Sequences| Sequences |
In General
$ Def: A sequence in a set X
is a subset A ⊂ X and a map \(\mathbb N\) → A,
n \(\mapsto\) an, which is
onto, so that A = {an |
n ∈ \(\mathbb N\)}.
> Online resources:
The On-Line Encyclopedia of Integer Sequences (OEIS) site;
Wikipedia page.
Operations on Sequences
$ Difference operator:
The mapping {an}
→ {bn}
defined by bn
= an+1 −
an for all n.
Types of Sequences
$ Cauchy sequence:
A sequence {an}
in a normed space such that, for all ε > 0, ∃
N such that for all n, m > N,
|| an−
am|| < ε.
$ Monotonic sequence: (Increasing)
One such that the difference operator produces a non-negative sequence.
$ Completely monotonic sequence:
(Increasing) One such that k iterations of the difference operator produce
a non-negative sequence, for all k.
Convergence > s.a. limit;
series.
$ In a topological space (X,
T): A sequence {xn}
converges to x0 iff for all U
∈ T, x0 ∈ U,
∃ N such that n > N implies
xn ∈ U.
$ In a metric space (X,
d): A sequence {xn}
converges to x0 iff for all
ε > 0, ∃ N such that n > N
implies d(xn,
x0) < ε.
$ In a normed vector space:
The sequence {xn}
converges strongly s to x0
iff || xn −
x0|| → 0 as n → ∞.
$ For random variables: A sequence
{xn} of random variables
in a metric space (X, d) converges in probability to a random
variable x if \(\forall\epsilon,\,\lim_{n\to\infty}P(d(x_n,x) >
\epsilon) = 0\) (this is stronger than convergence in distribution).
Fibonacci Sequence > s.a. Golden Ratio.
$ Def:
The sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987,
..., in which Fn
= Fn−2
+ Fn−1;
It appears in life patterns.
* Result: (Zeckendorf) Every
integer can be written uniquely as a sum of non-consecutive Fibonacci numbers.
@ General references:
Vorobiev 02 [IIb];
Posamentier & Lehmann 07;
Miller & Wang JCTA(12) [and central-limit-type theorems].
@ In nature:
news pw(07)apr [stress-related Fibonacci spirals];
Pennybacker & Newell PRL(13) [underlying physical and biochemical mechanisms in plant growth];
Faraoni & Atieh a2101
[continuous generalizations and cosmological analogies].
> Online resources:
See Wikipedia page.
Other Examples > see Wikipedia page on integer sequences.
Generalizations > s.a. projective family.
* Inductive family or system:
Given a directed set I, an inductive family on I is
a collection {Xi
| i ∈ I} of objects in a category, and immersions
{φij:
Xi →
Xi |
i ≥ j ∈ I}, such that
φii
= idXi, and
φij \(\circ\)
φjk =
φik; Remark:
In many (all?) categories each such family defines an inductive limit.
main page
– abbreviations
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– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 28 jan 2021