On an asymptotically log-periodic solution to the graphical curve shortening flow equation

<abstract><p>With the help of heat equation, we first construct an example of a graphical solution to the curve shortening flow. This solution $ y\left(x, t\right) \ $has the interesting property that it converges to a log-periodic function of the form</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ A\sin \left( \log t\right) +B\cos \left( \log t\right) $\end{document} </tex-math></disp-formula></p> <p>as$ \ t\rightarrow \infty, \ $where $ A, \ B $ are constants. Moreover, for any two numbers $ \alpha &lt; \beta, \ $we are also able to construct a solution satisfying the oscillation limits</p> <p><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \liminf\limits_{t\rightarrow \infty}y\left( x,t\right) = \alpha,\ \ \ \limsup\limits _{t\rightarrow \infty}y\left( x,t\right) = \beta,\ \ \ x\in K $\end{document} </tex-math></disp-formula></p> <p>on any compact subset$ \ K\subset \mathbb{R}. $</p></abstract>

Study of Dielectric, Polarization Properties of Cyclopentanol and Their Solutions in Cyclopentane by a New Variation Method

Results of calculating the theoretical principles of the variational method for measuring the dielectric parameters of polar liquids: cyclopentanol and its solutions in cyclopentane have been shown in the paper. Their dielectric constant ξ' and dielectric losses ξ'' are calculated. Solutions to the equations were found and a graphical solution method and an automated method for calculating ξ' and ξ'' were developed on the basis of this method. Comparison with the results of other methods revealed that these indicators are at the same time minimal within 1.5–2.0%.

Least-Squares Chebyshev Method for Solving Fractional Integro-diffrential Equations

The main purpose of this study gears towards finding numerical solution to fractional integro-differential equations. The technique involves the application of caputo properties and Chebyshev polynomials to reduce the problem to system of linear algebraic equations and then solved using MAPLE 18. To demonstrate the accuracy and applicability of the presented method some numerical examples are given. Numerical results show that the method is easy to implement and compares favorably with the exact results. The graphical solution of the method is displayed.

A new Approach for surveying the axis of cylindrical pipes using a reflectorless total station

To survey the pipe’s axis used in the installation of the cable-stayed bridge, a reflectorless total station and a proposed surveying observation technique were used. The approach includes a new method for observation and a graphical solution for the pipe’s axis determination. The technique shown in this paper can be applied for vertical and semi-vertical cylindrical pipes. A field test with real data and results is also presented.

COUPLING WITH FLAT SPRING PACKAGES: METHODS FOR CALCULATING THE MAIN PARAMETERS

Methods are presented that simplify the calculation of the main sizes of couplings with packages of flat springs for power drives of machines. The main parameter of such couplings is torsional stiffness, which must be provided during their design. Given a number of parameters, first of all, the size of the cross sections of elastic elements, their number in the package, the number of packages is graphically determined, the diameter of the circle on which they are located, the length of flat springs. At the same time, their bending strength is guaranteed. The graphical solution of the task allows you to purposefully adjust, if necessary, the design parameters and dimensions of the couplings.