Indian Journal of Pure and Applied Mathematics最新文献

Motivated by a recently observed bias in products of two prime numbers with congruence conditions by Dummit, Granville and Kisilevsky, we try to observe some bias in the distribution of integers which are product of two distinct primes taken from Beatty sequences with each one is in an arithmetic progression.

We consider the monomial expansion of the q-Whittaker polynomials and the modified Hall-Littlewood polynomials arising from specialization of the modified Macdonald polynomial. The two combinatorial formulas for the latter, due to Haglund, Haiman, and Loehr and Ayyer, Mandelshtam and Martin, give rise to two different parameterizing sets in each case. We produce bijections between the parameterizing sets, which preserve the content and major index statistics. We identify the major index with the charge or cocharge of appropriate words, and use descriptions of the latter due to Lascoux–Schützenberger and Killpatrick to show that our bijections have the desired properties.

In this article we derive both Hamilton type and Souplet–Zhang type gradient estimations for a system of semilinear equations along a geometric flow on a weighted Riemannian manifold.

This paper consists of certain congruence properties of Stirling numbers of the first and second kinds. Some congruence relations between s(n, k) and S(n, k) for different modulo are obtained through their generating functions. We also present some exact p-adic valuations of s(n, k) and S(n, k) for some cases, mainly when (n-k) is divisible by (p-1) for odd prime p. Some estimates of the p-adic valuation of these two numbers are also presented when (p-1) does not divide (n-k).

This paper investigates the point-wise time-space estimates for a class of oscillatory integrals given by (int _{mathbb R^{n} }e^{i<x,; xi >pm itP^{frac{1}{2} } (xi )} P^{-frac{alpha }{2} } (xi )dxi ), where P is a real non-degenerate elliptic polynomial of order (mge 4) on (mathbb R^{n} ). These estimates are applied to obtain time-space integrability estimates with regularity for solutions to higher order wave-type equations.

In this paper, we study the convergence rates for homogenization problems for solutions of partial differential equations with rapidly oscillating Neumann boundary data in the convex polygonal domains. As a consequence, we obtain the pointwise and (L^{p}) convergence results. Our techniques are based on using Fourier analysis method as well as Diophantine condition on the boundary

Let N be a sufficiently large real number. In this paper, we prove that for (2<c< frac{68}{33}) and for any arbitrary large number (E>0) , the Diophantine inequality

is solvable in prime variables (p_1,p_2,p_3,p_4,p_5) such that, each of the numbers (p_{i}+2,, (1le ile 5)) has at most (big [frac{214467}{136000-66000c}big ]) prime factors, counted with multiplicity.

In this note, two generalized partition functions (p_o^alpha (n)) and (p_e^beta (n)) are considered, where for any odd positive integer (alpha ), (p_o^alpha (n)) denotes the number of partitions of n into odd parts such that no parts is congruent to (alpha ) modulo (2alpha ), and for any even positive integer (beta ), (p_e^beta (n)) denotes the number of partitions of n into even parts such that no parts is congruent to (beta ) modulo (2beta ). Some divisibility properties of (p_o^alpha (n)) and (p_e^beta (n)) are discussed for some particular values of (alpha ) and (beta ).

By the Ostrowski theorem, the Riemann zeta-function (zeta (s)) does not satisfy any algebraic-differential equation. Voronin proved that the function (zeta (s)) does not satisfy algebraic-differential equation with continuous coefficients. In the paper, a joint generalization of the Voronin theorem is given, i. e., that a collection ((zeta (s_1), dots , zeta (s_r))) does not satisfy a certain algebraic-differential equation with continuous coefficients.

In this paper, we present a new family of higher-order numerical methods for solving non-linear fractional delay differential equations (FDDEs) along with the error analysis. Further, we solve various non-trivial systems of FDDEs to illustrate their applicability and utility. By using the proposed numerical methods, computational time is reduced drastically. These methods take only 5 to 10 percent of the time required for other methods such as the fractional Adams method (FAM). Furthermore, these methods converge for very small values of fractional derivative while FAM and the new predictor-corrector method (NPCM) introduced by Daftardar-Gejji et al. [1] do not converge. The order of convergence of the proposed methods is (r+alpha ), where r is the order of fractional backward difference formulae and (alpha ) denotes the order of the fractional derivative. Thus these methods have a higher order of accuracy than FAM or NPCM.