On a Class of Univalent Functions with Negative Coefficients

Донка Пашкулева - Предмет на тази статия е получаването на точни оценки за коефициентите и ръста на функциите за някои класове еднолистни функции с отрицателни коефициенти.

The structure and dynamics of saxicolous lichen communities

The saxicolous lichen vegetation on Ordovician slate rock at the mouth of the River Dovey, South Merionethshire, Wales was described in relation to several environmental variables which include aspect, slope angle, light intensity, rock porosity, rock microtopography and rock stability. Each of the measured environmental variables was shown to influence the lichen vegetation. A number of groups of species which were characteristic of certain environments were described. The data from the saxicolous lichen communities were analysed using multivariate analysis. Qualitative and quantitative data were ordinated, the qualitative data being easier to interpret ecologically, and site number (which reflects distance from the sea and altitude), rock porosity and light intensity were shown to be important environmental variables. A classification of the data was also carried out. The results of the ordination and classification were combined together and a model constructed which describes saxicolous lichen vegetation. A method which uses the model as an aid to the design and interpretation of field experiments is described. The model is applied to an experiment which investigates the effect on growth of transplanting four saxicolous lichens to different aspects. Growth was inhibited in Physcia orbicularis and Parmelia conspersa on rock surfaces of northwest aspect compared with growth on rock surfaces of southeast aspect. Growth was inhibited in Parmelia glabratula ssp. fuliginosa on rock surfaces of southeast aspect compared with rock surfaces of northwesr aspect. The growth of Parmelia saxatilis was similar at both southeast and northwesr aspects. Growth inhibition or stimulation in thalli of Physcia orbicularis, Parmelia conspersa and Parmelia glabratula ssp. fuliginosa after transplantation was consistent with the predictions of the model while the results for Parmelia saxatilis were not as expected. There was evidence that the frequency of Parmelia conspersa and Parmelia glabratula at a site is related to an effect of the environment on the growth of the thalli. There was also evidence that the frequency of Physcia orbicularis at a site is related to an effect of the environment on the establishment phase of the thalli and for the competitive exclusion of Parmelia saxatilis thalli from southeast facing rock surfaces. The distribution of lichens in relation to height on nine rock surfaces was investigated. It was suggested that the distribution of the lichens was influenced by microclimatic factors which are related to height on the rock, environmental variables which are associated with the rock substratum (e.g. rock porosity and rock microtopography) and by historical factors. The pattern of one crustose and one foliose lichen on four rock surfaces of different aspect and slope was investigated. On the vertically inclined surface the density of small thalli of Buellia aethalea and Parmelia glabratula ssp fuliginosa was correlated with the microtopography of the surface in transects horizontally across the rock surface but not in transects vertically down the rock surface. there were consitent differences in the scale and intensity of pattern horizontally and vertically and also a decrease in the intensity of pattern vertically as the slope of the rock surface decreased. These results were consistent with the suggestion of a gradient of microclimatic factors up the rock. The differences in the scale and intensity of pattern in different size classes in the population were consistent with the changes in pattern with time which have been shown to occur during succession in sand dune and salt marsh vegetation. The relationship between thallus size and height on a rock surface and between the radial growth rate and location of a thallus on a rock surface were investigated. Thalli of Parmelia glabratula ssp. fuliginosa were larger at the top of the rock surface than at the bottom and the data were consistent with the suggestion that the colonisation of the rock surface began at the top and, in time, spread downwards. The radial growth rate of the thalli could not be related to variation in slope, porosity, microtopography or directly to height on the rock but could be related to the horizontal location of the thalli on the rock. These results were consistent with the suggestion that here is a gradient of microclimatic factors across the rock surface which is also modified by height on the rock surface. The succession of lichen communities was described by relating the vegetation to rock porosity, rock microtopography, species diversity and rock stability. An initial stage dominated by crustose lichens leads to communities dominated by crustose, foliose and fruticose species. In the late stages of the succession on some rock surfaces crustose species again become dominant. The occurrence of the climax state and cyclic vegetation change in lichen communities are discussed. A mthod of estimating the age structure of a lichen population by relating thallus size to growth rate is described. The sources of error in the method are discussed in some detail and several refinements suggested to increase the accuracy of the method. The population dynamics of Parmelia glabratula ssp. fuliginosa was investigated by applying life tables to the age structures of eight different populations. The data were consistent with a period of relatively constant recruitment of thalli into the populations. Mortality in lichen populations was divided into deaths which occur after fragmentation of the thallus and deaths which occur after catastrophic environmental events. THe data suggest that the rate of fragmenting death is dependent on the age of the thallus while the rate of catastrophic death is dependent on the number of thalli established in an age class. A comparison of the numbers of thalli in each age class in the eight populations suggested that population density is controlled firstly, by climate and secondly, by variables related to the local rock surface environment. The rate of fragmenting death is related to the diversity of the community and the influence of diversity together with environmental variables in fluctuating or cyclic changes in population number.

