Analysis is almost
algebra
Analysis is almost
algebra
Linear algebra meets coding theory
Bjørn Ian Dundas
Analysis is almost algebra
Bjørn Ian Dundas
Almost algebra
Almost algebra
Ali defined
Ali defined
Alis add up
You can order alis
Alis add up
1
Alis multiply
Bjørn Ian Dundas
2
Half the Ali
A retraction
Forum for matematiske perler (og kuriositeter), NTNU April 15, 20161
Coding theory: Really? On the nose all the time ?2
Half the Ali
Compromise: allow some slack.
A retraction
Bounds for Ali
Ali is real
References
References
Fun facts
Fun facts
Pf of bounds
Pf of bounds
2
“C’mon! these functions grow without bounds. You can’t expect to keep track of the very last
of billions of decimal places!”
Analysis is almost
algebra
Linear algebra meets coding theory
Bjørn Ian Dundas
Almost algebra
Bjørn Ian Dundas
Examples
Ali defined
f
cr : Z × Z → Z,
Half the Ali
Bounds for Ali
Ali is real
f is almost linear (ali) if
crf
is bounded.
Two ali f and g are almost the same (f ≈ g ) if f − g is bounded.
References
Fun facts
Alis add up
You can order alis
h(x) = 7x + (−1)x is ali and h ≈ f .
Alis multiply
A retraction
Ali defined
g (x) = 7x − 6 is ali and g ≈ f
You can order alis
cr (a, b) = f (a + b) − f (a) − f (b)
Almost algebra
f (x) = 7x is linear, and so ali.
Alis add up
f
Alis multiply
Half the Ali
This is something we shouldn’t be naı̈ve about: there are significantly more
exotic alis3 than even . . . 4 can imagine.
Reality5 is quite exotic: also some of the alis are irrational.
This talk is about them.
“almost the same” f ≈ g : bounded difference.
5
Analysis is almost
algebra
An estimate
Bjørn Ian Dundas
Almost algebra
Ali defined
A true generalization
Alis add up
You can order alis
Ali defined
Two useful facts
If f ali and n a natural number, then
Alis multiply
Hom(Z, Z) ֒→ Ali(Z, Z).
Z∼
= Hom(Z, Z),
Half the Ali
Bounds for Ali
n 7→ {x 7→ nx};
Ali is real
and are almost the same :f (nx)
References
Pf of bounds
You can order alis
Alis multiply
Half the Ali
≈ nf (x)
A retraction
Proof: There is an M such that for all a, b we have |crf (a, b)| < M,7
AND
gives by induction on n that
6
“almost the same” f ≈ g : bounded difference.
You can add alis8
Fun facts
Pf of bounds
|f (nx) − nf (x)| < nM .
7
Analysis is almost
algebra
crf (a, b) = f (a + b) − f (a) − f (b)
You can order alis9
Analysis is almost
algebra
Bjørn Ian Dundas
We can add linear functions. Can we add alis?
If f , g alis, then f + g is ali:
crf +g (a, b) = (f (a + b) + g (a + b)) − (f (a) + g (a)) − (f (b) + g (b))
g
= cr (a, b) + cr (a, b)
f ≈ f ′ and g ≈ g ′ implies f + g ≈ f ′ + g ′
Ali(Z, Z) is an abelian group.
Almost algebra
Ali defined
Ali defined
Alis add up
Alis add up
You can order alis
Ali(Z, Z) is an ordered abelian group
You can order alis
Alis multiply
If f is ali, exactly one of the following is true:
Alis multiply
Half the Ali
1
A retraction
Bounds for Ali
2
Ali is real
3
References
Fun facts
Z∼
= Hom(Z, Z) ⊆ Ali(Z, Z) = {ali f : Z → Z}/ ≈.
Can you multiply alis?10
Bjørn Ian Dundas
Almost algebra
Pf of bounds
8
A retraction
f is positive: {f (x) | x > 0} has no upper bound
Bounds for Ali
f is negative: {f (x) | x > 0} has no lower bound
Ali is real
References
We write f ≻ g if f − g is positive.
