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On a one-dimensional uniform finite element mesh of spacing $h$, let $w$ be an affine function and ${\varphi }_i$ the interior piecewise-linear hat functions. The weighted mass matrix has entriesand zero entries for nonadjacent indices. The diagonal product ${\varphi }_i^2$ is symmetric about $x_i$ and integrates to $2h/3$; the neighbouring product is symmetric about $(x_i+x_{i+1})/2$ and integrates to $h/6$. The odd part of an affine function integrates to zero around each centre, giving the formulas. This evaluates a variable reaction term exactly without substituting a constant coefficient or a quadrature approximation.

If a martingale has increments bounded by a single deterministic constant, then $M_n/n\to 0$ almost surely. Its increments form a uniformly $L^2$-bounded martingale difference sequence; apply the strong law for uniformly L2-bounded uncorrelated random variables to their partial sums and note $M_0/n\to 0$. No square-integrability of the initial value is needed beyond the usual martingale integrability.

A time-indexed increment $D_j$ with $\mathbb{E}(D_j\mid {\mathcal{F}}_{j-1})=0$ is a martingale difference relative to its history. Square-integrable differences at different indices are orthogonal, allowing their conditional variances to accumulate. They need not be independent.

Let $V_t=\mathbb{E}[P_t{P}_t^{\top }\mid {\mathcal{F}}_{t-1}]$ be positive definite. Market completeness restricts each parent atom to at most $n$ successors, while positive definiteness forces exactly $n$ independent successor price vectors. The positive martingale deflator supplied by the fundamental theorem of asset pricing then has the unique one-step ratio $Z_t=P_t^{\top }{V}_t^{-1}{P}_{t-1}>0$. Its product deflates the prices into martingales.

On the symmetric group $S_n$, this Markov chain chooses uniformly from the identity and the $n-1$ adjacent transpositions, and multiplies on the right. Its stationary distribution is the uniform distribution on a finite set, because the generators are self-inverse. A fixed label's position is a lumped Markov chain on a path graph with off-diagonal transition probabilities $1/n$ to existing neighbours.

Choose two positions independently and uniformly from $\{1,\dots ,n\}$ and swap them. This Markov chain on the symmetric group holds with probability $1/n$ and assigns probability $2/n^2$ to each unordered transposition. Its stationary distribution is uniform, and its ordinary spectral gap is $2/n$ for $n\ge 2$. The latter follows from the Aldous spectral gap theorem: the single-label continuous-time Markov chain has rate $2/n^2$ between every pair of positions, whose mean-zero eigenvalues are all $-2/n$.

The Ehrenfest urn model moves one uniformly chosen ball between two boxes at each step. For $k\ge 1$ balls, the count in one box is a Markov chain on $\{0,\dots ,k\}$. Its stationary distribution is $2^{-k}\left(\genfrac{}{}{0ex}{}{k}{i}\right)$, and detailed balance makes it a reversible Markov chain. The chain has period two, but its mean recurrence time to the all-in-one-box state is still $2^k$.

A Harris recurrent chain visits every measurable set of positive irreducibility measure almost surely. Positive Harris recurrence additionally supplies an invariant probability measure. The ergodic theorem for a positive Harris recurrent Markov chain gives consistency of time averages of integrable functions.

A discrete-time model describes individual transitions between states at successive steps. State occupancy counts, first-entry counts and transition counts are distinct quantities, especially when a state has a self-loop. With independent individuals and fixed transition probabilities, destination counts conditional on the source count have a multinomial distribution.

A phi-irreducible Markov chain is Harris recurrent when every set of positive irreducibility measure is hit almost surely from every state. With an invariant probability, it becomes a positive Harris recurrent Markov chain.

A partition of a Markov chain state space is strongly lumpable when, for every target block $C$, the sum ${\sum }_{j\in C}{p}_{ij}$ is identical for all source states $i$ in a given source block. These sums define the transition matrix of the block-valued Markov chain, independently of the initial law. Failure of this condition means the projection is not guaranteed to be a homogeneous Markov chain for every initial law, although special initial laws may still produce a Markov process.

On the nonnegative integers, every positive state decreases by one, while zero jumps to $j\ge 1$ with probability $p{q}^{j-1}$, $p=1-q\in (0,1)$. Its return time from zero is one plus this geometric jump, so it has mean $(1+p)/p$. Return lengths two and three occur, establishing aperiodicity; deterministic downward motion and positive jumps establish irreducibility.

In a Directed acyclic graph, let $B$ contain the parents, children, and other parents of children of vertex $k$. Then $k$ is D-separated from all remaining vertices by $B$. Every path out of $k$ meets a conditioned noncollider either at a parent, at a child, or at another parent just after a conditioned child-collider. The global Markov property for a directed acyclic graph gives $Z_k\perp Z_{(B\cup \{k\}{)}^c}\mid Z_B$.

Under faithfulness of a directed acyclic graph, every conditioning set screening a vertex from all remaining vertices contains its graphical Markov blanket. A missing adjacent vertex leaves a one-edge active path; a missing other parent of a child leaves the active path $k\to h\leftarrow u$, since the child $h$ is necessarily conditioned. Thus the graphical blanket is the unique smallest screening set.

For a nondegenerate market return $R_M$, ${\beta }_i=\mathrm{Cov}(R_i,R_M)/\mathrm{Var}(R_M)$. It measures the asset's market-related exposure and enters the capital asset pricing model.

The expected excess return obeys $\mathbb{E}(R_i-r)={\beta }_i\mathbb{E}(R_M-r)$ in the mean-variance market model. The beta of an asset is a covariance-to-market-variance ratio.

A mapping class is an isotopy class of self-homeomorphisms of a surface. Orientation-preserving classes form the mapping class group. Acting on markings changes a point of Teichmüller space while preserving the underlying unmarked Riemann surface.

A representative scaled amplitude model has complex $A$ and real $B$ withPeriodic boundary conditions conserve $\int BdX$, because the right-hand side is a total second derivative. The slow field can change long-wave convection roll stability and permit localized pulses. For zero mean $B$, the uniform convection rolls are $A=R{e}^{iqX}$, $R^2=1-q^2$, $B=0$.

The magnetohydrodynamic total pressure is $\Pi =p+B^2/(2{\mu }_0)$, the sum of gas pressure and magnetic pressure. The remaining Lorentz force is magnetic tension, so this scalar does not include the entire magnetic stress.

Magnetic diffusivity is the coefficient of magnetic diffusion. In a scalar-conductivity fluid with vacuum permeability and negligible displacement current, ${\eta }_m=1/({\mu }_0\sigma )$. Its dimensions are length squared per time, and a diffusion time on scale $L$ is $L^2/{\eta }_m$.