The [[presentation of a group]] defined the generators and their identities.
The [[presentation of a group]] is defined by the generators of the groups involved and their identities.
The sphere group is obtained by tending ''k'' to infinity for two cyclic groups ''g'' and ''h'', order ''4k''.
The sphere group is obtained by tending ''k'' to infinity for two cyclic groups ''g'' and ''h'', order ''4k''.
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Balancing a disc
Balancing a circle
Let the triple 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 means that the sum of perpendiculars from P to is zero. Let be a distinct other axis with the same property.
Prove that all 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 weight after the first 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.
If is the original angle, and is the rotated amount, then we have:
and summing:
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 , . Then:
Consider , . Then:
So need to prove:
or
or, dividing by :
So:
Catalan q-polynomials
The Catalan q-polynomials count the number of blocks present in the diagram under the diagonal height k, and start
The sum of the coefficients of give .
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 , where 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.
For example, .
Cubic polynomial
Let
The local minima and maxima are given by the zeroes of the differential of .
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
and
and we have counted the single x twice.
Coordinates of circumcentre
Let be coordinates . Then the midpoints of AB and AC are
The perpendiculars are
and the equations of the perpendiculars through the midpoints are
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.
Base conversion
An integer in base b can be written as
and the same integer in base d is
If , then the first equation becomes
Expand this using the binomial formula to get
and collect by like terms in d to get the second equation.
Presentation of the sphere group
The presentation of a group is defined by the generators of the groups involved and their identities.
The sphere group is obtained by tending k to infinity for two cyclic groups g and h, order 4k.
There are two identities, the polar identity, from the fact that both poles are stable points, hence given , , and the equator identities, from the two routes from pole to pole, .