9
Analysis is almost
algebra
Half the Ali
f is bounded (f ≈ 0)
Fun facts
Pf of bounds
Ali(Z, Z) = {ali f : Z → Z}/ ≈.
Is the composite of two alis an ali?11
Analysis is almost
algebra
Bjørn Ian Dundas
The product of two (almost) linear functions is usually not (almost) linear:
f (x) = x, f (x) · f (x) = x 2 ,
crf ·f (a, b) = (a + b)2 − a2 − b 2 = 2ab
Almost algebra
Ali defined
Ali defined
Alis add up
Alis add up
You can order alis
You can order alis
Expanding, we see that
Alis multiply
composition in Hom(Z, Z).
Z∼
= Hom(Z, Z) ⊆ Ali(Z, Z) = {ali f : Z → Z}/ ≈.
Alis multiply
Half the Ali
Half the Ali
cr
A retraction
f ◦g
f
f
g
References
g
(a, b) = cr (g (a), g (b)) − cr (g (a) + g (b), cr (a, b)) − f (cr (a, b)),
consequently, if f and g are ali, so is the composite fg = f ◦ g .
Ali is real
↔
Is the composite of two alis an ali?
10
Bjørn Ian Dundas
Almost algebra
Bounds for Ali
is without bounds.
The multiplication in Z
Ali is real
crf (a, b) = f (a + b) − f (a) − f (b)
f is almost linear if crf is bounded.
f
Bounds for Ali
References
f ((n + 1)x) − (n + 1)f (x) = crf (nx, x) + f (nx) − nf (x)
Fun facts
what about the almost linear?
Alis add up
f (nx) (coarser sampling) and nf (x) (scaled output) are ali,
A retraction
Linear functions are easy to understand:
References
name of politician deleted
pun intended
Almost algebra
Ali(Z, Z) = { ali f : Z → Z}/ ≈
Ali is real
crf (a, b) = f (a + b) − f (a) − f (b)
Bjørn Ian Dundas
Definition
Bounds for Ali
Pf of bounds
4
Analysis is almost
algebra
A retraction
Fun facts
3
Pf of bounds
f : Z → Z is almost linear if crf is bounded.
Ali defined6
Alis multiply
Ali is real
Analysis is almost
algebra
Definition
A function f : Z → Z has cross effect
You can order alis
Bounds for Ali
1
Nothing of what follows is due to me. See the list of references at the end. Thanks to whoever
brought this pearl to my attention
Compromise: allow some slack
3
Algebra: A linear function f : Z → Z satisfies f (a + b) = f (a) + f (b) !
A retraction
Bounds for Ali
Ali is real
References
Fun facts
Fun facts
Pf of bounds
Pf of bounds
11
Z∼
= Hom(Z, Z) ⊆ Ali(Z, Z) = {ali f : Z → Z}/ ≈.
Analysis is almost
algebra
Two more elementary estimates
Bjørn Ian Dundas
fg ≈ gf
Analysis is almost
algebra
Bjørn Ian Dundas
Almost algebra
If f ali12 and n a positive integer. We proved by induction that
|f (nx) − nf (x)| < nM
Ali defined
Alis add up
You can order alis
13
Alis multiply
Half the Ali
Similarly, there are constants A, B, C , D such that
Almost algebra
Ali defined
If f and g ali, then fg ≈ gf .
Alis multiply
|xfg (x) − g (x)f (x)| < (|x| + |g (x)|)C + D.
A retraction
Bounds for Ali
Ali is real
You can order alis
Proof: Let x 6= 0. Setting y = g (x) in |xf (y ) − yf (x)| < (|x| + |y |)C + D gives
A retraction
|f (x)| < A|x| + B
Alis add up
Swapping f and g and adding the resulting equations give
References
|xf (y ) − yf (x)| < (|x| + |y |)C + D.
Half the Ali
Bounds for Ali
Ali is real
References
|x||fg (x) − gf (x)| < (2|x| + |f (x)| + |g (x)|)C + 2D.
Fun facts
Pf of bounds
Fun facts
Pf of bounds
Using that alis are bounded by linear functions and dividing by |x| we are done.
