# Journal of Prime Research in Mathematics

Journal of Prime Research in Mathematics (JPRM) ISSN: 1817-3462 (Online) 1818-5495 (Print) is an HEC recognized, Scopus indexed, open access journal which provides a plate forum to the international community all over the world to publish their work in mathematical sciences. JPRM is very much focused on timely processed publications keeping in view the high frequency of upcoming new ideas and make those new ideas readily available to our readers from all over the world for free of cost. Starting from 2020, we publish one Volume each year containing two issues in June and December. The accepted papers will be published online immediate in the running issue. All issues will be gathered in one volume which will be published in December of every year.

Latest Published Articles

### New recurrence relationships between orthogonal polynomials which lead to new lanczos-type algorithms

JPRM-Vol. 1 (2012), Issue 1, pp. 61 – 75 Open Access Full-Text PDF
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.

### Multivariable and scattered data interpolation for solving multivariable integral equations

JPRM-Vol. 1 (2012), Issue 1, pp. 51 – 60 Open Access Full-Text PDF
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.

### On the power mean inequality of the hyperbolic metric of unit ball

JPRM-Vol. 1 (2012), Issue 1, pp. 45 – 50 Open Access Full-Text PDF
Barkat Ali Bhayo
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.

### Boyd indices for quasi-normed rearrangement invariant spaces

JPRM-Vol. 1 (2012), Issue 1, pp. 36 – 44 Open Access Full-Text PDF
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.

### Gaps in binary expansions of some arithmetic functions and the irrationality of the Euler constant

JPRM-Vol. 1 (2012), Issue 1, pp. 28 – 35 Open Access Full-Text PDF
Jorge Jimenez Urroz , Florian Luca, Michel Waldschmidt
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.

### The Banach-Saks index of intersection

JPRM-Vol. 1 (2012), Issue 1, pp. 22 – 27 Open Access Full-Text PDF
Novikova A. I
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.