We examine functions of two variables whose all vertical sections are equiderivatives. In particular we show that a bounded function whose horizontal sections are strongly measurable and vertical sections are equiderivatives, is strongly measurable. The theorems we prove are generalizations of the results of Z. Grande [3]. Let R and N denote the real line and the set of positive integers, respe...

*Correspondence: [email protected] 1School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM, Penang, Malaysia Full list of author information is available at the end of the article Abstract The estimates for the second Hankel determinant a2a4 – a3 of the analytic function f (z) = z + a2z + a3z + · · · , for which either zf ′(z)/f (z) or 1 + zf ′′(z)/f ′(z) is subordinate to a certai...

an r-module m is called strongly noncosingular if it has no nonzero rad-small (cosingular) homomorphic image in the sense of harada. it is proven that (1) an r-module m is strongly noncosingular if and only if m is coatomic and noncosingular; (2) a right perfect ring r is artinian hereditary serial if and only if the class of injective modules coincides with the class of (strongly) noncosingula...

A. Proof of Lemma 1 We need the following lemma that characterizes the property of the extra-gradient descent. Lemma 8 (Lemma 3.1 in (Nemirovski, 2005)). Let Z be a convex compact set in Euclidean space E with inner product 〈·, ·〉, let ‖ · ‖ be a norm on E and ‖ · ‖∗ be its dual norm, and let ω(z) : Z 7→ R be a α-strongly convex function with respect to ‖ · ‖. The Bregman distance associated wi...

Let Z:={Zt,t≥0} be a stationary Gaussian process. We study two estimators of E[Z02], namely fˆT(Z):=1 T ∫0TZt2dt, and f˜n(Z):=1 n ∑i=1nZti2, where ti=iΔn, i=0,1,…,n, Δn→0 Tn:=nΔn→∞. prove that the are strongly consistent establish Berry-Esseen bounds for central limit theorem involving fˆT(Z) f˜n(Z). apply these results to asymptotically processes estimate drift parameter Ornstein-Uhlenbeck pro...

The submodules with the property of the title ( a submodule $N$ of an $R$-module $M$ is called strongly dense in $M$, denoted by $Nleq_{sd}M$, if for any index set $I$, $prod _{I}Nleq_{d}prod _{I}M$) are introduced and fully investigated. It is shown that for each submodule $N$ of $M$ there exists the smallest subset $D'subseteq M$ such that $N+D'$ is a strongly dense submodule of $M$ and $D'bi...

in this paper, we introduce the dual notion of strongly top modules and study some of the basic properties of this class of modules.