scholarly journals On Generalized Step-Functions and Superposition Operators

2004 ◽  
Vol 11 (4) ◽  
pp. 753-758
Author(s):  
A. Kharazishvili
Keyword(s):  
Step Functions ◽  
Functions Of Two Variables ◽  
Superposition Operators

Abstract For a given σ-ideal of sets, the notion of a generalized stepfunction is introduced and investigated in connection with the problem of sup-measurability of certain functions of two variables, regarded as superposition operators.

Uniform continuity of nonautonomous superposition operators in ΛBV-spaces

2019 ◽  
Vol 31 (3) ◽  
pp. 713-726
Author(s):  
Jacek Gulgowski
Keyword(s):  
Sufficient Conditions ◽  
Uniform Continuity ◽  
Regular Functions ◽  
Superposition Operators

Abstract In this paper we investigate the problem of uniform continuity of nonautonomous superposition operators acting between spaces of functions of bounded Λ-variation. In particular, we give the sufficient conditions for nonautonomous superposition operators to continuously map a space of functions of bounded Λ-variation into itself. The conditions cover the generators being functions of {C^{1}} -class (in view of two variables), but also allow for less regular functions, including discontinuous generators.

scholarly journals A study of Babylonian planetary theory III. The planet Mercury

2021 ◽  
Author(s):  
Teije de Jong
Keyword(s):  
Direct Method ◽  
Step Function ◽  
Computational Effort ◽  
Stationary Points ◽  
Planetary Motion ◽  
Time Intervals ◽  
Outer Planets ◽  
Step Functions ◽  
System A ◽  
Computational Systems

AbstractIn this series of papers I attempt to provide an answer to the question how the Babylonian scholars arrived at their mathematical theory of planetary motion. Papers I and II were devoted to system A theory of the outer planets and of the planet Venus. In this third and last paper I will study system A theory of the planet Mercury. Our knowledge of the Babylonian theory of Mercury is at present based on twelve Ephemerides and seven Procedure Texts. Three computational systems of Mercury are known, all of system A. System A1 is represented by nine Ephemerides covering the years 190 BC to 100 BC and system A2 by two Ephemerides covering the years 310 to 290 BC. System A3 is known from a Procedure Text and from Text M, an Ephemeris of the last evening visibility of Mercury for the years 424 to 403 BC. From an analysis of the Babylonian observations of Mercury preserved in the Astronomical Diaries and Planetary Texts we find: (1) that dates on which Mercury reaches its stationary points are not recorded, (2) that Normal Star observations on or near dates of first and last appearance of Mercury are rare (about once every twenty observations), and (3) that about one out of every seven pairs of first and last appearances is recorded as “omitted” when Mercury remains invisible due to a combination of the low inclination of its orbit to the horizon and the attenuation by atmospheric extinction. To be able to study the way in which the Babylonian scholars constructed their system A models of Mercury from the available observational material I have created a database of synthetic observations by computing the dates and zodiacal longitudes of all first and last appearances and of all stationary points of Mercury in Babylon between 450 and 50 BC. Of the data required for the construction of an ephemeris synodic time intervals Δt can be directly derived from observed dates but zodiacal longitudes and synodic arcs Δλ must be determined in some other way. Because for Mercury positions with respect to Normal Stars can only rarely be determined at its first or last appearance I propose that the Babylonian scholars used the relation Δλ = Δt −3;39,40, which follows from the period relations, to compute synodic arcs of Mercury from the observed synodic time intervals. An additional difficulty in the construction of System A step functions is that most amplitudes are larger than the associated zone lengths so that in the computation of the longitudes of the synodic phases of Mercury quite often two zone boundaries are crossed. This complication makes it difficult to understand how the Babylonian scholars managed to construct System A models for Mercury that fitted the observations so well because it requires an excessive amount of computational effort to find the best possible step function in a complicated trial and error fitting process with four or five free parameters. To circumvent this difficulty I propose that the Babylonian scholars used an alternative more direct method to fit System A-type models to the observational data of Mercury. This alternative method is based on the fact that after three synodic intervals Mercury returns to a position in the sky which is on average only 17.4° less in longitude. Using reduced amplitudes of about 14°–25° but keeping the same zone boundaries, the computation of what I will call 3-synarc system A models of Mercury is significantly simplified. A full ephemeris of a synodic phase of Mercury can then be composed by combining three columns of longitudes computed with 3-synarc step functions, each column starting with a longitude of Mercury one synodic event apart. Confirmation that this method was indeed used by the Babylonian astronomers comes from Text M (BM 36551+), a very early ephemeris of the last appearances in the evening of Mercury from 424 to 403 BC, computed in three columns according to System A3. Based on an analysis of Text M I suggest that around 400 BC the initial approach in system A modelling of Mercury may have been directed towards choosing “nice” sexagesimal numbers for the amplitudes of the system A step functions while in the later final models, dating from around 300 BC onwards, more emphasis was put on selecting numerical values for the amplitudes such that they were related by simple ratios. The fact that different ephemeris periods were used for each of the four synodic phases of Mercury in the later models may be related to the selection of a best fitting set of System A step function amplitudes for each synodic phase.

scholarly journals Comparing different wavelet transforms on removing electrocardiogram baseline wanders and special trends

2020 ◽  
Vol 20 (S11) ◽  
Author(s):  
Chao-Chen Chen ◽  
Fuchiang Rich Tsui
Keyword(s):  
Wavelet Transforms ◽  
Low Frequency ◽  
Clinical Decision Support Systems ◽  
Clinical Decision ◽  
Temporal Position ◽  
Important Indicator ◽  
Real Time Processing ◽  
Step Functions ◽  
Baseline Wander ◽  
Ecg Data

Abstract Background Electrocardiogram (ECG) signal, an important indicator for heart problems, is commonly corrupted by a low-frequency baseline wander (BW) artifact, which may cause interpretation difficulty or inaccurate analysis. Unlike current state-of-the-art approach using band-pass filters, wavelet transforms can accurately capture both time and frequency information of a signal. However, extant literature is limited in applying wavelet transforms (WTs) for baseline wander removal. In this study, we aimed to evaluate 5 wavelet families with a total of 14 wavelets for removing ECG baseline wanders from a semi-synthetic dataset. Methods We created a semi-synthetic ECG dataset based on a public QT Database on Physionet repository with ECG data from 105 patients. The semi-synthetic ECG dataset comprised ECG excerpts from the QT database superimposed with artificial baseline wanders. We extracted one ECG excerpt from each of 105 patients, and the ECG excerpt comprised 14 s of randomly selected ECG data. Twelve baseline wanders were manually generated, including sinusoidal waves, spikes and step functions. We implemented and evaluated 14 commonly used wavelets up to 12 WT levels. The evaluation metric was mean-square-error (MSE) between the original ECG excerpt and the processed signal with artificial BW removed. Results Among the 14 wavelets, Daubechies-3 wavelet and Symlets-3 wavelet with 7 levels of WT had best performance, MSE = 0.0044. The average MSEs for sinusoidal waves, step, and spike functions were 0.0271, 0.0304, 0.0199 respectively. For artificial baseline wanders with spikes or step functions, wavelet transforms in general had lower performance in removing the BW; however, WTs accurately located the temporal position of an impulse edge. Conclusions We found wavelet transforms in general accurately removed various baseline wanders. Daubechies-3 and Symlets-3 wavelets performed best. The study could facilitate future real-time processing of streaming ECG signals for clinical decision support systems.

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