Perturbation analysis for the positive definite solution of the nonlinear matrix equation X-\sum_{i=1}^{m} A_{i}^{*}X^{-1}A_{i}=Q

Consider the perturbation analysis for positive definite solution of the nonlinear matrix equation $X-\sum_{i=1}^{m}A_{i}^{*}X^{-1}A_{i}=Q$ which arises in an optimal interpolation problem. Two perturbation bounds for the unique positive definite solution are obtained, and an explicit expression of the condition number for the unique positive definite solution is derived. The theoretical results are illustrated by several numerical examples.     

Ramanujan summation and the exponential generating function \sum_{k=0}^{\infty}\frac{z^{k}}{k!}\zeta^{\prime}(-k)

In the sixth chapter of his notebooks, Ramanujan introduced a method of summing divergent series which assigns to the series the value of the associated Euler-MacLaurin constant that arises by applying the Euler-MacLaurin summation formula to the partial sums of the series. This method is now called the Ramanujan summation process. In this paper we calculate the Ramanujan sum of the exponential generating functions ∑ _n≥1log?n e ^nz and $\sum_{n\geq 1}H_{n}^{(j)}~e^{-nz}$ where $H_{n}^{(j)}=\sum_{m=1}^{n}\frac{1}{m^{j}}$ . We find a surprising relation between the two sums when j=1 from which follows a formula that connects the derivatives of the Riemann zeta-function at the negative integers to the Ramanujan sum of the divergent Euler sums ∑ _n≥1 n ^k H _n , k ≥ 0, where $H_{n}=H_{n}^{(1)}$ . Further, we express our results on the Ramanujan summation in terms of the classical summation process called the Borel sum.     

On the recursive sequencex_{n + 1} = \frac{{ax_{n - 2m + 1}^p }}{{b + cx_{n - 2k}^{p - 1} }}

The boundedness, global attractivity, oscillatory and asymptotic periodicity of the nonnegative solutions of the difference equation $$x_{n + 1} = \frac{{ax_{n - 2m + 1}^p }}{{b + cx_{n - 2k}^{p - 1} }}, n = 0, 1,...$$ wherem, k ∈ N, 2k > 2m?1,a, b, c are nonnegative real numbers andp < 1, are investigated.     

On the diophantine equation $$\varvec{y^{2} = \prod _{i \le 8}(x + k_i)}$$

This paper improves the result of Tengely (Periodica Math. Hung., 72(1) (2016) 23–28).     

Clifford Cauchy type integrals on Ahlfors-David regular surfaces in $$\mathbb{R}^{m + 1} $$

The main goal of this paper is centred around the study of the behavior of the Cauchy type integral and its corresponding singular version, both over nonsmooth domains in Euclidean space. This approach is based on a recently developed quaternionic Cauchy integrals theory [1, 5, 7] within the three-dimensional setting. The present work involves the extension of fundamental results of the already cited references showing that the Clifford singular integral operator has a proper invariant subspace of generalized H?lder continuous functions defined in a surface of the (m+1)-dimensional Euclidean space.     

On the regularity of the locally integrable solutions of the functional equations $$\sum\limits_{i = 1}^k {a_i \left( {x,t} \right)f\left( {x + \varphi _i \left( t \right)} \right) = b\left( {x,t} \right)} $$

Aequationes mathematicae -