Mathematics

This paper proves that a binary operation {\star} on {[0, 1]} , ensuring that the binary operation {\curlywedge} is a {t} -norm or {\curlyvee} is a {t} -conorm, is a {t} -norm, where {\curlywedge} and {\curlyvee} are special convolution operations defined by {(f\curlywedge g)(x)=\sup\left\{f(y)\star g(z): y\vartriangle z=x\right\},} {(f\curlyvee g)(x)=\sup\left\{f(y)\star g(z): y\ \triangledown\ z=x\right\},} for any {f, g\in Map([0, 1], [0, 1])} , where {\vartriangle} and {\triangledown} are a continuous {t} -norm and a continuous {t} -conorm on {[0, 1]} , answering negatively an open problem posed in \cite{HCT2015}. Besides, some characteristics of {t} -norm and {t} -conorm are obtained in terms of the binary operations {\curlywedge} and {\curlyvee} .

We describe a numerical algorithm for evaluating the numbers of roots minus the number of poles contained in a region based on the argument principle with the function of interest being written as a Mellin transformation of a usually simpler function. Because the function to be transformed may be simpler than its Mellin transform whose roots are to be sought we express the final integrals in terms of the former accepting higher dimensional integrals. Nonlinear terms are expressed as convolutions approximating reciprocal values by exponential sums. As an example the final expression is applied to the Riemann Zeta function. The procedure is very inefficient numerically. However, depending on the function to be investigated it may be possible to find analytical estimates of the resulting integrals.

We consider a method for the approximation of iterated stochastic integrals of arbitrary multiplicity k (k∈N) with respect to the infinite-dimensional Q -Wiener process using the mean-square approximation method of iterated Ito stochastic integrals with respect to the scalar standard Wiener processes based on generalized multiple Fourier series. The case of multiple Fourier-Legendre series is considered in details. The results of the article can be applied to construction of high-order strong numerical methods (with respect to the temporal discretization) for the approximation of mild solution for non-commutative semilinear stochastic partial differential equations.

We show a possibility to apply certain philosophical concepts to the analysis of concrete mathematical structures. Such application gives a clear justification of topological and geometric properties of considered mathematical objects.

A 4-regular matchstick graph is a planar unit-distance graph whose vertices have all degree 4. Examples of 4-regular matchstick graphs are currently known for all number of vertices ≥ 52 except for 53, 55, 56, 58, 59, 61, and 62. In this article we present 38 different examples with 50 - 62 vertices which contain two, three, or four distances which differ slightly from the unit length. These graphs should show why this subject is so extraordinarily difficult to deal with and should also be an incentive for the interested reader to find solutions for the missing numbers of vertices.

In this paper, we will be delving deeper into the connection between the rough theory and the ring theory precisely in the principle and maximal ideal. The rough set theory has shown by Pawlak as good formal tool for modeling and processing incomplete information in information system. The rough theory is based on two concepts the upper approximation of a given set is the union of all the equivalence classes, which are subsets of the set, and the lower approximation is the union of all the equivalence classes, which are intersection with set non-empty. Many researchers develop this theory and use it in many areas. Here, we will apply this theorem in the one of the most important branches of mathematics that is ring theory. We will try to find the rough principal and maximal ideal as an extension of the notion of a principal maximal ideal respectively in ring theory. In addition, we study the properties of the upper and lower approximation of a principal maximal ideal. However, some researchers use the rough theory in the group and ring theory. Our work, is Shaw there are rough maximal and principle ideal as an extension of the maximal and principle ideal respectively. Our result will introduce the rough maximal ideal as an extended notion of a classic maximal ideal and we study some properties of the lower and the upper approximations a maximal ideal.

In this paper a spline based integral approximation is utilized to propose a sequence of approximations to the error function that converge at a significantly faster manner than the default Taylor series. The approximations can be improved by utilizing the approximation erf(x) approximately equal to one for x>>1. Two generalizations are possible, the first is based on demarcating the integration interval into m equally spaced sub-intervals. The second, it based on utilizing a larger fixed sub-interval, with a known integral, and a smaller sub-interval whose integral is to be approximated. Both generalizations lead to significantly improved accuracy. Further, the initial approximations, and the approximations arising from the first generalization, can be utilized as the inputs to a custom dynamical system to establish approximations with better convergence properties. Indicative results include those of a fourth order approximation, based on four sub-intervals, which leads to a relative error bound of 1.43 x 10-7 over the positive real line. Various approximations, that achieve the set relative error bounds of 10-4, 10-6, 10-10 and 10-16, over the positive real, are specified. Applications include, first, the definition of functions that are upper and lower bounds, of arbitrary accuracy, for the error function. Second, new series for the error function. Third, new sequences of approximations for exp(-x2) which have significantly higher convergence properties that a Taylor series approximation. Fourth, the definition of a complementary demarcation function eC(x) which satisfies the constraint eC(x)^2 + erf(x)^2 = 1. Fifth, arbitrarily accurate approximations for the power and harmonic distortion for a sinusoidal signal subject to a error function nonlinearity. Sixth, approximate expressions for the linear filtering of a step signal that is modelled by the error function.

We will study the asymptotic behavior of summation functions of a natural argument, including the asymptotic behavior of summation functions of a prime argument in the paper. A general formula is obtained for determining the asymptotic behavior of the sums of functions of a prime argument based on the asymptotic law of primes. We will show, that under certain conditions: ∑ p≤n f(p)= ∑ n k=2 f(k) log(k) (1+o(1)) , where p is a prime number. In the paper, the necessary and sufficient conditions for the fulfillment of this formula are proved.

The paper substantiates the conjecture of the asymptotic behavior of the largest distance between consecutive primes: su p p i ≤x ( p i+1 − p i )∼2 e −γ log 2 (x) , where γ is the Euler constant. The Hardy-Littlewood conjecture about the number of prime tuples is investigated and the rationale for this conjecture is given, taking into account the fact that a large natural number is not divisible by primes. It also substantiates why the accuracy of this conjecture is not affected by another assumption about the probability of a natural number being prime, although such a probability does not exist. The paper also considers the distribution of prime tuples using a mathematical model based on the Hardy-Littlewood conjecture.

This paper deals with system with n identical elements and one repairing device. While one element working other ones stay in reserve. The distribution of element working and repairing times are supposed to be exponential. Here we obtain the asymptotic distribution of the system lifetime under conditions of its high reliability.