scholarly journals Connectedness locus for pairs of affine maps and zeros of power series

2015 ◽  
Vol 2 (3) ◽  
pp. 281-308 ◽  
Author(s):  
Boris Solomyak
Keyword(s):  
Power Series ◽  
Affine Maps ◽  
Connectedness Locus

scholarly journals On a family of self-affine sets: Topology, uniqueness, simultaneous expansions

2015 ◽  
Vol 37 (1) ◽  
pp. 193-227 ◽  
Author(s):  
KEVIN G. HARE ◽  
NIKITA SIDOROV
Keyword(s):  
Hausdorff Dimension ◽  
Positive Lebesgue Measure ◽  
Compact Set ◽  
Annual Conference ◽  
Topological Properties ◽  
Empty Interior ◽  
Affine Maps ◽  
Simply Connected ◽  
Set Of Points ◽  
Connectedness Locus

Let$\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{2}>1$and$T_{i}(x,y)=((x+i)/\unicode[STIX]{x1D6FD}_{1},(y+i)/\unicode[STIX]{x1D6FD}_{2}),i\in \{\pm 1\}$. Let$A:=A_{\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{2}}$be the unique compact set satisfying$A=T_{1}(A)\cup T_{-1}(A)$. In this paper, we give a detailed analysis of$A$and the parameters$(\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{2})$where$A$satisfies various topological properties. In particular, we show that if$\unicode[STIX]{x1D6FD}_{1}<\unicode[STIX]{x1D6FD}_{2}<1.202$, then$A$has a non-empty interior, thus significantly improving the bound from Dajaniet al[Self-affine sets with positive Lebesgue measure.Indag. Math. (N.S.)25(2014), 774–784]. In the opposite direction, we prove that the connectedness locus for this family studied in Solomyak [Connectedness locus for pairs of affine maps and zeros of power series.Preprint, 2014, arXiv:1407.2563] is not simply connected. We prove that the set of points of$A$which have a unique address has positive Hausdorff dimension for all$(\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{2})$. Finally, we investigate simultaneous$(\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{2})$-expansions of reals, which were the initial motivation for studying this family in Güntürk [Simultaneous and hybrid beta-encodings.Information Sciences and Systems, 2008. CISS 2008. 42nd Annual Conference2008, pp. 743–748].

The Software «MMI–calibration 3.0»

Metrologiya ◽  
2020 ◽  
pp. 16-24
Author(s):  
Alexandr D. Chikmarev
Keyword(s):  
Power Series ◽  
Data Processing ◽  
Statistical Treatment ◽  
Correction Function ◽  
Calibration Function ◽  
Measuring Instrument ◽  
Working Standard ◽  
Calibration Protocol ◽  
Error Probabilities ◽  
Continuous Power

A single program has been developed to ensure that the final result of the data processing of the measurement calibration protocol is obtained under normal conditions. The calibration result contains a calibration function or a correction function in the form of a continuous sedate series and a calibration chart based on typical additive error probabilities. Solved the problem of the statistical treatment of the calibration protocol measuring in normal conditions within a single program “MMI–calibration 3.0” that includes identification of the calibration function in a continuous power series of indications of a measuring instrument and chart calibration. An example of solving the problem of calibration of the thermometer by the working standard of the 3rd grade with the help of the “MMI-calibration 3.0” program.

Identification of the string boundary conditions using natural frequencies

2016 ◽  
Vol 11 (1) ◽  
pp. 38-52
Author(s):  
I.M. Utyashev ◽  
A.M. Akhtyamov
Keyword(s):  
Inverse Problems ◽  
Power Series ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Natural Frequencies ◽  
Boundary Value ◽  
External Environment ◽  
Direct And Inverse Problems ◽  
Symmetric Potential ◽  
Modified Method

The paper discusses direct and inverse problems of oscillations of the string taking into account symmetrical characteristics of the external environment. In particular, we propose a modified method of finding natural frequencies using power series, and also the problem of identification of the boundary conditions type and parameters for the boundary value problem describing the vibrations of a string is solved. It is shown that to identify the form and parameters of the boundary conditions the two natural frequencies is enough in the case of a symmetric potential q(x). The estimation of the convergence of the proposed methods is done.

scholarly journals Approximation by a power series summability method of Kantorovich type Szász operators including Sheffer polynomials

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Valdete Loku ◽  
Naim L. Braha ◽  
Toufik Mansour ◽  
M. Mursaleen
Keyword(s):  
Power Series ◽  
Summability Method ◽  
Approximation Properties ◽  
Type Result ◽  
Szász Operators ◽  
Sheffer Polynomials

AbstractThe main purpose of this paper is to use a power series summability method to study some approximation properties of Kantorovich type Szász–Mirakyan operators including Sheffer polynomials. We also establish Voronovskaya type result.

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