The novelty and originality of the work consists in the fact that in order to obtain asymptotic expansions with optimal values of the remainder terms in the zone of large deviations, along with the cumulant method the classical method of characteristic functions has to be used. In addition, when solving the problems stated in the work, other than the well known results in the problems of limit theorems of the probability theory and mathematical statistics, we have to estimate constants. Technically it is frequently rather a complicated task. The results obtained in the work have good opportunities to be applied in probability theory, mathematical statistics, econometric, etc. That is illustrated in the last section of the work in which theorems of large deviations are proved in the summation of weighted random variables with weights as well as discounted limit theorems.

Magistro darbe nustatyta, kad dzeta funkcijos φ(s, F1),…, φ(s, Fr) turi jungtinį ribinį pasiskirstymą kompleksinėje plokštumoje. Tai reiškia, kad tikimybinis matas silpnai konverguoja į kurį nors tikimybinį matą erdvėje (C^{r},B(C^{r})). / We prove a joint limit theorem for the functions φ(s, F1),…, φ(s, Fr) on the complex plane. Then we prove that the probability measure converges weakly to some probability measure on (C^{r},B(C^{r})).

This work was carried out to study the effect of shot-peening on the fatigue behaviour of carbon steels. Differently heat treated medium and high carbon steel specimens were selected. Medium carbon steels, AISI 1141 and AISI 1151, were respectively air cooled and quenched-tempered. A high carbon steel, C70S6 (AISI 1070), was air cooled. The other material was a powder metal (0.5% C) steel. Each group of steels was divided into two. One was shot-peened. The other half remained in their original conditions. All were fatigue tested under fully reversed (R=-1) tension-compression loading conditions. Microhardness tests were carried out on both the grip and gage sections of selected non shot-peened and shot-peened specimens to determine the hardness profile and effect of cycling. Shot-peening was found to be deeper on one side of each specimen. Compressive residual stress profiles and surface roughness measurements were provided. Shot-peening increased the surface roughness from 0.26±0.03µm to 3.60±0.44µm. Compressive residual stresses induced by shot-peening reached a maximum of -463.9MPa at a depth of 0.1mm.The fatigue limit (N≈106 cycles) and microhardness profiles of the non shot-peened and shot-peened specimens were compared to determine the material behaviour changes after shot-peening and cycling. Also their fatigue properties were related to the manufacturing process including heat and surface treatments. Comparing the grip and gage microhardness profiles of each steel showed that neither cyclic softening nor hardening occurred in the non shot-peened condition. Cyclic softening was apparent in the shot-peened regions of all steels except powder metal (PM) steel. The amount of softening in the shot-peened region was 55.0% on the left side and 73.0% on the right in the AISI 1141 steel , 46.0% on the left side and 55.0% on the right in the C70S6AC steel and 31.0% on the right side in AISI 1151QT steel. Softening was accompanied by a decrease in the depth of surface hardness. It is suggested that although the beneficial effects of shot peening, compressive residual stresses and work hardening, were offset by surface roughness, crack initiation was more likely to occur below the surface. Surface roughness was not a significant factor in controlling the fatigue lives of AISI 1141AC and C70S6 steels, since they were essentially the same for the non shot-peened and shot-peened conditions. Shot-peening had very little effect on the push-pull fatigue limit of C70S6 steel (-2.1%), and its effect on AISI 1141AC steel was relatively small (6.0%). However, the influence of shot-peening on the AISI 1151QT and PM steels was more apparent. The fatigue limit of the PM steel increased 14.0% whereas the fatigue limit of the AISI 1151QT steel decreased 11.0% on shot peening.

