagrange equation but also the additional Weierstrass-Erdmann corner conditions. Hence non-local stability criteria are introduced. The proposed variational procedure is applied to any branch of physics, mechanics, biomechanics, etc. The present analysis is applied to the Λ-fractional refraction of light.

The energy parameters obtained during the tests of turbines of power units on the stand differ from those in the product. The research data, which results are presented in the materials of the paper, are aimed at analyzing the discrepancies between the parametric indicators of power units and bench tests of the turbines. The novelty of the obtained results reveals the direct and inverse relationship of changes in the amplitude-frequency characteristics of the elements of the stand (depending on the realized turbine power) with the obtained results of measuring the turbine power on the stand. We developed and described an algorithm for constructing dynamic analysis during the formation of the wave field of the test bench for turbines both in transient modes and in stationary modes corresponding to a constant number of turbine revolutions. It is shown that, by using the algorithm of modal deduction and conditions of dynamic excitation of vibrations from the tested turbine with elements of studying its power, it is possible to construct transients with certain reliability when the number of revolutions of the turbine changes, i.e. its power. The diagnostic model has a novelty since it allows not only to assess of the influence of the elements of the stand on the nature of the transient process when measuring turbine power in transient modes, taking into account the frequency adjustment when changing the revolutions of the turbine and the elements of the stand but also to form requirements for the frequency tuning of the stand. To clarify the transfer function of the "stand–turbine" system, a modal analysis was applied, which made it possible to clarify the structure of the transfer function in the frequency range of the natural (partial) frequencies of the elements of the stand when the restructuring of the wave field during the transient operation of the turbine, but also when the turbine reaches the specified power.

The SK theory provides a deeper insight into the magnetic properties of celestial bodies. In this study, the magnetic field calculated of the parent body of asteroid 4 Vesta, could facilitate deeper insight into the formation of planets or the Universe.

Two methods are presented for determining advanced combinatorial identities. The first is based on extending the original identity so that it can be expressed in terms of hypergeometric functions whereupon tabulated values of the functions can be used to reduce the identity to a simpler form. The second is a computer method based on Koepf's version of the Wilf-Zeilberger approach that has been implemented in a suite of intrinsic routines in Maple. As a consequence, some new identities are presented.

We demonstrate that the model of zero-range potentials can be successfully employed for the description of attached electrons in atomic and molecular anions, for example, negatively charged carbon clusters. To illustrate the capability of the model we calculate the energies of the attached electron for the family of carbon cluster anions consisting of two-, three- (equilateral triangle) and four (tetrahedron) carbon atoms equidistant from each other as well as for a C3 molecule having a chain structure. The considered approach can be easily extended to carbon clusters containing an arbitrary number of atoms arranged in an arbitrary configuration.

It is shown how changing only one word in the usual interpretation of quantum mechanics makes it possible to turn its puzzles and miracles into obvious trivialities

Based on the Bezout approach we propose a simple algorithm to determine the gcd of two polynomials that don't need division, like the Euclidean algorithm, or determinant calculations, like the Sylvester matrix algorithm. The algorithm needs only n steps for polynomials of degree n. Formal manipulations give the discriminant or the resultant for any degree without needing division or determinant calculation.

Accurate results are needed to confirm the experimental results of various atomic processes and analyze the solar and astrophysical observations of intensities of emission lines to infer plasma parameters like electron density, electron temperature and element abundance. A number of theories have been developed over the years to calculate phase shifts when electrons and positrons are scattered from targets. We discuss in this article the recent hybrid theory which has been applied to scattering processes, resonances and photoabsorption process, which is a bound-free transition.

Traditional deterministic modeling of epidemics is usually based on a linear system of differential equations in which compartment transitions are proportional to their population, implicitly assuming an exponential process for leaving a compartment as happens in radioactive decay. Nonetheless, this assumption is quite unrealistic since it permits a class transition such as the passage from illness to recovery that does not depend on the time an individual got infected. This trouble significantly affects the time evolution of epidemy computed by these models. This paper describes a new deterministic epidemic model in which transitions among different population classes are described by a convolutional law connecting the input and output fluxes of each class. The new model guarantees that class changes always take place according to a realistic timing, which is defined by the impulse response function of that transition, avoiding model output fluxes by the exponential decay typical of previous models. The model contains five population compartments and can take into consideration healthy carriers and recovered-to-susceptible transition. The paper provides a complete mathematical description of the convolutional model and presents three sets of simulations that show its performance. A comparison with predictions of the SIR model is given. Outcomes of simulation of the COVID-19 pandemic are discussed which predicts the truly observed time changes of the dynamic case-fatality rate. The new model foresees the possibility of successive epidemic waves as well as the asymptotic instauration of a quasi-stationary regime of lower infection circulation that prevents a definite stopping of the epidemy. We show the existence of a quadrature function that formally solves the system of equations of the convolutive and the SIR models and whose asymptotic limit roughly matches the epidemic basic reproduction number.

N continuous prime numbers can combine a group of continuous even numbers. If an adjacent prime number is followed, the even number will continue. For example, if we take the prime number 3, we can get an even number 6. If we follow an adjacent prime number 5, we can get even numbers by using 3 and 5: 6, 8 and 10. If a group of continuous prime numbers 3, 5, 7, 11, ..., P, we can get a group of continuous even numbers 6, 8, 10, 12, 2n. Then if an adjacent prime number q is followed, the Original group of even numbers 6, 8, 10, 12, 2n will be finitely extended to 2(n + 1) or more adjacent even numbers. My purpose is to prove that the continuity of prime numbers will lead to even continuity as long as 2(n + 1) can be extended. If the continuity of even numbers is Discontinuous, it violates the Bertrand Chebyshev theorem of prime Numbers. Because there are infinitely many prime numbers: 3, 5, 7, 11, We can get infinitely many continuous even numbers: 6, 8, 10, 12,Get: Gold Bach conjecture holds. 2020 Mathématiques Subjectif Classification: 11P32, 11U05, 11N05, 11P70. Research ideas: If the prime number is continuous and any pairwise addition can obtain even number continuity, then Gold Bach’s conjecture is true. Human even number experiments all get (prime number + prime number). I propose a new topic: the continuity of prime numbers can lead to even continuity. I designed a continuous combination of prime numbers and got even continuity. If the prime numbers are combined continuously and the even numbers are forced to be discontinuous, a breakpoint occurs. It violates Bertrand Chebyshev's theorem. It is proved that prime numbers are continuous and even numbers are continuous. The logic is: if Gold Bach's conjecture holds, it must be that the continuity of prime numbers can lead to the continuity of even numbers. Image interpretation: turn Gold Bach’s conjecture into a ball, and I kick the ball into Gold Bach’s conjecture channel. There are several paths in this channel and the ball is not allowed to meet Gold Bach’s conjecture conclusion in each path. This makes the ball crazy, and the crazy ball must violate Bertrand Chebyshev's theorem.