Best polynomial approximation

U některých matematických úloh nedokážeme nalézt řešení exaktně, ale pouze přibližně metodami numerické matematiky. Klíčovým konceptem je pak nahrazení neboli aproximace. Mezi nejčastěji používané metody patří například polynomiální interpolace, nicméně chyba této metody může být v některých bodech intervalu velmi vysoká. Aproximace, která v každém bodě daného intervalu minimalizuje maximum chyby, se nazývá nejlepší stejnoměrná aproximace (best approximation). Obecně existuje právě jedno řešení úlohy nalezení nejlepší polynomiální aproximace, ale nelze ji vypočítat přesně. K přibližnému řešení se lze dopracovat například použitím Remezova algoritmu. Překvapivě dobré (near-best approximation) řešení nám také nabízí interpolace v Čebyševových uzlech. Cílem naší práce je zejména nastudovat a naimplementovat Remezův algoritmus a otestovat jeho funkčnost. Remezův algoritmus nakonec porovnáme s Lagrangeovou interpolací v Čebyševových uzlech.For some mathematical problems we are not able to find solutions exactly, but only approximately by using methods of numerical mathematics. The key concept is then replacement or in other words approximation. One of the most used methods is, for example, polynomial interpolation, however at some points of the interval the error of this method might be significant. The approximation which minimizes the maximum error at every point of the interval is called the best approximation. Generally, there is a unique solution to the problem of finding the best polynomial approximation, but it is not possible to calculate it exactly. The approximate solution can be obtained, for example, by using the Remez algorithm. The interpolation at Chebyshev nodes also yields surprisingly good results, generally being near-best approximations. The goal of our thesis is namely to study and implement the Remez algorithm and test its functionality. Finally, we compare the Remez algorithm with Lagrange interpolation at Chebyshev nodes.470 - Katedra aplikované matematikyvýborn

Stlačitelnost intermetalické sloučeniny

Názov práce: Stlačiteľnosť intermetalickej zlúčeniny Autor: Daniel Staško Katedra: Katedra fyziky kondenzovaných látok Vedúci bakalárskej práce: RNDr. Jiří Prchal, Ph.D. Abstrakt: Predmetom tejto práce je štúdium metódy určenia zmien objemu materiálu pôsobením hydrostatického tlaku pomocou miniatúrnych tenzometrov. Medená vzorka s vysokou čistotou bola použitá ako referenčný materiál na zistenie tlakových a teplotných korekcii pre konkrétny typ tenzometra použitého v tejto práci. Na demonštráciu tejto metódy bola zmeraná tlaková závislosť predĺženia vzorky CeCuAl3 pozdĺž 2 kryštalografických osí v rozsahu do 2,3GPa. CeCuAl3 je intermetalická zlúčenina už dlho študovaná kvôli jej zaujímavým magnetickým vlastnostiam. Táto práca je súčasťou komplexného štúdia tohto materiálu. Zároveň bola zmeraná teplotná závislosť predĺženia v nízkom a vysokom tlaku. Kľúčové slová: vysoký tlak, stlačiteľnosť, tenzometer, CeCuAl3Title: Compressibility of an intermetallic compound Author: Daniel Staško Department: Department of Condensed Matter Physics Supervisor: RNDr. Jiří Prchal, Ph.D., Department of condensed matter physics Abstract: This work focuses on the method of using miniature strain gages for the measurement of volume change and strain in hydrostatic pressure. Two state variables are involved in the changes - temperature and pressure. A copper reference material was used to determine the pressure and temperature corrections for the specific type of strain gage used it this work. To demonstrate this method, a pressure dependence of strain was measured up to 2.3GPa for the CeCuAl3 single crystalline sample along particular crystallographic directions. It is studied for its interesting magnetic properties and this work aims to contribute to its complex study. Temperature dependencies of strain under low and high pressure were also measured. Keywords: high pressure, compressibility, strain gage, CeCuAl3Katedra fyziky kondenzovaných látekDepartment of Condensed Matter PhysicsMatematicko-fyzikální fakultaFaculty of Mathematics and Physic

