In this thesis we are investigating identities regarding Fibonacci sequences. In particular we are examiningthe so called Pisano period, which is the period for the Fibonacci sequence considered modulo n to repeatitself. The theory shows that it suces to compute Pisano periods for primes. We are also looking atthe same problems for the generalized Pisano period, which can be described as the Pisano period forthe generalized Fibonacci sequence.

Prie jau egzistuojančių įvairių skaičių sekų generatorių siūlomas dar vienas naujas netiesinis sekos generatorius. Šiuo generatorium naujo tipo sveikųjų skaičių sekos formuojamos netiesiniu, prieš tai esančių narių sukarpymo ir naujo montavimo metodu. Formuojami nauji sekos nariai, randamos jų priklausomybės nuo sekos pagrindo P ir dviejų pradinių sekos narių. Taip pat ištiriamos naujų sekų savybės: Periodiškumas. Ribotumas (sekos narių reikšmės ieškomos intervale [0; P^2-1]). Maksimalus periodas. / To the existing number of different sequences of generators proposed another new non-linear sequence generator. In this new type of generator integer non-linear sequence formed after the members sukarpymo installation of a new method. Forming a new sequence of members found in their dependence on base sequence of P and two members of the original sequence. It also examined the new series features: Periodicity. Limitations (the sequence of interest sought in the interval [0, P ^ 2-1]). Maximum period.

In previous research, the author developed a novel PPDM method – Data Unrealization – that preserves both data privacy and utility of discrete-value training samples. That method transforms original samples into unrealized ones and guarantees 100% accurate decision tree mining results. This dissertation extends their research scope and achieves the following accomplishments: (1) it expands the application of Data Unrealization on other data mining algorithms, (2) it introduces data compression methods that reduce storage requirements for unrealized training samples and increase data mining performance and (3) it adds a second-level privacy protection that works perfectly with Data Unrealization. From an application perspective, this dissertation proves that statistical information (i. e. counts, probability and information entropy) can be retrieved precisely from unrealized training samples, so that Data Unrealization is applicable for all counting-based, probability-based and entropy-based data mining models with 100% accuracy. For data compression, this dissertation introduces a new number sequence – J-Sequence – as a mean to compress training samples through the J-Sampling process. J-Sampling converts the samples into a list of numbers with many replications. Applying run-length encoding on the resulting list can further compress the samples into a constant storage space regardless of the sample size. In this way, the storage requirement of the sample database becomes O(1) and the time complexity of a statistical database query becomes O(1). J-Sampling is used as an encryption approach to the unrealized samples already protected by Data Unrealization; meanwhile, data mining can be performed on these samples without decryption. In order to retain privacy preservation and to handle data compression internally, a column-oriented database management system is recommended to store the encrypted samples. / Graduate / 0984 / fong_bee@hotmail.com

Efficiency in algorithms is important, especially in terms of execution time, as it directly impacts user experience. For example, when a customer visits a website, even a mere one-second delay can significantly reduce their patience, and the likelihood of them abandoning the site increases. This principle applies to search algorithms as well. This project is about implementing a time-efficient tree-based search algorithm that focuses on finding similarities between search input and stored data. The objective is to achieve an execution time as close to O(1) regardless of the data size. The implemented algorithm will be compared with a linear search algorithm, which has an execution time that grows along with the data size. By measuring the executiontimes of both search methods, the project aims to demonstrate the superiority of the tree-based search algorithm in terms of time efficiency.