Covariogram of non-convex sets

The covariogram of a compact set A contained in R^n is the function that to each x in R^n associates the volume of A intersected with (A+x). Recently it has been proved that the covariogram determines any planar convex body, in the class of all convex bodies. We extend the class of sets in which a planar convex body is determined by its covariogram. Moreover, we prove that there is no pair of non-congruent planar polyominoes consisting of less than 9 points that have equal discrete covariogram.Comment: 15 pages, 7 figures, accepted for publication on Mathematik

Using of laserinterferometer Renishaw for straightness and flatness measurement

Tato diplomová práce se zabývá měřením přímosti a rovinnosti a jeho vyhodnocením. K měření je použit laserinterferometr firmy Renishaw. Cílem této práce je naměřit hodnoty přímosti na pojezdu měřícího mikroskopu, dále rozkódovat výpočetní metody vedoucí ke grafům přímosti, přepočítat z naměřených hodnot dle ČSN a porovnat výsledky softwaru a výsledky dosažené dle ČSN. Měření rovinnosti je aplikováno na desce před lapováním (povrchová úprava) a po měn a tyto výsledky jsou vzájemně porovnány.This diploma dissertation deals with straightness and flatness measurement and data evaluation. For all measurements laserinterferometer Renishaw has been used. The main purpose of this diploma work is to get values of straightness on the travel of microscope, decrypt computing methods leading to graphs of straightness, re-count it by ČSN standards and compare both sets of graphs. Flatness measurement is performed on a desk before and after lapping and these results are compared.

Characterization of the first operating period of a two-unit standby redundant system with three states of units

summary:A two-unit cold-standby redundant system with one repair facility is considered. Each unit can be in three states: good (I), degraded (II), and failed (III). We suppose that only the following state-transitions af a unit are possible: $I\rightarrow II, II\rightarrow III, II\rightarrow I, III\rightarrow I$. The paper is devoted to the problems which arise only provided that the units of the redundant system can be in more than two states (i.e. in operating and failed states). The following characteristics dealing with a single operating period of the system are studied under the condition that at its starting instant both units are new: the whole time of operation of units in state $I$ (or $II$), the whole time of repairs of units of the type $II\rightarrow I$ (or $III\rightarrow I$) and the number of finished repairs of units of the type $II\rightarrow I$ (or $III\rightarrow I$)

Some examples of non-monotonicities in a two-unit redundant system

summary:A cold-standby redundant sytem with two identical units and one repair facility is considered. Units can be in three states: good $(I)$, degraded $(II)$, and failed $(III)$. It is supposed that only the following state-transitions of a unit are possible: $I\rightarrow II$, $II\rightarrow III$, $II\rightarrow I$, $III\rightarrow I$. The paper deals with the comparison of some initial situations of the system and with a stochastical improvement of units (stochastical increase of time of work in state $I$ and/or stochastical decrease of times of repairs of the types $II\rightarrow I$ and/or $III\rightarrow I$) and shows on examples that some surprising non-monotonicities can take place

Analysis of a two-unit standby redundant system with three states of units

summary:A cold-standby redundant system with two identical units and one repair facility is considered. Units can be in three states> good (I), degraded (II), and failed (III). We suppose that only the following state-transitions of a unit are possible: $I\rightarrow II, II\rightarrow III, II\rightarrow I, III\rightarrow I$. The repair of a unit of the type $II \rightarrow I$ can be interpreted as a preventive maintenance. Its realization depends on the states of both units. Several characteristics of the system (probabilities, distribution functions or their Laplace-Stieltjes transforms and mathematical expectations) are derived, e.g. time to system failure, time of non-operating period of the system and stationary-state probabilities of the couple of units of the system