For jump rates $q_{ij}=r_j$ when $i\ne j$, assume $r_j\ge 0$ and $0<R={\sum }_j{r}_j<\infty$. The positive-rate states form the unique Closed communicating class; every zero-rate state is an open singleton. The stationary law is ${\pi }_i=r_i/R$, with Detailed balance ${\pi }_i{q}_{ij}=r_i{r}_j/R$. When at least two rates are positive, the embedded jump-chain law is proportional to $r_i(R-r_i)$. With only one positive rate, that state is absorbing and the genuine sequence of jump times stops there; an extended discrete chain can use an absorbing self-loop.

A Markov jump process is a nonexplosive, piecewise-constant continuous-time Markov process. Transition rates $q(x,y)$ give exponential holding time of rate ${\sum }_{y\ne x}q(x,y)$ and next-state probabilities proportional to these rates. Being Markov relative to a specified filtration requires the conditional future law given that filtration to depend only on the present state.

// Target: probability-and-statistics.bigb

A continuous-time Markov chain is nonexplosive if it makes only finitely many jumps on every bounded time interval. A uniformly bounded total jump intensity permits domination by a Poisson process. More generally, if a population process stays in a finite random set of states on each finite interval and every individual state has finite exit rate, localizing to fixed finite sets gives the same conclusion. In an open migration process, external Poisson immigration bounds the total population while internal moves preserve it and exits reduce it, giving this localization.

A linear functional on a normed vector space is continuous exactly when its kernel is closed. The forward direction follows because zero is closed in the scalar field. Conversely, if $l\ne 0$, choose $v$ with $l(v)=1$ and write $M=\ker l$. Closedness gives $d=\mathrm{dist}(v,M)>0$. Since $x/l(x)-v\in M$ when $l(x)\ne 0$, $|l(x)|\le \Vert x\Vert /d$. The zero functional is continuous separately. Thus the criterion does not require completeness or the closed graph theorem.

For bounded self-adjoint $a_i,a$, no uniform bound on $\Vert a_i\Vert$ is needed. The resolvent identity gives $(a_i-i{)}^{-1}-(a-i{)}^{-1}=(a_i-i{)}^{-1}(a-a_i)(a-i{)}^{-1}$, and the first factor has norm at most one. Thus both resolvents at $i$ and $-i$ converge strongly. Products preserve strong convergence for uniformly bounded nets; uniform approximation by the algebra of these resolvent functions then gives the displayed statement for every continuous function vanishing at infinity.

For a Positive element of a C-star algebra $x$, the displayed improper integral converges in norm. Near zero its integrand has norm at most $t^{-1/2}$; near infinity it has norm at most $\Vert x\Vert t^{-3/2}$. The scalar identity follows by setting $t=s{u}^2$ for $s>0$ and using ${\int }_0^{\infty }2/(1+u^2)du=\pi$; for $s=0$ both sides vanish. The bounds are uniform on the compact spectrum of an element, so continuous functional calculus gives the operator identity.

For a normal C-star element $x$ in a nonunital algebra, take its spectrum of an element in the C-star unitization. Zero belongs to that spectrum. Continuous functions on this compact spectrum that vanish at zero form an ideal canonically identified with the displayed function algebra. Their values under the unital continuous functional calculus belong to the original algebra because the scalar quotient evaluates them at zero. Star-polynomial approximation with the constant term removed gives exactly the image $C^{\ast }(x)$.

The Takagi function is a continuous nowhere differentiable function. The triangular summands are continuous and the geometric tail is bounded uniformly. On dyadic intervals of length $2^{-N}$, all terms of index at least $N$ vanish at both endpoints, while every lower summand has slope $+1$ or $-1$. Each refinement therefore changes the secant slope by exactly one in absolute value. The secants straddling a differentiability point lemma rules out a derivative at every real point.

For $f:D\to \mathbb{R}$ and $a\in D$, continuity at $a$ is equivalent to the following condition: every sequence $(x_n)$ in $D$ with $x_n\to a$ satisfies $f(x_n)\to f(a)$. This is useful when a numerical method or a proof approaches a point through successively refined function inputs: applying a continuous function preserves the intended limit.

For the forward direction, choose $\delta$ from the epsilon-delta definition of continuity. A convergent sequence eventually lies in that input window, so its function outputs eventually lie in the requested output band. Conversely, if continuity fails, there is some ${\epsilon }_0>0$ for which every input window contains an allowed $x$ with $|f(x)-f(a)|\ge {\epsilon }_0$. Choosing such an $x_n$ within $1/n$ of $a$ gives $x_n\to a$ while $f(x_n)$ does not tend to $f(a)$. This converse relies on the countable shrinking windows available in metric spaces; a general topological space needs a different formulation.

