Conditional on a common latent variable $\Theta$, annual averages $X_j$ are independent, have common conditional expectation $\mu (\Theta )$, and have conditional variance ${\sigma }^2(\Theta )/m_j$ for known positive exposures $m_j$. Its structural parameters are $m_0=\mathbb{E}\mu (\Theta )$, expected process variance $v=\mathbb{E}{\sigma }^2(\Theta )$ and variance of hypothetical means $a=\mathrm{Var}(\mu (\Theta ))$. It generalizes the equal-exposure Bühlmann model.

If $u$ is a bounded harmonic function on a domain $D$, continuous on its closure, and the Brownian exit time $T_D$ is almost surely finite from $z$, then $u(z)={\mathbb{E}}_z[u(B_{T_D})]$. The Itô formula and compact localization make $u(B_{t\wedge T_D})$ a bounded martingale. bounded convergence theorem at the exit time proves the representation even if the domain is unbounded. Removing boundedness of $u$ can invalidate this conclusion through loss of uniform integrability.

For a bounded domain whose boundary points are all regular boundary points, continuous Dirichlet boundary data $f$ have a unique solutionHere ${\tau }_D$ is the Brownian exit time. The Strong Markov property gives the mean value property for harmonic functions; a barrier for the Dirichlet problem gives continuity at the boundary. This is an existence theorem as well as the Brownian representation of the Dirichlet problem for an already known solution.

Independent runs of a BQP algorithm with fresh ancilla qubits can be combined by a classical threshold rule. If completeness exceeds soundness by $\gamma$, a Hoeffding inequality bounds the error after $r$ runs by $e^{-r{\gamma }^2/2}$. Polynomial repetition handles inverse-polynomial gaps. This does not automatically preserve the restricted gate and measurement rules of a stoquastic circuit.

For a real-valued function $H$ on a nonempty finite Cartesian product, let $T_{H}v(x)={\mathbb{E}}_{y}H(x,y)v(y)$ with uniform inner products. The box norm obeys $\Vert H{\Vert }_{\square }^4=\mathrm{Tr}((T_H{T}_H^{\ast }{)}^2)={\sum }_j{\sigma }_j^4$, where ${\sigma }_j$ are its singular values. For a biregular graph this separates the constant component ${\gamma }^4$ from the other fourth powers.

With mechanically and thermally equivalent upper and lower boundaries, the Boussinesq approximation permits reflection through the layer midplane together with reversal of the temperature perturbation and vertical velocity. In a reduced convection temperature equation this induces temperature inversion. A quadratic term which changes differently from the linear terms breaks this symmetry; a cubic term can preserve it. The Boussinesq approximation alone does not impose symmetry on unequal boundary conditions.

The number and energy integrals of a zero-chemical-potential photon gas give $n=2\zeta (3)T^3/{\pi }^2$ and $\rho ={\pi }^2{T}^4/15$ in natural units. Their ratio is the mean energy per photon. It is useful for converting a relic number-density ratio into an energy-density ratio.

If a Borel summation contour meets a simple pole, an upper or lower indentation fixes the exponentially small contribution. For the transform $(1-t^2{)}^{-1}$ on the positive ray, the relevant pole is $t=1$. A lower indentation gives the Cauchy principal value minus $i\pi e^{-z}/2$. Finite geometric expansion gives an exact remainder integral, which enables optimal truncation and error-function smoothing of a Stokes multiplier.

For arbitrary events $A_n$ in one probability space,Equivalently, only finitely many events occur almost surely; no independence is needed.^[ref]

Proof outline: put $B_m={\bigcup }_{n\ge m}{A}_n$. The union bound givesThe tail of a convergent nonnegative series tends to zero.

For the Boolean hypercube, the initial segment of the simplicial order on the discrete cube minimizes the size of the closed vertex neighbourhood among families of a given size. Equivalently, it minimizes the external vertex boundary. This is a vertex assertion, distinct from the edge-isoperimetric inequality in the discrete cube.

Order subsets first by increasing cardinality and, within each level, by lexicographic order, with the smallest differing coordinate belonging to the earlier set. This convention makes simplicial initial segments the extremizers in Harper theorem.

A face is obtained by fixing some coordinates of the Boolean hypercube and letting the remaining $d$ coordinates vary freely. It is isomorphic to $Q_d$. A two-dimensional face is a square with four vertices, and a face is contained in a set family when all its vertices belong to that family.

Let $E_q={\max }_{(a,q)=1}|\pi (x;q,a)-\mathrm{Li}(x)/\varphi (q)|$. For $Q\le x^{1-\eta }$, the Brun–Titchmarsh theorem and Cauchy-Schwarz inequality giveWrite each summand as $\sqrt{E_q}{3}^{\omega (q)}\sqrt{E_q}$, then use $E_q{\ll }_{\eta }x/(\varphi (q)\log x)$.

The Bloch sphere is the unit sphere of Bloch vectors, corresponding to pure qubit states through $\rho =(I+\mathbf{r}\cdot \mathbf{\sigma })/2$. A local unitary change of qubit basis induces a rotation of this sphere, with its common phase physically irrelevant. Mixed states occupy the interior of the unit ball.

A biregular graph is a bipartite graph whose vertices have constant degree on each part separately. The two degree constants can differ. If its parts are $X,Y$ and its edge density of a bipartite graph is $\gamma$, the two degrees are $\gamma |Y|$ and $\gamma |X|$. Thus its adjacency function has constant row and column averages $\gamma$.

Equip the function spaces of the two nonempty parts of a bipartite graph with uniform inner products. Its normalized adjacency operator is $({\theta }_{G}v)(x)={\mathbb{E}}_{y}G(x,y)v(y)$. In orthonormal coordinate bases, its matrix is $G/\sqrt{|X||Y|}$, not the unscaled adjacency matrix. For a biregular graph of edge density of a bipartite graph $\gamma$, constant functions give a largest singular value $\gamma$, and the remaining singular values belong to the centered kernel $G-\gamma$.

For a bipartite graph with ordered parts $X,Y$, its biadjacency matrix has rows indexed by $X$ and columns indexed by $Y$, with entry one for an edge and zero otherwise. It has size $|X|$ by $|Y|$, rather than the square full bipartite adjacency matrix on $X\bigsqcup Y$, which is the block matrix $\left(\begin{array}{cc}0& M\\ M^{\mathsf{T}}& 0\end{array}\right)$. Its singular values describe the normalized adjacency operator of a bipartite graph after division by $\sqrt{|X||Y|}$.

For nonnegative integers $r,n$, the alternating binomial moment satisfies $S_r(n)=0$ when $r<n$ and $S_n(n)=(-1{)}^{n}n!$. Expand the polynomial $X^r$ in falling factorials and differentiate the binomial theorem to prove these identities. Equivalently, alternating binomial coefficients implement a finite difference that annihilates polynomials of degree less than $n$.

For the binomial random graph $G(n,1/2)$, the chromatic number is $(1+o(1))n/(2{\log }_{2}n)$ with high probability. A first moment method bounds the independence number above. Edge-disjoint clique packing in the complement graph, followed by an edge-exposure martingale and a union bound over moderately large vertex sets, supplies independent sets for greedy colouring by removing independent sets.

A coeval binary has components formed at the same time. A population of such binary stars can still have a distribution of system ages. If component masses have independence and the state at age $t$ is determined separately for each component, the states have conditional independence given $t$. If $g(t)$ and $w(t)$ are the conditional probabilities of two states, the unconditional unordered pair fractions are $\mathbb{E}[g(t{)}^2]$ and $2\mathbb{E}[g(t)w(t)]$, not generally products of marginal state fractions.