Let $\mathfrak B$ be the class of all bornological algebras. Let $A$ be in $\mathfrak B$. We show that an ideal $I$ of $A$ is non-removable if and only if it consists of joint bounding elements (déf.2).

In this paper we establish convergence results for a family $\mathbb{T}$ of nonlinear integral operators of the form: $$(T_{w}f)(s) = \int_0^{+\infty}K_{w}(t,f(st))dt = \int_0^{+\infty} L_{w}(t)H_{w}(f(st))dt, \ s \in \mathbb{R}_0^+,$$ where $f \in Dom \mathbb{T}$, $Dom \mathbb{T}$ being the class of all the measurable functions $f:\mathbb{R}_0^+ \rightarrow \mathbb{R}$ such that $T_{w}f$ is well defined as Lebesgue integral for every $s \in \mathbb{R}_0^+$. For the above family of nonlinear Mellin type operators, under suitable singularity assumptions on the kernels $\mathbb{K} = \{K_{w}\}$, we state a convergence result of type $\mathrm{lim}_{w \rightarrow + \infty}V_{\varphi}[\mu(T_{w}f - f)] = 0$, for some constant $\mu > 0$ and for every $f$ belonging to a suitable subspace of $BV_\varphi$-functions.

In this paper we investigate the existence of solutions to an hyperbolic functional differential inclusion in Banach spaces with nonlocal conditions. We shall rely on a. fixed point theorem for condensing maps due to of Martelli.

We show that the convexified topology of a locally bounded algebra can be detennined by an explicit $A$-semi-norm. We also exhibit relationship between some properties of $A$-$p$-normed algebras and the associated $A$-semi-normed algebras obtained by convexification.

We show that unitary and complete $l. m. c. a. $'s endowed with certain orders are actually locally $C^*$-algebras's or even reduce to the complex field.

The Durrmeyer modification $\tilde{F}_{n}f$ of the generalized Favard operators in some weighted function spaces are considered. The rate of convergence of $\tilde{F}_{n}f(x)$ at the Lebesgue points $x$ of $f$ is estimated. In particular, a corresponding estimate in the class of functions $f$ of bounded $p$-th power variation is deduced.

Following [4] and [l] we describe the analytic character of some function $f(z)$ connected with the Euler's $\varphi$ function being for $\mathrm{Im} \ z > 0$ a series over all non-trivial zeros of the Riemann zeta - function.

We give an approach to treating inequalities of Bernstein type for trigonometric polynomials. Some necessary and sufficient conditions of their validity are established.

The paper deals with linear and semi-linear integral equations in Sobolev spaces $H_{p}^{s}(\mathbb{R}^{n})$. The main aim is to extend the resulting a.e. validity of the corresponding equations with respect to the Lebesgue measure to an $\mu$-a.e. validity for some Radon measures $\mu$ in $\mathbb{R}^n$.

In the present paper we derive an upper bond for the number of primes $p$, for which $a$ is a primitive root mod $p$.