In this diploma thesis I dealt with proposal of optimization security algorithms calculations, that will be implemented on smartcards. This project is based on knowledge of modular arithmetic, that is used in many encryption like public keys, digital signatures and has function. The result of this project is a design of changes for accelerate the security algorithms calculations.

<p>U disertaciji je predložen algoritam za proračun kratkih spojeva zasnovan na korelisanim intervalima. U savremenim distributivnim mrežama postoje različiti tipovi generatora koji proizvode električnu energiju iz energije obnovljivih izvora. Za takve generatore, kao i za potrošače, karakteristično je to što je njihova proizvodnja i potrošnja neizvesna. Predloženi algoritam u disertaciji uvažava te neizvesnosti, kao i korelacije između navedenih elemenata. Neizvesnosti su modelovane intervalima i direktno su uvažene u predloženom algoritmu za proračun kratkih spojeva. Algoritam je prvenstveno razvijen za proračun kratkih spojeva savremenih distributivnih mreža sa velikim brojem distribuiranih generatora i potrošača. NJime je moguće proračunavati režime sa kratkim spojevima distributivnih mreža velikih dimenzija, što je numerički verifikovano u disertaciji. Predloženim algoritmom se dobija režim distributivne mreže sa kratkim spojem koji je realističniji od režima dobijenih algoritmima sa determinističkim pristupom. Proračuni kao što su: koordinacija, podešenje i provera osetljivosti relejne zaštite, provera kapaciteta prekidača i osigurača, lokacija kvara itd. mogu na osnovu realističnijeg režima, dobijenog predloženim algoritmom, da daju kvalitetnije rezultate, što je numerički potvrđeno na primeru koordinacije prekostrujne zaštite</p> / <p>In this dissertation an algorithm for correlated intervals-based short-circuit calculation is<br />proposed. In modern distribution networks there are various types of generators that produce<br />electric energy from renewable energy resources. For these generators, as well as loads, uncertain<br />production and consumption is characteristic. The proposed algorithm in the dissertation deals<br />with above-mentioned uncertainties, as well as correlations among them. The uncertainties are<br />modeled with intervals and directly taken into account in the proposed algorithm for short-circuit<br />calculation. The algorithm is primarily developed for short-circuit calculation in modern<br />distribution networks with a great number of distributed generators and consumers. The proposed<br />algorithm enables calculation of short circuits states of large-scale distribution networks, which is<br />numerically verified in the dissertation. The proposed algorithm provides short circuit state of<br />distribution network which is more realistic than the one obtained with algorithms with<br />deterministic approach. Calculations such as: coordination, settings and sensitivity check of relay<br />protection, breaker and fuse capacity check, fault location, etc. can give better results, on the basis<br />of the more realistic state obtained by the proposed algorithm for short circuit calculation, which<br />is numerically confirmed by the example of coordination of overcurrent protection.</p>

This thesis deals with the concepts of numerical integrator using floating point arithmetic for solving partial differential equations. The integrator uses Euler method and Taylor series. Thesis shows parallel and serial approach to computing with exponents and significands. There is also a comparison between modern parallel systems and the proposed concepts.

In this thesis we consider the Lanczos algoritm and its behaviour in finite precision. Having summarized theoretical properties of the algorithm and its connection to orthogonal polynomials, we recall the idea of the Lanczos method for approximating the matrix eigenvalues. As the behaviour of the algorithm is strongly influenced by finite precision arithmetic, the linear independence of the Lanczos vectors is usually lost after a few iterations. We use the most im- portant results from analysis of the finite precision Lanczos algorithm according to Paige, Greenbaum, Strakos and others. Based on that, we study formulation and properties of the mathematical model of finite presicion Lanczos computati- ons suggested by Greenbaum. We carry out numerical experiments in Matlab, which support the theoretical results.

