### $\mathcal{I}$-convergence and extremal $\mathcal{I}$-limit points

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In [5] and [10], statistical-conservative and $\sigma $-conservative matrices were characterized. In this note we have determined a class of statistical and $\sigma $-conservative matrices studying some inequalities which are analogous to Knopp’s Core Theorem.

L. Olivier proved in 1827 the classical result about the speed of convergence to zero of the terms of a convergent series with positive and decreasing terms. We prove that this result remains true if we omit the monotonicity of the terms of the series when the limit operation is replaced by the statistical limit, or some generalizations of this concept.

The fundamental semigroup $({a}^{t}{)}_{t\>0}$ of the heat equation for the real line has an analytic extension $({a}^{t}{)}_{\mathrm{Re}\phantom{\rule{0.166667em}{0ex}}t\>0}$ to the right-hand open half plane which satisfies $\parallel {a}^{t}\parallel \le \sqrt{\left|t\right|}$ for Re$\phantom{\rule{0.166667em}{0ex}}t\ge 1$. Using the Ahlfors-Heins theorem for bounded analytic functions on a half-plane we show that the Wiener tauberian theorem for ${L}^{1}\left(\mathbf{R}\right)$ follows from the above inequality.

Converging sequences in metric space have Hausdorff dimension zero, but their metric dimension (limit capacity, entropy dimension, box-counting dimension, Hausdorff dimension, Kolmogorov dimension, Minkowski dimension, Bouligand dimension, respectively) can be positive. Dimensions of such sequences are calculated using a different approach for each type. In this paper, a rather simple formula for (lower, upper) metric dimension of any sequence given by a differentiable convex function, is derived....

An infinite series which arises in certain applications of the Lagrange-Bürmann formula to exponential functions is investigated. Several very exact estimates for the Laplace transform and higher moments of this function are developed.