On Locating Clusters Of Zeros Of Analytic Functions

Import 03/11/2016Táto práca sa zaoberá hľadaním koreňov holomorfných funkcií. V prvej časti práce pripomenieme niektoré pojmy z komplexnej analýzy, ktoré budú v práci často používané. Potom sa zameriame na známe vety z komplexnej analýzy, týkajúce sa koreňov holomorfných funkcií. Z týchto viet bude v druhej časti práce dôležitý dôsledok Reziduovej vety. Prvá časť práce taktiež obsahuje niektoré dôkazy Základnej vety algebry. Druhá časť práce popisuje metódy hľadania koreňov holomorfných funkcií - metódu Newtonových súm a metódu založenú na formálnych ortogonálnych polynómoch.This thesis deals with the locating zeros of holomorfic functions. In the first part of thesis some concepts of complex analysis are mentioned, which will be frequently used in this thesis. Then, we devote to known theorems of complex analysis relating to zeros of holomorfic functions. From these theorems will be important the consequence of Residue theorem, but it will be shown in the second part of thesis. The first part of thesis also contains some proofs of Fundamental theorem of algebra. The second part of thesis describes methods for locating zeros of analytic functions - method of Newton sums and method based on formal orthogonal polynomials.470 - Katedra aplikované matematikyvýborn

Primality testing

Import 05/08/2014Táto práca popisuje najznámejšie testy prvočíselnosti. Prvé dve kapitoly sú venované základným pojmom, definíciám a dôkazom. Tretia kapitola popisuje Fermatovu-Eulerovu vetu, ktorá sa neskôr využíva pri odvodení Fermatovho a Miller-Rabinovho testu. Zvyšné kapitoly sú venované pravdepodobnostným a deterministickým testom prvočíselnosti, a to konkrétne ich teoretickému základu, odvodeniu a popisu. Tieto kapitoly obsahujú aj algoritmus naimplementovaný v Matlabe aj v Maple.This thesis describes the most famous primality tests. The first two chapters are about basic terms, definitions and proofs. The third chapter describes Fermat-Euler theorem, which was later used in the derivation of Fermat and Miller-Rabin primality test. The other chapters are about probabilistic and deterministic primality tests, specifically about their theoretical basis, derivation and description. These chapters also contain algorithm, which is deployed in Matlab and Maple.470 - Katedra aplikované matematikyvýborn

Finding zeros of analytic functions and local eigenvalue analysis using contour integral method in examples

A numerical method for computing zeros of analytic complex functions is presented. It relies on Cauchy's residue theorem and the method of Newton's identities, which translates the problem to finding zeros of a polynomial. In order to stabilize the numerical algorithm, formal orthogonal polynomials are employed. At the end the method is adapted to finding eigenvalues of a matrix pencil in a bounded domain in the complex plane. This work is based on a series of papers of Professor Sakurai and collaborators. Our aim is to make their work available by means of a systematic study of properly chosen examples

Matching point clouds with STL models by using the principle component analysis and a decomposition into geometric primitives

While repairing industrial machines or vehicles, recognition of components is a critical and time-consuming task for a human. In this paper, we propose to automatize this task. We start with a Principal Component Analysis (PCA), which fits the scanned point cloud with an ellipsoid by computing the eigenvalues and eigenvectors of a 3-by-3 covariant matrix. In case there is a dominant eigenvalue, the point cloud is decomposed into two clusters to which the PCA is applied recursively. In case the matching is not unique, we continue to distinguish among several candidates. We decompose the point cloud into planar and cylindrical primitives and assign mutual features such as distance or angle to them. Finally, we refine the matching by comparing the matrices of mutual features of the primitives. This is a more computationally demanding but very robust method. We demonstrate the efficiency and robustness of the proposed methodology on a collection of 29 real scans and a database of 389 STL (Standard Triangle Language) models. As many as 27 scans are uniquely matched to their counterparts from the database, while in the remaining two cases, there is only one additional candidate besides the correct model. The overall computational time is about 10 min in MATLAB.Web of Science115art. no. 226