scholarly journals N-Valid trees in wavelet theory on Vilenkin groups

2015 ◽  
Vol 13 (05) ◽  
pp. 1550037 ◽  
Author(s):  
S. F. Lukomskii ◽  
G. S. Berdnikov
Keyword(s):  
Rooted Tree ◽  
Step Function ◽  
Elementary Step ◽  
Vilenkin Group ◽  
Wavelet Theory ◽  
Vilenkin Groups ◽  
Step Functions

We consider a class of (N, M)-elementary step functions on the p-adic Vilenkin group. We prove that (N, M)-elementary step function generates a MRA on p-adic Vilenkin group if and only if it is generated by a special N-valid rooted tree on the set of vertices {0,1,…p - 1} with the vector (0,…,0) ∈ ℤN as a root.

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 The boundedness of multilinear commutators on locally compact Vilenkin groups

2006 ◽  
Vol 4 (3) ◽  
pp. 261-273 ◽  
Author(s):  
Canqin Tang
Keyword(s):  
Integral Operator ◽  
Hardy Spaces ◽  
Fractional Integral ◽  
Fractional Integral Operator ◽  
Vilenkin Group ◽  
Lebesgue Spaces ◽  
Locally Compact ◽  
Vilenkin Groups ◽  
Multilinear Commutators

LetGbe a locally compact Vilenkin group. In this paper, the authors investigate the boundedness of multilinear commutators of fractional integral operator on Lebesgue spaces onG. Furthermore, the boundedness on Hardy spaces are also obtained in this paper.

A Liquid-Medium Step-Function Pressure Calibrator

1964 ◽  
Vol 86 (4) ◽  
pp. 723-727 ◽  
Author(s):  
R. O. Smith
Keyword(s):  
Theoretical Prediction ◽  
Liquid Medium ◽  
High Frequency ◽  
Step Function ◽  
Wave Generation ◽  
Step Functions ◽  
Pressure Calibrator

Some characteristics of a step-function pressure calibrator of the type devised by D. P. Johnson and J. L. Cross of NBS have been investigated. Theoretical prediction of the wave generation appears to be impracticable. It has been found possible, by empirically determined adjustments, to produce monotonic step functions having 95 percent rise times in about 10 millisec suitable for calibration of transducers. Calibration accuracies are around 5 to 7 percent. The monotonic step-function pressure is also useful in evaluating creep or hysteresis in high frequency transducers which previously could not be inspected for that characteristic.

On the Consistency of Bayesian Function Approximation Using Step Functions

2007 ◽  
Vol 19 (11) ◽  
pp. 2871-2880 ◽  
Author(s):  
Heng Lian
Keyword(s):  
Bayesian Approach ◽  
Function Approximation ◽  
Step Function ◽  
Consistent Estimate ◽  
Piecewise Constant ◽  
Unknown Number ◽  
Step Functions ◽  
The Bayesian Approach

We consider the problem of estimating a step function with an unknown number of jumps under noisy observations on a grid. Under mild assumptions, the Bayesian approach is shown to produce a consistent estimate, even when the underlying true function is not piecewise constant. A simple prior is constructed to illustrate our assumptions.

PROPAGATION OF A PULSE IN A FLUID SPHERE

Geophysics ◽  
1964 ◽  
Vol 29 (2) ◽  
pp. 259-287 ◽  
Author(s):  
Z. Alterman ◽  
P. Kornfeld
Keyword(s):  
Geometrical Optics ◽  
Step Function ◽  
Surface Velocity ◽  
Fluid Sphere ◽  
Function Type ◽  
Arrival Times ◽  
Step Functions ◽  
The Difference ◽  
Deep Focus ◽  
Theoretic Solution

The exact solution obtained in a previous paper for the motion of a uniform compressible fluid sphere due to a pressure pulse from a point source situated below the surface is applied to a source at a distance of one‐eighth of the radius below the surface. Taking the sphere as a simplified model of the earth, this corresponds to a source at a depth of about 800 km, which is not far from the depth of a deep‐focus earthquake. The time variation of pressure due to the source is represented by the difference between two step functions with rounded shoulders. The surface velocity due to sources of different durations has been evaluated for eight angular distances. The solution exhibits step function type and “logarithmic” reftected pulses, which one would anticipate from the “steepest descents” analysis of Jeffreys and Lapwood. In addition, the solution reveals single diffracted pulses and groups of diffracted pulses which have no counterpart in ray theory. When geometrical optics allows a ray to appear only after a minimum range [Formula: see text] from the epicenter, the complete wave‐theoretical solution shows that these pulses show up earlier in the forbidden zones. Similarly, in the case where the geometrical optics predicts that a certain ray should appear only for ranges [Formula: see text], and should not appear for [Formula: see text], the wave‐theoretic solution shows that such a ray does appear, by diffraction, for some range of [Formula: see text]. Arrival times of the diffracted pulses of the first group increase with increasing θ, while for the second group they decrease with θ.

