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Balancing a disc
Balancing a circle
Let the triple
(
θ
i
,
d
i
,
w
i
)
{\displaystyle (\theta _{i},d_{i},w_{i})}
be the angle, distance and weight of a particle from the centre of the disc.
Let P be some particles on a unit disk such that a given diameter axis
d
1
{\displaystyle d_{1}}
means that the sum of perpendiculars from P to
d
1
{\displaystyle d_{1}}
is zero. Let
d
2
{\displaystyle d_{2}}
be a distinct other axis with the same property.
Prove that all
d
{\displaystyle d}
have this property.
Consider two axis orthogonal to each other and a set of weights with the above property (which can be achieved by calculating the position of the
n
t
h
{\displaystyle n^{th}}
weight after the first
n
−
1
{\displaystyle n-1}
weights have been placed).
Note that the perpendiculars to a new rotated orthogonal pair of axes from a given weight all lie on the circumference of the circle centred halfway between the origin and the weight, radius half the full distance from O to W.
This applies to each weight, and so there are a set of circles each with a common point on the circumference of the origin and the new axes intersect with these circles at the new distances.
From trigonometric identities , we have:
sin
(
α
+
β
)
=
sin
α
cos
β
+
cos
α
sin
β
cos
(
α
+
β
)
=
cos
α
cos
β
−
sin
α
sin
β
{\displaystyle {\begin{aligned}\sin(\alpha +\beta )&=\sin \alpha \cos \beta +\cos \alpha \sin \beta \\\cos(\alpha +\beta )&=\cos \alpha \cos \beta -\sin \alpha \sin \beta \end{aligned}}}
We know:
∑
w
i
sin
α
i
=
0
∑
w
i
cos
α
i
=
0
{\displaystyle {\begin{aligned}\sum w_{i}\sin \alpha _{i}&=0\\\sum w_{i}\cos \alpha _{i}&=0\end{aligned}}}
If
α
i
{\displaystyle \alpha _{i}}
is the original angle, and
β
{\displaystyle \beta }
is the rotated amount, then we have:
w
i
sin
(
α
i
+
β
)
=
w
i
sin
α
i
cos
β
+
w
i
cos
α
i
sin
β
w
i
cos
(
α
i
+
β
)
=
w
i
cos
α
i
cos
β
−
w
i
sin
α
i
sin
β
{\displaystyle {\begin{aligned}w_{i}\sin(\alpha _{i}+\beta )&=w_{i}\sin \alpha _{i}\cos \beta +w_{i}\cos \alpha _{i}\sin \beta \\w_{i}\cos(\alpha _{i}+\beta )&=w_{i}\cos \alpha _{i}\cos \beta -w_{i}\sin \alpha _{i}\sin \beta \end{aligned}}}
and summing:
∑
w
i
sin
(
α
i
+
β
)
=
∑
w
i
sin
α
i
cos
β
+
∑
w
i
cos
α
i
sin
β
∑
w
i
cos
(
α
i
+
β
)
=
∑
w
i
cos
α
i
cos
β
−
∑
w
i
sin
α
i
sin
β
{\displaystyle {\begin{aligned}\sum w_{i}\sin(\alpha _{i}+\beta )&=\sum w_{i}\sin \alpha _{i}\cos \beta +\sum w_{i}\cos \alpha _{i}\sin \beta \\\sum w_{i}\cos(\alpha _{i}+\beta )&=\sum w_{i}\cos \alpha _{i}\cos \beta -\sum w_{i}\sin \alpha _{i}\sin \beta \end{aligned}}}
∑
w
i
sin
(
α
i
+
β
)
=
cos
β
∑
w
i
sin
α
i
+
sin
β
∑
w
i
cos
α
i
=
0
+
0
=
0
∑
w
i
cos
(
α
i
+
β
)
=
cos
β
∑
w
i
cos
α
i
−
sin
β
∑
w
i
sin
α
i
=
0
−
0
=
0
{\displaystyle {\begin{aligned}\sum w_{i}\sin(\alpha _{i}+\beta )&=\cos \beta \sum w_{i}\sin \alpha _{i}+\sin \beta \sum w_{i}\cos \alpha _{i}=0+0=0\\\sum w_{i}\cos(\alpha _{i}+\beta )&=\cos \beta \sum w_{i}\cos \alpha _{i}-\sin \beta \sum w_{i}\sin \alpha _{i}=0-0=0\end{aligned}}}
The resultant weighted vector at the origin is therefore zero, and this is invariant over axis rotation.
