A note on factorization of the Fermat numbers and their factors of the form 3h2\sp n+1

summary:We show that any factorization of any composite Fermat number $F_m={2^{2}}^m+1$ into two nontrivial factors can be expressed in the form $F_m=(k2^n+1)(\ell2^n+1)$ for some odd $k$ and $\ell, k\geq 3, \ell \geq 3$, and integer $n\geq m+2, 3n<2^m$. We prove that the greatest common divisor of $k$ and $\ell$ is 1, $k+\ell\equiv 0\ mod 2^n,\ max(k,\ell)\geq F_{m-2}$, and either $3|k$ or $3|\ell$, i.e., $3h2^{m+2}+1|F_m$ for an integer $h\geq 1$. Factorizations of $F_m$ into more than two factors are investigated as well. In particular, we prove that if $F_m=(k2^n+1)^2(\ell2^j+1)$ then $j=n+1,3|\ell$ and $5|\ell$

On simplicial red refinement in three and higher dimensions

summary:We show that in dimensions higher than two, the popular "red refinement" technique, commonly used for simplicial mesh refinements and adaptivity in the finite element analysis and practice, never yields subsimplices which are all acute even for an acute father element as opposed to the two-dimensional case. In the three-dimensional case we prove that there exists only one tetrahedron that can be partitioned by red refinement into eight congruent subtetrahedra that are all similar to the original one

There are only two nonobtuse binary triangulations of the unit nn-cube

Triangulations of the cube into a minimal number of simplices without additional vertices have been studied by several authors over the past decades. For $3\leq n\leq 7$ this so-called simplexity of the unit cube $I^n$ is now known to be $5,16,67,308,1493$, respectively. In this paper, we study triangulations of $I^n$ with simplices that only have nonobtuse dihedral angles. A trivial example is the standard triangulation into $n!$ simplices. In this paper we show that, surprisingly, for each $n\geq 3$ there is essentially only one other nonobtuse triangulation of $I^n$, and give its explicit construction. The number of nonobtuse simplices in this triangulation is equal to the smallest integer larger than $n!({\rm e}-2)$.Comment: 17 pages, 7 figure

Why quintic polynomial equations are not solvable in radicals

summary:We illustrate the main idea of Galois theory, by which roots of a polynomial equation of at least fifth degree with rational coefficients cannot general be expressed by radicals, i.e., by the operations $+,\,-,\,\cdot,\,:\,$, and \root n \of{\cdot}. Therefore, higher order polynomial equations are usually solved by approximate methods. They can also be solved algebraically by means of ultraradicals

The design of methodology and creating of chosen SW modules for setting and exploring of lathe tools defects using laser measuring device

Tato diplomová práce se zabývá možností využití bezkontaktního měření korekcí a detekcí poškození nástrojů na soustružnických centrech. Za tímto účelem byla vytvořena metodika měření a její následná realizace v NC řídícím programu. K otestování programu v praxi bylo použito soustružnické centrum SPM 16 s řídicím systémem Sinumerik 840D. Jako měřící zařízení byla použita laserová sonda NC4 firmy Renishaw.This diploma thesis deals with possibility of a non-contact tool corrections measurement and broken tool detection on CNC lathes. For this purpose was made a methodology for measuring and application in NC program. For testing the program in praxis was used CNC lathe SPM 16 with Sinumerik 840D controller. As measuring equipment was used a laser probe NC4 by Renishaw manufacturer.

Generalization of the Zlámal condition for simplicial finite elements in Rd{\Bbb R}^d

summary:The famous Zlámal's minimum angle condition has been widely used for construction of a regular family of triangulations (containing nondegenerating triangles) as well as in convergence proofs for the finite element method in $2d$. In this paper we present and discuss its generalization to simplicial partitions in any space dimension

Kleinova čtyřgrupa

summary:The article deals with properties of the Klein Four-Group and its occurrences not only in mathematics

Wireless Power Transmission System for Powering Rotating Parts of Automatic Machineries

This paper deals with the analysis of a suitable compensation topology of a wireless power transmission system for powering the rotating parts of modern automatic machine tools. It summarizes the important properties of the serio-parallel compensation topology suitable for this application and demonstrates a detailed mathematical derivation using the first harmonic approximation. The paper details the industrial implementation of the system in a specific automatic machine tool application and demonstrates the strong technical advantages of the proposed design. Important theoretical conclusions and technical assumptions made when considering the system layout are verified by experimental laboratory measurements and the final deployment of the technology in the professional tool DMU 40 eVo linea