Schwartz - Zippelov teorem i neke njegove primjene

U ovom radu izložen je rezultat poznat pod nazivom Schwartz - Zippelova lema ili Schwartz - Zippelov teorem. Tema rada pripada pretežno algebri, ali ima značajne primjene u drugim matematičkim područjima kao što je na primjer teorija algoritama, te kombinatorika. Rad se sastoji od tri poglavlja. U uvodu je ukratko opisana tema i cilj rada. Prvo poglavlje sadrži kratku povijest nastanka teorema, različite oblike rezultata pojedinih autora, te kratki opis pojmova korištenih u samom radu. Drugo poglavlje sastoji se od iskaza i dokaza Schwartz - Zippelovog teorema, a u trećem poglavlju izložene su neke primjene tog teorema na probleme koji se mogu svesti na testiranje jednakosti polinoma. Takvi su, primjerice, problem postojanja savršenog sparivanja u grafu i ispitivanje svojstva asocijativnosti u grupoidu. Uz svaku primjenu navedeni su i prikladni primjeri.In this diploma thesis we present the result usually called the Schwartz-Zippel lemma or the Schwartz-Zippel theorem. The nature of this theorem is basically algebraic, but it has significant applications in other areas of mathematics, such as the theory of algorithms and combinatorial theory. The thesis consists of three chapters. The main theme and objective are briefly described in the introduction. The first chapter contains a short history of the theorem’s origins, various forms of the main results by different authors and some comments of basic concepts related to this topic. The statement and a proof of the Schwartz-Zippel theorem are given in the second chapter, together with the general outline of its applications. The third and final chapter consists of some applications to problems which can be reduced to polynomial identity testing, including the existence of a perfect matching in a graph and testing of the associativity property in a groupoi

Naupliar and Metanaupliar development of Thysanoessa raschii (Malacostraca, Euphausiacea) from Godthåbsfjord, Greenland, with a reinstatement of the ancestral status of the free-living Nauplius in Malacostracan evolution

The presence of a characteristic crustacean larval type, the nauplius, in many crustacean taxa has often been considered one of the few uniting characters of the Crustacea. Within Malacostraca, the largest crustacean group, nauplii are only present in two taxa, Euphauciacea (krill) and Decapoda Dendrobranchiata. The presence of nauplii in these two taxa has traditionally been considered a retained primitive characteristic, but free-living nauplii have also been suggested to have reappeared a couple of times from direct developing ancestors during malacostracan evolution. Based on a re-study of Thysanoessa raschii (Euphausiacea) using preserved material collected in Greenland, we readdress this important controversy in crustacean evolution, and, in the process, redescribe the naupliar and metanaupliar development of T. raschii. In contrast to most previous studies of euphausiid development, we recognize three (not two) naupliar (= ortho-naupliar) stages (N1-N3) followed by a metanauplius (MN). While there are many morphological changes between nauplius 1 and 2 (e.g., appearance of long caudal setae), the changes between nauplius 2 and 3 are few but distinct. They involve the size of some caudal spines (largest in N3) and the setation of the antennal endopod (an extra seta in N3). A wider comparison between free-living nauplii of both Malacostraca and non-Malacostraca revealed similarities between nauplii in many taxa both at the general level (e.g., the gradual development and number of appendages) and at the more detailed level (e.g., unclear segmentation of naupliar appendages, caudal setation, presence of frontal filaments). We recognize these similarities as homologies and therefore suggest that free-living nauplii were part of the ancestral malacostracan type of development. The derived morphology (e.g., lack of feeding structures, no fully formed gut, high content of yolk) of both euphausiid and dendrobranchiate nauplii is evidently related to their non-feeding (lecithotrophic) status

Seasonality of bivalve larvae within a high Arctic fjord

Paid Open Acces

Linear and nonlinear optical properties of Si nanocrystals in SiO2 deposited by plasma-enhanced chemical-vapor deposition