Magnesium, with a Hexagonal Close-Packed (HCP) structure, is the eighth most abundant element in the earth’s crust and the third most plentiful element dissolved in the seawater. Magnesium alloys exhibit the attractive characteristics of low densities and high strength-to-weight ratios along with good castability, recyclability, and machinability. Replacing the steel and/or aluminum sheet parts with magnesium sheet parts in vehicles is a great way of reducing the vehicles weight, which results in great savings on fuel consumption. The lack of magnesium sheet components in vehicle assemblies is due to magnesium’s poor room-temperature formability. In order to successfully form the sheets of magnesium at room temperature, it is necessary to understand the formability of magnesium at room temperature controlled by various plastic deformation mechanisms. The plastic deformation mechanisms in pure magnesium and some of its alloys at room temperature are crystallographic slip and deformation twinning. The slip systems in magnesium at room temperature are classified into primary (first generation), secondary (second generation), and tertiary (third generation) slip systems. The twinning systems in magnesium at room temperature are classified into primary (first generation) and secondary (second generation, or double) twinning systems. A new comprehensive rate-dependent elastic-viscoplastic Crystal Plasticity Constitutive Model (CPCM) that accounts for all these plastic deformation mechanisms in magnesium was proposed. The proposed model individually simulates slip-induced shear in the parent as well as in the primary and secondary twinned regions, and twinning-induced shear in the primary and secondary twinned regions. The model also tracks the texture evolution in the parent, primary and secondary twinned regions. Separate resistance evolution functions for the primary, secondary, and tertiary slip systems, as well as primary and secondary twinning systems were considered in the formulation. In the resistance evolution functions, the interactions between various slip and twinning systems were accounted for. The CPCM was calibrated using the experimental data reported in the literature for pure magnesium single crystals at room temperature, but needs further experimental data for full calibration. The partially calibrated model was used to assess the contributions of various plastic deformation mechanisms in the material stress-strain response. The results showed that neglecting secondary slip and secondary twinning while simulating plastic deformation of magnesium alloys by crystal plasticity approach can lead to erroneous results. This indicates that all the plastic deformation mechanisms have to be accounted for when modelling the plastic deformation in magnesium alloys. Also, the CPCM in conjunction with the Marciniak–Kuczynski (M–K) framework were used to assess the formability of a magnesium single crystal sheet at room temperature by predicting the Forming Limit Diagrams (FLDs). Sheet necking was initiated from an initial imperfection in terms of a narrow band. A homogeneous deformation field was assumed inside and outside the band, and conditions of compatibility and equilibrium were enforced across the band interfaces. Thus, the CPCM only needs to be applied to two regions, one inside and one outside the band. The FLDs were simulated under two conditions: a) the plastic deformation mechanisms are primary slip systems alone, and b) the plastic deformation mechanisms are primary slip and primary twinning systems. The FLDs were computed for two grain orientations. In the first orientation, primary extension twinning systems had favourable orientation for activation. In the second orientation, primary contraction twinning systems had favourable orientation for activation. The effects of shear strain outside the necking band, rate sensitivity, and c/a ratio on the simulated FLDs in the two grain orientations were individually explored.

This paper presents the theory of Secondary Representation of modules over a commutative ring and their Attached Primes; introduced in 1973 by I. MacDonald as a dual to the important theory of associated primes and primary decomposition in commutative algebra. The paper explores many of the basic aspects of the theory of primary decomposition and associated primes of modules in the hopes to delineate and motivate the construction of a secondary representation, when possible. The thesis discusses the results of the uniqueness of representable modules and their attached primes, and, in particular, the existence of a secondary representation for Artinian modules. It concludes with some interesting examples of both secondary and representable modules, highlighting the consequences of the results thus established.

Darbo tikslas – sukonstruoti dvimatį skirstinį, kai duoti vienmačiai (marginalieji) skirstiniai, atlikti maksimumų asimptotinę analizę ir ištirti konvergavimo greitį. Dvimatis skirstinys konstruojamas dviem atvejais: kai vektorių komponentės yra priklausomos ir nepriklausomos. Detalesnė konvergavimo greičio analizė atlikta, kai komponentės yra priklausomos. Tyrimui buvo pasirinktas Pareto skirstinys. Pirmoje tiriamosios dalies ir rezultatų dalyje yra konstruojamas dvimatis skirstinys, skaičiuojamos jo pagrindinės charakteristikos, tiriama, ar prie visų parametrų reikšmių jos egzistuoja. Taip pat generuojami atsitiktiniai dydžiai, kurių skirstiniai yra sukonstruotosios skirstinio funkcijos marginalieji skirstiniai, ir eksperimentiškai bandoma pagrįsti gautus rezultatus. Antroje dalyje atliekama asimptotinė analizė. Apibrėžiami dvimačiai maksimumai, ieškomas ribinis skirstinys. Juos suradus, apibrėžiamas apytikslis konvergavimo greičio įvertis, atliekama jo bei paklaidų kompiuterinė analizė, ieškoma, kokioms sąlygoms esant jie yra mažiausi. Sukonstruoto dvimačio skirstinio skaitinių charakteristikų tyrimas atliekama programiniu paketu MathCAD. Kompiuterinė konvergavimo greičio įverčių analizė atliekama programinio paketo Matlab pagalba. Jo aplinkoje buvo sukurta programa vartotojui, kuri nubraižo konvergavimo greičio įvertį bei paklaidas. / The aim of this paper is to construct two-dimensional random variables, having one-dimensional ones, carry out the asymptotical analysis and study the speed of convergence. Two-dimensional distribution is constructed in two ways: when the components of random variables are independent and dependent. As in the last few years Pareto distribution is popular in financial models, it was chosen for the analyses. It was proved, that in both cases of independent and dependent components of the vector, the limit distribution is the same. This means that although the components of the vector are dependent, the maxima are asymptotically independent. Besides, the errors are smaller than the approximate estimate. Although, the approximate estimate in the case of independent components is smaller than in the case of dependent components, the errors are on the contrary: they are smaller when the components are dependent than when the components are independent.

