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Period of the power generator and small values of Carmichael's function

John B. Friedlander, Carl Pomerance, Igor E. Shparlinski
2000 Mathematics of Computation  
And so, in this case, the power generator is uniformly distributed.  ...  We show rigorously that for almost all choices of p, l it is the case that for almost all choices of ϑ, e, the period of the power generator exceeds (pl) 1−ε .  ...  Period of the power generator Here we apply Theorem 5 to obtain a lower bound for the largest possible period of the power generator. Recall that P denotes the set of primes. Theorem 6.  ... 
doi:10.1090/s0025-5718-00-01282-5 fatcat:2fokg2wlajdqrg5ohxd4ieczua

Corrigendum to ``Period of the power generator and small values of Carmichael's function''

John B. Friedlander, Carl Pomerance, Igor E. Shparlinski
2002 Mathematics of Computation  
As in [1] we consider the power generator with the initial value u 0 = ϑ (an integer coprime to m) and exponent e (an integer at least 2).  ...  If p a is the largest power of the prime p which divides n, and a is at least 1, we write p a n. The letters p, q, l always denote prime numbers.  ...  (In case p = 2, these are not quite the same.) ϕ(m)ϕ(λ(m)) τ (λ(m)) exp(∆/3) + τ (λ(λ(m))) exp(∆/3) ≤ mλ(m ) exp (−∆/4) exceptional pairs (ϑ, e), the period t of the power generator satisfies t ≥ λ(λ(m  ... 
doi:10.1090/s0025-5718-02-01519-3 fatcat:z44wvntd2fav5mv3karhirk53e

The Structure of Cosmic-Ray Air Showers

Konrad L. Kingshill, Lloyd G. Lewis
1946 Physical Review  
The chambers used in this investigation had a cross-sectional area intermediate between Carmichael’s small and large chambers, and recorded an intermediate value of the average particle density at a given  ...  The point C—S gives the slope for Carmichael’s small chamber; C—L for Carmichael’s large chamber; L for Lapp’s chamber; and K&L for the chamber used in this experiment.  ... 
doi:10.1103/physrev.69.159 fatcat:md54xxv7qjasferpg7zbh23yjq

Fast Digital Convolutions using Bit-Shifts [article]

Shekhar S. Chandra
2010 arXiv   pre-print
The transform is an extension of the Rader Transforms via Carmichael's Theorem.  ...  An exact, one-to-one transform is presented that not only allows digital circular convolutions, but is free from multiplications and quantisation errors for transform lengths of arbitrary powers of two  ...  CARMICHAEL'S THEOREM This section presents a new and more general theory of Number Theoretic Transforms (NTTs) utilising the concept of primitive roots from Carmichael's Theorem [13] (see Appendix B)  ... 
arXiv:1005.1497v1 fatcat:fonjxomvnjaqrgdvhgscaofa5m

Page 4703 of Mathematical Reviews Vol. , Issue 2002G [page]

2002 Mathematical Reviews  
(S-MCQR-CP; North Ryde) Period of the power generator and small values of Carmichael’s function. (English summary) Math. Comp. 70 (2001), no. 236, 1591-1605 (electronic).  ...  We show rigorously that for almost all choices of p,/ it is the case that for almost all choices of ¥, e, the period of the power generator exceeds (pl)'-€.  ... 

Page 6584 of Mathematical Reviews Vol. , Issue 2003i [page]

2003 Mathematical Reviews  
(S5-MCQR-CP; North Ryde) Corrigendum to: “Period of the power generator and small values of Carmichael’s function” |Math. Comp. 70 (2001), no. 236, 1591-1605; MR 2002g:11112]. Math.  ...  Let b > 2 be a natural number, let d'"’(x) be the m-th digit of the decimal part of the real number x > 0 in its base-b expansion, and let m xX," (x,a) = Sod "(x +na) (mod b) k=] be a {0 b —1}-valued function  ... 

Page 3186 of Mathematical Reviews Vol. , Issue 90F [page]

1990 Mathematical Reviews  
Gurak, Cubic and biquadratic pseudoprimes of Lucas type (pp. 330-347); J. L. Hafner and A. Ivié, On some mean value results for the Riemann zeta-function (pp. 348-358).  ...  Pétermann, On the distribution of values of an error term related to the Euler function (pp. 785-797); Patrice Philippon and Michel Waldschmidt, Formes linéaires de logarithmes elliptiques et mesures de  ... 