Linear and nonlinear optical properties of silicon suboxide SiOx films deposited by plasma-enhanced chemical-vapor deposition have been studied for different Si excesses up to 24¿at.¿%. The layers have been fully characterized with respect to their atomic composition and the structure of the Si precipitates. Linear refractive index and extinction coefficient have been determined in the whole visible range, enabling to estimate the optical bandgap as a function of the Si nanocrystal size. Nonlinear optical properties have been evaluated by the z-scan technique for two different excitations: at 0.80¿eV in the nanosecond regime and at 1.50¿eV in the femtosecond regime. Under nanosecond excitation conditions, the nonlinear process is ruled by thermal effects, showing large values of both nonlinear refractive index (n2 ~ ¿10¿8¿cm2/W) and nonlinear absorption coefficient (ß ~ 10¿6¿cm/W). Under femtosecond excitation conditions, a smaller nonlinear refractive index is found (n2 ~ 10¿12¿cm2/W), typical of nonlinearities arising from electronic response. The contribution per nanocrystal to the electronic third-order nonlinear susceptibility increases as the size of the Si nanoparticles is reduced, due to the appearance of electronic transitions between discrete levels induced by quantum confinement

Si nanocrystal-based LEDs fabricated by ion implantation and plasma-enhanced chemical vapour deposition

Dept. ElectrònicaAn in-depth study of the physical and electrical properties of Si-nanocrystal-based MOSLEDs is presented. The active layers were fabricated with different concentrations of Si by both ion implantation and plasma-enhanced chemical vapour deposition. Devices fabricated by ion implantation exhibit a combination of direct current and field-effect luminescence under a bipolar pulsed excitation. The onset of the emission decreases with the Si excess from 6 to 3 V. The direct current emission is attributed to impact ionization and is associated with the reasonably high current levels observed in current–voltage measurements. This behaviour is in good agreement with transmission electron microscopy images that revealed a continuous and uniform Si nanocrystal distribution. The emission power efficiency is relatively low, ~10−3%, and the emission intensity exhibits fast degradation rates, as revealed from accelerated ageing experiments. Devices fabricated by chemical deposition only exhibit field-effect luminescence, whose onset decreases with the Si excess from 20 to 6 V. The absence of the continuous emission is explained by the observation of a 5 nm region free of nanocrystals, which strongly reduces the direct current through the gate. The main benefit of having this nanocrystal-free region is that tunnelling current flow assisted by nanocrystals is blocked by the SiO2 stack so that power consumption is strongly reduced, which in return increases the device power efficiency up to 0.1%. In addition, the accelerated ageing studies reveal a 50% degradation rate reduction as compared to implanted structures

High quality coupled ring resonators based on silicon clusters slot waveguide

High Q factor ring resonators based on Si clusters sandwich slot-waveguide structure have been fabricated by standard DUV lithography. These devices are considered an important step towards the demonstration of an all optical logic gate

Optically active Er3+ ions in SiO2 codoped with Si nanoclusters

International audienceOptical properties of directly excited erbium (Er3+) ions have been studied in silicon rich silicon oxide materials codoped with Er3+. The spectral dependence of the direct excitation cross section (σdir) of the Er3+ atomic 4I152→4I112 transition (around 0.98 μm) has been measured by time resolved µ-photoluminescence measurements. We have determined that σdir is 9.0±1.5 x10−21 cm2 at 983 nm, at least twice larger than the value determined on a stoichiometric SiO2 matrix. This result, in combination with a measurement of the population of excited Er3+ as a function of the pumping flux, has allowed quantifying accurately the amount of optically active Er3+. This concentration is, in the best of the cases, 26% of the total Er population measured by secondary ion mass spectrometry, which means that only this percentage could provide optical gain in an eventual optical amplifier based on this material

Deep-UV Lithography Fabrication of Slot Waveguides and Sandwiched Waveguides for Nonlinear Applications

Associated to silicon nanocrystals, slot waveguides are promising components for the fabrication of fast optical switches. We report here on the fabrication of slot waveguides with industrial microelectronic tools, especially DUV lithography. The filling of slot waveguides with silicon nanocrystals within SiO2 by PECVD and LPCVD deposition results in almost filled waveguides owing to incorrect silicon etching profiles. We also report on slot waveguide alternative structure: the sandwiched waveguide. This structure eliminates the slot filling step and allows slot thickness as thin as desired. The association of silicon nanocrystals and slot waveguides, if used in a Mach-Zender Interferometer structure, will pave the way to the first compact alloptical XOR logic gate entirely fabricated with CMOS tools

Non linear optical properties of silicon nanocrystals for applications in photonic logic gates devices

A systematic study of Silicon nanocrystals (Si-ncs) nonlinearities at 1550nm was carried out in view of the realization of an all optical Mach-Zehender Interferometer (MZI)