In this work, we provide a $q$-generalization of flexible algebras and related bialgebraic structures, including center-symmetric (also called anti-flexible) algebras, and their bialgebras. Their basic properties are derived and discussed. Their connection with known algebraic structures, previously developed in the literature, is established. A $q$-Myung theorem is given. Main properties related to bimodules, matched pairs and dual bimodules as well as their algebraic consequences are investigated and analyzed. Besides, the equivalence between $q$-generalized flexible algebras, their Manin triple and bialgebras is established. Finally, various remarkable identities are established for the octonion algebra.

Symplectic-Haantjes manifolds are constructed for several Hamiltonian systems following P. Tempesta and G. Tondo [14], which yield the complete integrability of systems.

Let $f$ be an operator convex function on $I$ and $A,B\mathrm{\hspace{0.17em}ϵ\hspace{0.17em}}S{A}_I\left(H\right),$ the convex set of selfadjoint operators with spectra in $I$. If $A\ne B$ and $f$, as an operator function, is Gâteaux differentiable on

$$\left[A,B\right] := \left\{\left(1-t\right)A+tB\text{|}t\in \left[0,1\right]\right\}\mathrm{,}$$

while $p : \left[0,1\right]\to [0,\infty )$ is Lebesgue integrable satisfying the condition

$$0\le {\displaystyle {\int }_0^{\tau }p}(s)ds\le {\displaystyle {\int }_0^{1}p}(s)ds\text{ for all }\tau \in [0,1]$$

and symmetric, namely $p\left(1-t\right)$ $=p\left(t\right)$ for all $tϵ\left[0,1\right]$ then

$$\begin{array}{l}-\frac{1}{{\displaystyle {\int }_0^{1}p}(\tau )d\tau }{\displaystyle {\int }_0^1\left({\displaystyle {\int }_0^{\tau }p}(s)ds\right)}(1-\tau )d\tau [\nabla f_B(B-A)-\nabla f_A(B-A)]\\ \le \frac{1}{{\displaystyle {\int }_0^{1}p}(\tau )d\tau }{\displaystyle {\int }_0^{1}p}(\tau )f((1-\tau )A+\tau B)d\tau -{\displaystyle {\int }_0^{1}f}((1-\tau )A+\tau B)d\tau \\ \le \frac{1}{{\displaystyle {\int }_0^{1}p}(\tau )d\tau }{\displaystyle {\int }_0^1\left({\displaystyle {\int }_0^{\tau }p}(s)ds\right)}(1-\tau )d\tau [\nabla f_B(B-A)-\nabla f_A(B-A)].\end{array}$$

Some particular examples of interest are also given.

This paper provides a new procedure for testing the null hypothesis of multivariate elliptical symmetry. A test for uniformity over the Stiefel manifold based on modified degenerate $V$-statistics is employed since the test statistic proposed in this paper consists of independent random matrices, formed by the scaled residuals (or the Studentized residuals), which are uniformly distributed over the Stiefel manifold under the null hypothesis. Also, Monte Carlo simulation studies are carried out to evaluate the type I error and power of the test. Finally, the procedure is illustrated using the Iris data.

Let $m\mathrm{,}q\mathrm{,}r$ be positive integers. Then we show that the equation ${(3m^2+1)}^x+{(q{m}^2-1)}^y=(rm{)}^z$ has only the positive integer solution $(x,y,z)=(1,1,2)$ under some conditions. The proof is based on elementary methods and Baker’s method.

The current article purports to investigate $\alpha$-almost Ricci solitons in the framework of Kenmotsu manifolds. Among others, we prove that an $\alpha$-almost Ricci solitons on a Kenmotsu manifold is expanding. Furthermore, we take into account $\alpha$-almost Ricci solitons on Kenmotsu manifolds with Codazzi type of Ricci tensor and cyclic parallel Ricci tensor. Also, Kenmotsu manifolds satisfying the curvature condition $R.R=Q(S\mathrm{,}R)$ is studied. Ultimately, we consider an example to prove the non-existence of proper $\alpha$-almost Ricci solitons on Kenmotsu manifolds and verify some results.

Let $\mathcal{O}$ be a complete discrete valuation ring with a residue field $\mathit{\text{k}}=\mathcal{O}/J\left(\mathcal{O}\right)$ of characteristic $p$, $G$ a finite group, and $b$ a block of $kG$ with lift $\widehat{b}$. In this paper, we show that any indecomposable Brauer-friendly $kGb$-module satisfying certain condition is liftable to an indecomposable Brauer-friendly $\mathcal{O}G\hat{b}$-module.