12
13
|crf (a, b)| < M where
crf (a, b) = f (a + b) − f (a) − f (b)
we used this to deduce that f (nx) ≈ nf (x)
fg ≈ gf 14
Analysis is almost
algebra
Bjørn Ian Dundas
1/2 ∈ Ali(Z, Z) = {ali f : Z → Z}/ ≈
Analysis is almost
algebra
Bjørn Ian Dundas
Almost algebra
Almost algebra
Ali defined
Ali defined
Alis add up
Corollary
You can order alis
Ali(Z, Z) is a commutative ring: pointwise addition and with composition as
multiplication.
Proof: Zero and one from Z.
Associativity is just associativity of composition.
Since (f1 + f2 )g = f1 g + f2 g , distributivity follows by commutativity fg ≈ gf .
Alis multiply
Alis add up
Let f (x) = 2x (representing 2 ∈ Z ∼
= Hom(Z, Z) ֒→ Ali(Z, Z)), and define
g (2n) = n = g (2n + 1).
Half the Ali
Ali is real
References
Fun facts
A retraction
ali15
Then g is
and gf (x) = x, so that g represents 1/2!
Similarly, we get that Ali(Z, Z) contains the rationals Q. . .
if q is a positive integer, 1/q is represented by x 7→ ⌊x/q⌋.
Pf of bounds
14
Z∼
= Hom(Z, Z) ⊆ Ali(Z, Z) = {ali f : Z → Z}/ ≈.
Analysis is almost
algebra
Lemma
Assume the ali f is positive. Then the odd function given by
Ali defined
Ali(Z, Z) is a field, II
is ali and is the almost inverse of f .
Half the Ali
g (n) = min{q > 0 | f (q) ≥ n} for n > 0
A retraction
“g ≈
By definition f (g (m)) ≥ m ≥ f (g (m) − 1), giving
Ali is real
0 ≤ fg (m) − m ≤ fg (m) − f (g (m) − 1),
Fun facts
References
Alis add up
You can order alis
Half the Ali
A retraction
fg (m+n)−f (g (m)−1)−f (g (n)−1) > 0 > f (g (m+n)−1)−fg (m)−fg (n).
References
Bounds for Ali
Ali is real
Fun facts
Since f ali, both sides have bounded difference to
Pf of bounds
Pf of bounds
f (g (m + n) − g (m) − g (n)),
g ali: next slide
which itself consequently is bounded. Since f is positive, this means that
the input crg (m, n) = g (m + n) − g (m) − g (n) is bounded.
Analysis is almost
algebra
A retraction
Ali(Z, Z) has LUB
Analysis is almost
algebra
Bjørn Ian Dundas
Identifying n ∈ Z with the ali nx, we see that if f is ali,
Almost algebra
Ali defined
Ali defined
Alis add up
You can order alis
Half the Ali
is well defined (the set is is nonempty and bounded above).
Bjørn Ian Dundas
Almost algebra
Alis multiply
⌊f ⌋ = max{n ∈ Z | f n}
Alis add up
Theorem
If S ⊆ Ali(Z, Z) is nonempty and bounded above, then S has a least upper
bound given by the odd function
fS (n) = max{⌊ng ⌋ | g ∈ S},
Ali is real
Ali(Z, Z)
f 7→⌊f ⌋
/ Z.
References
a
n ≥ 0.a
⌊h⌋ = max{n ∈ Z | h n}
Fun facts
⌊f + g ⌋ − ⌊f ⌋ − ⌊g ⌋ ∈ {0, 1}.
Pf of bounds
Ali(Z, Z) is real
Analysis is almost
algebra
History and references
fS (n) = max{⌊ng ⌋ | g ∈ S},
n≥
⌊h⌋ = max{n ∈ Z | h n}
Corollary
Ali(Z, Z) is a complete ordered field, and hence by Hölder:
Ali(Z, Z) is isomorphic to R.
