published_or_final_version / Mathematics / Master / Master of Philosophy

Ανασκόπηση του κανόνα ολοκλήρωσης του gauss. Αναπαραστάσεις και εκτιμήσεις του υπολοίπου του. Τέλος περί της σύγκλισης του κανόνα ολοκλήρωσης. / A survey on gaussian quqdrature rules. Representation and estimates of its remainder. And about its convergence.

Šiais laikais vis daugiau domimasi daugiamačių duomenų aproksimavimo teorija. Daugiamatėje erdvėje aproksimavimo teorija palčiai kur taikoma, pavyzdžiui, skaitinių metodų analizėje, bangų analizėje, signalų apdorojime, įvairiose informacinių technologijų sistemose, kompiuterių grafikoje, astronomijoje, naftos klodų tyrinėjime. Ši sritis viliojanti, nes didelė dalis klasikinės matematikos sunkiai pritaikoma daugiamačiams uždaviniams analizuoti. Taigi senoms problemoms spręsti reikalingi nauji įrankiai. Funkcijų aproksimavimo uždavinių gausu įvairiose matematikos, fizikos ir technikos srityse. Gausu ir jų sprendimų būdų bei metodų. Nesunkiai šie uždaviniai sprendžiami, kai funkcija priklauso nuo vieno ar dviejų kintamųjų. Tačiau realiame gyvenime naudojamos funkcijos turi daug daugiau nežinomųjų. Didėjant kintamųjų skaičiui uždavinio sudėtingumas taip pat auga. Pavyzdžiui, kai funkcija priklauso nuo vieno kintamojo, ją galima pavaizduoti plokštumoje kaip kreivę. Dviejų kintamųjų funkciją atitinka paviršius, nubrėžtas trimatėje erdvėje. Funkcijų, kurios priklauso nuo trijų ir daugiau kintamųjų, vaizdavimas jau sukelia problemų, nes žmogus nebegali suvokti didesnio matumo erdvės. Kadangi trimatę erdvę galima pavaizduoti plokštumoje, manoma, kad panašiu principu keturmatę erdvę galima pavaizduoti trimatėje, o šią vėl plokštumoje. Jei pavyktų sugalvoti tokį metodą, erdvės matumas nebesukeltų problemų. Visgi trijų kintamųjų funkciją bandoma vaizduoti dviem būdais: 1. pateikti... [toliau žr. visą tekstą] / This Master‘s work covers a mathematical analysis system which can visualize multivariate data layers, approximate multi-dimensional functions by polynomials, estimate approximation accuracy and present few the most effective aproximation models. Multivariate approximation theory is an increasingly active research area today. It encompasses a wide range of tools for multivariate approximation such as multi-dimensional splines and finite elements, shift-invariant spaces and radial-basis functions. Approximation theory in the multivariate setting has many applications including numerical analysis, wavelet analysis, signal processing, geographic information systems, computer aided geometric design and computer graphics. The field is fascinating since much of the mathematics of the classical univariate theory does not straightforwardly generalize to the multivariate setting, so new tools are required. Graphs of one variable functions are frequantly displayed as curves, bivariate functions - as contour plots. In generally it is very hard to display or realize function in the multivariate setting. However, some efforts have been made to render functions of precisely three variables. Two obvious approaches suggest themselves: 1. Display a number of cross sections where one of the variables is held constant, or, 2. display contour surfaces where the value of function equals some constant. We will use the first method modification in this Master‘s work. All function variables except... [to full text]

No description available.

The Generalized Minimal Residual method (GMRES) is often used to solve a nonsymmetric linear system Ax = b. But its convergence analysis is a rather difficult task in general. A commonly used approach is to diagonalize A = XΛX-1 and then separate the study of GMRES convergence behavior into optimizing the condition number of X and a polynomial minimization problem over As spectrum. This artificial separation could greatly overestimate GMRES residuals and likely yields error bounds that are too far from the actual ones. On the other hand, considering the effects of both As spectrum and the conditioning of X at the same time poses a difficult challenge, perhaps impossible to deal with in general but only possible for certain particular linear systems. This thesis will do so for a (nonsymmetric) tridiagonal Toeplitz system. Sharp error bounds on and sometimes exact expressions for residuals are obtained. These expressions and/or bounds are in terms of the three parameters that define A and Chebyshev polynomials of the first kind or the second kind.

No description available.

This project is concerned with the set of primes modulo which some monic, irreducible polynomial over the integers has a root, called the Prime Set of the polynomial. We completely characterise these sets for degree 2 polynomials, and develop sufficient machinery from algebraic number theory to show that if the Galois group of a monic, irreducible polynomial over the integers is abelian, then its Prime Set can be written as the union of primes in some congruence classes modulo some integer.

No description available.

The generalized moment of order k of a mass distribution sigma for a natural number k is given by integral of lambda to the power k with respect to mass distribution sigma and variable lambda. In extended moment problem, given a sequence of real numbers, it is required to find a mass distribution whose generalized moment of order k is k&#039 / th term of the sequence. The conditions of existence and uniqueness of the solution obtained by Hamburger are studied in this thesis by the use of orthogonal polynomials determined by a measure on real line. A chapter on the study of asymptotic behaviour of orthogonal functions on compact subsets of complex numbers is also included.

This thesis is a contribution to determine the absolute irreducibility of polynomials via their Newton polytopes. For any field F / a polynomial f in F[x1, x2,..., xk] can be associated with a polytope, called its Newton polytope. If the polynomial f has integrally indecomposable Newton polytope, in the sense of Minkowski sum, then it is absolutely irreducible over F / i.e. irreducible over every algebraic extension of F. We present some new results giving integrally indecomposable classes of polytopes. Consequently, we have some new criteria giving infinitely many types of absolutely irreducible polynomials over arbitrary fields.