<content> nullspace </content>

4e5021be-c744-4edc-b3ab-9cf88b05fae6 14 Alvaro Diaz Falconi 4 5 false false 0 decimal <content> nullspace </content>

SVD Decomposition using dn library A tol MSVD S U V' 13 mat A dn_LinAlgSVD transpose : U - S V' - transpose 13 mat eval 21 line :

Null space A tol null U S V 13 mat A tol MSVD : r S tol > vectorize sum eval : V 1 V rows range r 1 + (V cols range el 31 line :

A null A TOL null 11 line :

X# product P# 1 : x# X# P# x# P# * : for P# 31 line :

gcd of two numbers a b gcd x a abs : y b abs : 12 mat x 0 ≠ (y 0 ≠ (& x y > x x y mod : y y x mod : if while x y + 31 line :

gcd of an array. A gcd g A 1 el : k 2 A length range g g A k el gcd : for g 31 line :

lcm of two numbers a b lcm a b * a b gcd / 11 line :

lcm of an array A lcm l A 1 el : k 2 A length range l l A k el lcm : for l 31 line :

rat returns two integer matrices so that vectorize(N/D) is close to X in the sense that abs(N./D - X) <= tol*abs(X).The rational approximations are generated by truncating continued fraction expansions. X tol rat N X X - : D N : T tol X normi * : 13 mat j 1 X length range x X j el : k 0 : C 2 identity : 13 mat Iters 110000 range k k 1 + : d x trunc : x x d - : C C d 121 mat * C 1 col augment eval : C 11 el C 21 el / X j el - abs T < (x 0 ≡ (| break x 1 x / : if 51 line for N j el C 21 el sign C 11 el * : D j el C 21 el abs : 12 mat 31 line for N D 12 mat eval 31 line :

X rat X TOL rat 11 line :

Try to find integers values from a vector of rationals. A tol ratint X A abs vectorize : m X min eval : (tol > 0 m X max eval : k 1 X length range m X k el > (X k el tol > (& m X k el : continue if for 21 line if N D 12 mat A m / rat : D lcm N D / vectorize * eval 41 line :

A ratint A TOL ratint 11 line :

TOL 109 -^:

<content> Dimensional analysis </content>

Dim [T] [L] [M] [I] [Θ] [J] [N] s m kg A K cd mol 27 mat :

u# k# DimAux dummy# Clear ans# u# UnitsOf dummy# Dim 2 k# el : diff : s# 1 - (ans# ans# UnitsOf / 0 <^: ans# u# UnitsOf (s#^: 12 mat n# 199 range ans# ans# dummy# Dim 2 k# el : diff : (0 ≡ break continue if for s# n# 1 - (* 51 line :

x# Dim D# 1 : k# 1 Dim cols range D# D# Dim 1 k# el (x# k# DimAux^* : for D# 31 line :

v# x# Dim D# 1 : K# 1 : 12 mat λ# 1 x# length range k# 1 Dim cols range D# k# λ# el x# λ# el k# DimAux : for for D# D# null ratint : k# 1 v# length range K# K# v# k# el (D# k# el^* : for K K# ≡ 51 line :

5 J * Dim [M] [L] 2^* [T] 2^/

4 V * Dim [M] [L] 2^* [I] [T] 3^* /

6 W * Dim [M] [L] 2^* [T] 3^/

h Dim [M] [L] 2^* [T] /

Speed of free fall

A particle of mass mass m falling at speed v from h in gravity of strength g

eq h g v m 14 mat m g.e m s / kg 14 mat Dim : K v 2^h g * / ≡

with

K 2 ≡

Free of m

Period of an hamronic oscilator

Period T of a mass m attached to an ideal linear spring of constant k suspended in gravity of strength g

eq T m k g 14 mat s kg kg s 2^/ g.e 14 mat Dim : K m T 2^k * / ≡

with

K 2 π * (2^≡

Free of g

The Trinity Test

Geoffrey Taylor was able to determine how much energy was released for the first atomic bomb explosion just by looking at the photograph doing dimmensional analysis: the time after detonation t, the radius of the explosion R and the density of air ρ

eq t R ρ E 14 mat s m kg m 3^/ J 14 mat Dim : K t 2^E * R 5^ρ * / ≡

with

K something ≡

Black hole temperature

Black body radiation

K.1 K T k ħ R c 15 mat K k h 2 π * / m c 15 mat Dim at : ħ c * T k * R * /

Scape speed (Schwarzschild radius)

K.2 K G M R c 14 mat G.N kg m c 14 mat Dim at : R c 2^* G M * /

Joining

eq K K.1 K.2 * ≡ : K ħ c 3^* T k * G * M * / ≡

with

K 8 π * ≡

(Hawking)

Alvaro