If a real square matrix satisfies $M^2=-I$, split the absolutely convergent matrix exponential into even and odd powers to obtain $e^{tM}=I\cos t+M\sin t$. It is the fundamental matrix for ${\mathbf{x}}^{\prime }=M\mathbf{x}$. Consequently every solution has period $2\pi$, and the eigenvalues over the complex numbers belong to $\{i,-i\}$.

For inequivalent chosen unitary irreducible representations of a finite group, uniform expectation givesThe Kronecker deltas in the first case refer to entries in the same chosen basis. Averaging an intertwiner and applying the Schur lemma proves the formula; summing diagonal entries gives character orthogonality.

For real $m\ne 0$ and constant real $\alpha$, the Fourier symbol of $-\Delta -\alpha \cdot \nabla +m^2$ has modulus at least $|\xi {|}^2+m^2$. Its inverse therefore maps $L^2$ into $H^2$ with norm at most $\max (1,m^{-2})$, uniformly in the drift.

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.

Consider an arbitrage-free complete finite-state market with $n$ traded assets, strictly positive physical probabilities on all successor states, and conditional second-moment matrixpositive definite. Completeness permits at most $n$ successors from each parent atom, while positive definiteness forces at least $n$ independent successor price vectors. Thus there are exactly $n$.

The fundamental theorem of asset pricing supplies strictly positive one-step state prices. If their vector is $q$, their physical probabilities are $p_s$, and the independent successor price vectors are the columns of $C$, then $P_{t-1}=Cq$ and $V_t=C\mathrm{diag}(p)C^{\top }$. ConsequentlyThese unique ratios satisfy $\mathbb{E}[Z_t{P}_t\mid {\mathcal{F}}_{t-1}]=P_{t-1}$. Their successive product is a martingale deflator. The no-arbitrage hypothesis is essential: completeness and an invertible second-moment matrix alone do not make the regression expression positive.^[ref]

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.