scholarly journals Achievement sets and sum ranges with ideal supports

Filomat ◽  
2018 ◽  
Vol 32 (14) ◽  
pp. 4911-4922
Author(s):  
Jacek Marchwicki
Keyword(s):  
Convergent Series ◽  
Topological Properties ◽  
Real Numbers ◽  
Small Set ◽  
Conditionally Convergent Series

We introduce the notion of ideally supported achievement sets for a series of real numbers. We analize their complexity and topological properties. We compare the notion of ideal achievement sets with the notion of ideally supported sum range of real series, considered by Filip?w and Szuca. We complete Filip?w and Szuca characterization of ideal sum ranges, [R. Filip?w, P. Szuca, Rearrangement of conditionally convergent series on a small set, J. Math. Anal. Appl. 362 (2010), no. 1, 64-71.], and we obtain some generalization of Riemann?s Theorem.

scholarly journals Changes of signs in conditionally convergent series on a small set

2011 ◽  
Vol 24 (11) ◽  
pp. 1831-1834
Author(s):  
Teresa Bermúdez ◽  
Antonio Martinón
Keyword(s):  
Convergent Series ◽  
Small Set ◽  
Conditionally Convergent Series

Rearrangements that Preserve Rates of Divergence

1982 ◽  
Vol 34 (4) ◽  
pp. 916-920
Author(s):  
Elgin H. Johnston
Keyword(s):  
Real Number ◽  
Infinite Series ◽  
Special Interest ◽  
Convergent Series ◽  
Divergent Series ◽  
Classical Theorem ◽  
Real Numbers ◽  
Conditionally Convergent Series ◽  
Positive Integers

Let Σak be an infinite series of real numbers and let π be a permutation of N, the set of positive integers. The series Σaπ(k) is then called a rearrangement of Σak. A classical theorem of Riemann states that if Σak is a conditionally convergent series and s is any fixed real number (or ± ∞), then there is a permuation π such that Σaπ(k) = s. The problem of determining those permutations that convert any conditionally convergent series into a convergent rearrangement (such permuations are called convergence preserving) has received wide attention (see, for example [6]). Of special interest is a paper by P. A. B. Pleasants [5] in which is shown that the set of convergence preserving permutations do not form a group.In this paper we consider questions similar to those above, but for rearrangements of divergent series of positive terms. We establish some notation before stating the precise problem.

scholarly journals Singular perturbations of the unicritical polynomials with two parameters

2016 ◽  
Vol 37 (6) ◽  
pp. 1997-2016 ◽  
Author(s):  
YINGQING XIAO ◽  
FEI YANG
Keyword(s):  
Julia Set ◽  
Dynamical Behavior ◽  
Rational Maps ◽  
Topological Properties ◽  
Fatou Set ◽  
The Family ◽  
Image Position ◽  
Two Parameters ◽  
Unicritical Polynomials

In this paper, we study the dynamics of the family of rational maps with two parameters $$\begin{eqnarray}f_{a,b}(z)=z^{n}+\frac{a^{2}}{z^{n}-b}+\frac{a^{2}}{b},\end{eqnarray}$$ where $n\geq 2$ and $a,b\in \mathbb{C}^{\ast }$. We give a characterization of the topological properties of the Julia set and the Fatou set of $f_{a,b}$ according to the dynamical behavior of the orbits of the free critical points.

scholarly journals Orderability of topological spaces by continuous preferences

2001 ◽  
Vol 27 (8) ◽  
pp. 505-512 ◽  
Author(s):  
José Carlos Rodríguez Alcantud
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Topological Properties ◽  
Linear Orders ◽  
Mathematical Economics

We extend van Dalen and Wattel's (1973) characterization of orderable spaces and their subspaces by obtaining analogous results for two larger classes of topological spaces. This type of spaces are defined by considering preferences instead of linear orders in the former definitions, and possess topological properties similar to those of (totally) orderable spaces (cf. Alcantud, 1999). Our study provides particular consequences of relevance in mathematical economics; in particular, a condition equivalent to the existence of a continuous preference on a topological space is obtained.

scholarly journals Distinct partitions and overpartitions

2021 ◽  
Vol 38 (1) ◽  
pp. 149-158
Author(s):  
MIRCEA MERCA ◽  
Keyword(s):  
Partition Function ◽  
Recurrence Relation ◽  
Recurrence Relations ◽  
Convergent Series ◽  
The Other ◽  
Large Integer ◽  
Linear Recurrence ◽  
Fast Computation ◽  
Real Numbers ◽  
Very High

In 1963, Peter Hagis, Jr. provided a Hardy-Ramanujan-Rademacher-type convergent series that can be used to compute an isolated value of the partition function $Q(n)$ which counts partitions of $n$ into distinct parts. Computing $Q(n)$ by this method requires arithmetic with very high-precision approximate real numbers and it is complicated. In this paper, we investigate new connections between partitions into distinct parts and overpartitions and obtain a surprising recurrence relation for the number of partitions of $n$ into distinct parts. By particularization of this relation, we derive two different linear recurrence relations for the partition function $Q(n)$. One of them involves the thrice square numbers and the other involves the generalized octagonal numbers. The recurrence relation involving the thrice square numbers provide a simple and fast computation of the value of $Q(n)$. This method uses only (large) integer arithmetic and it is simpler to program. Infinite families of linear inequalities involving partitions into distinct parts and overpartitions are introduced in this context.

scholarly journals PTM-Shepherd: analysis and summarization of post-translational and chemical modifications from open search results

2020 ◽  
pp. mcp.TIR120.002216
Author(s):  
Daniel J. Geiszler ◽  
Andy T. Kong ◽  
Dmitry M Avtonomov ◽  
Fengchao Yu ◽  
Felipe da Veiga Leprevost ◽  
...  
Keyword(s):  
Shotgun Proteomics ◽  
Chemical Modifications ◽  
Bioinformatics Tool ◽  
Site Specific ◽  
Search Results ◽  
Formalin Fixed Paraffin ◽  
Formalin Fixed Paraffin Embedded ◽  
Small Set ◽  
Formalin Fixed

Open searching has proven to be an effective strategy for identifying both known and unknown modifications in shotgun proteomics experiments. Rather than being limited to a small set of user-specified modifications, open searches identify peptides with any mass shift that may correspond to a single modification or a combination of several modifications. Here we present PTM-Shepherd, a bioinformatics tool that automates characterization of PTM profiles detected in open searches based on attributes such as amino acid localization, fragmentation spectra similarity, retention time shifts, and relative modification rates. PTM-Shepherd can also perform multi-experiment comparisons for studying changes in modification profiles, e.g. in data generated in different laboratories or under different conditions. We demonstrate how PTM-Shepherd improves the analysis of data from formalin-fixed paraffin-embedded samples, detects extreme underalkylation of cysteine in some datasets, discovers an artefactual modification introduced during peptide synthesis, and uncovers site-specific biases in sample preparation artifacts in a multi-center proteomics profiling study.

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