Or, the lines midway between two are balanced, and this process continues ad infinitum.
Convex function proofs
Consider
f
(
x
)
=
x
2
{\displaystyle f(x)=x^{2}}
,
x
≤
y
,
0
≤
t
≤
1
{\displaystyle x\leq y,0\leq t\leq 1}
. Then:
f
(
t
x
+
(
1
−
t
)
y
)
−
t
f
(
x
)
−
(
1
−
t
)
f
(
y
)
{\displaystyle f(tx+(1-t)y)-tf(x)-(1-t)f(y)}
=
t
2
x
2
+
2
t
(
1
−
t
)
x
y
+
(
1
−
t
)
2
y
2
−
t
x
2
−
(
1
−
t
)
y
2
{\displaystyle =t^{2}x^{2}+2t(1-t)xy+(1-t)^{2}y^{2}-tx^{2}-(1-t)y^{2}}
=
t
(
t
−
1
)
x
2
+
2
t
(
1
−
t
)
x
y
+
(
1
−
t
)
(
−
t
)
y
2
{\displaystyle =t(t-1)x^{2}+2t(1-t)xy+(1-t)(-t)y^{2}}
=
−
t
(
1
−
t
)
(
x
2
−
2
x
y
+
y
2
)
{\displaystyle =-t(1-t)\left(x^{2}-2xy+y^{2}\right)}
=
−
t
(
1
−
t
)
(
x
−
y
)
2
{\displaystyle =-t(1-t)(x-y)^{2}}
≤
0
{\displaystyle \leq 0}
Consider
f
(
x
)
=
log
(
x
)
{\displaystyle f(x)=\log(x)}
,
0
<
x
≤
y
,
0
≤
t
≤
1
{\displaystyle 0<x\leq y,0\leq t\leq 1}
. Then:
f
(
t
x
+
(
1
−
t
)
y
)
−
t
f
(
x
)
−
(
1
−
t
)
f
(
y
)
{\displaystyle f(tx+(1-t)y)-tf(x)-(1-t)f(y)}
=
log
(
t
x
+
(
1
−
t
)
y
)
−
(
t
log
(
x
)
+
(
1
−
t
)
log
(
y
)
)
{\displaystyle =\log(tx+(1-t)y)-(t\log(x)+(1-t)\log(y))}
=
log
(
t
x
+
(
1
−
t
)
y
)
−
log
(
x
t
y
1
−
t
)
{\displaystyle =\log(tx+(1-t)y)-\log(x^{t}y^{1-t})}
=
log
(
t
x
+
(
1
−
t
)
y
x
t
y
1
−
t
)
{\displaystyle =\log({\frac {tx+(1-t)y}{x^{t}y^{1-t}}})}
So need to prove:
t
x
+
(
1
−
t
)
y
x
t
y
1
−
t
≥
1
{\displaystyle {\frac {tx+(1-t)y}{x^{t}y^{1-t}}}\geq 1}
or
t
x
+
(
1
−
t
)
y
≥
x
t
y
1
−
t
{\displaystyle tx+(1-t)y\geq x^{t}y^{1-t}}
or, dividing by
y
{\displaystyle y}
:
t
(
x
y
−
1
)
+
1
≥
(
x
y
)
t
{\displaystyle t\left({\frac {x}{y}}-1\right)+1\geq \left({\frac {x}{y}}\right)^{t}}
So:
t
(
x
y
−
1
)
+
1
−
(
x
y
)
t
{\displaystyle t\left({\frac {x}{y}}-1\right)+1-\left({\frac {x}{y}}\right)^{t}}
=
(
x
y
−
1
)
(
t
−
(
x
y
)
t
−
1
x
y
−
1
)
{\displaystyle =\left({\frac {x}{y}}-1\right)\left(t-{\frac {\left({\frac {x}{y}}\right)^{t}-1}{{\frac {x}{y}}-1}}\right)}
=
(
x
y
−
1
)
(
t
−
(
1
+
x
y
+
⋯
+
(
x
y
)
t
−
1
)
)
{\displaystyle =\left({\frac {x}{y}}-1\right)\left(t-(1+{\frac {x}{y}}+\dots +\left({\frac {x}{y}}\right)^{t-1})\right)}
≥
0
{\displaystyle \geq 0}
Proof of Euler's Theorem
Euler's Theorem
We have
a
p
−
1
≡
1
(
mod
p
)
{\displaystyle a^{p-1}\equiv 1{\pmod {p}}}
Therefore
(
a
p
−
1
)
p
