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For strictly positive finite holding rates $q_i$, a measure $\pi$ satisfies $\pi Q=0$ exactly when ${\mu }_i=q_i{\pi }_i$ is invariant for the jump chain. Thus ${\pi }_i={\mu }_i/q_i$ when normalization is possible.

On $\{0,1,2,\dots \}$, let the chain jump from every $i\ge 1$ to $i+1$ with probability $p\in (0,1)$ and to $i-1$ with probability $1-p$, and jump from zero to one. It is recurrent exactly when $p\le 1/2$; at $p=1/2$ it is null recurrent.

A nilpotent Jordan block has zeros on its diagonal and ones on its superdiagonal. Its $k$th power has ones on the $k$th superdiagonal, and a block of size $d$ satisfies $J_d(0{)}^d=0$.

For exobase number density $n_e$ and $v_{\mathrm{th}}=\sqrt{2k_{B}T/m}$, the collisionless Jeans number flux is

The Jacobian determinant is the determinant of the square Jacobian matrix. Its absolute value is the local volume-scaling factor in the change of variables formula.

The Jacobian matrix and determinant describe respectively the linear derivative and its signed volume factor.

If $A=D+L+L^T$ is symmetric positive definite and tridiagonal, set $S=\mathrm{diag}(1,-1,1,-1,\dots )$. Thenis positive definite. Applying the Householder-John theorem to $M=D$ and $N=-(L+L^T)$ proves that the Jacobi iteration converges.

For Jacobi coordinates $\mathbf{a},\mathbf{b},\mathbf{c}$ and their corresponding reduced masses $\alpha ,\beta ,\gamma$, total angular momentum is

For $p=c\log n/n$, the number $N$ of isolated vertices satisfiesThe upper side follows from the first-moment bound. On the lower side, $\mathbb{E}N\to \infty$ and $\mathrm{var}(N)/(\mathbb{E}N{)}^2\to 0$, so the second-moment method applies.

An entropy solution of a scalar conservation law is a weak solution satisfying additional inequalities that select the admissible continuation through shock waves. For a bounded solution of $u_t+f(u{)}_x=0$ on the real line with $f\in C^1$, require every Kruzhkov entropy inequalityin the sense of distributions, together with the initial condition as a local $L^1$ trace. Equivalently, the interior integral of $|u-k|{\phi }_t+q_k(u){\phi }_x$ is nonnegative for every nonnegative compactly supported test function away from $t=0$.

For a smooth convex function $\eta$, the corresponding entropy flux for a scalar conservation law satisfies $q^{\prime }={\eta }^{\prime }{f}^{\prime }$. A viscous scalar conservation law yields ${\partial }_t\eta (u)+{\partial }_{x}q(u)=\epsilon {\partial }_{xx}\eta (u)-\epsilon {\eta }^{\prime \prime }(u)u_x^2$; the final term is nonpositive. Suitable bounded strongly convergent vanishing viscosity approximations therefore motivate the inequalities. Convexity of the flux itself is unnecessary for the local L1 contraction for scalar conservation laws and the resulting uniqueness; for strictly convex fluxes, the inequalities exclude expansion jumps and select compressive entropy shocks.

Maps from a module $X$ to ${\underleftarrow{\lim }}_i{N}_i$ correspond naturally to compatible families of maps $X\to N_i$.

If $A$ is real symmetric, $s$ is not an eigenvalue, one eigenvalue ${\lambda }_{\ast }$ is uniquely closest to $s$, and the initial vector has a nonzero component in its eigenspace, fixed-shift inverse iteration converges to that eigenspace. Its asymptotic direction-error ratio iswhere ${\lambda }_{\mathrm{next}}$ is second closest to $s$.

For a morphism of ringed spaces, the pullback of an ${\mathcal{O}}_Y$-module is

The inverse image functor supplies the sheaf $f^{-1}\mathcal{E}$; tensoring with the source structure sheaf supplies the module scalar extension.