A function $f:D\to \mathbb{R}$ is continuous at $a\in D$ whenFirst choose the required output tolerance $\epsilon$; then choose an input tolerance $\delta$ that works for every allowed function input in that window. Picking a different $\delta$ separately after seeing each $x$ would reverse the quantifiers and lose this guarantee. There is no need to exclude $x=a$, because the output difference is then zero.

At an isolated point of the domain of a function, choose a small enough window that its only allowed function input is $a$. This proves continuity there directly. At a point with arbitrarily close other inputs, the same definition is the limit of a function condition $f(x)\to f(a)$. For metric spaces, replace the two absolute values by their distance functions; for arbitrary topological spaces, the corresponding definition is a continuous map.

The input tolerance may depend on $a$. A single choice valid everywhere on the domain of a function is the stronger requirement of uniform continuity. The continuous function $x\mapsto x^2$ on $\mathbb{R}$ shows the difference: for any proposed $\delta >0$, the points $a=1/\delta$ and $b=a+\delta /2$ have $|b-a|<\delta$, yet $b^2-a^2=1+{\delta }^2/4>1$. Thus the fixed output tolerance $\epsilon =1$ admits no global input tolerance on the real line.

The earliest approach within distance $d$ is a convex optimization problem with a norm constraint and the same segment and time bounds as contact. An elementary alternative enumerates whether each segment parameter is free or at an endpoint. In each nonsingular active case the free closest-point parameters are affine functions of time, leaving a quadratic distance inequality on a valid time interval. Rank-deficient cases need their null-space constraints retained. Unlike contact, the two segment parameters and time are no longer determined by a vector equality.

With recovery rate one, the number of infected sites in a nearest-neighbour contact process on ${\mathbb{Z}}^d$ has death rate equal to its size and birth rate at most $2d\lambda$ times its size. It is dominated by a linear birth-death branching process with these per-particle rates. In particular, from one infection its expected size is at most $e^{(2d\lambda -1)t}$, so its Survival probability is zero when $2d\lambda <1$. This proves the displayed lower bound on the critical rate.

For the Contact process on a countable undirected bounded-degree graph with recovery rate one and symmetric edge infection rate $\lambda$, the all-healthy law $\underline{\nu }={\delta }_0$ is invariant. The graphical representation preserves coordinatewise order. Since every configuration is below the all-infected configuration, the laws ${\delta }_1{P}_t$ decrease in stochastic domination as $t$ grows. Their weak limit $\overline{\nu }$ exists and is invariant by the Feller property.

Every invariant probability law lies between $\underline{\nu }$ and $\overline{\nu }$ in stochastic order. These are the least and greatest invariant laws; classifying all extremal invariant laws requires further hypotheses. The greatest law is also an extreme point: in a convex decomposition into invariant laws, both components are dominated by it, and equality of all increasing finite-cylinder expectations forces both to equal it.

Self-duality identifies the one-site density with survival from that site:On a transitive graph the right side is independent of $x$ and is conventionally written $\theta (\lambda )$. On a general graph its site dependence must be retained. The two bounding invariant laws coincide exactly when ${\theta }_x(\lambda )=0$ for every site: countability then forces the upper law to assign probability one to the empty configuration. The lattice version of the maximal-law and duality construction is given by the two basic contact-process theorems in Rick Durrett's survey.^[ref]

Consider the Contact process started from a finite infected set $A$ on a degree-$(d+1)$ regular tree, with $d\ge 2$, recovery rate one and per-oriented-edge infection rate $\lambda >0$. Choose $0<\rho <1$ satisfying $\rho \lambda (d-1)\ge 1$. The bounded functional $f(A)={\rho }^{|A|}$ has generatorRecoveries decrease the infected count by one, whereas infections across the edge boundary increase it by one; summing these changes gives the formula.

A finite nonempty set induces a forest. If it has $c(A)$ connected components, edge counting gives $|{\partial }_{E}A|=(d-1)|A|+2c(A)$. Therefore the stated parameter condition makes $Lf\le 0$. At the absorbing empty state its generator is zero. Localizing the finite-particle process and using boundedness yields that ${\rho }^{|A_t|}$ is a supermartingale.