=
(
1
+
k
p
)
p
≡
1
(
mod
p
2
)
{\displaystyle (a^{p-1})^{p}=(1+kp)^{p}\equiv 1{\pmod {p^{2}}}}
Also, if
p
,
q
{\displaystyle p,q}
are primes, then
(
a
p
−
1
)
q
−
1
=
a
(
p
−
1
)
(
q
−
1
)
≡
1
(
mod
q
)
{\displaystyle (a^{p-1})^{q-1}=a^{(p-1)(q-1)}\equiv 1{\pmod {q}}}
(
a
q
−
1
)
p
−
1
=
a
(
p
−
1
)
(
q
−
1
)
≡
1
(
mod
p
)
{\displaystyle (a^{q-1})^{p-1}=a^{(p-1)(q-1)}\equiv 1{\pmod {p}}}
These are the same, and, by the Chinese Remainder Theorem
a
(
p
−
1
)
(
q
−
1
)
≡
1
(
mod
p
q
)
{\displaystyle a^{(p-1)(q-1)}\equiv 1{\pmod {pq}}}
Catalan q-polynomials
The Catalan q-polynomials
C
k
(
q
)
{\displaystyle C_{k}(q)}
count the number of blocks present in the diagram under the diagonal height k , and start
C
0
(
q
)
=
1
{\displaystyle C_{0}(q)=1}
C
1
(
q
)
=
1
{\displaystyle C_{1}(q)=1}
C
2
(
q
)
=
1
+
q
{\displaystyle C_{2}(q)=1+q}
C
3
(
q
)
=
1
+
q
+
2
q
2
+
q
3
{\displaystyle C_{3}(q)=1+q+2q^{2}+q^{3}}
C
4
(
q
)
=
1
+
q
+
2
q
2
+
3
q
3
+
3
q
4
+
3
q
5
+
q
6
{\displaystyle C_{4}(q)=1+q+2q^{2}+3q^{3}+3q^{4}+3q^{5}+q^{6}}
The sum of the coefficients of
C
k
(
q
)
{\displaystyle C_{k}(q)}
give
C
k
{\displaystyle C_{k}}
.
To generate the next level, we add a horizontal and vertical step. The horizontal step is always placed at the origin, and the vertical step can be placed anywhere off the bounding diagonal, and is the first time the path touches the diagonal.
The first path from the convolution is inserted along the (k-1) -th diagonal created by the two endpoints of the new steps, and the second is placed along the k -th diagonal.
With Dyck words, where the new template is
X
(
k
)
Y
(
n
−
k
)
{\displaystyle X(k)Y(n-k)}
, where
(
k
)
{\displaystyle (k)}
is a Dyck word of length k . The X step is always first, and the Y step comes after a k -length Dyck word.
The new polynomial is therefore the heights of the previous polynomials, plus the rectangle created by the insertion.
C
n
(
q
)
=
∑
i
=
0
n
−
1
q
i
(
n
−
i
)
C
i
(
q
)
C
n
−
i
(
q
)
{\displaystyle C_{n}(q)=\sum _{i=0}^{n-1}q^{i(n-i)}C_{i}(q)C_{n-i}(q)}
For example,
C
4
(
q
)
=
q
0
(
1
)
(
1
+
q
+
2
q
2
+
q
3
)
+
q
3
(
1
)
(
1
+
q
)
+
q
4
(
q
+
1
)
(
1
)
+
q
3
(
1
+
q
+
2
q
2
+
q
3
)
(
1
)
{\displaystyle C_{4}(q)=q^{0}(1)(1+q+2q^{2}+q^{3})+q^{3}(1)(1+q)+q^{4}(q+1)(1)+q^{3}(1+q+2q^{2}+q^{3})(1)}
.
