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

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. ...

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

2002
*
Mathematics of Computation
*

As in [1] we consider

doi:10.1090/s0025-5718-02-01519-3
fatcat:z44wvntd2fav5mv3karhirk53e
*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 ...##
###
The Structure of Cosmic-Ray Air Showers

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. ...

##
###
Fast Digital Convolutions using Bit-Shifts
[article]

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) ...

##
###
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

2022
*
Proverbium
*

This essay explores some

doi:10.29162/pv.39.1.64
fatcat:ytetouzpfnfh5o63ihlelepuye
*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. ...##
###
The general theory of factorial series

1930
*
Bulletin of the American Mathematical Society
*

Replace

doi:10.1090/s0002-9904-1930-04923-x
fatcat:ruon5bbv4veo5ajkbi75yvz7ga
*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. ...##
###
On the Relation between Fourier Frequency and Period for Discrete Signals, and Series of Discrete Periodic Complex Exponentials

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. ...

##
###
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. ...

##
###
Digital Star Dust: The Hoagy Carmichael Collection at Indiana University

2000
*
First Monday
*

Carmichael

doi:10.5210/fm.v5i6.753
fatcat:abkvlubu5ffq7msmc5fzg3f2ry
*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 ...##
###
On factorisation, with a suggested new approach

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*. ...

##
###
On Factorisation, with a Suggested New Approach

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*. ...

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