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

Age and growth, reproduction and diet of a sublittoral population of the rock goby Gobius paganellus (Teleostei, Gobiidae)

Copyright © 2000 Kluwer Academic Publishers. Printed in the Netherlands.Basic biological information for a sublittoral population of the rock goby Gobius paganellus Linnaeus, 1758 is presented based on a 2-year study involving 1680 specimens. The length-weight relationship was given by TW = 0.0089 * TL^3,163 (where TW= total weight in g; TL= total length in mm). Age at length data were inferred by modal analysis of the monthly length-frequency distributions. The parameters of the fitted Von Bertalanffy growth equation (with seasonal component, birth date on the 1st of January) were L1 = 13.8 cm; K = 0.73 yr^-1; to = -0.22 yr; C = 0.95; W = 0.07. This growth rate is much higher than that described for northern Europe populations (where K is about 0.3 yr^-1) and is probably associated with a shorter life span. Macroscopic examination of the gonads, and analysis of the monthly values of the gonadosomatic index, indicated that reproduction occurs in winter and early spring, with a maximum in February and March, when water temperatures are lowest. Individuals become sexually mature around 6-7 cm TL, a size that can be reached in less than 1 year. By contrast, individuals of this goby in the British Isles mature in their second or third year. Stomach contents were mainly small benthic invertebrates, predominantly crustaceans

Complete larval development of the crab Ilia nucleus (Linnaeus, 1758) (Decapoda, Brachyura, Leucosiidae) reared under laboratory conditions

The complete larval development of the crab Ilia nucleus (Linnaeus, 1758) reared under laboratory conditions was obtained. The four zoeal stages and the megalopa are described and illustrated in detail. The larval features observed in I. nucleus fit into the characteristics of the family proposed by Rice (1980) for the zoeal stages and by Quintana (1986) for the megalopa. The morphological characters of larval stages of I. nucleus are compared with previous descriptions, and with those of other known larvae of Leucosiidae (only for subfamilies Ebaliinae and Leucosiinae). The present work supports the hypothesis that the subfamily Ebaliinae is a heterogeneous group. For the correct identification of a zoeal stage of a leucosiid crab, besides counting the number of setae on maxilliped exopods, the antennule as well as the pereiopods development should be used as additional characters

Larval morphology of the family Parthenopidae, with the description of the megalopa stage of Derilambrus angulifrons (Latreille, 1825) (Decapoda: Brachyura), identified by DNA barcode

Although Parthenopidae is a brachyuran decapod family comprising almost 140 species, there is little knowledge about its larval morphology. There are only two complete larval developments reared in the laboratory and some larval stages described for seven species. In the present work these data are compared and analysed. A summary is made of the larval features that characterize parthenopids that can be used to distinguish them from other brachyuran larvae. In addition, the megalopa stage of Derilambrus angulifrons and Parthenopoides massena was collected from plankton and identified by DNA barcodes. The morphology of the megalopa of D. angulifrons is described for the first time, and that of P. massena is compared with a previous description

Post-larval development and sexual dimorphism of the spider crab <i>Maja brachydactyla</i> (Brachyura: Majidae)

12 pages, 13 figures, 2 tables.-- Published online 8 Sept. 2009.[EN] The post-larval development of the majid crab Maja brachydactyla Balss, 1922 was studied using laboratory-reared larvae obtained from adult individuals collected in the NE Atlantic. The morphology of the first juvenile stage is described in detail, while the most relevant morphological changes and sexual differentiation are highlighted for subsequent juvenile stages, until juvenile 8. The characteristic carapace spines of the adult phase are present in the first juvenile stage, though with great differences in the degree of development and relative size. The carapace shows a high length/weight ratio, which becomes similar to that of adults at stage 7-8. Males and females can be distinguished from juvenile stage 4, based on sexual dimorphism in the pleopods and the presence of gonopores. In addition, the allometric growth of the pleon is sex-dependent from juvenile stage 4, with females showing a positive allometry (b=1.23) and males an isometric allometry (b=1.02).[ES] El desarrollo postlarvario del májido Maja brachydactyla ha sido estudiado en el laboratorio después del cultivo larvario realizado a partir de individuos adultos capturados en el del Atlántico. La morfología del primer estadio juvenil se ha descrito en detalle y el desarrollo juvenil, hasta el estadio 8, ha sido estudiado en lo referente a los cambios morfológicos y la diferenciación sexual. En el primer juvenil, las espinas del cefalotórax características de los adultos están presentes, pero existen grandes diferencias en el grado de desarrollo y el tamaño relativo. cociente longitud/anchura muestra valores elevados que tienden a parecerse a los de los adultos a partir del juvenil 7-8. Machos y hembras se pueden diferenciar a partir del cuarto estadio juvenil, según el dimorfismo sexual en los pleópodos y la presencia de gonoporos. Además, existen diferencias sexuales en el crecimiento alométrico del pleon a partir del cuarto estadio juvenil: las hembras muestran una alometría positiva (b= 1.23) y los machos muestran un crecimiento isométrico (b= 1.02).Financial support was provided by the Ministry of Science and Research to GG (post-doctoral fellowship; INIA) and GR (Ramón y Cajal contract). Bench fees were funded by JACUMAR (REPES project) to GR. The authors would like to thank Jose Manuel Fortuño of the ICM (CSIC) for his assistance in SEM study and Glòria Macià, Miriam Matas, Isolda Banus, Sandra Molas and Joaquín Canoura for their support as hatchery and laboratory technicians at the IRTA, Sant Carles de la Ràpita.Peer reviewe

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

Tracking seasonal changes in North Sea zooplankton trophic dynamics using stable isotopes

Trophodynamics of meso-zooplankton in the North Sea (NS) were assessed at a site in the southern NS, and at a shallow and a deep site in the central NS. Offshore and neritic species from different ecological niches, including Calanus spp., Temora spp. and Sagitta spp., were collected during seven cruises over 14 months from 2007 to 2008. Bulk stable isotope (SI) analysis, phospholipid-derived fatty acid (PLFA) compositions, and δ 13CPLFA data of meso-zooplankton and particulate organic matter (POM) were used to describe changes in zooplankton relative trophic positions (RTPs) and trophodynamics. The aim of the study was to test the hypothesis that the RTPs of zooplankton in the North Sea vary spatially and seasonally, in response to hydrographic variability, with the microbial food web playing an important role at times. Zooplankton RTPs tended to be higher during winter and lower during the phytoplankton bloom in spring. RTPs were highest for predators such as Sagitta sp. and Calanus helgolandicus and lowest for small copepods such as Pseudocalanus elongatus and zoea larvae (Brachyura). δ 15NPOM-based RTPs were only moderate surrogates for animals’ ecological niches, because of the plasticity in source materials from the herbivorous and the microbial loop food web. Common (16:0) and essential (eicosapentaenoic acid, EPA and docosahexaenoic acid, DHA) structural lipids showed relatively constant abundances. This could be explained by incorporation of PLFAs with δ 13C signatures which followed seasonal changes in bulk δ 13CPOM and PLFA δ 13CPOM signatures. This study highlighted the complementarity of three biogeochemical approaches for trophodynamic studies and substantiated conceptual views of size-based food web analysis, in which small individuals of large species may be functionally equivalent to large individuals of small species. Seasonal and spatial variability was also important in altering the relative importance of the herbivorous and microbial food webs