This article provides an overview of some recent results and ideas related to the study of finite groups depending on the restrictions on some systems of their sections. In particular, we discuss some properties of the lattice of all subgroups of a finite group related with conditions of permutability and generalized subnormality for subgroups. The paper contains more than 30 open problems which were posed, at different times, by some mathematicians working in the discussed direction.

For finding the real roots of a polynomial, we propose a clipping algorithm called SLEFE clipping and an isolation algorithm called SLEFE isolation algorithm. At each iterative step, the SLEFE clipping algorithm generates two broken lines bounding the given polynomial. Then, a sequence of intervals can be obtained by computing the intersection of the sequence of broken lines with the abscissa axis. The sequence of these intervals converges to the root with a convergence rate of 2. Numerical examples show that SLEFE clipping requires fewer iterations and less computation time than current algorithms, and the SLEFE isolation algorithm can compute all intervals that contain the roots rapidly and accurately.

The main goal of this paper is to introduce necessary efficiency conditions for a class of multi-time vector fractional variational problems with nonlinear equality and inequality constraints involving higher-order partial derivatives. We consider the multi-time multiobjective variational problem (MFP) of minimizing a vector of path-independent curvilinear integral functionals quotients subject to PDE and/or PDI constraints, developing an optimization theory on the higher-order jet bundles.

Sometimes, people with interest in measuring quality of education take into account level in academic performance and various associated factors. Usually, an average academic performance is an accustomed way of assessment; however, this study examines on individual basis different factors that might have an impact on the academic performance of undergraduate students. Data on the semester weighted average of class of 2012 mathematics students were acquired from the Quality Assurance and Planning Unit and the Examination Office of the Department of Mathematics, Kwame Nkrumah University of Science and Technology. The main factors considered for this research were entry age, gender, entry aggregate, Ghana education service graded level of senior high school attended and geographical location. The statistical method considered was random effect. Since the interaction or variation around the slope was highly insignificant, the random intercept model was the better alternative ahead of the random intercept and slope model. Statistically, not all the parameter estimates are significant at $\alpha =0.05$ level of significance. It was observed that the difference in geographical location was not significant in the main effect model. Hence where a student comes from has no influence on their academic performance. However, entry aggregate, entry age and gender were all significant. Nevertheless, the geographical location with regard to the Northern Belt was significant in the linear trend with a standard deviation of approximately 0.712.

Consider the following nonlocal integro-differential operator: for $\alpha \in (0,2)$,

where $\sigma {:}{\mathbb {R}}^d\rightarrow {\mathbb {R}}^d\otimes {\mathbb {R}}^d$ and $b{:}{\mathbb {R}}^d\rightarrow {\mathbb {R}}^d$ are smooth and have bounded first-order derivatives, and p.v. stands for the Cauchy principal value. Let $B_1(x):=\sigma (x)$ and $B_{j+1}(x):=b(x)\cdot \nabla B_j(x)-\nabla b(x)\cdot B_j(x)$ for $j\in {\mathbb {N}}$. Under the following Hörmander’s type condition: for any $x\in {\mathbb {R}}^d$ and some $n=n(x)\in {\mathbb {N}}$, by using the Malliavin calculus, we prove the existence of the heat kernel $\rho _t(x,y)$ to the operator ${\mathcal {L}}^{(\alpha )}_{\sigma ,b}$ as well as the continuity of $x\mapsto \rho _t(x,\cdot )$ in $L^1({\mathbb {R}}^d)$ as a density function for each $t>0$. Moreover, when $\sigma (x)=\sigma $ is constant and $B_j\in C^\infty _b$ for each $j\in {\mathbb {N}}$, under the following uniform Hörmander’s type condition: for some $j_0\in {\mathbb {N}}$, we also show the smoothness of $(t,x,y)\mapsto \rho _t(x,y)$ with $\rho _t(\cdot ,\cdot )\in C^\infty _b({\mathbb {R}}^d\times {\mathbb {R}}^d)$ for each $t>0$.

For the solution to $\partial ^2_tu(x,t)-\triangle u(x,t)+q(x)u(x,t)=\delta (x,t)$ and $u\mid _{t<0}=0$, consider an inverse problem of determining $q(x), x\in \Omega $ from data $f=u\mid _{S_T}$ and $g=(\partial u/\partial \mathbf {n})\mid _{S_T}$. Here $\Omega \subset \{(x_1,x_2,x_3)\in \mathbb {R}^3\mid x_1>0\}$ is a bounded domain, $S_{T}=\{(x,t)\mid x\in {\partial \Omega },\vert {x}\vert<t<T+\vert {x}\vert \}$, $\mathbf {n}=\mathbf {n}(x)$ is the outward unit normal $\mathbf {n}$ to $\partial \Omega $, and $T>0$. For suitable $T>0$, prove a Lipschitz stability estimation:

provided that $q_1$ satisfies a priori uniform boundedness conditions and $q_2$ satisfies a priori uniform smallness conditions, where $u_k$ is the solution to problem (1.1) with $q = q_k, k = 1, 2$.