Non-Euclidean geometry

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= £
{synonym}
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= Pound
{synonym}


Pound sterling

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{title2=$\Psi$}
{title2=$\psi$}
{wiki}

= Psi
{synonym}

As if it weren't enough, there's also a <Cyrillic script> psi that is slightly different. Life's great.


Psi (Greek)

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<Multivariate polynomial> where each term has degree 2, e.g.:
$$
f(x,y) = 2y^2 + 10yx + x^2
$$
is a quadratic form because each term has degree 2:
* $y^2$
* $xy$
* $x^2$
but e.g.:
$$
f(x,y) = 2y^2 + 10yx + x^3
$$
is not because the term $x^3$ has degree 3.

More generally for any number of variables it can be written as:
$$
f(x_1, x_2, \ldots, x_n) = \sum_{i,j} a_i a_j x_i x_j
$$

There is a <1-to-1> relationship between <quadratic forms> and <symmetric bilinear forms>. In matrix representation, this can be written as:
$$
\vec{x}^T B \vec{x}
$$
where $\vec{x}$ contains each of the variabes of the form, e.g. for 2 variables:
$$
\vec{x} = [x, y]
$$

Strictly speaking, the associated <bilinear form> would not need to be a <symmetric bilinear form>, at least for the <real numbers> or <complex numbers> which are <commutative>. E.g.:
$$
\begin{bmatrix}x y\end{bmatrix}
\begin{bmatrix}0 & 1 \\ 2 & 0 \\ \end{bmatrix}
\begin{bmatrix}x \\ y \\ \end{bmatrix}
=
\begin{bmatrix}x y\end{bmatrix}
\begin{bmatrix}y \\ 2x \\\end{bmatrix}
= xy + 2yx
= 3xy
$$
But that same matrix could also be written in symmetric form as:
$$
\begin{bmatrix}0 & 1.5 \\ 1.5 & 0 \\ \end{bmatrix}
$$
so why not I guess, its simpler/more restricted.


Quadratic form

IP stack fingerprinting

Quadratically closed field

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<Ciro Santilli> appreciates this concept of "remembering the suffering of others" a lot due to <Ciro Santilli's self perceived compassionate personality> and <Ciro Santilli's cheapness>.


Ramadan

Graphviz example

Lagrangian (field theory)

* https://en.wikipedia.org/wiki/List_of_small_groups


List of finite groups

List of image file formats

Poisson bracket

This is one of the first examples in most <quantum field theory>.

It usually does not involve any forces, just the interpretation of what the <quantum field> is.

https://www.youtube.com/watch?v=zv94slY6WqY&list=PLSpklniGdSfSsk7BSZjONcfhRGKNa2uou&index=2 Quantization Of A Free Real Scalar Field by Dietterich Labs (2019)


Ciro Santilli @cirosantilli 40