scholarly journals Abstract integral spaces and minimal extensions

1971 ◽  
Vol 12 (2) ◽  
pp. 166-178 ◽  
Author(s):  
M. J. Maron
Keyword(s):  
Measurable Function ◽  
Linear Functional ◽  
Vector Lattice ◽  
Measure Theory ◽  
Elementary Step ◽  
Summable Function ◽  
Step Functions ◽  
Positive Linear Functional ◽  
Almost Everywhere ◽  
Pointwise Limit

The development of the theory of absolute integrals derives from certain key facts. Among them are:(I) An integral is a positive linear functional on a vector lattice, which is continuous in a certain sense.(II) A function equal almost everywhere to a summable function is itself summable.(III) Every measurable function is the pointwise limit of a sequence of elementary step functions.A device that often plays an important role in measure theory, but which has not beenfully exploited in the theory of abstract integrals is that of(IV) the smallest class containing a given class and having a certain property(such as being a σ-ring of sets). It is our purpose in this paper to examine the theory of abstract real-valued absolute integrals axiomatically, in such a way as to isolate and clarify the roles of (I) through (IV).

scholarly journals Uncertainty product for Vilenkin groups

2018 ◽  
Vol 16 (05) ◽  
pp. 1850036
Author(s):  
Ivan Kovalyov ◽  
Elena Lebedeva
Keyword(s):  
Uncertainty Principle ◽  
Product Formula ◽  
Vilenkin Group ◽  
Vilenkin Groups ◽  
Haar Basis ◽  
Uncertainty Product ◽  
Heisenberg Uncertainty

We study a localization of functions defined on Vilenkin groups. To measure the localization, we introduce two uncertainty products [Formula: see text] and [Formula: see text] that are similar to the Heisenberg uncertainty product. [Formula: see text] and [Formula: see text] differ from each other by the metric used for the Vilenkin group [Formula: see text]. We discuss analogs of a quantitative uncertainty principle. Representations for [Formula: see text] and [Formula: see text] in terms of Walsh and Haar basis are given.

scholarly journals On the Poisson Integral of Step Functions and Minimal Surfaces

2002 ◽  
Vol 45 (1) ◽  
pp. 154-160 ◽  
Author(s):  
Allen Weitsman
Keyword(s):  
Minimal Surface ◽  
Lower Bounds ◽  
Minimal Surfaces ◽  
Step Function ◽  
Poisson Integral ◽  
Harmonic Mappings ◽  
Step Functions ◽  
Number Of Zeros ◽  
Univalent Harmonic Mappings

AbstractApplications of minimal surface methods are made to obtain information about univalent harmonic mappings. In the case where the mapping arises as the Poisson integral of a step function, lower bounds for the number of zeros of the dilatation are obtained in terms of the geometry of the image.

The Research of Using Step Function to Simulate the Step-Movement of Forklift Machine in Mechanical Engineering Based on ADAMS

2013 ◽  
Vol 644 ◽  
pp. 212-215
Author(s):  
Qun Feng Cui ◽  
Xu Rong Li ◽  
Jian Zhang Wang
Keyword(s):  
Mechanical Engineering ◽  
Solid Modeling ◽  
Step Function ◽  
Step Movement ◽  
Step Functions

the movements of forklift mainly include the forward and backward of body,driving of boom cylinder to stick cylinder, flipping and lifting of stick cylinder to shovel,and the recovery of them after the completion of these actions.The paper puts forward using step function in ADAMS to simulate these movements. First, the solid modeling of a forklift established in ADAMS. And then, the step function was utilized to carry on emulation step by step. The concret form of step function is step(parameter,time1,position1,time2,position2),the parameter here is time.The component is in position1 at time1.The component is in position2 at time2.At the same time, the utilization of combined step functions ,which means multiply step functions are add to components , can realize various motions at different time.The step movement is achieved consequently

scholarly journals Lp-multipliers of mixed-norm type on locally compact Vilenkin groups

1998 ◽  
Vol 65 (3) ◽  
pp. 370-387 ◽  
Author(s):  
C. W. Onneweer ◽  
T. S. Quek
Keyword(s):  
Dual Group ◽  
Mixed Norm ◽  
Vilenkin Group ◽  
Locally Compact ◽  
Vilenkin Groups ◽  
Multiplier Theorems

AbstractLet G be a locally compact Vilenkin group with dual group Γ. We prove Littlewood-Paley type inequalities corresponding to arbitrary coset decompositions of Γ. These inequalities are then applied to obtain new Lp(G) multiplier theorems. The sharpness of some of these results is also discussed.

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