Algorithm 25: real zeros of an arbitrary function

1960 ◽  
Vol 3 (11) ◽  
pp. 602
Author(s):  
B. Leavenworth
Keyword(s):  
Arbitrary Function ◽  
Real Zeros

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

SOME EXTENSIONS OF A LEMMA OF KOTLARSKI

2012 ◽  
Vol 28 (4) ◽  
pp. 925-932 ◽  
Author(s):  
Kirill Evdokimov ◽  
Halbert White
Keyword(s):  
Characteristic Function ◽  
Characteristic Functions ◽  
Real Zeros ◽  
Number Of Zeros ◽  
Weaker Conditions

This note demonstrates that the conditions of Kotlarski’s (1967, Pacific Journal of Mathematics 20(1), 69–76) lemma can be substantially relaxed. In particular, the condition that the characteristic functions of M, U1, and U2 are nonvanishing can be replaced with much weaker conditions: The characteristic function of U1 can be allowed to have real zeros, as long as the derivative of its characteristic function at those points is not also zero; that of U2 can have an isolated number of zeros; and that of M need satisfy no restrictions on its zeros. We also show that Kotlarski’s lemma holds when the tails of U1 are no thicker than exponential, regardless of the zeros of the characteristic functions of U1, U2, or M.

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.

scholarly journals Real Zeros of a Class of Hyperbolic Polynomials with Random Coefficients

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Mina Ketan Mahanti ◽  
Amandeep Singh ◽  
Lokanath Sahoo
Keyword(s):  
Random Variables ◽  
Random Coefficients ◽  
Expected Number ◽  
Hyperbolic Polynomial ◽  
Gaussian Random Variables ◽  
Hyperbolic Polynomials ◽  
Asymptotic Value ◽  
Real Zeros ◽  
Number Of Real Zeros

We have proved here that the expected number of real zeros of a random hyperbolic polynomial of the formy=Pnt=n1a1cosh⁡t+n2a2cosh⁡2t+⋯+nnancosh⁡nt, wherea1,…,anis a sequence of standard Gaussian random variables, isn/2+op(1). It is shown that the asymptotic value of expected number of times the polynomial crosses the levely=Kis alson/2as long asKdoes not exceed2neμ(n), whereμ(n)=o(n). The number of oscillations ofPn(t)abouty=Kwill be less thann/2asymptotically only ifK=2neμ(n), whereμ(n)=O(n)orn-1μ(n)→∞. In the former case the number of oscillations continues to be a fraction ofnand decreases with the increase in value ofμ(n). In the latter case, the number of oscillations reduces toop(n)and almost no trace of the curve is expected to be present above the levely=Kifμ(n)/(nlogn)→∞.

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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