Iracionalita, Transcendence a Lineární nezávislost nekonečných řad

Mgr. Pavel RUCKI

Doctoral thesis

Irrationality, Transcendence and Linear Independence of Infinite Series

Anotácia:

Prvn´? tˇri kapitoly pojedn´avaj´? o iracion´aln´?ch, transcendentn´?ch resp. Liouvillov ´ych ˇc´?slech, uv´adˇej´? historick´y pˇrehled, d°uleˇzit´e vˇety, kter´e doplˇnuj´? n´azorn´ymi pˇr´?klady. Posledn´? ˇctvrt´a kapitola obsahuje v´ysledky t´ykaj´?c´? se line´arn´? nez´avislosti nekoneˇcn´ych ˇrad. Druh´a ˇc´ast pˇredstavuje autorovy vlastn´? v´ysledky t´ykaj´?c´? se nekoneˇcn´ych ˇrad kladn´ych racion´aln´?ch ˇc´?sel konverguj´?c´?ch ke speci´aln´?m re´aln´ym ˇc´?sl°um. Tato kapitola je rozdˇelena do ˇctyˇr podkapitol. Prvn´? tˇri z nich pojedn´avaj´? o nekoneˇcn´ych ˇrad´ach, jejichˇz souˇcet je iracion´aln´?, transcendentn´? resp. Liouvillovo ˇc´?slo. Posledn´? podkapitola se zab´yv´a line´arn´? nez´avislost´? nekoneˇcn´ych ˇrad. Hlavn´? myˇslenka d°ukaz°u vˇsech vˇet je zaloˇzena na S´andorovˇe vˇetˇe (Theorem 6), viz [31]. Vˇety, kter´e d´avaj´? postaˇcuj´?c´? podm´?nky pro transcendenci souˇctu nekoneˇcn´e ˇrady, jsou nav´?c zaloˇzeny na tvrzen´? Rothovy vˇety (Theorem 11), viz [30]. Z nˇekter´ych vˇet jsou speci´aln´? volbou urˇcit´ych konstant odvozeny d°usledky a ke vˇsem vˇet´am a d°usledk°um je pˇripojen jeden n´azorn´y pˇr´?klad. Nˇekter´a z tˇechto krit´eri´? byla publikov´ana v mezin´arodn´?ch ˇcasopisech, napˇr. Rocky Mountain Journal of Mathematics, Saitama Journal of Mathematics, Mathematica Slovaca a Annali dell?Universita? di Ferrara

Abstract:

Abstract This thesis deals with the irrationality, transcendence and linear independence of infinite series consisting of rational numbers in a special form. It is divided into two chapters. In the first one there is introduced the concept of irrational and transcendent and Liouville numbers, a history overview of this, important theorems and some examples. The fourth chapter contains results regarding of linear independence of infinite series. The second one presents author?s main results concerning of infinite series of positive rationals converging to special real numbers. This chapter is divided into four subchapters. The first, second resp. third one focuses in infinite series whose sum is an irrational, a transcendental resp. a Liouville number. The last subchapter represents the results regarding linear independence of infinite series. The main idea of proof all of theorems is analogous to one used in Sandor?s Theorem (Theorem 6), see [31]. The theorems giving sufficient condition for transcendence of the sum of series are moreover based to the statement of Roth?s Theorem (Theorem 11), see [30]. Setting special value of given constants there are derived corollaries from some theorems and an illustrative example is added to each of them. Some of these criteria are published or going to publish in known international journals and magazines, e.g. Rocky Mountain Journal of Mathematics, Saitama Journal of Mathematics, Mathematica Slovaca and Annali dell?Universita? di Ferrara.

Kľúčové slová

Iracionalita, iracionální čí?slo, nekonečná řada, transcendence, transcendentní čí?slo, Liouvillovo čí?slo, line´arnˇe nez´avisl´e nekoneˇcn´e ˇrady, line´arn´? nez´avislost

Jazyk práce: Czech

Datum vytvoření / odevzdání či podání práce: 20. 5. 2006

Zverejniť od: 20. 5. 2006

Obhajoba závěrečné práce

• Obhajoba proběhla 20. 6. 2006
• Vedúci: prof. RNDr. Jaroslav Hančl, CSc.

University of Ostrava

Faculty of Science

Doctoral programme / odbor:
Applied Mathematics / Applied Mathematics