The Darboux Problem for a Class of 3-D Weakly Hyperbolic Equations

Недю Попиванов, Цветан Христов - Изследвани са някои тримерни аналози на задачата на Дарбу в равнината. През 1952 М. Протер формулира нови тримерни гранични задачи както за клас слабо хиперболични уравнения, така и за някои хиперболично-елиптични уравнения. За разлика от коректността на двумерната задача на Дарбу, новите задачи са некоректни. За слабо хиперболични уравнения, съдържащи младши членове, ние намираме достатъчни условия както за съществуване и единственост на обобщени решения с изолирана степенна особеност, така и за единственост на квази-регулярни решения на задачата на Протер.

Exact Solutions of Nonlocal Boundary Value Problems for One- and Two-Dimensional Heat Equation

Иван Хр. Димовски, Юлиан Ц. Цанков - Предложен е метод за намиране на явни решения на клас двумерни уравнения на топлопроводността с нелокални условия по пространствените променливи. Методът е основан на директно тримерно операционно смятане. Класическата дюамелова конволюция е комбинирана с две некласически конволюции за операторите ∂xx и ∂yy в една тримерна конволюция. Съответното операционно смятане използва мултипликаторни частни. Мултипликаторните частни позволяват да се продължи принципът на Дюамел за пространствените променливи и да се намерят явни решения на разглежданите гранични задачи. Общите разглеждания са приложени в случая на гранични условия от типа на Йонкин. Намерени са експлицитни решения в затворен вид.

Singular Solutions of Protter’s Problem for a Class of 3-D Hyperbolic Equations

Недю Иванов Попиванов, Алексей Йорданов Николов - През 1952 г. М. Протър формулира нови гранични задачи за вълновото уравнение, които са тримерни аналози на задачите на Дарбу в равнината. Задачите са разгледани в тримерна област, ограничена от две характеристични конуса и равнина. Сега, след като са минали повече от 50 години, е добре известно, че за безброй гладки функции в дясната страна на уравнението тези задачи нямат класически решения, а обобщеното решение има силна степенна особеност във върха на характеристичния конус, която е изолирана и не се разпространява по конуса. Тук ние разглеждаме трета гранична задача за вълновото уравнение с младши членове и дясна страна във формата на тригонометричен полином. Дадена е по-нова от досега известната априорна оценка за максимално възможната особеност на решенията на тази задача. Оказва се, че при по-общото уравнение с младши членове възможната сингулярност е от същия ред като при чисто вълновото уравнение.

Applying Associative Classifier PGN for Digitised Cultural Heritage Resource Discovery

Resource discovery is one of the key services in digitised cultural heritage collections. It requires intelligent mining in heterogeneous digital content as well as capabilities in large scale performance; this explains the recent advances in classification methods. Associative classifiers are convenient data mining tools used in the field of cultural heritage, by applying their possibilities to taking into account the specific combinations of the attribute values. Usually, the associative classifiers prioritize the support over the confidence. The proposed classifier PGN questions this common approach and focuses on confidence first by retaining only 100% confidence rules. The classification tasks in the field of cultural heritage usually deal with data sets with many class labels. This variety is caused by the richness of accumulated culture during the centuries. Comparisons of classifier PGN with other classifiers, such as OneR, JRip and J48, show the competitiveness of PGN in recognizing multi-class datasets on collections of masterpieces from different West and East European Fine Art authors and movements.

A Method for Solving a Class of Multiple-Criteria Analysis Problems

AMS subject classification: 90C29.