Cubic and biquadratic reciprocity

Hlavní motivací pro zkoumání kubické a bikvadratické reciprocity je rozhod- nout, zda mají kongruence x3 ≡ a (p) nebo x4 ≡ a (p), kde a ∈ Z, p prvočíslo, nějaké celočíselné řešení. Jádrem této práce je prostřednictvím postupně vybudo- vané teorie v okruzích Eisensteinových a Gaussových celých čísel dokázat zákony kubické a bikvadratické reciprocity. U obou těchto tvrzení se navíc podrobněji podíváme na speciální případy, ve kterých je nelze použít. To nás povede k od- vození tzv. doplňku k zákonu kubické (resp. bikvadratické) reciprocity. Nakonec ukážeme, jak lze tyto výsledky aplikovat na problém řešitelnosti zmíněných kon- gruencí. 1The main motivation for studying cubic and biquadratic reciprocity is to de- cide, whether the congruences x3 ≡ a (p) or x4 ≡ a (p), where a ∈ Z, p prime, have any integer solution. The core of this thesis will be to prove the laws of cubic and biquadratic reciprocity through gradually built theory in the rings of Eisen- stein and Gaussian integers. In addition, for both of these theorems, we will take a closer look at the special cases, in which they cannot be used. This will lead us to the derivation of the supplement to the law of cubic (or biquadratic) re- ciprocity. Finally, we will show how these results can be applied to the problem of solvability of mentioned congruences. 1Department of AlgebraKatedra algebryFaculty of Mathematics and PhysicsMatematicko-fyzikální fakult

Neutron scattering study of the Tm2 Ir2 O7 pyrochlore iridate

The newly synthesized Tm2 Ir2 O7 iridate is investigated using neutron scattering techniques. Powder neutron diffraction patterns confirm that Tm2 Ir2 O7 crystallizes in a cubic structure of pyrochlore type, consistent with the rest of the rare-earth A2 Ir2 O7 series, and the crystal structure is preserved down to 0.2 K. The thermal expansion of the crystal lattice is well described by a Debye model of lattice vibrations. A weak magnetic signal consistent with the so-called all-in-all-out magnetic order (AIAO) is traced at low temperature; however, it cannot serve as unambiguous evidence of the AIAO structure in the compound. Nondispersive magnetic excitations in energy-momentum space are observed in inelastic neutron scattering spectra of Tm2 Ir2 O7. The crystal-field scheme and parameters of the thulium iridate are refined and discussed with respect to the specific heat and magnetization data. The results are supported by mean-field calculations

Návrh mechanizované technologie svařování montážních svarů uzavíracích vrat plavební komory vodního díla

Import 20/04/2006Prezenční výpůjčkaFakulta strojní a elektrotechnická VŠB (Ostrava). Katedra mechanické technologie (345

Creation of dashboards in the tool QlikView

This diploma thesis deals with performance dashboards and QlikView tool which serves for creation of these dashboards. The work is divided into seven parts. The first and the second part together focus on defining the theoretical foundations of Business Intelligence and Corporate Performance Management, which create basis for performance dashboards. The focus of the third part lies in definition of term performance dashboard and clarifying its major advantages and benefits for a company. An important part is also the description of performance dashboard composition in terms of their usability, layered structure and division into various types. The forth part analyses existing criteria for evaluating of tools dedicated for creation of dashboards. Its goal is, according to the forgone analysis, to make the own list of criteria for evaluating these tools. The aim of the fifth part of this work is to describe QlikView platform. It provides a brief overview of the platform in terms of its most important components and their purpose and also provides an insight into basic architecture of this platform and compares it with traditional BI solutions. The sixth part describes basic principles of work in QlikView. Its goal is to demonstrate the basic procedure of dashboards creation in this environment on sample examples, which should also serve as a simple guide for beginner users of this tool. The last seventh part deals with evaluation of QlikView solution. Its main purpose is to review possibilities and capabilities of this tool for dashboard creation according to the predetermined criteria which are applied to the tool during its evaluation