Quasi-optimal reception of the random pulse with arbitrary-function envelope and unknown time and power parameters

2016 ◽  
Vol 10 ◽  
pp. 2611-2626
Author(s):  
O.V. Chernoyarov ◽  
B. Dobrucky ◽  
D.N. Shepelev ◽  
B.I. Shakhtarin
Keyword(s):  
Arbitrary Function ◽  
Random Pulse ◽  
Optimal Reception

Testing Pulse Timing Devices Using Arbitrary Function Generators

2020 ◽  
Vol 54 (5) ◽  
pp. 466-473
Author(s):  
V. A. Bespal’ko ◽  
I. Burak ◽  
A. S. Rybakov
Keyword(s):  
Arbitrary Function ◽  
Pulse Timing ◽  
Function Generators

scholarly journals Reducing a Class of Two-Dimensional Integrals to One-Dimension with an Application to Gaussian Transforms

Atoms ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 53
Author(s):  
Jack C. Straton
Keyword(s):  
Quantum Theory ◽  
Arbitrary Function ◽  
Plane Waves ◽  
Wave Functions ◽  
One Dimension ◽  
Two Dimensional ◽  
Transition Amplitudes ◽  
Multidimensional Integrals ◽  
Coulomb Potentials

Quantum theory is awash in multidimensional integrals that contain exponentials in the integration variables, their inverses, and inverse polynomials of those variables. The present paper introduces a means to reduce pairs of such integrals to one dimension when the integrand contains powers multiplied by an arbitrary function of xy/(x+y) multiplying various combinations of exponentials. In some cases these exponentials arise directly from transition-amplitudes involving products of plane waves, hydrogenic wave functions, and Yukawa and/or Coulomb potentials. In other cases these exponentials arise from Gaussian transforms of such functions.

XXIII.— Dual Series Relations. V. A Generalized Schlömilch Series and the Uniqueness of the Solution of Dual Equations involving Trigonometric Series

1964 ◽  
Vol 66 (4) ◽  
pp. 258-268
Author(s):  
R. P. Srivastav
Keyword(s):  
Trigonometric Series ◽  
Arbitrary Function ◽  
Fractional Integration ◽  
Series Expansions ◽  
Uniqueness Of The Solution ◽  
Special Case

SynopsisThe methods employed in papers I–IV of this series are modified to provide the solution of certain dual equations involving trigonometric series. It is necessary to introduce a modified form of the conventional operators of fractional integration and to discuss their relation with generalized Schlömilch series expansions of an arbitrary function. These general methods are illustrated by detailed reference to a particular special case.

scholarly journals Quantifying Noninvertibility in Discrete Dynamical Systems

10.37236/9475 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Colin Defant ◽  
James Propp
Keyword(s):  
Dynamical Systems ◽  
Real Number ◽  
Arbitrary Function ◽  
Discrete Dynamical Systems ◽  
Open Problems ◽  
Natural Measure ◽  
Stack Sorting ◽  
Finite Set

Given a finite set $X$ and a function $f:X\to X$, we define the \emph{degree of noninvertibility} of $f$ to be $\displaystyle\deg(f)=\frac{1}{|X|}\sum_{x\in X}|f^{-1}(f(x))|$. This is a natural measure of how far the function $f$ is from being bijective. We compute the degrees of noninvertibility of some specific discrete dynamical systems, including the Carolina solitaire map, iterates of the bubble sort map acting on permutations, bubble sort acting on multiset permutations, and a map that we call "nibble sort." We also obtain estimates for the degrees of noninvertibility of West's stack-sorting map and the Bulgarian solitaire map. We then turn our attention to arbitrary functions and their iterates. In order to compare the degree of noninvertibility of an arbitrary function $f:X\to X$ with that of its iterate $f^k$, we prove that \[\max_{\substack{f:X\to X\\ |X|=n}}\frac{\deg(f^k)}{\deg(f)^\gamma}=\Theta(n^{1-1/2^{k-1}})\] for every real number $\gamma\geq 2-1/2^{k-1}$. We end with several conjectures and open problems.  

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