For an explosive chain, a normalized solution of the formal balance equations $\pi Q=0$ need not be an invariant probability for the minimal transition semigroup. Explosion sends Probability mass to a cemetery state, and the generator equations on the original states do not include that boundary loss.

A concrete example uses the reflected jump chain on $\{0,1,2,\dots \}$ which moves from zero to one, and from $i\ge 1$ moves up with probability $2/3$ and down with probability $1/3$. Give state $i$ holding rate $q_i=4^i$. The embedded chain has invariant weights ${\mu }_0=1$, ${\mu }_i=32^{i-1}$ for $i\ge 1$. Dividing by the holding rates and normalizing yieldsThese weights even satisfy formal Detailed balance: ${\pi }_0{q}_{01}={\pi }_1{q}_{10}=2/5$, and for $i\ge 1$ both opposing edge flows equal $(2/5)2^i$. Thus $\pi Q=0$ holds at each state.

Nevertheless the process explodes. Starting from zero, the embedded walk has expected visit count two at zero and three at every positive level. The zero-state return probability is $1/2$; at a positive state it is $2/3$, giving those geometric visit means. Therefore the expected total elapsed holding time isThe lifetime is finite almost surely. The same bounded expected-visit argument works from every initial state, so the killed minimal semigroup loses all its mass as time tends to infinity. The displayed $\pi$ is consequently a formal generator-balance solution, not a stationary probability of that killed evolution. A conservative continuation after explosion would require an extra boundary rule.

A terminal boundedness requirement can select the free constant of an unstable first-order linear differential equation. Suppose $y^{\prime }-a(t)y=f(t)$ on $[0,\infty )$, with continuous $a(t)\ge a_0>0$ and bounded continuous $f$. The unique bounded solution isThe bound $|y(t)|\le \Vert f{\Vert }_{\infty }/a_0$ proves convergence and boundedness. Differentiation verifies $y^{\prime }=a(t)y+f(t)$. Any difference of two solutions is $C\exp ({\int }_0^{t}a)$ and is unbounded unless $C=0$.

For $y^{\prime }-y=1$, boundedness selects $y=-1$ from the family $-1+C{e}^t$. The tail integral accounts for cancellation of the growing homogeneous component; an arbitrary finite-time initial condition generally fails to produce that cancellation. If the forcing is unbounded or the lower growth bound is removed, existence of a bounded solution needs separate hypotheses.

A matrix is totally unimodular when every square subdeterminant is $0$, $1$, or $-1$; linear programs with such a constraint matrix and integer right side have integral vertices.

If a graded ring $A$ is a graded extension of a graded subring $R$, then the integral closure of $R$ in $A$ is graded: every homogeneous component of an integral element is integral. This follows by separating the extremal degrees in a monic integral equation and inducting on the number of nonzero homogeneous components.

For $a=-1/(H_{\inf }\tau )$, the rescaled modesatisfies

The Nambu–Goto actionis minus the string tension times the Lorentzian area of the worldsheet.

For two spin-one particles, the spin tensor product has a six-dimensional symmetric subspace and a three-dimensional antisymmetric subspace. A symmetric spatial state combines with symmetric spin, while an antisymmetric spatial state combines with antisymmetric spin, so that the total bosonic state remains symmetric.

A fermion has an antisymmetric complete multiparticle state under exchange of identical particles. Exchanging two full single-particle labels changes the amplitude's sign. Under the usual relativistic locality and positivity assumptions in three spatial dimensions, the Spin-statistics theorem relates this behavior to half-integer spin.

If two identical fermions occupy the same complete single-particle state, antisymmetrization gives zero. This is the Pauli exclusion principle, which restricts full states rather than merely positions; different spin states can share a spatial orbital. Electrons in atoms and a degenerate gas illustrate its physical consequences. Filling available modes produces a Fermi energy and strong occupation effects even without a classical repulsive force between all pairs. Interactions and finite temperature modify distributions without removing exchange antisymmetry.