Chinese Remainder Theorem
If p and q are prime, then the solution to
a
mod
p
,
b
mod
q
{\displaystyle a\mod p,b\mod q}
is given by
a
q
−
b
p
q
−
p
=
a
−
(
b
+
a
)
p
q
−
p
=
b
−
(
b
−
a
)
q
q
−
p
{\displaystyle {\frac {aq-bp}{q-p}}=a-{\frac {(b+a)p}{q-p}}=b-{\frac {(b-a)q}{q-p}}}
as the last two expressions are
a
mod
p
{\displaystyle a\mod p}
and
b
mod
q
{\displaystyle b\mod q}
respectively.
For example,
3
mod
7
,
6
mod
11
{\displaystyle 3\mod 7,6\mod {11}}
gives
33
−
42
11
−
7
=
−
9
4
{\displaystyle {\frac {33-42}{11-7}}={\frac {-9}{4}}}
, and
17
×
4
=
68
≡
−
9
mod
77
{\displaystyle 17\times 4=68\equiv -9\mod 77}
For a practical solution when p and q are coprime, create an array from 0 to p-1. Populate a[kp%q]=kp. Read a[1].
Bezout's identity
Bezout's identity
If
p
,
q
{\displaystyle p,q}
are primes, then there exist integers
a
,
b
{\displaystyle a,b}
such that
a
p
−
b
q
=
1
{\displaystyle ap-bq=1}
.
Let
a
=
p
q
−
2
{\displaystyle a=p^{q-2}}
, then
a
p
≡
1
(
mod
q
)
,
{\displaystyle ap\equiv 1{\pmod {q}},}
and choose b accordingly.
Cubic polynomial
Let
f
(
x
)
=
a
x
3
+
b
x
2
+
c
x
+
d
{\displaystyle f(x)=ax^{3}+bx^{2}+cx+d}
The local minima and maxima are given by the zeroes of the differential of
f
(
x
)
{\displaystyle f(x)}
.
f
′
(
x
)
=
3
a
x
2
+
2
b
x
+
c
{\displaystyle f'(x)=3ax^{2}+2bx+c}
f
′
(
x
)
=
0
{\displaystyle f'(x)=0}
i.e., by the quadratic formula , when
x
=
−
2
b
±
4
b
2
−
12
a
c
6
a
=
−
b
±
b
2
−
3
a
c
3
a
{\displaystyle x={\frac {-2b\pm {\sqrt {4b^{2}-12ac}}}{6a}}={\frac {-b\pm {\sqrt {b^{2}-3ac}}}{3a}}}
Let the two roots be
r
1
<
r
2
{\displaystyle r_{1}<r_{2}}
.
Let w.l.o.g.
f
(
r
1
)
{\displaystyle f(r_{1})}
is the local maxima and
f
(
r
2
)
{\displaystyle f(r_{2})}
is the local minima.
f
(
r
1
)
{\displaystyle f(r_{1})}
and :
f
(
r
2
)
{\displaystyle f(r_{2})}
have several terms equal, so
f
(
r
1
)
−
f
(
r
2
)
=
−
2
a
27
a
3
(
3
b
2
b
2
−
3
a
c
+
b
2
−
3
a
c
3
)
+
−
2
b
9
a
2
(
2
b
b
2
−
3
a
c
)
2
+
−
2
c
3
a
(
b
2
−
3
a
c
)
{\displaystyle f(r_{1})-f(r_{2})={\frac {-2a}{27a^{3}}}(3b^{2}{\sqrt {b^{2}-3ac}}+{\sqrt {b^{2}-3ac}}^{3})+{\frac {-2b}{9a^{2}}}(2b{\sqrt {b^{2}-3ac}})^{2}+{\frac {-2c}{3a}}({\sqrt {b^{2}-3ac}})}
which is essentially a quadratic.
