In recent years the field of real numbers expanded by a multiplicative subgroup has been studied extensively. In this thesis, the known results will be extended to expansions of the real field. I will consider the structure R consisting of the field of real numbers and an irrational power function. Using Schanuel conditions, I will give a first-order axiomatization of expansions of R by a dense multiplicative subgroup which is a subset of the real algebraic numbers. It will be shown that every definable set in such a structure is a boolean combination of existentially definable sets and that these structures have o-minimal open core. A proof will be given that the Schanuel conditions used in proving these statements hold for co-countably many real numbers. The results mentioned above will also be established for expansions of R by dense multiplicative subgroups which are closed under all power functions definable in R. In this case the results hold under the assumption that the Conjecture on intersection with tori is true. Finally, the structure consisting of R and the discrete multiplicative subgroup 2^{Z} will be analyzed. It will be shown that this structure is not model complete. Further I develop a connection between the theory of Diophantine approximation and this structure.

The principal focus of this thesis is the study of the real numbers regarded as a structure endowed with its usual addition and multiplication and the operations of raising to real powers. For our first main result we prove that any statement in the language of this structure is equivalent to an existential statement, and furthermore that this existential statement can be chosen independently of the concrete interpretations of the real power functions in the statement; i.e. one existential statement will work for any choice of real power functions. This result we call uniform model completeness. For the second main result we introduce the first order theory of raising to an infinite power, which can be seen as the theory of a class of real closed fields, each expanded by a power function with infinite exponent. We note that it follows from the first main theorem that this theory is model-complete, furthermore we prove that it is decidable if and only if the theory of the real field with the exponential function is decidable. For the final main theorem we consider the problem of expanding an arbitrary o-minimal expansion of a field by a non-trivial exponential function whilst preserving o-minimality. We show that this can be done under the assumption that the structure already defines exponentiation on a bounded interval, and a further assumption about the prime model of the structure.

O problema de controlar o nível significância do teste FBST(Full BayesianSignificantTest) é estudado no contexto de modelos bayesianos para densidade. Assim, é mostrado um método bayesiano que trabalha com estimação da densidade e como deve ser conduzido o FBST com este método quando deseja-se testar se uma população pode ser dita de determinada distribuição ou testar igualdade de duas populações. Para isso é apresentada a definição do e-valor modificado que é uma maneira alternativa de cálculo da medida de evidência do FBST. Por fim, é feito um estudo de simulação com diferentes distribuições de densidade e analisado o comportamento da função poder do teste nos casos de uma e duas populações. / The problem of controlling the significan celevelof the FBST (FullBayesianSignificantTest) test is studied in the contex of Bayesian models for density, thus, a Bayesian method is shown that works with density estimation estimation and how the FBST should be conducted in that situation with this method when it is desired to test if one population has certain density distribution or equality test of two populations. For this, a modified e-value definition is presented that is an alternative to calculate the FBST measure. At end a simulation study with different density distributions and analysis the power function of the test in cases of one and two populations.

\par The thesis deals with the statistical evaluation of data gained by the sensory analysis of the foodstuff. It brings a selection of the suitable statistical tests, a detailed analysis of these tests and their comparision based on the particular power functions for given parameters. As an important part of the thesis, there is a creating of custom software for the evaluating of sensorial data.