Esant dideliam informacijos srautui, svarbiausias mokymo tikslas bendrojo lavinimo mokyklose tampa mokyti moksleivį mokytis, skatinti mokymo metodus, padedančius moksleiviui atrasti žinomus reiškinius ir dėsningumus. Kadangi matematika yra vienas iš sudėtingiausių mokslų būtina tobulinti matematinių žinių perteikimą įvairiais būdais. Šiuo magistriniu darbu siekiama supaprastinti sveikųjų skaičių aritmetikos mokymą, tobulinti, daryti įdomesnį, vizualesnį įprastą matematikos dėstymą, palengvinant teigiamųjų ir neigiamųjų skaičių veiksmų suvokimą, algebrinių reiškinių sudarymą žemesnėse pagrindinio ugdymo klasėse. Darbe išanalizuotos laisvai platinamos mokomosios kompiuterinės programos (MKP), skirtos matematikai, išsiaiškintas moksleivių požiūris į kompiuterizuotą mokymą. Atsižvelgiant į MKP analizės ir moksleivių apklausos išvadas, išnagrinėjus galimas mokomosios priemonės kūrimo technologijas, sukurta mokomoji aplinka skyriui „Teigiamųjų ir neigiamųjų skaičių veiksmai“: parengtos ir suprogramuotos įvairios užduotys (teorija, savikontrolės testai, praktinės užduotys ir interaktyvūs HTML žaidimai). Mokomoji aplinka skirta tiek mokytojams, norintiems paįvairinti matematikos mokymą pamokose, tiek moksleiviams, savarankiškai atlikti užduotis namuose. / According to increasing data flow the most important aim of education is teaching the students to learn, to stimulate the methods of education, which help them to discover known phenomens at school. Mathematics is the most elaborate of studies. Because of this it needs perfect mathematical knowledge of various methods. This master work helps simplify the teaching of number arithmetics, whole perfects common mathematical training of positive and negative numbers and their operations in lower classes of secondary school. There are traversed free educational computer programs (ECP) in this work which help the teaching of mathematics. Also there is an ennanced pupil side of educational computer programs and paperless teaching. Considering how pupils respondent to ECP and paperless teaching, this work explored the likely creation of teaching measure technologies. There was made the analysis of computer programs and the teaching application was made using three technologies: Macromedia Flash, Microsoft FrontPage 2003 and the Hot Potatoes, which suite includes six applications, enabling to create interactive multiple-choice, short-answer, jumbled-sentence, crossword, matching/ordering and gap-fill exercises for the World Wide Web. It also explored teaching interacting numbers and their operations. This teaching application is designed for teaches who want to add the variety of usual mathematical teaching to their classes and also for pupils to do various self-control tests at home... [to full text]

Tato práce popisuje vznik obrazu v transmisním světelném mikroskopu s proměnnou aperturou kondenzoru. Tento způsob pořizování obrazu je výhodný při pozorování tlustých vzorků, což jsou objekty, u kterých je v mikroskopickém měřítku významná výška. Klasické transmisní mikroskopy pro tento výzkum vhodné nejsou, protože výsledný obraz, který produkují, obsahuje patrné informace z velkého objemu vzorku mimo oblast, která je vyšetřovaná. Tento problém byl vyřešen po vynálezu konfokálního mikroskopu, který je ovšem daleko dražší a má i některé nevýhody navíc. Cílem je zpracovat obrazy pořízené pomocí kondenzoru s rotující aperturou tak, aby došlo k redukci podílu nežádoucí informace a obraz se tak stal "čistším". Tato metoda zpracování obrazu nemohla být použita v minulosti, protože tehdejší počítače neměly dostatečnou paměť a výkon. Na tuto metodu se dá nahlížet jako na převod množiny výstupů z mikroskopu s vylepšenou osvětlovací soustavou na odpovídající množinu výstupů konfokálního mikroskopu.

In the Light of Intuitionism: Two Investigations in Proof Theory This dissertation focuses on two specific interconnections between the clas- sical and the intuitionistic proof theory. In the first part, we will propose a formalization for Gödel's informal reading of the BHK interpretation, using the usual classical arithmetical proofs. His provability interpretation of the propositional intuitionistic logic, first appeared in [1], in which he introduced the modal system, S4, as a formalization of the intuitive concept of prov- ability and then translated IPC to S4 in a sound and complete manner. His work suggested the search for a concrete provability interpretation for the modal logic S4 which itself leads to a concrete provability interpretation for the intutionistic logic. In the first chapter of this work, we will try to solve this problem. For this purpose, we will generalize Solovay's provabil- ity interpretation of the modal logic GL to capture other modal logics such as K4, KD4 and S4. Then, using the mentioned Gödel's translation, we will propose a formalization for the BHK interpretation via classical proofs. As a consequence, it will be shown that the BHK interpretation is powerful enough to admit many different formalizations that surprisingly capture dif- ferent propositional logics, including...

This thesis examines the method of forcing in set theory and focuses on aspects that are set aside in the usual presentations or applications of forcing. It is shown that forcing can be formalized in Peano arithmetic (PA) and that consis- tency results obtained by forcing are provable in PA. Two ways are presented of overcoming the assumption of the existence of a countable transitive model. The thesis also studies forcing as a method giving rise to interpretations between theories. A notion of bi-interpretability is defined and a method of forcing over a non-standard model of ZFC is developed in order to argue that ZFC and ZF are not bi-interpretable. 1

The thesis deals with arithmetics of elliptic curves over finite fields and methods to improve those calculations. In the first part, algebraic geometry helps to define elliptic curves and derive their basic properties including the group law. The second chapter seeks ways to speed up these calculations by means of time-memory tradeoff, i.e. adding redundancy. At last, the third part introduces a wholly new curve form, which is particularly effective for such purposes.

The thesis deals with arithmetics of elliptic curves over finite fields and methods to improve those calculations. In the first part, algebraic geometry helps to define elliptic curves and derive their basic properties including the group law. The second chapter seeks ways to speed up these calculations by means of time-memory tradeoff, i.e. adding redundancy. At last, the third part introduces a wholly new curve form, which is particularly effective for such purposes.