We require
f
(
r
1
)
>
0
>
f
(
r
2
)
{\displaystyle f(r_{1})>0>f(r_{2})}
and if X is the common terms of each side, then this becomes
f
(
r
1
)
−
X
>
−
X
>
f
(
r
2
)
−
X
{\displaystyle f(r_{1})-X>-X>f(r_{2})-X}
Round Robin
6
12 34 56
13 26 45
14 25 36
15 23 46
16 24 35
8
12 34 58 67
13 24 57 68
14 25 36 78
15 26 37 48
16 27 38 45
17 28 35 46
18 23 47 56
GF for cubes
1
+
4
x
+
x
2
(
1
−
x
)
4
=
1
+
8
x
+
27
x
2
+
64
x
3
+
125
x
4
+
…
{\displaystyle {\frac {1+4x+x^{2}}{(1-x)^{4}}}=1+8x+27x^{2}+64x^{3}+125x^{4}+\dots }
https://www.wolframalpha.com/input?i=%281%2B4x%2Bx%5E2%29%2F%281-x%29%5E4
The general relation is given by the Eulerian numbers .
Binomials
Prove
(
m
j
)
(
n
k
)
≤
(
m
+
n
j
+
k
)
{\displaystyle {\binom {m}{j}}{\binom {n}{k}}\leq {\binom {m+n}{j+k}}}
=
m
!
n
!
j
!
k
!
(
m
−
j
)
!
(
n
−
k
)
!
≤
(
m
+
n
)
!
(
j
+
k
)
!
(
m
+
n
−
j
−
k
)
!
{\displaystyle ={\frac {m!n!}{j!k!(m-j)!(n-k)!}}\leq {\frac {(m+n)!}{(j+k)!(m+n-j-k)!}}}
Consider
(
m
+
n
)
!
m
!
n
!
j
!
k
!
(
j
+
k
)
!
(
m
−
j
)
!
(
n
−
k
)
!
(
m
+
n
−
j
−
k
)
!
{\displaystyle {\frac {(m+n)!}{m!n!}}{\frac {j!k!}{(j+k)!}}{\frac {(m-j)!(n-k)!}{(m+n-j-k)!}}}
where
(
a
+
b
)
!
a
!
b
!
=
(
a
+
b
a
)
=
(
a
+
1
)
⋯
(
a
+
b
)
b
!
=
∏
i
=
1
b
(
1
+
a
i
)
{\displaystyle {\frac {(a+b)!}{a!b!}}={\binom {a+b}{a}}={\frac {(a+1)\cdots (a+b)}{b!}}=\prod _{i=1}^{b}\left(1+{\frac {a}{i}}\right)}
so
∏
i
=
1
m
(
1
+
n
i
)
∏
i
=
1
j
(
1
+
k
i
)
∏
i
=
1
m
−
j
(
1
+
n
−
k
i
)
{\displaystyle {\frac {\prod _{i=1}^{m}\left(1+{\frac {n}{i}}\right)}{\prod _{i=1}^{j}\left(1+{\frac {k}{i}}\right)\prod _{i=1}^{m-j}\left(1+{\frac {n-k}{i}}\right)}}}
and
(
m
+
n
m
)
(
j
+
k
j
)
(
m
+
n
−
j
−
k
m
−
j
)
{\displaystyle {\frac {\binom {m+n}{m}}{{\binom {j+k}{j}}{\binom {m+n-j-k}{m-j}}}}}
The meaning of logic
TRUTH
0000 (0) FALSE
0110 (6) XOR
1001 (9) NXOR
1111 (F) TRUE
IDEMPOTENT
0001 (1) AND
0011 (3) A
0101 (5) B
0111 (7) OR
INJECTIVE
0100 (4) LT
1100 (C) NB
1101 (D) LTE
1110 (E) NAND
SURJECTIVE
0010 (2) GT
1000 (8) NOR
1011 (B) GTE
1010 (A) NA
Partial sums of binomials
∑
i
=
0
k
(
n
−
k
−
1
+
i
i
)
2
n
−
k
−
i
{\displaystyle \sum _{i=0}^{k}{\binom {n-k-1+i}{i}}2^{n-k-i}}
n
=
6
,
k
=
3
:
1
+
6
+
15
+
20
=
42
{\displaystyle n=6,k=3:1+6+15+20=42}
(
2
0
)
2
3
+
(
3
1
)
2
2
+
(
4
2
)
2
+
(
5
3
)
=
8
+
12
+
12
+
10
=
4
{\displaystyle {\binom {2}{0}}2^{3}+{\binom {3}{1}}2^{2}+{\binom {4}{2}}2+{\binom {5}{3}}=8+12+12+10=4}
2.