Introduction Performance in cross-country skiing is influenced by the skier’s ability to continuously produce propelling forces and force magnitude in relation to the net external forces. A surrogate indicator of the “power supply” in cross-country skiing would be a physiological variable that reflects an important performance-related capability, whereas the body mass itself is an indicator of the “power demand” experienced by the skier. To adequately evaluate an elite skier’s performance capability, it is essential to establish the optimal ratio between the physiological variable and body mass. The overall aim of this doctoral thesis was to investigate the importance of body-mass exponent optimization for the evaluation of performance capability in cross-country skiing. Methods In total, 83 elite cross-country skiers (56 men and 27 women) volunteered to participate in the four studies. The physiological variables of maximal oxygen uptake (V̇O2max) and oxygen uptake corresponding to a blood-lactate concentration of 4 mmol∙l-1 (V̇O2obla) were determined while treadmill roller skiing using the diagonal-stride technique; mean oxygen uptake (V̇O2dp) and upper-body power output (Ẇ) were determined during double-poling tests using a ski-ergometer. Competitive performance data for elite male skiers were collected from two 15-km classical-technique skiing competitions and a 1.25-km sprint prologue; additionally, a 2-km double-poling roller-skiing time trial using the double-poling technique was used as an indicator of upper-body performance capability among elite male and female junior skiers. Power-function modelling was used to explain the race and time-trial speeds based on the physiological variables and body mass. Results The optimal V̇O2max-to-mass ratios to explain 15-km race speed were V̇O2max divided by body mass raised to the 0.48 and 0.53 power, and these models explained 68% and 69% of the variance in mean skiing speed, respectively; moreover, the 95% confidence intervals (CI) for the body-mass exponents did not include either 0 or 1. For the modelling of race speed in the sprint prologue, body mass failed to contribute to the models based on V̇O2max, V̇O2obla, and V̇O2dp. The upper-body power output-to-body mass ratio that optimally explained time-trial speed was Ẇ ∙ m-0.57 and the model explained 63% of the variance in speed. Conclusions The results in this thesis suggest that V̇O2max divided by the square root of body mass should be used as an indicator of performance in 15-km classical-technique races among elite male skiers rather than the absolute or simple ratio-standard scaled expression. To optimally explain an elite male skier’s performance capability in sprint prologues, power-function models based on oxygen-uptake variables expressed absolutely are recommended. Moreover, to evaluate elite junior skiers’ performance capabilities in 2-km double-poling roller-skiing time trials, it is recommended that Ẇ divided by the square root of body mass should be used rather than absolute or simple ratio-standard scaled expression of power output. / <p>Incorrect ISBN in printed thesis: 973-91-7601-270-3</p>

Vienas iš svarbiausių šiuolaikinės kriptografijos uždavinių yra saugių vienkrypčių funkcijų paieška. Dabartiniai mokslininkai skiria šiam klausimui ypatingą demėsį. Šiame darbe yra nagrinėjama viena iš naujausių vienkrypčių funkcijų – matricinio laipsnio funkcija. Ši funkcija yra panaudota netiesinės algebrinės lygčių sistemos sudarymui. Pagrindinis demėsys darbe yra skirtas šios lygčių sistemos analizei bei jos praktiniam taikymui. Nustatysime ar matricinio laipsnio funkcija gali būti panaudota kriptografijoje. Taip pat nustatysime lygčių sistemos sprendinių skaičiaus priklausomybę nuo jos parametrų: matricų eilės m bei grupės Z_p parametro p. / Since the introduction of Diffie-Hellman key agreement protocol in 1976 computer technology has made a giant step forward. Nowadays there is not much time left before quantum computers will be in every home. However it was theoretically proven that discrete logarithm problem which is the basis for Diffie-Hellman protocol could be solved in polynomial time using such computers. Such possibility would make D-H protocol insecure. Thus cryptologists are searching for different ways to improve the security of the protocol by using hard problems. One of the ways to do so is to introduce secure one-way functions (OWF). In this paper a new kind of OWF called the matrix power function will be analyzed. Professor Eligijus Sakalauskas introduced this function in 2007 and later used this function to construct a Diffie-Hellman type key agreement protocol using square matrices. This protocol is not only based on matrix power function but also on commutative matrices which are defined in finite fields or rings. Thus an algebraic non-linear system of equations is formed. The security of this system will be analyzed. It will be shown that we can use matrix power function in cryptography. We will also be analyzing how does the solution of the system depend on system parameters: the order of matrices and a parameter p which defines a finite group Z_p. We will also briefly discuss the usage of this system in real life and the algebraic properties of the suggested OWF.

Hlavním zájmem diplomové práce je studium zobecněné nabla Laplaceové transformace na časových škálach a její jednoznačnosti, včetně důkazu jednoznačnosti a aplikace jednoznačnosti v zlomkovém kalkulu na časových škálach.

