Abstract:The Taylor expansion is a widely used and powerful tool in all branches of Mathematics, both pure and applied. In Probability and Mathematical Statistics, however, a stronger version of Taylor’s classical theorem is often needed, but only tacitly assumed. In this note, we provide an elementary proof of this measurable Taylor’s theorem, which guarantees that the interpolating point in the Lagrange form of the remainder can be chosen to depend measurably on the independent variable.

Abstract:In this paper, we establish sharp inequalities for trigonometric functions. We prove in particular for \(0 < x < \frac{\pi}{2}\) and any \(n \geq 5\) \[0 < P_n(x)\ <\ (\sin x)^2- x^3\cot x < P_{n-1}(x) + \left[\left(\frac{2}{\pi}\right)^{2n} - \sum_{k=3}^{n-1} a_k \left(\frac{2}{\pi}\right)^{2n-2k}\right] x^{2n} \] where \(P_n(x) = \sum_{3=k}^n a_k x^{2k+1}\) is a \(n\)-polynomial, with positive coefficients (\(k \geq 5\)), \(a_{{k}}=\frac{{2}^{2\,k-2}}{\ \left( 2\,k-2 \right) ! } \left( \left| {B}_{ 2\,k-2} \right| +{\frac { \left( -1\right) ^{k+1}}{ \left( 2\,k-1 \right) k}} \right),\) \( B_{2k} \) are Bernoulli numbers. This improves a lot of lower bounds of \( \frac{\sin(x)}{x}\) and generalizes inequalities chains. Moreover, bounds are obtained for other trigonometric inequalities as Huygens and Cusa inequalities as well as for the function \[g_n(x) = \left(\frac{\sin(x)}{x}\right)^2 \left( 1 - \frac{2\left(\frac{2 x}{\pi}\right)^{2n+2}}{1-(\frac{2x}{\pi})^2}\right) +\frac{\tan(x)}{x}, \ n\geq 1 \].

Abstract:In this manuscript, our primary focus revolves around extending the inequalities associated with the Quadratic \(\varphi(\delta_{1},\delta_{2})-\)function. Our approach involves leveraging the general quadratic functional equation encompassing \(2k\)-variables within the context of the fuzzy Banach space. Our main contribution lies in the expansion of these inequalities, representing a significant result within this study.

Abstract:We classify particle paths for systems in thermal equilibrium satisfying the usual relations and prove that the only solutions are given by straight line parallel paths with speed \(c\).