Trichotomy
If
x
≤
y
{\displaystyle x\leq y}
and
y
≤
x
{\displaystyle y\leq x}
then
x
=
y
{\displaystyle x=y}
.
Proof: We have
y
−
x
=
a
,
x
−
y
=
b
{\displaystyle y-x=a,x-y=b}
, where both
a
{\displaystyle a}
and
b
{\displaystyle b}
are non-negative.
So
0
=
a
+
b
{\displaystyle 0=a+b}
, and therefore
a
=
b
=
0
{\displaystyle a=b=0}
.
Pythagorean and 2 squares
Let
(
a
,
b
,
c
)
{\displaystyle (a,b,c)}
be a Pythagorean triple , with a even.
Then both
c
−
a
{\displaystyle c-a}
and
c
+
a
{\displaystyle c+a}
are square numbers with roots r and s.
Then
c
=
(
r
−
s
2
)
2
+
(
r
+
s
2
)
2
{\displaystyle c=\left({\frac {r-s}{2}}\right)^{2}+\left({\frac {r+s}{2}}\right)^{2}}
e.g. (48, 55, 73) ->73-48=25 (5) and 73+48=121 (11)
therefore 73=3^2+8^2
GF for Catalan numbers
Catalan number
Start with the GF for the central binomial coefficient .
1
1
−
4
x
=
∑
k
=
0
∞
(
2
k
k
)
x
k
{\displaystyle {\frac {1}{\sqrt {1-4x}}}=\sum _{k=0}^{\infty }{\binom {2k}{k}}x^{k}}
Integrating and setting the constant to
1
2
{\displaystyle {\frac {1}{2}}}
from the case
x
=
0
{\displaystyle x=0}
yields
1
−
1
−
4
x
2
=
∑
k
=
0
∞
1
k
+
1
(
2
k
k
)
x
k
+
1
{\displaystyle {\frac {1-{\sqrt {1-4x}}}{2}}=\sum _{k=0}^{\infty }{\frac {1}{k+1}}{\binom {2k}{k}}x^{k+1}}
Divide by x to get
1
−
1
−
4
x
2
x
=
∑
k
=
0
∞
C
k
x
k
{\displaystyle {\frac {1-{\sqrt {1-4x}}}{2x}}=\sum _{k=0}^{\infty }C_{k}x^{k}}
Primitive roots
Primitive root modulo n
The order of an element is the smallest k where
a
k
≡
1
(
mod
n
)
{\displaystyle a^{k}\equiv 1{\pmod {n}}}
.
k divides
ϕ
(
n
)
{\displaystyle \phi (n)}
. Testing a candidate against each possible k returns negative if and only if the candidate is a primitive root.
a/k
1
2
3
4
5
6
1
1
1
1
1
1
1
2
2
4
1
2
4
1
3
3
2
6
4
5
1
4
4
2
1
4
2
1
5
5
4
6
2
3
1
6
6
1
6
1
6
1
Fermat's theorem on sums of two squares
(5) can be proved by Euler's criterion . It also tells us that if
p
=
4
k
+
1
{\displaystyle p=4k+1}
,
p
|
∏
i
=
1
4
k
(
i
2
k
+
(
i
−
1
)
2
k
)
{\displaystyle p|\prod _{i=1}^{4k}\left(i^{2k}+(i-1)^{2k}\right)}
which is the sum of two squares by (1). (2), (3) and (4) can then be used to find it.
Finding primitive roots for primes
For each prime p of p-1, remove k^p from 1,..,p-1 as k runs over the same range. Only primitive roots remain.
