Journal of Prime Research in Mathematics

Abstract: Lanczos methods for solving \(Ax = b\) consist in constructing a sequence of vectors \((x_k)\), \(k = 1, …\) such that \(r_k = b − Ax_k = P_k(A)r_0\), where \(P_k\) is the orthogonal polynomial of degree at most k with respect to the linear functional c defined as c(ξ^i) = (y, A^ir_0)\). Let \(P^(1)_k\) be the regular monic polynomial of degree k belonging to the family of formal orthogonal polynomials (FOP) with respect to \(c^(1)\) defined as c^(1)(ξ ^{i}) = c^{(ξi+1)}\). All Lanczos-type algorithms are characterized by the choice of one or two recurrence relationships, one for \(P_k\) and one for \(P^{(1)}_k\). We shall study some new recurrence relations involving these two polynomials and their possible combinations to obtain new Lanczos-type algorithms. We will show that some recurrence relations exist, but cannot be used to derive Lanczos-type algorithms, while others do not exist at all.

Abstract: In this paper we use radial basis functions in one of the projection methods to solve integral equations of the second kind with two or more variables. This method implemented without needing any introductory algorithms. Relatively good error bound and the numerical experiments show the accuracy of the method.

Abstract: The hyperbolic distances from the origin are changed under the radial selfmapping \(x → |x|^{1/K−1}x\), \(K > 1\) of the unit ball. Here author gives the power mean inequality of the hyperbolic metric under the radial mapping.

Abstract: We calculate the Boyd indices for the sum and intersection of two quasi-normed rearrangement invariant spaces. An application to Lorentz type spaces is also given.

Abstract: We show that if \(F_n = 22^n+ 1\) is the nth Fermat number, then the binary digit sum of \(π(F_n)\) tends to infinity with \(n\), where \(π(x)\) is the counting function of the primes \(p ≤ x\). We also show that if \(F_n\) is not prime, then the binary expansion of \(φ(F_n)\) starts with a long string of 1’s, where \(φ\) is the Euler function. We also consider the binary expansion of the counting function of irreducible monic polynomials of degree a given power of 2 over the field \(\mathbb{F}_{2}\). Finally, we relate the problem of the irrationality of Euler constant with the binary expansion of the sum of the divisor function.

Abstract: In this paper we estimate Banach-Saks index of intersection of two spaces with symmetric bases from below by indices of these spaces. We also show on example of Orlicz spaces that we can’t estimate Banach-Saks index of intersection in the same way from above.