A compound Poisson process with rate $\lambda$ and mark $Y$ is a martingale exactly when $\mathbb{E}|Y|<\infty$ and $\mathbb{E}Y=0$.

An Iwasawa algebra is a completed group algebra such as ${\mathbb{Z}}_p[[G]]$ or ${\mathbb{F}}_p[[G]]$ for a compact p-adic analytic group $G$.

A geodesic $\gamma :I\to M$ is maximal when it has no geodesic extension to a strictly larger interval. Every initial position and velocity determine a unique maximal geodesic through the geodesic equation.

If $f:[1,\infty )\to [0,\infty )$ is decreasing, then ${\sum }_{n\ge 1}f(n)$ and ${\int }_1^{\infty }f(x)dx$ either both converge or both diverge. This follows by bounding the area under the graph between adjacent upper and lower rectangle sums.

Two finite CW complexes can have isomorphic integral cohomology rings but different cohomology rings modulo $p$. For example, attaching a three-cell of degree $p$ to the projective line in ${\mathbb{CP}}^2$ preserves the nonzero square of the degree-two class modulo $p$, whereas the corresponding Moore space wedged with $S^4$ has zero square. The spaces therefore cannot be homotopy equivalent.

For positive $n$, let $x,y$ be the pullbacks of the orientation classes from the two factors. Then $x^2=y^2=0$, $xy$ generates degree $2n$, and $yx=(-1{)}^{n}xy$. When $n$ is even, every graded ring automorphism restricts on $H^n$ to a signed permutation matrix; when $n$ is odd the ring admits every matrix in $G{L}_2(\mathbb{Z})$.

The only regular functions defined on all of a nonempty projective space over a field $k$ are the constants:For ${\mathbb{P}}^1$, this also follows by gluing $k[t]$ and $k[t^{-1}]$ inside $k[t,t^{-1}]$: their intersection is $k$.

The CYK algorithm decides membership in a context-free language given by a grammar in Chomsky normal form. It records, for every substring, the nonterminals that derive that substring and combines adjacent substrings using binary productions. For a word of length $n$ and a fixed grammar, it runs in $O(n^3)$ time.

To convert a context-free grammar to Chomsky normal form, first add a fresh start symbol. Remove epsilon productions, retaining the exceptional start production only when the empty word is to remain in the language; then remove unit productions and symbols that cannot participate in a terminal derivation. Replace each terminal occurring in a right-hand side of length at least two by a new nonterminal that produces that terminal. Finally split every right-hand side of length greater than two into binary productions using fresh nonterminals.

If $K_X=(\omega )$ is represented by a nonzero rational differential, multiplication by $\omega$ identifies the Riemann-Roch space $L(K_X)$ with the vector space of holomorphic differential forms. The Riemann-Roch theorem gives $\ell (K_X)=g$ for a smooth projective curve of genus $g$.

If $t$ is a uniformizer at a point $p$ of a smooth projective curve and a nonzero rational differential is $\omega =f(t)dt$, then ${\nu }_p(\omega )={\nu }_p(f)$. This integer is independent of the chosen uniformizer. Its positive and negative values are respectively the orders of the zero and pole of $\omega$ at $p$.

A rational differential on a smooth projective curve is a rational section of its canonical bundle. In a local coordinate $t$ it has the form $f(t)dt$ for a rational function $f$.

The BRST current is the conserved worldsheet current whose contour integral is the BRST operator. For the bosonic string it is built from the reparameterization ghost and the matter and ghost stress tensors; its self-OPE is nonsingular only when the total central charge vanishes.

The transition density of standard Brownian motion in ${\mathbb{R}}^d$ is the heat kernelThus ${\mathbb{P}}_x(X_t\in A)={\int }_A{p}_t(x,y)dy$.

A face-centered cubic lattice has lattice points at every corner and face center of a conventional cube. One primitive cell has one quarter of the conventional cube's volume.

If $S_i$ are the simple modules of a split group algebra and $P_i$ are their projective covers, then

Two probability measures supported on one compact interval are equal if all their moments are equal. Indeed, equality on monomials gives equality on polynomials, the Weierstrass approximation theorem extends it to every continuous function on the interval, and compactly supported continuous functions determine a finite Borel measure.

For independent binomial counts $Y_i\sim \mathrm{Bin}(m_i,p_i)$ with observed proportions $y_i=Y_i/m_i$ and fitted probabilities ${\hat{p}}_i$, the deviance from the saturated model iswith the usual convention $0\log 0=0$.

If an event occurs with probability $p$ and is independently retained with conditional probability $q$, its retained indicator is Bernoulli with probability $pq$. Applying this independently to a binomial count turns $\mathrm{Binomial}(n,p)$ into $\mathrm{Binomial}(n,pq)$.

For reparameterization ghosts of weights two and minus one, the canonical anticommutators imply $b(z)c(w)\sim 1/(z-w)$. This operator-product expansion fixes the ghost contractions and the ghost stress tensor.