Darbe nagrinėjami nepriklausomų ir vienodai pasiskirsčiusių atsitiktinių dydžių maksimumai su skirstinio funkcija F. Skaičiuojami maksimumų vidurkiai Pareto ir Buro skirstinių atveju, palyginami su tiksliomis reikšmėmis ir žinomu įverčiu. Kai imties didumas n yra didelis, naudojamos ribinės teoremos, Pareto skirstinio atveju randamas konvergavimo greičio įvertis. Taip pat skaičiuojami Buro atsitiktinių dydžių maksimumų vidurkiai, kai imties didumas N yra pasiskirstęs pagal geometrinį skirstinį. / In this work maxima of independent and identically distributed random variables with distribution function F are analyzed. We calculate maxima means for Pareto and Buro distributions and compare theoretical values with known estimates. We use limit theorems for maxima means when the set size n is large and find the estimate of convergence rate for Pareto random variables. When the set size N is geometric random number maxima means for Buro random variables are calculated.

The lower oxygen limit (LOL) in plants marks the oxygen (O2) level where the metabolism shifts from being predominantly aerobic to anaerobic; recent work has shown that respiratory-based indicators of this metabolic shift are well-correlated with changes in chlorophyll a fluorescence signals. The physiological and biochemical changes at the root of this relationship have not been well-described in the literature. The processes involved are spatially separated: chlorophyll fluorescence is associated with the lightdependent reactions and emanates from the chloroplasts whereas aerobic respiration and fermentation occurs in the mitochondria and cytosol, respectively. Evidences outlined in this thesis are used to suggest the mechanistic link between these three regions of the cell is a fluid exchange of cellular reductant. When mitochondrial respiration is inhibited as a result of inadequate O2, used as a terminal electron acceptor, glycolytic reductant in the form of NADH accumulates in the cytosol. Reductant imbalances between the cytosol and organelles can be adjusted indirectly using translocators. Excess chloroplastic reductant is used to reduce the plastoquinone (PQ) pool via NADPH-dehydrogenase, a component of the chlororespiratory pathway, effectively decreasing the photochemical quenching (qP) capacity thereby inducing a switch from minimum fluorescence (Fo) to a higher relative fluorescence (F) value where qP < 1. Subjecting dark-adapted photosystems to low-intensity light increased Fo to a slightly higher F value due to a lightinduced reduction of the oxidized PQ pool when the O2 was above the LOL, but decreased F as a result of a PSI-driven oxidation of the already over-reduced PQ pool when the O2 was below the LOL. Low O2 was also shown to increase violaxanthin deepoxidation and non-photochemical quenching (qN), likely a reflection of the overreduced state of the photosystems and associated pH decrease. Dynamic controlled atmosphere (DCA) is a fluorescence-based controlled atmosphere (CA) system that sets the optimum atmosphere for fruits and vegetables based on a product’s fluorescence response. Experiments in this thesis on the relationship between O2, temperature, light, metabolism, pigmentation and chlorophyll fluorescence were used to interpret the physiology behind fluorescence changes, suggest improved DCA techniques and outline potentially profitable avenues for future research.

Darbe įrodyta, kad Hurvico dzeta funkcijai su algebriniu parametru yra teisinga ribinė teorema su svoriu kompleksinėje plokštumoje. Pagrindinis šio darbo rezultatas yra suformuluotas teorema. / Proof of limit theorem with weight for the Hurwitz zeta-function with an algebraic irrational parameter. The rezult formulated by theorem.

Axicons have been proposed for imaging applications since they offer the advantage of extended depth of field (DOF). This enhanced DOF comes at the cost of degraded image quality. Image processing has been proposed to improve the image quality. Initial efforts were focused on the use of an axicon in a borescope thereby extending depth of focus and eliminating the need for a focusing mechanism. Though promising, it is clear that image processing would lead to improved image quality. This would also eliminate the need, in certain applications, for a fiber optic imaging bundle as many modern day video borescopes use an imaging sensor coupled directly to the front end optics. In the present work, three types of refractive axicons are examined: a linear axicon, a logarithmic axicon and a Fresnel axicon. The linear axicon offers the advantage of simplicity and a significant amount of scientific literature including the application of image restoration techniques. The Fresnel axicon has the advantage of compactness and potential low cost of production. As no physical prior examples of the Fresnel axicons were available for experimentation until recently, very little literature exists. The logarithmic axicon has the advantage of nearly constant longitudinal intensity distribution and an aspheric design producing superior pre-processed images over the aforementioned elements. Point Spread Functions (PSFs) for each of these axicons have been measured. These PSFs form the basis for the design of digital image restoration filters. The performance of these three optical elements and a number of restoration techniques are demonstrated and compared.