S-boxes are key components of many symmetric cryptographic primitives. Among them, some block ciphers and hash functions are vulnerable to attacks based on differential cryptanalysis, a technique introduced by Biham and Shamir in the early 90’s. Resistance against attacks from this family depends on the so-called differential properties of the S-boxes used. When we consider S-boxes as functions over finite fields of characteristic 2, monomials turn out to be good candidates. In this Master’s Thesis, we study the differential properties of a particular family of monomials, namely those with exponent 2ͭᵗ-1 In particular, conjectures from Blondeau’s PhD Thesis are proved. More specifically, we derive the differential spectrum of monomials with exponent 2ͭᵗ-1 for several values of t using a method similar to the proof Blondeau et al. made of the spectrum of x -<img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Crightarrow" /> x⁷. The first two chapters of this Thesis provide the mathematical and cryptographic background necessary while the third and fourth chapters contain the proofs of the spectra we extracted and some observations which, among other things, connect this problem with the study of particular Dickson polynomials.

In certain areas such as Pharmacokinetic(PK) and Pharmacodynamic(PD), the hazard rate function, denoted by ??, plays a central role in modeling the instantaneous risk of failure time data. In the context of assessing the appropriateness of a given parametric hazard rate model, Huh and Hutmacher [22] showed that their hazard-based visual predictive check is as good as a visual predictive check based on the survival function. Even though Huh and Hutmacher’s visual method is simple to implement and interpret, the final decision reached there depends on the personal experience of the user. In this thesis, our primary aim is to develop nonparametric goodness-ofit tests for hazard rate functions to help bring objectivity in hazard rate model selections or to augment subjective procedures like Huh and Hutmacher’s visual predictive check. Toward that aim two nonparametric goodnessofit (g-o) test statistics are proposed and they are referred to as chi-square g-o test and kernel-based nonparametric goodness-ofit test for hazard rate functions, respectively. On one hand, the asymptotic distribution of the chi-square goodness-ofit test for hazard rate functions is derived under the null hypothesis ??0 : ??(??) = ??0(??) ??? ? R + as well as under the fixed alternative hypothesis ??1 : ??(??) = ??1(??) ??? ? R +. The results as expected are asymptotically similar to those of the usual Pearson chi-square test. That is, under the null hypothesis the proposed test converges to a chi-square distribution and under the fixed alternative hypothesis it converges to a non-central chi-square distribution. On the other hand, we showed that the power properties of the kernel-based nonparametric goodness-ofit test for hazard rate functions are equivalent to those of the Bickel and Rosenblatt test, meaning the proposed kernel-based nonparametric goodness-ofit test can detect alternatives converging to the null at the rate of ???? , ?? < 1/2, where ?? is the sample size. Unlike the latter, the convergence rate of the kernel-base nonparametric g-o test is much greater; that is, one does not need a very large sample size for able to use the asymptotic distribution of the test in practice.

The application of varnish, metal coats, and paint is a common practice for modifying or enhancing material properties. Metal coats are frequently used as protective layers against corrosion, heat, and wear, while also influencing characteristics like conductivity, weight, and production costs. Achieving the optimal thickness of the coating is critical, as a too-thin layer may not offer sufficient protection, while an overly thick layer adds unnecessary weight and increases expenses. Therefore, it is crucial to accurately measure the coating thickness without causing any damage. This project focuses on utilising flash pulse thermography, a non-invasive and non-destructive measuring technique, with three algorithms — Dynamical Thermal Tomography, Power Function, and Pulse Phase Thermography — to measure and differentiate between plates with known variations in the number of coating layers. The study also aims to identify the limiting factors associated with the experimental equipment and the characteristics of the thermography algorithms. The thickness calculations were performed both individually for each plate and simultaneously for multiple plates. The results demonstrate that Dynamical Thermal Tomography exhibits superior precision and strong linear correlation when measuring individual plates. On the other hand, the Power Function algorithm outperforms in effectively distinguishing between two plates simultaneously, while providing decent precision for individual plates. It is worth noting that the framerate of the camera significantly affects the performance and serves as the primary limiting factor in this specific experimental setup.Further investigations are necessary to obtain more conclusive results and determine the limitations of accuracy when measuring coating thickness.