The lower extreme asymptotic is analised in this master’s work. I have analysed the marginal term case, when sample size N is accidental. The ordinary accidental sample is taken from general set, what is spreaded along logistic low. I have searched for logistic dimensions minimum limiting distribution function in the investigative part. Than I’ve practised transfering theorem. My task is to find such normalization, along what logistic dimensions lower extreme distribution functions are geometrically ministable or asymptotically k-stable. I have proved in my job, that first lower extreme distribution function is geometrically ministable, and other distribution functions – asymptotically k-stable.

We consider k maximum’s multiplication where factors are independently distributed random extremes of independent identically distributed random variables which are distributed uniformly over (0, 1). We find distribution function, limiting distribution function and estimate convergence rate.

Herein work is researching extremes asymptotic of Pareto random values. Here is analyzing geometrically maximum (minimum) stability tasks, also asymptotically tasks, when succession value is geometrical and geometrically stability of lower extremes. Aim of this work is to check if Pareto distribution values are stable maximum and minimum distributions and to continue researches in the area of lower extremes structures. It was proved that maximum (minimum) distribution (when ) is geometrically stable maximum (minimum) distribution, while others – asymptotically k-stable. When , maximum (minimum) distribution is asymptotically stable, only maximum distribution is also Pareto distribution, but with the displacement, while other - asymptotically k-stable.

Magistro darbe nagrinėjamas katenoidų uždavinys, kuris sprendžiamas variaciniu metodu. Ištirta, kada egzistuoja šio uždavinio sprendinys su simetrinėmis kraštinėmis sąlygomis: y(-a)=y=(a)=A>0. Ištirti atvejai, kai uždavinys neturi sprendinio glodžių funkcijų klasėje. Ištirta, kada egzistuoja šio uždavinio sprendinys su nesimetrinėmis kraštinėmis sąlygomis y(a)=A, y(b)=B, A≠B. Uždavinys buvo sprendžiamas kompiuterinės programos Maple pagalba. / In this master thesis the catenoid problem is discussed using the variational methods. The existence of the solution in the case of symmetrical boundary value conditions y(-a)=y=(a)=A>0 is established. An example is given in the case, when catenoid problem doesn’t have any solution in the class of smooth functions. Besides, an example of this variational problem with non-symmetrical boundary value conditions of the type y(a)=A, y(b)=B, A≠B, is considered. All problems were solved using technical computing sofware Maple.