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The Riemann theorem and divergent permutations

Roman Wituła (1996)

Colloquium Mathematicae

In this paper the fundamental algebraic propeties of convergent and divergent permutations of ℕ are presented. A permutation p of ℕ is said to be divergent if at least one conditionally convergent series ∑ a n of real terms is rearranged by p to a divergent series ∑ a p ( n ) . All other permutations of ℕ are called convergent. Some generalizations of the Riemann theorem about the set of limit points of the partial sums of rearrangements of a given conditionally convergent series are also studied.

The splitting number can be smaller than the matrix chaos number

Heike Mildenberger, Saharon Shelah (2002)

Fundamenta Mathematicae

Let χ be the minimum cardinality of a subset of ω 2 that cannot be made convergent by multiplication with a single Toeplitz matrix. By an application of a creature forcing we show that < χ is consistent. We thus answer a question by Vojtáš. We give two kinds of models for the strict inequality. The first is the combination of an ℵ₂-iteration of some proper forcing with adding ℵ₁ random reals. The second kind of models is obtained by adding δ random reals to a model of M A < κ for some δ ∈ [ℵ₁,κ). It...

Three open problems and a conjecture

Huizeng Qin, Ovidiu Furdui (2015)

Open Mathematics

In this paper we solve three open problems and a conjecture related to the calculations of some classes of multiple series posed by Furdui in [1].

Translativity of absolute weighted mean summability

Cihan Orhan (1998)

Czechoslovak Mathematical Journal

In this paper, we give necessary and sufficient conditions on ( p n ) for | R , p n | k , k ≥ 1 , to be translative. So we extend the known results of Al-Madi [1] and Cesco 4 to the case k > 1 .

Two geometric constants for operators acting on a separable Banach space.

E. Martín Peinador, E. Induráin, A. Plans Sanz de Bremond, A. A. Rodes Usan (1988)

Revista Matemática de la Universidad Complutense de Madrid

The main result of this paper is the following: A separable Banach space X is reflexive if and only if the infimum of the Gelfand numbers of any bounded linear operator defined on X can be computed by means of just one sequence on nested, closed, finite codimensional subspaces with null intersection.

Two valued measure and summability of double sequences

Pratulananda Das, Santanu Bhunia (2009)

Czechoslovak Mathematical Journal

In this paper, following the methods of Connor [connor], we extend the idea of statistical convergence of a double sequence (studied by Muresaleen and Edely [moe]) to μ -statistical convergence and convergence in μ -density using a two valued measure μ . We also apply the same methods to extend the ideas of divergence and Cauchy criteria for double sequences. We then introduce a property of the measure μ called the (APO 2 ) condition, inspired by the (APO) condition of Connor [jc]. We mainly investigate...

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