Half the Ali
Bounds for Ali
Ali is real
References
Pf of bounds
Analysis is almost
algebra
Bjørn Ian Dundas
Almost algebra
0.a
Alis multiply
Fun facts
Follows by inequality manipulations similar to what we’ve done
Bjørn Ian Dundas
Theorem
If S ⊆ Ali(Z, Z) is nonempty and bounded above, then S has a least upper
bound given by the odd function
You can order alis
A retraction
A retraction
Bounds for Ali
This gives a retraction
a
Ali defined
“g ≈ f −1 ”: ok
g ali: By definition f (g (n)) ≥ n ≥ f (g (n) − 1), giving
and since f is ali, the upper limit is bounded. Hence fg (x) ≈ 1 · x.
f 7→ ⌊f ⌋ is “ali”:
Almost algebra
Alis multiply
is ali and is the almost inverse of f .
Bounds for Ali
f −1 ”:
Fun facts
Bjørn Ian Dundas
Alis add up
Alis multiply
References
Analysis is almost
algebra
Lemma
Assume the ali f is positive. Then the odd function given by
You can order alis
g (n) = min{q > 0 | f (q) ≥ n} for n > 0
Ali is real
|crg (a, b)| ≤ 1
Bjørn Ian Dundas
Almost algebra
Bounds for Ali
Pf of bounds
15
Ali(Z, Z) is a field, I
Alis multiply
Half the Ali
A retraction
Bounds for Ali
You can order alis
Almost algebra
Ali defined
Ali defined
Alis add up
Steven Schanuel (unpublished, early 80’s). Independently Richard Lewis?
Alis add up
You can order alis
Ross Street, An efficient construction of the real numbers. Gazette
Australian Math. Soc., 12:5758, 1985 (later improved).
You can order alis
Alis multiply
Half the Ali
A retraction
Bounds for Ali
Ali is real
References
Fun facts
Pf of bounds
John Harrison. Theorem Proving with the Real Numbers. Technical report,
University of Cambridge Computer Laboratory, 1996.
Norbert ACampo. A natural construction for the real numbers.
arXiv:math.GN/0301015 v1, 3 January 2003.
R.D. Arthan, The Eudoxus Real Numbers, arXiv:math/0405454 v1, 24 May
2004.
Alis multiply
Half the Ali
A retraction
Bounds for Ali
Ali is real
References
Fun facts
Pf of bounds
Analysis is almost
algebra
Fun facts
Bjørn Ian Dundas
Almost algebra
Ali defined
Any ali is represented by an f : Z → Z with
Alis add up
|crf (a, b)| = |f (a + b) − f (a) − f (b)| ≤ 1.
f 7→ lim f (n)/n
n→∞
is an isomorphism of complete ordered fields.
∼ Mn×m (R).
Ali(Zm , Zn ) =
AliZ ≃ VectR .
Alis multiply
a
A retraction
Bounds for Ali
Ali is real
References
Fun facts
Pf of bounds
Bjørn Ian Dundas
Almost algebra
Ali defined
Alis add up
fS (n) = max{⌊ng ⌋ | g ∈ S},
You can order alis
Half the Ali
The asymptotic slope function
Ali(Z, Z) → R,
Analysis is almost
algebra
Theorem
If S ⊆ Ali(Z, Z) is nonempty and bounded above, then S has a least upper
bound given by the odd function
n ≥ 0.a
⌊h⌋ = max{n ∈ Z | h n}
Proof. Let F be an upper bound for S. {⌊ng ⌋ | g ∈ S} is nonempty and
bounded by ⌊nF ⌋, so fS (n) is well defined.
Must show that fS is ali!
Given a there is an ga ∈ S s.t. fS (a) = ⌊aga ⌋, and for any g ∈ S, ⌊ag ⌋ ≤ ⌊aga ⌋.
Let G = max{ga , gb , ga+b }.
Notice: fS (a) = ⌊aG ⌋, fS (b) = ⌊bG ⌋ and fS (a + b) = ⌊(a + b)G ⌋ = ⌊aG + bG ⌋.
But then crfS (a, b) = ⌊aG + bG ⌋ − ⌊aG ⌋ − ⌊bG ⌋ ∈ {0, 1}!
You can order alis
Alis multiply
Half the Ali
A retraction
Bounds for Ali
Ali is real
References
Fun facts
Pf of bounds