Preferencje
Język
Widoczny [Schowaj] Abstrakt
Liczba wyników
Tytuł książki

## Complex series and connected sets

Autorzy
Seria
Rozprawy Matematyczne tom/nr w serii: 52 wydano: 1966
Zawartość
Warianty tytułu
Abstrakty
EN

CONTENTS
PREFACE..........................................................................................................................................................................3
INTRODUCTION............................................................................................................................................................. 4
1. Notation. 2. Subject of the paper.
Chapter I. DECOMPOSITION OF Σ INTO $Σ_1$, $Σ_2$, $Σ_3$, $Σ_4$ INESSENTIAL RESTRICTION
OF GENERALITY ............................................................................................................................................................ 6
1. Families $Σ_k$, k = 1, 2, 3, 4. 2. Families $Σ^0$ and $Σ^0_k$, k = 1, 2, 3, 4.
Chapter II. FURTHER AUXILIARY THEOREMS....................................................................................................... 10
1. Chains of order n. 2. Further notations. 3. A sufficient condition for
Ʌ(S) = Γ. Property (.). 4. A lemma on complex numbers. 5. Properties
(..), (...) and (....). 6 A necessary and sufficient condition for Ʌ (S) = Γ.
Chapter III. CASES: $S∈∑^0_4$ and $S∈∑^0_1$................................................................................................ 20
1. Case: $S∈∑^0_4$. 2. Case: $S∈∑^0_1$.
Chapter IV. CASES: $S∈∑^0_2$ and $S∈∑^0_3$ FAMILIES ɸ(S)..................................................................... 22
1. Notations. 2. Preliminary remarks on ɸ(S) for S from $∑^0_2$. 3. General
theorems on ɸ(S) for S from $∑^0_2◡∑^0_3$. 4. Detailed remarks on ɸ(S). 5. The
structure of $ɸ_0(S)$ for a special S from $∑^0_3$
Chapter V. CASE: $S∈∑^0_3$, FAMILIES Ω(S)...................................................................................................... 34
1. Definitions of the families Ω, Ω(S), $Ω_k$ and $Ω_k(S)$, k = 0, 1, 2, 3, 4.
2. Families $Ω^n_k$, k = 0, 1, 2, 3, 4 and $Ω^n$. 3. A sufficient condition for
L(S) = C in the case $S∈Ω_4$. 4. Regions F_j(z, p; e), j = 1, 2, 3, 4. 5.
Families $Ω_4(S)$. 6. Families $Ω_3(S)$ and Ω(S).
Chapter VI. CASE: $S∈∑^0_2◡∑^0_3$ VARIOUS PROBLEMS........................................................................... 42
1. Property (—). 2. An example of the equality Λ(S) = Γ for S from $∑^0_3$
3. An open problem concerning $Λ_0(S)$
REFERENCES................................................................................................................................................................ 46
Słowa kluczowe
Tematy
Miejsce publikacji
Warszawa
Seria
Rozprawy Matematyczne tom/nr w serii: 52
Liczba stron
47
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom LII
Daty
wydano
1966
Twórcy
autor
• Department of Mathematics, Technical University, Wrocław (Katedra Matematyki Politechniki Wrocławskiej)
• Institute of Mathematics of the Polish Academy of Sciences (Instytut Matematyczny PAN)
Bibliografia
• [1] E. Calabi and A. Dvoretzky, Convergence-and sum-factors of series of complex numbers, Trans. Amer. Math. Soc. 70(1951), pp. 177-194.
• [2] A. Dvoretzky et H. Hanani, Sur les changements des signes des termes d'une serie à termes complexes, C. R. Acad. Sci., Paris 225 (1947), pp. 516-518.
• [3] H. Hanani, On sums of series of complex numbers, Pacific Journal of Math. 3 (1953), pp. 695-709.
• [4] H. Hornich, Über beliebige Teilsummen absolut Jconvergenter Reihen, Monats-hefte Math. Phys. 49 (1941).
• [5] B. Jasek, Transformations of complex series, Colloq. Math. 9 (1962), pp. 266-275.
• [6] G. Polya und G. Szegö, Aufgaben und Lehrsätze aus der Analysis, B. I.
• [7] T. Salàt, K'absolútne konvergentným řadom, Matematicko Fysikálny Čàsopis 7(3) (1957), pp. 139-142.
• [8] E. Steinitz, Bedingt konvergente Reihen und konvexe Systeme, Journal für reine und angewandte Math. 143 (1913), pp. 128-179; 144 (1914), pp. 1-40; 146 (1916), pp. 1-32.
• [I] B. Jasek, Complex series and connected sets (I), Bull. Acad. Polon. Sci. XI (1963).
• [II] B. Jasek, Complex series and connected sets (II), Bull. Acad. Polon. Sci. XI (1963).
Języki publikacji
 EN
Uwagi