In mathematics, the Dehn–Sommerville equations are a complete set of linear relations between the numbers of faces of different dimension of a simplicial polytope. For polytopes of dimension 4 and 5, they were found by Max Dehn in 1905. Their general form was established by Duncan Sommerville in 1927. The Dehn–Sommerville equations can be restated as a symmetry condition for the h-vector of the simplicial polytope and this has become the standard formulation in recent combinatorics literature. By duality, analogous equations hold for simple polytopes.
== Statement ==
Let P be a d-dimensi...
My mentor tommy1729 told me :
Let $b_1 = 1.$
Let $ b_2 = x.$
Also $ \frac{1}{2} < x < 2 $
Define for $ n>2 $:
$$ b_n = \frac{\exp(\ln(b_{n-1})^3)}{b_{n-2}} $$
Then
$$ \sup_{n>2} b_n = x , \inf_{n>2} b_n = \frac{1}{x} $$
Is this true ?
How to show that ?
We can rewrite the sequence $b_n$ ...
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In mathematics, a hyperbolic angle is a geometric figure that defines a hyperbolic sector. The relationship of a hyperbolic angle to a hyperbola parallels the relationship of an "ordinary" angle to a circle.
The magnitude of the hyperbolic angle is the area of the corresponding sector of the hyperbola xy = 1. This hyperbola is rectangular with a semi-major axis of
2
{\displaystyle {\sqrt {2}}}
, analogous to the magnitude of a circular angle corresponding to the area of a circular sector in a circle with radius...
In relativity, rapidity is commonly used as a measure for relativistic velocity. Mathematically, rapidity can be defined as the hyperbolic angle that differentiates two frames of reference in relative motion, each frame being associated with distance and time coordinates.
For one-dimensional motion, rapidities are additive whereas velocities must be combined by Einstein's velocity-addition formula. For low speeds, rapidity and velocity are proportional, but for higher velocities, rapidity takes a larger value, the rapidity of light being infinite.
Using the inverse hyperbolic function artanh, the...