Monty Hall problem
Monty Hall problem
If Monty reveals a goat *before* the contestant chooses a door, the odds are 50:50.
In the original version, once the host reveals a goat, the contestant knows with 100% certainty that the other card is opposite to the one they have.
Plane partition GF
Plane partition
The GF is
∏
i
=
1
∞
1
(
1
−
x
i
)
i
{\displaystyle \prod _{i=1}^{\infty }{\frac {1}{(1-x^{i})^{i}}}}
Consider instead the much simpler GF of
1
1
−
x
∏
i
=
2
∞
1
(
1
−
x
i
)
2
{\displaystyle {\frac {1}{1-x}}\prod _{i=2}^{\infty }{\frac {1}{(1-x^{i})^{2}}}}
The n -th term is the number of partitions of n into at least 0 parts + partitions of n-1 into at least 1 part + partitions of n-2 into at least 2 parts + etc...
Proof
The GF for at least k parts is
L
P
k
=
x
k
∏
i
=
1
k
1
(
1
−
x
i
)
∏
i
=
1
∞
1
(
1
−
x
i
)
{\displaystyle LP_{k}=x^{k}\prod _{i=1}^{k}{\frac {1}{(1-x^{i})}}\prod _{i=1}^{\infty }{\frac {1}{(1-x^{i})}}}
and
∑
k
=
0
∞
L
P
k
=
∏
i
=
1
∞
1
(
1
−
x
i
)
2
{\displaystyle \sum _{k=0}^{\infty }LP_{k}=\prod _{i=1}^{\infty }{\frac {1}{(1-x^{i})^{2}}}}
and we have counted the single x twice.
Coordinates of circumcentre
Let
△
A
B
C
{\displaystyle \triangle ABC}
be coordinates
(
x
1
,
y
1
)
,
(
x
2
,
y
2
)
,
(
x
3
,
y
3
)
{\displaystyle (x_{1},y_{1}),(x_{2},y_{2}),(x_{3},y_{3})}
. Then the midpoints of AB and AC are
(
x
1
+
x
2
2
,
y
1
+
y
2
2
)
{\displaystyle \left({\frac {x_{1}+x_{2}}{2}},{\frac {y_{1}+y_{2}}{2}}\right)}
(
x
1
+
x
3
2
,
y
1
+
y
3
2
)
{\displaystyle \left({\frac {x_{1}+x_{3}}{2}},{\frac {y_{1}+y_{3}}{2}}\right)}
The perpendiculars are
(
y
2
−
y
1
x
1
−
x
2
)
{\displaystyle {\binom {y_{2}-y_{1}}{x_{1}-x_{2}}}}
(
y
3
−
y
1
x
1
−
x
3
)
{\displaystyle {\binom {y_{3}-y_{1}}{x_{1}-x_{3}}}}
and the equations of the perpendiculars through the midpoints are
(
x
y
)
=
(
x
1
+
x
2
2
y
1
+
y
2
2
)
+
s
(
y
2
−
y
1
x
1
−
x
2
)
{\displaystyle {\binom {x}{y}}={\binom {\frac {x_{1}+x_{2}}{2}}{\frac {y_{1}+y_{2}}{2}}}+s{\binom {y_{2}-y_{1}}{x_{1}-x_{2}}}}
(
x
y
)
=
(
x
1
+
x
3
2
y
1
+
y
3
2
)
+
t
(
y
3
−
y
1
x
1
−
x
3
)
{\displaystyle {\binom {x}{y}}={\binom {\frac {x_{1}+x_{3}}{2}}{\frac {y_{1}+y_{3}}{2}}}+t{\binom {y_{3}-y_{1}}{x_{1}-x_{3}}}}
with equality when
x
1
+
x
2
2
+
s
(
y
2
−
y
1
)
=
x
1
+
x
3
2
+
t
(
y
3
−
y
1
)
{\displaystyle {\frac {x_{1}+x_{2}}{2}}+s(y_{2}-y_{1})={\frac {x_{1}+x_{3}}{2}}+t(y_{3}-y_{1})}
