Let $w=w(x_1,\ldots,x_r)$ be a lower central word or a derived word. We show that the word $w(u_1,\ldots,u_r)$ is concise whenever $u_1,\ldots,u_r$ are non-commutator words in disjoint sets of variables, thus proving a generalized version of a conjecture of Azevedo and Shumyatsky. This applies in particular to words of the form $w(x_1^{n_1},\ldots,x_r^{n_r})$, where the $n_i$ are non-zero integers. Our approach is via the study of values of $w$ on normal subgroups, and in this setting we obtain the following result: if $N_1,\ldots,N_r$ are normal subgroups of a group $G$ and the set of all values $w(g_1,\ldots,g_r)$ with $g_i\in N_i$ is finite then also the subgroup generated by these values, i.e.\ $w(N_1,\ldots,N_r)$, is finite.

Let $w=w(x_1,\ldots,x_r)$ be an outer commutator word‎. ‎We show that the word $w(u_1,\ldots,u_r)$ is concise whenever $u_1,\ldots,u_r$ are non-commutator words in disjoint sets of variables‎. ‎This applies in particular to words of the form $w(x_1^{n_1},\ldots,x_r^{n_r})$‎, ‎where the $n_i$ are non-zero integers‎. ‎Our approach is via the study of values of $w$ on normal subgroups‎, ‎and in this setting, ‎we obtain the following result‎: ‎if $N_1,\ldots,N_r$ are normal subgroups of a group $G$ and the set of all‎ ‎values $w(g_1,\ldots,g_r)$ with $g_i\in N_i$ is finite, ‎then also the subgroup generated by these values‎, ‎i.e. $w(N_1,\ldots,N_r)$‎, ‎is finite‎.

Total elbow arthroplasty (TEA) is increasingly used, especially in patients with rheumatoid arthritis (RA) and distal humerus fractures (DHFs). This narrative review of the recent literature published in 2022 reached the following conclusions: 1) Age greater than 80 is not a contraindication for TEA. 2) The estimated 10-year survival reported for linked TEAs is 92%, and unlinked TEAs 84%. 3) For DHFs in the elderly, regarding the flexion/extension arc, TEA gives substantially better results than open reduction and internal fixation (ORIF). The reoperation and elbow stiffness rates are substantially lower in TEA than in ORIF. 4) Comparing distal humeral hemiarthroplasty (DHH) vs. TEA in individuals over 65 years with a non-reconstructable DHF favors DHH regarding the range of motion but with a similarly high rate of adverse events in the two surgical techniques. 5) The rate of eradication of periprosthetic joint infection (PJI) is 69-76% with two-stage, 71% with resection arthroplasty, 67% with one-stage, 58% with DAIR, and 40% with elbow arthrodesis.

Recent works done by nano-engineers and nano-sciences about mechanical behavior of nano-plates including bending, buckling and vibration response were reviewed. The authors used non-classical elasticity theories to explain these behaviors of plate structures. Some of them employed Hamilton’ s principle along with stain gradient theory, nonlocal theory, surface theory and couple stress theory to derive the governing equation of nanostructures. Also, the authors have used various plate theories such as classical plate theory (CPT), first-order shear deformation theory (FSDT) and higher-order shear deformation plate theory (HSDT) to explain the linear and nonlinear behavior of nano-plates. Few researchers utilized molecular dynamics or experimental tests to explain size-dependent behavior of nano-plates. Investigated nano-plates were made of homogeneous and functionally graded materials (FGM) and were under mechanical and/or thermal loads. The effect of the magnetic field was considered, in other few researches. Governing equations solved using numerical methods such as differential quadrature method (DQM). The results of recent researches were presented and discussed.