The paper presents general solutions for fractional state-space equations. The analysis of the fractional electrical circuit in the transient state is described by the equation of the state and space equations. The results are presented for the voltage of a capacitor and current in a coil, for different alpha values. The Caputo and conformable fractional derivative definitions have been considered. At the end, the results have been obtained.
W 1900 roku, na II międzynarodowym kongresie matematyków, wybitny matematyk niemiecki David Hilbert, ogłaszając listę 23 ważnych problemów do rozwiązania, snuł wizję matematyki jako nauki uniwersalnej, pewnej i oczywistej, która rozwiąże każdy problem. Była to odpowiedź uczonego na III kryzys podstaw matematyki. Rodzi się jednak pytanie o to, czy ocena ta odnosi się do matematyki dzisiejszej. Celem mojego artykułu jest nawiązanie do badań tzw. klasycznych kierunków filozofii matematyki (formalizm, intuicjonizm, logicyzm), zwrócenie uwagi na znaczenie twierdzeń K. Gödla dla rozwoju filozofii matematyki oraz na powstanie nowych nurtów (kierunek kulturowy, quasi-empiryzm, społeczny konstruktywizm, etnomatematyka), nadto uwzględnienie tendencji zachodzących w najnowszej nauce i filozofii matematyki, jak i wskazanie na niektóre cele i zadania stojące przed „nową filozofią matematyki”.
An analysis of a given electrical circuit using a fractional derivative. The statespace equation was developed. The dynamics of tensions described by Kirchhoff’s laws equations. The paper used the definition of the integral derivative Caputo and CDF conformable fractional definition. An electrical circuit solution using Caputo and CDF defini- tions for rectangular with zero initial conditions was developed. The results obtained using the Caputo and CDF definitions were compared. The solutions are shown for capacitor voltages, for fractional derivative orders of 0.6, 0.8, 1. The results were compared using graphs.
The research was conducted at the Kwiatków site,1 in the Koło Basin (Central Poland). It included a fragment of a low terrace and the valley floor of the Warta river valley. The archaeological investigation documented over 100 wells that archaeological material indicates are associated with the Przeworsk culture. Geomorphological, lithological and geochemical studies were carried out at the archaeological sites and their surroundings. Selected for the presentation were two wells whose fillings were carefully tested and subjected to geochemical and lithological analyses. The wells showed a slightly different content of artifacts, as well as differences in their grain-size distributions, the structure of their filling deposits, and their geochemistry. This allows us to conclude that the two wells were used differently, but also probably about a different course for how each well was filled after the end of its operation.
The paper presents the results of investigations concerning the application of zinc oxide - a wideband gap semiconductor in optical planar waveguide structures. ZnO is a promising semiconducting material thanks to its attractive optical properties. The investigations were focused on the determination of the technology of depositions and the annealing of ZnO layers concerning their optical properties. Special attention was paid to the determination of characteristics of the refractive index of ZnO layers and their coefficients of spectral transmission within the UV-VIS-NIR range. Besides that, also the mode characteristics and the attenuation coefficients of light in the obtained waveguide structures have been investigated. In the case of planar waveguides, in which the ZnO layers have not been annealed after their deposition, the values of the attenuation coefficient of light modes amount to a~ 30 dB/cm. The ZnO layers deposited on the heated substrate and annealed by rapid thermal annealing in an N2 and O2 atmosphere, are characterized by much lower values of the attenuation coefficients: a~ 3 dB/cm (TE0 and TM0 modes). The ZnO optical waveguides obtained according to our technology are characterized by the lowest values of the attenuation coefficients a encountered in world literature concerning the problem of optical waveguides based on ZnO. Studies have shown that ZnO layers elaborated by us can be used in integrated optic systems, waveguides, optical modulators and light sources.