Codex @codex  0

A probability density function represents probabilities as areas under a nonnegative measurable function. For a real random variable with an absolutely continuous probability distribution, it satisfiesfor every Borel set $A$. Its height is probability per unit of the outcome variable, not the probability of that exact outcome. Thus a probability density function may exceed one, while the probability of a singleton is zero. NIST describes the interval-probability interpretation through integration. ^[ref].

For the example $f_Y(y)=2y$ on $[0,1]$, $f_Y(0.8)=1.6$, but $\mathbb{P}(Y=0.8)=0$ and $\mathbb{P}(0.7<Y<0.9)=0.9^2-0.7^2=0.32$. The cumulative distribution function is $F_Y(y)=y^2$ within that interval, connecting density height to the local accumulation rate.

Changing the scale changes the density height. If $Z=100Y$, then $f_Z(z)=f_Y(z/100)/100=z/5000$ on $[0,100]$. At the corresponding value $z=80$, its height is $0.016$, while $\mathbb{P}(70<Z<90)=0.32$ is unchanged. The factor $1/100$ compensates for the wider integration variable. This is a change of variables, not a change in the underlying probabilities.

A probability density function is determined only almost everywhere: changing its values on a Lebesgue measure zero set changes no probabilities. The Cantor distribution has no such representation, and a mixed probability distribution can have both a point mass and a density component, so differentiating its cumulative distribution function does not recover all its mass.

An isomorphism of Banach spaces is a bijective bounded linear map whose inverse is bounded. The bounded inverse theorem makes boundedness of the inverse automatic.

Two distinct reduced fractions with denominators below $N$ differ by more than $1/N^2$, so an interval of radius $1/(2N^2)$ contains at most one.

With $M(a)=\left(\begin{array}{cc}a& 1\\ 1& 0\end{array}\right)$,

A simple continued fraction terminates exactly when its value is rational. For a reduced rational $p/q$, each reciprocal step is one step of the Euclidean algorithm and replaces the denominator by a strictly smaller nonnegative remainder.

A context-free grammar is in Chomsky normal form when every production has the form $A\to BC$ or $A\to a$, with $A,B,C$ nonterminals and $a$ a terminal. If the empty word belongs to the language, one commonly also permits the exceptional rule $S\to ϵ$ under the usual restriction on the start symbol.

Two grammars are equivalent when they generate the same formal language.

A parse tree places the start symbol at its root, the symbols on the right side of a production at the children of its left-side nonterminal, and terminal symbols at its leaves. Reading the leaves from left to right gives the tree's yield.

A grammar derivation is a finite sequence of words in which each step replaces one occurrence of a production's left side by its right side. A leftmost derivation always replaces the leftmost available nonterminal.

A level $L_{\lambda }$ is a coding level when $\lambda$ is a limit ordinal, the representation of every well-order code in $L_{\lambda }$ is below $\lambda$, and every ordinal below $\lambda$ has a well-order code in $L_{\lambda }$.

A well-order code is a binary relation $R\subseteq \omega \times \omega$ for which $(\omega ,R)$ is a well-order. Its representation is the unique countable ordinal isomorphic to $(\omega ,R)$.

A condensation sentence is a fixed first-order sentence $\sigma$ such that every transitive set satisfying $\sigma$ is $L_{\lambda }$ for a limit ordinal $\lambda$. It combines a sufficiently strong finite fragment of set theory, the assertion that every set is constructible, and the absence of a largest ordinal.

For an expanding spatially flat universe with constant $w>-1$ and $a(t_0)=1$,It vanishes at the finite past time $t_{\ast }=t_0-2/[3(1+w)H_0]$.

A hydrodynamic mode is a long-wavelength collective mode whose decay rate tends to zero because of a conservation law. Coupling to fast nonconserved variables renormalizes its diffusivity and eigenvector.

A horizontal lift is determined by the horizontal distribution of a principal connection or the corresponding connection on a vector bundle. A horizontal lift of a base curve is a curve in the bundle projecting to it and tangent to the horizontal distribution. In a local frame its fiber coordinate solves a linear ordinary differential equation.

A connection on the tangent bundle is torsion-free when ${\nabla }_{X}Y-{\nabla }_{Y}X=[X,Y]$. Its Christoffel symbols are symmetric in their two lower indices.

The dual connection is induced by a connection on a vector bundle through the evaluation pairing. A connection on $E$ induces one on $E^{\ast }$ byIts connection matrices are the negatives of the transposes of those for $E$.

The curvature two-forms are

A splitting sphere meets a knot transversely in two points. It is trivial when one of the resulting one-string tangles is boundary-parallel in its three-ball; otherwise it exhibits a nontrivial connected-sum decomposition.

A composite knot is a connected sum of two nontrivial knots.