Mirror and Circular Symmetry of Autofocusing Beams

This article demonstrates the crucial importance of the symmetrization method for the formation of autofocusing beams. It is possible to impart autofocusing properties to rather arbitrary distributions, for example, truncated and inverted classical modes (such as Hermite–Gaussian, Laguerre–Gaussian, and Bessel modes) or shift the fundamental Gaussian beam by inserting mirror or circular symmetry. The most convenient for controlling autofocusing characteristics is the truncated sinus function with a power-law argument dependence. In this case, superlinear chirp beams (with power q > 2) exhibit sudden and more abrupt autofocusing than sublinear chirp beams (with power 1 < q < 2). Comparison of the different beams’ propagation is performed using fractional Fourier transform, which allows obtaining the field distribution in any paraxial region (both in the Fresnel and Fraunhofer diffraction regions). The obtained results expand the capabilities of structured beams in various applications in optics and photonics.

The problem of extreme decomposition of the plane

Considered here is one quite general problem about the description of extremal configurations maximizing the product of inner radii mutually non-overlapping domains the next following form: \begin{equation} J_{n}(\gamma)=\left[r\left(B_0,0\right)r\left(B_\infty,\infty\right)\right]^{\gamma}\prod\limits_{k=1}^n r\left(B_k,a_k\right),\end{equation} where $\gamma\in\mathbb{R^{+}}$, $A_{n}=\{a_{k}\}_{k=1}^{n}$ -- $n$-radial points system, $B_0$, $B_\infty$, $\{B_{k}\}_{k=1}^{n}$ -- set of systems of mutually disjoint domains, \textup{(}$a_{0}=0\in B_0\subset{\mathbb{C}}$, $\infty\in B_\infty\subset\overline{\mathbb{C}}$, $a_k\in B_k\subset{\mathbb{C}}$\textup{)}, achieved for some configuration of domains $B_{0}$, $B_{k}$, $B_{\infty}$ and points $a_{0}$, $a_{k}$, $\infty$, $k=\overline{1, n}$. The functional (1) evaluation for the first time was obtained in 1988 by V.M. Dubinin \cite{4} for {$\gamma=\frac{1}{2}$ и $n\geqslant2$} for systems of disjoint domains using symmetrization method. A special case, when domains are univalent, was examined by G.V. Kuzmina in \cite{KYZMINA-01} After the result of V.M. Dubinin in the general formulation for the arbitrary multiply-concted domains was not until 2017. In 2017 in the work of I. Dengi, A. Targonsky \cite{iratarg} was the result for $\gamma_{n}=0.08n^{2}$, $n\geq7$. The result was obtained through a lower bound of $min_t (n-1)x_{1}+x_{2},$ where $x_{1}(t)+x_{2}(t)$ is the equation $F^\prime (x)=t$ solution, $F^\prime (x)=4x\ln(x)-2(x-1)\ln|x-1|-2(x+1)\ln(x+1)+\frac{2}{x}, $ $y_0 \leqslant t<0$, $y_{0}\approx-1,0599$. In this paper, a much better estimate of $min_t (n-1)x_{1}+x_{2}$ was obtained through a lower bound with the specified parameters. On the basis of this, the article succeeded in obtaining an estimate of the maximum of the functional (1) over a larger interval of the parameter $\gamma$, $\gamma\in(0,\gamma_{n}]$, $n\geq7$. Received the result for for any points of the circle $|a_k|=R$, $k=\overline{1,n}$, and any pairwise disjoint systems of domains $B_k$, $a_{0}=0\in B_0\subset{\mathbb{C}}$, $\infty\in B_\infty\subset\overline{\mathbb{C}}$, $a_k\in B_k\subset{\mathbb{C}}$, $k=\overline{1,n}.$

The Minimum Number of Triangular Edges and a Symmetrization Method for Multiple Graphs

We give an asymptotic formula for the minimum number of edges contained in triangles among graphs with n vertices and e edges. Our main tool is a generalization of Zykov's symmetrization method that can be applied to several graphs simultaneously.

Analysis of a continuous 24-hour glycemic curve by the symmetrization method

The study included 18 patients (10 females and 8 males) with a not less than 2 year-history of type 1 diabetes (T1D), who had received insulin therapy since its diagnosis was established. The patients ’ mean age was 32.9± 13.0 years; the mean duration of TID was 15.1+11.5 years; the mean daily dose of insulin was 40.1+16.0 units; the mean level of glycosylated hemoglobin (HbAJ was 9.4±2.1% (the normal value 4.4-4.9%). The glycemic curve symmetrization method proposed for statistical analysis of glycemic self-control is also quite suitable for the statistical monitoring of a continuous daily glycemic curve. The high and low glycemic indices calculated from the symmetrized glycemic data correlate well with the level of HbAk and with the duration of hypoglycemia and hyperglycemia and hence they may be used as additional criteria for a risk of diabetes complications. The criteria, calculated from the symmetrized data of glycemia for the risk of hyper- and hypoglycemia, adequately reflect the behavior of a continuous glycemic curve and may be used as integral indices of the efficiency of glucose-reducing therapy in clinical practice.