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For a positive semidefinite matrix, a trace bound is a bound on the sum of its nonnegative eigenvalues.

In a strictly stationary spacetime, the timelike Killing flow gives a preferred positive-frequency splitting. The corresponding vacuum is annihilated by every annihilation operator associated with a positive-frequency mode.

LetThenThus the one-soliton Schrodinger operator is paired with the free Schrodinger operator. Its normalizable state annihilated by $A$ is proportional to $\mathrm{sech}(\chi x)$.

In a neutrally stratified polytropic atmosphere, vertically trapped polynomial solutions of degree $n\ge 1$ describe acoustic pressure modes. The degree-zero member is the surface-gravity mode.

A surface gravito-inertial wave combines free-surface gravity restoration with Coriolis restoration. In a uniformly rotating deep atmosphere its incompressible dispersion relation can be written

A polytropic atmosphere is neutrally stratified when $1+1/m=\gamma$, so its equilibrium pressure-density relation matches the adiabatic relation and its buoyancy frequency vanishes.

A polynomial $f\in R[t]$ is a zero divisor exactly when some nonzero scalar $a\in R$ annihilates it. One proof chooses a nonzero polynomial $g$ of minimal degree with $fg=0$ and successively kills its leading coefficient.

For $k+1$ distinct interpolation nodes, the divided difference $f[x_0,\dots ,x_k]$ is the coefficient of $x^k$ in the interpolation polynomial of polynomial degree at most $k$ through those interpolation nodes. It is the leading coefficient when that polynomial degree is exactly $k$, and zero when it is smaller. Explicitly,starting with $f[x_i]=f(x_i)$. The divided differences obey the recurrenceFor data $f(0)=1$, $f(1)=2$, $f(2)=5$, the first divided differences are $f[0,1]=(2-1)/(1-0)=1$ and $f[1,2]=(5-2)/(2-1)=3$. The second divided difference is $f[0,1,2]=(3-1)/(2-0)=1$, the leading coefficient of $x^2+1$. The first divided difference is the slope of a secant line; higher divided differences supply successive coefficients for the Newton polynomial.

For a monic separable polynomial over a field of characteristic other than two, its Galois group is contained in the alternating group exactly when its discriminant is a square in the base field. The Galois action multiplies the Vandermonde product by the sign of the induced root permutation.

The depressed cubic $x^3+ax+b$ has discriminant $-4a^3-27b^2$.

The Vandermonde product ${\prod }_{i<j}(r_i-r_j)$ is alternating, and its square is the discriminant of the monic polynomial with roots $r_i$.

A primitive polynomial has unit content.

Gauge fixing worldsheet diffeomorphism and Weyl symmetry introduces an anticommuting vector ghost $c^{\alpha }$ and a symmetric traceless antighost $b_{\alpha \beta }$. In complex coordinates they form holomorphic and antiholomorphic $bc$ systems.

Conformal gauge uses worldsheet diffeomorphisms and Weyl transformations to set $g_{\alpha \beta }=e^{2\varphi }{\eta }_{\alpha \beta }$ locally.

A Weyl transformation locally rescales a metric tensor, $g\mapsto e^{2\omega }g$, while retaining the same points and coordinates. Applied to a worldsheet metric, it leaves the embedding $X^{\mu }$ unchanged, and the classical Polyakov action is invariant under these Weyl transformations. Applied to a spacetime metric, a Weyl rescaling can relate different frame conventions, such as a string metric and an Einstein metric; the accompanying field normalizations must be transformed as well. It is not itself a coordinate transformation.

A worldsheet diffeomorphism is a smooth reparameterization of the worldsheet coordinates under which $X^{\mu }$ is a scalar and $g_{\alpha \beta }$ is a rank-two tensor.

With $J_0=0$, the age $A(t)=t-J_{N_t}$ of a rate-$\lambda$ Poisson process has meanGiven $N_t=n$, the last arrival is the maximum of $n$ uniform points and $\mathbb{E}[A(t)\mid N_t=n]=t/(n+1)$, including $n=0$.

Let $J_i$ be the points of a rate-$\lambda$ Poisson process and let independent identically distributed marks $X_i$ be independent of the process. ThenCondition on $N_t$, use uniform order statistics, and sum the resulting exponential series.

For independent Poisson processes of rates ${\lambda }_1$ and ${\lambda }_2$, the number $K$ of type-two arrivals before the first type-one arrival satisfiesThis follows by marking the superposed process by its arrival type.

Conditioned on $N_t=n$, the arrival times are distributed as the order statistics of $n$ independent uniform points on $[0,t]$. Their conditional joint density onis $n!/t^n$, obtained by dividing the Poisson-process density ${\lambda }^n{e}^{-\lambda t}$ by $\mathbb{P}(N_t=n)$.