Efficient resummation of high post-Newtonian contributions to the binding energy

A factorisation property of Feynman diagrams in the context the Effective Field Theory approach to the compact binary problem has been recently employed to efficiently determine the static sector of the potential at fifth post-Newtonian (5PN) order. We extend this procedure to the case of non-static diagrams and we use it to fix, by means of elementary algebraic manipulations, the value of more than one thousand diagrams at 5PN order, that is a substantial fraction of the diagrams needed to fully determine the dynamics at 5PN. This procedure addresses the redundancy problem that plagues the computation of the binding energy with respect to more “efficient” observables like the scattering angle, thus making the EFT approach in harmonic gauge at least as scalable as the others methods.

PRIME NUMBER LAW. DEPENDENCE OF PRIME NUMBERS ON THEIR ORDINAL NUMBERS AND GOLDBACH – EULER BINARY PROBLEM USING COMPUTER

The article considers the methods of defining and finding the distribution of composite numbers CN, prime numbers PN, twins of prime numbers Tw and twins of composite numbers TwCN that do not have divisors 2 and 3 in the set of natural numbers - ℕ based on a set of numbers like Θ = {6∙κ ± 1, κ ∈ ℕ}, which is a semigroup in relation to multiplication. There has been proposed a method of obtaining primes by using their ordinal numbers in the set of primes and vice versa, as well as a new algorithm for searching and distributing primes based on a closedness of the elements of the set Θ. It has been shown that a composite number can be presented in the form of products (6x ± 1) (6y ± 1), where x, y ℕ - are positive integer solutions of one of the 4 Diophantine equations: . It has been proved that if there is a parameter λ of prime twins, then none of Diophantine equations P (x, y, λ) = 0 has positive integer solutions. There has been found the new distribution law of prime numbers π(x) in the segment [1 ÷ N]. Any even number is comparable to one of the numbers i.e. . According to the above remainders m, even numbers are divided into 3 types, each type having its own way of representing sums of 2 elements of the set Θ. For any even number in a segment [1 ÷ ν], where ν = (ζ−m) / 6, , there is a parameter of an even number; it is proved that there is always a pair of numbers that are elements of the united sets of parameters of prime twins and parameters of transition numbers , i.e. numbers of the form with the same λ, if the form is a prime number, then the form is a composite number, and vice versa.

Fuzzy Modelling and Chaos Control in the Photogravitational Magnetic Binary Problem With Potential From a Belt

Control of chaotic systems has led to many fruitful results, such as the famous OGY and feedback control. Of course, controlling chaos is not limited to the approaches above and is not specifically reviewed one by one here. The representation of chaotic systems using fuzzy models has a unified approach. The fuzzy logic controllers had been proposed for a long time and were successfully applied to control the chaos in many systems. The main purpose of this study is to presents the new fuzzy model for the photogravitational magnetic-binary problem (PMBP) where the bigger primary is a source of radiation and the smaller primary is an oblate body; and they are encompassed by a homogeneous circular cluster of material points centred at the mass centre of the system (belt). It was shown that using a new fuzzy controller, it is possible to control of chaotic behaviour of the photogravitational magnetic-binary problem (PMBP). The simulation results have demonstrated that the proposed method can satisfy the control object and enhanced stability.

A Metaheuristic Approach and Mathematical Programming for Packing Objects in a Rectangular Container

The problem of packing non-congruent circles within bounded regions is considered. The aim is to maximize the number of circles placed into a rectangular container or minimize the waste. The circle is considered as a set of points that are all the same distance (not necessarily Euclidean) from a given point. An integer programming model is proposed using a dotted-board approximating the container and considering the dots as potential positions for assigning centers of the circles. The packing problem is then stated as a large scale linear 0–1 optimization problem. Binary decision variables are associated with each discrete point of the board (a dot) and with each object. Then, the same grid is used to prove a population-based metaheuristic. This metaheuristic is inspired by the monkeys' behavior. The resulting binary problem is then solved by using Gurobi Solver and Python Programming Language as Interface

Beyond Newton

The post-Newtonian approximation is explained and the main issues arising in the description of inspiralling binary systems are considered. The ideas behind the effective-one-body approach to the binary problem are discussed.

Branch and Bound method for binary problems with the procedure that reduces dimension of problems

The relationships between elements aij of coefficient matrix, elements di of vector d and elements cj of vector c in general binary problem are considered. Some of them allow us to establish the values of selected elements of feasible or optimal vector x. This procedure reduces the dimension of basic problem and can be install in branch and bound method. It gives positive effects.