A partial wave is a component of a scattering state with definite orbital angular momentum. A central potential preserves each angular-momentum channel, so the scattering problem decomposes into independent partial waves.

For a localized target and a plane wave incident along the $z$-axis, the large-radius scattering wavefunction has the formwhere $f$ is the scattering amplitude.

In the Cox--Ross--Rubinstein model, a stock changes at each discrete time by an up factor $U$ or a down factor $D$, while the money-market account changes by a deterministic factor $R$. After discounting by the money-market account, the factors are $u=U/R$ and $d=D/R$. The model is arbitrage-free exactly when $D<R<U$, equivalently $d<1<u$.

For a nonnegative random variable $X$, its Laplace transform is

Wald's equation turns a random sum into a product of two expected values: the average contribution per step and the average number of steps. Its surprising feature is that the number of steps may depend on what has already happened. A procedure can stop on its first successful Bernoulli trial, for example, while the identity still accounts for its total cost. The crucial restriction is that the decision to take the next step cannot use that step's unobserved outcome.

Let $X_1,X_2,\dots$ be real independent and identically distributed random variables with $\mathbb{E}|X_1|<\infty$, let $\mu =\mathbb{E}[X_1]$, and put $S_0=0$ and $S_n={\sum }_{i=1}^n{X}_i$. Suppose $N$ is a nonnegative integer-valued stopping time for the natural filtration ${\mathcal{F}}_n=\sigma (X_1,\dots ,X_n)$, with $\mathbb{E}[N]<\infty$. Then $S_N$ is an integrable random variable andThese hypotheses allow statistical dependence between $N$ and the increments; they do not allow access to future increments. The finite-expected value requirement is stronger than stopping almost surely.^[ref]

For the proof idea, count step $i$ only on $\{N\ge i\}$. This event is determined by the first $i-1$ observations, so its indicator function is independent of $X_i$. The Proof of Wald's equation by stopping indicators combines this observation with the tail-sum formula for expectation. The same calculation with $|X_i|$ gives $\mathbb{E}{\sum }_{i=1}^N|X_i|=\mathbb{E}|X_1|\mathbb{E}[N]<\infty$, which justifies interchanging the expected value and the infinite series. This is why the result survives a data-dependent end point without assuming an independent count.

The identity is useful in randomized algorithms and repeated sampling: one can translate an average amount of work into an average number of attempts, or conversely, without first finding the full probability distribution of the total. The examples below separate a permissible stopping rule from a rule that peeks ahead and from a valid rule whose average waiting time is infinite.

For events $A$ and $B$, the Fréchet inequalities are

A finitely additive probability measure on a set $X$ is a function $m:\mathcal{P}(X)\to [0,1]$ such that $m(X)=1$ and $m(A\bigsqcup B)=m(A)+m(B)$ for disjoint subsets $A,B\subseteq X$. Unlike a probability measure, it need not be countably additive.

The probability of an event is its value under a probability measure, a real number between zero and one. It compares the mass assigned to that event with the unit mass of the whole sample space. For a finite sample space with equally likely outcomes, $\mathbb{P}(A)=|A|/|\Omega |$. For unequal outcome masses, add their individual probabilities instead.

For example, selecting a coin with head probability $1/2$ with chance $4/5$, or a coin with head probability $3/4$ with chance $1/5$, gives head probability $(4/5)(1/2)+(1/5)(3/4)=11/20$. This is the law of total probability: averaging over the hidden selection is part of calculating the outcome's probability.

A two-dimensional point source of strength $q$ has velocity potential $\varphi =q\log r/(2\pi )$ and outward velocity $q{\mathbf{e}}_r/(2\pi r)$. Every closed contour enclosing it has volume flux $q$.

A cap set is a subset $A\subseteq {\mathbb{F}}_3^n$ containing no three distinct points $x,y,z$ satisfying $x+y+z=0$. Equivalently, it contains no nonconstant three-term arithmetic progression.

Let $P\in {\mathbb{F}}_p[x_1,\dots ,x_n]$ have total degree of a polynomial less than $p$. If the associated polynomial function vanishes on all of ${\mathbb{F}}_p^n$, then $P$ is the zero polynomial. Induct on $n$: writeFor each fixed $(x_1,\dots ,x_{n-1})$, the resulting univariate polynomial of degree less than $p$ has all $p$ field elements as roots, so every coefficient $P_j$ vanishes everywhere. The induction hypothesis makes every $P_j$ the zero polynomial.

A discriminant is a polynomial expression in the coefficients that vanishes when the associated polynomial has a repeated root.

A zero of a function $f$ is an argument $a$ for which $f(a)=0$.

The real univariate polynomials form a vector space with the countable Hamel basisThe subspace of polynomials of degree at most $n$ has dimension $n+1$.

A polynomial function is the function obtained by evaluating a polynomial on a specified ring or field. Distinct polynomials can induce the same polynomial function over a finite field; for example, $x^p$ and $x$ agree on ${\mathbb{F}}_p$.

Over ${\mathbb{F}}_2$, the nonzero formal polynomial $X^2-X$ evaluates to zero at both possible inputs, zero and one. It therefore induces the same polynomial function as the zero polynomial, despite having different coefficients and polynomial degree. Over an infinite field, equality at every input forces equality of polynomials by the root bound for a polynomial. The distinction matters whenever a representation is supposed to identify the function uniquely.

A planar periodic orbit, oriented by the flow, has Poincaré index $+1$. The index theorem equates this with the sum of the indices of the isolated equilibria enclosed by the orbit. A hyperbolic equilibrium has index $+1$ when its Jacobian matrix has positive determinant and index $-1$ when it is a saddle equilibrium.

The Euclidean plane is ${\mathbb{R}}^2$ with its standard Euclidean norm and the resulting distance, angles, and Riemannian metric.

This reflection-symmetric planar system has equilibria $(0,\pm 1)$ and, for $|a|\le 1$, $(\pm \sqrt{1-a^2},a)$. Its divergence is the constant $-2a$, excluding periodic orbits when $a\ne 0$ by the Bendixson-Dulac criterion. At $a=0$ it is Hamiltonian withand the nonzero equilibria are nonlinear centers. At $a=\pm 1$ the symmetric equilibria undergo pitchfork bifurcations.

A second-order partial differential equation contains derivatives of the unknown function of order at most two and at least one second derivative. Its principal symbol determines its characteristic directions and elliptic, parabolic or hyperbolic type.

The parity of a two-particle centre-of-mass state with intrinsic parities ${\eta }_1,{\eta }_2$ and relative orbital angular momentum $\ell$ is