On extinction by time $t$, its value is one, so ${\mathbb{P}}_A(T_{\varnothing }\le t)\le {\mathbb{E}}_A{\rho }^{|A_t|}\le {\rho }^{|A|}$. Letting $t\to \infty$ proves the extinction bound ${\mathbb{P}}_A(T_{\varnothing }<\infty )\le {\rho }^{|A|}$. A strict choice $\rho <1$ exists when $\lambda (d-1)>1$, giving a sufficient survival condition from every nonempty finite seed. The range of $\rho$ is essential to both boundedness and the sign argument.

The obstruction to proving the consistency of nonconstructibility by a uniform inner model construction is that, inside a universe satisfying $V=L$, every transitive inner model containing all ordinals has $L$ as its own constructible universe and therefore is the entire ambient universe. The absoluteness of constructible levels proves this by induction over successor definability and limit unions. Thus a construction valid in every model of ZF cannot always produce such an inner model satisfying $V\ne L$, and in particular cannot always produce one violating axiom of choice.

For $N\ge 3$, a regular polygon with $N$ sides is a constructible regular polygon exactly when $N=2^a{p}_1\cdots p_r$, where $a$ is a nonnegative integer and the $p_j$ are distinct Fermat primes; the product may be empty. Thus seventeen and fifteen qualify, while nine does not. Carl Friedrich Gauss established sufficiency, and Pierre Wantzel proved necessity in 1837.^[ref]

A thin isotropic hyperelastic material subject to incompressibility, with equal in-plane principal stretches $\lambda$ has transverse stretch ${\lambda }^{-2}$. With unloaded transverse faces, $N_{33}=W_3+q{\lambda }^2=0$. Hence $N_{11}=N_{22}=W_1-{\lambda }^{-3}{W}_3$, using isotropy to set $W_1=W_2$. The derivative along the constrained path is $W_1+W_2-2{\lambda }^{-3}{W}_3$, proving the displayed formula. The two equal nominal tensions do work $2Nd\lambda$ per reference volume, exactly the change in strain energy density.

In a finite box, retain the constant mode $M$ while integrating all nonzero field modes using the Wilsonian coarse-grained statistical Hamiltonian. Below the smallest nonzero momentum, the remaining effective energy is $V{U}_{\Lambda }(M)$ and its Boltzmann weight is precisely the constrained order-parameter free energy weight. Thus the displayed equality holds with consistent field-independent normalization. It is not an identity between a bare statistical Hamiltonian and an unconstrained equilibrium free energy. An analytic polynomial Landau free energy further assumes a homogeneous branch and regular effective coefficients; long-distance fluctuations near and below an upper critical dimension can invalidate that analytic approximation.

Draw one random variable $R$, taking values zero and two with equal probabilities. Conditional on $R$, generate a Poisson point process on the real line with constant rate $R$. This is a Cox process directed by the random measure $Rdx$: the same rate is shared by every window. It is a locally finite simple point process and a stationary point process. The law of total expectation gives $\mu (A)=\mathbb{E}[R]|A|=|A|$; the law of total variance and law of total covariance give, for bounded Borel sets $A,B$ with $A\cap B=\varnothing$,The conditional covariance vanishes by independence, but the shared conditional means fluctuate together. This distinguishes conditional from unconditional independence.^[ref]

For a window of length $L$, the exact probability mass function isIn particular, its void probability functional is $(1+e^{-2L})/2$, compared with $e^{-L}$ for the unit-rate Poisson point process. Whole realizations can be empty or comparatively crowded, while their mean density remains one. This example demonstrates shared-rate statistical dependence, rather than a model of short-range offspring clusters.

Let $U$ have the uniform distribution on $[0,1)$ and place a point at every $k+U$, $k\in \mathbb{Z}$. This is a simple point process, locally finite, and a stationary point process: a translation only changes the phase modulo one, preserving its uniform distribution. For every Borel set $A$, the Tonelli theorem givesIts consecutive points are exactly one unit apart, despite having the same intensity measure as a unit-rate Poisson point process. This is the one-dimensional translated-grid construction.^[ref]

Every half-open interval of length one contains exactly one point. For $A=[0,1/2)$ and $B=[1/2,1)$, $\Pi (A)+\Pi (B)=1$, each count has the Bernoulli distribution with parameter $1/2$, andMore generally, an interval of length $L=m+r$, with $m$ a nonnegative integer and $0\le r<1$, contains $m$ or $m+1$ points with probabilities $1-r$ and $r$. Its count variance is $r(1-r)$, while a unit-rate Poisson point process has count variance $L$.