y
1
+
y
2
2
+
s
(
x
1
−
x
2
)
=
y
1
+
y
3
2
+
t
(
x
1
−
x
3
)
{\displaystyle {\frac {y_{1}+y_{2}}{2}}+s(x_{1}-x_{2})={\frac {y_{1}+y_{3}}{2}}+t(x_{1}-x_{3})}
so
(
x
2
−
x
3
)
+
2
s
(
y
2
−
y
1
)
=
2
t
(
y
3
−
y
1
)
{\displaystyle (x_{2}-x_{3})+2s(y_{2}-y_{1})=2t(y_{3}-y_{1})}
(
y
2
−
y
3
)
+
2
s
(
x
1
−
x
2
)
=
2
t
(
x
1
−
x
3
)
{\displaystyle (y_{2}-y_{3})+2s(x_{1}-x_{2})=2t(x_{1}-x_{3})}
and then
(
x
2
−
x
3
)
(
x
1
−
x
2
)
+
2
s
(
x
1
−
x
2
)
(
y
2
−
y
1
)
=
2
t
(
x
1
−
x
2
)
(
y
3
−
y
1
)
{\displaystyle (x_{2}-x_{3})(x_{1}-x_{2})+2s(x_{1}-x_{2})(y_{2}-y_{1})=2t(x_{1}-x_{2})(y_{3}-y_{1})}
(
y
2
−
y
3
)
(
y
2
−
y
1
)
+
2
s
(
x
1
−
x
2
)
(
y
2
−
y
1
)
=
2
t
(
x
1
−
x
3
)
(
y
2
−
y
1
)
{\displaystyle (y_{2}-y_{3})(y_{2}-y_{1})+2s(x_{1}-x_{2})(y_{2}-y_{1})=2t(x_{1}-x_{3})(y_{2}-y_{1})}
so
(
x
2
−
x
3
)
(
x
1
−
x
2
)
−
(
y
2
−
y
3
)
(
y
2
−
y
1
)
=
2
t
(
x
1
y
3
−
x
2
y
3
+
x
2
y
1
−
x
1
y
2
+
x
3
y
2
−
x
3
y
1
)
{\displaystyle (x_{2}-x_{3})(x_{1}-x_{2})-(y_{2}-y_{3})(y_{2}-y_{1})=2t(x_{1}y_{3}-x_{2}y_{3}+x_{2}y_{1}-x_{1}y_{2}+x_{3}y_{2}-x_{3}y_{1})}
zeta log functions
Namely,
∏
p
∈
p
r
i
m
e
1
−
log
(
1
−
1
p
s
)
{\displaystyle \prod _{p\in prime}1-\log \left(1-{\frac {1}{p^{s}}}\right)}
e
∑
p
∈
p
r
i
m
e
1
p
s
{\displaystyle e^{\sum \limits _{p\in prime}{\frac {1}{p^{s}}}}}
Subset complements
Let
B
i
{\displaystyle B_{i}}
be a collection of sets such that
B
i
⊂
A
{\displaystyle B_{i}\subset A}
for all
i
{\displaystyle i}
.
Let
C
=
A
∩
⋂
i
B
i
′
{\displaystyle C=A\cap \bigcap _{i}B_{i}'}
.
Then set
C
{\displaystyle C}
is a single component with a discontinuous boundary.
Every deflated sphere is a disc, a torus is a circle.
A sphere, like the torus, uses two generators , say f and g .
Torus, sphere
<
f
,
g
|
f
∞
=
g
∞
=
1
>
{\displaystyle <f,g|f^{\infty }=g^{\infty }=1>}
With the sphere, the polar axis of either rotation group, say F, has a pole in G for which F is a stabilizer . So
g
h
g
−
1
=
g
(
h
g
−
1
)
=
g
g
−
1
=
e
{\displaystyle ghg^{-1}=g(hg^{-1})=gg^{-1}=e}
.
Hamiltonian cycles
Hamiltonian cycles
Generate the permutations of all vertices. Remove all those with non-adjacent vertices.
Binary divide
Given n and d , left-shift d until it is greater than n , and then single right shift. Subtract this value from n and repeat until remainder is less than d .