"Black power" and black rhetorical tradition

Raymond M. Summerville
2022 Proverbium  
This essay explores some of the proverbs, sayings, and proverbialexpressions used by Stokely Carmichael (1941-1998) that are foundin writings such as: Black Power the Politics of Liberation in America(  ...  In the process of analyzing traditional languageused by Carmichael, this essay also: (1) seeks to identify sourcesfrom which Carmichael's sayings are derived, (2) examine some of thecontexts in which they  ...  Our starting point in history must precede the period of colonialism and slavery; it must precede the Arabic and European invasions.  ... 
doi:10.29162/pv.39.1.64 fatcat:ytetouzpfnfh5o63ihlelepuye

The general theory of factorial series

Tomlinson Fort
1930 Bulletin of the American Mathematical Society  
Replace the denominators in the terms of (1) by general polynomials in z of successively ascending powers.  ...  Replace the terms of (1) by certain general functions so that the series will include both ordinary Dirichlet series and factorial series. This was done by me a few years ago in the Transactions.  ...  Carmichael assumes a function 1+ -+ -+ « ) z z 2 / analytic at least when z is in a sector including the axis of reals and sufficiently large in absolute value. P(z) and Q(z) are polynomials.  ... 
doi:10.1090/s0002-9904-1930-04923-x fatcat:ruon5bbv4veo5ajkbi75yvz7ga

On the Relation between Fourier Frequency and Period for Discrete Signals, and Series of Discrete Periodic Complex Exponentials

Alfredo Restrepo, Julian Quiroga, Jairo Alberto Hurtado
2021 IEEE Open Journal of Signal Processing  
The absolute value of the frequency is an increasing function of the subadditive functional of average variation.  ...  Discrete complex exponentials are almost periodic signals, not always periodic; when periodic, the frequency determines the period, but not viceversa, the period being a chaotic function of the frequency  ...  For small values of N, i.e. for N = 2, 3, 4 and 6, they coincide; see Fig. 11 , below.  ... 
doi:10.1109/ojsp.2021.3064760 fatcat:gm3la3wuxndudblfwn5jfe6ehe

Page 6477 of Mathematical Reviews Vol. , Issue 95k [page]

1995 Mathematical Reviews  
The authors give a formula of arithmetic nature for the coefficients of the Fourier cosine series of an even step function f of period 2z.  ...  In number theory, the exponent of the multiplicative group (Z/LZ)~ of reduced residue classes mod L is known as Carmichael’s function A(L).  ... 

IRELAND

1904 The Lancet  
Carmichael's will, which enjoins as its subject the state of the medical profession in Great Britain and Ireland and of the various hospitals and schools and the modes of examination adopted by the licensing  ...  He is not-like his recent predecessors-a clergyman but he has for a long period taken a strong and active interest in the public affairs of the Church of Ireland.  ... 
doi:10.1016/s0140-6736(01)73064-4 fatcat:5zvzhvh4bzgnzkf4hkh5c5zlde

Digital Star Dust: The Hoagy Carmichael Collection at Indiana University

Kristine R. Brancolini, Jon W. Dunn, John A. Walsh
2000 First Monday  
Carmichael valued his association with the community and University, and as a result, he and his heirs have contributed to Indiana University many materials pertaining to Carmichael's career.  ...  Throughout the years since the collection came to Indiana University following Carmichael's death in 1981, Indiana University, in general, and the Archives of Traditional Music, specifically, have benefited  ... 
doi:10.5210/fm.v5i6.753 fatcat:abkvlubu5ffq7msmc5fzg3f2ry

On factorisation, with a suggested new approach

J. C. P. Miller
1975 Mathematics of Computation  
The most onerous problem lies in the finding of a sufficient number of congruences (at least fe) and in the choice of a suitable value of k.  ...  Western's method for finding indices and primitive roots, based on congruences N = a + b, where N is the number whose exponent is required, and both a and b are .  ...  The other generators will all have exponent dividing X(A); that is, X(A) is a period for every generator.  ... 
doi:10.1090/s0025-5718-1975-0366796-6 fatcat:zsxijdhfdnbbdbbncopz3ndmuy

On Factorisation, with a Suggested New Approach

J. C. P. Miller
1975 Mathematics of Computation  
The most onerous problem lies in the finding of a sufficient number of congruences (at least fe) and in the choice of a suitable value of k.  ...  Western's method for finding indices and primitive roots, based on congruences N = a + b, where N is the number whose exponent is required, and both a and b are .  ...  The other generators will all have exponent dividing X(A); that is, X(A) is a period for every generator.  ... 
doi:10.2307/2005472 fatcat:iguy6bonmje7xm7r7pm5yprhbi
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