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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 $L$.

For radiation in cosmology, the comoving-gauge density contrast satisfies ${\Delta }_k^{\prime \prime }+(k^2/3-2/{\eta }^2){\Delta }_k=0$. The displayed basis solves it. The first mode is regular and grows as $A{x}^2/3$ outside the sound horizon; the second diverges as $B/x$ toward the initial singularity. At large $x$, the regular mode approaches $-A\cos x$ and has constant-amplitude acoustic oscillations.

For fixed real $c$, the number of isolated vertices in a binomial random graph with $p=(\log n+c)/n$ converges to a Poisson distribution of mean $e^{-c}$. Its fixed factorial moments tend to $(e^{-c}{)}^k$. A union bound for small connected components, together with exclusion of intermediate components, shows that isolated vertices are asymptotically the only obstruction to connectivity. Thus the displayed limit is the zero-mass of that Poisson distribution.

A disconnected binomial random graph has a connected component of a graph with at most half its vertices. The union bound over its possible vertex sets givesThe final geometric series bound tends to zero for fixed $p>0$.

If $E$ is connective spectrum in the sense of a connective spectrum, ${\pi }_{0}E\cong \mathbb{Z}$, and $X$ is $(n-1)$-connected with $n\ge 2$, then ${\tilde{E}}_n(X)\cong {\pi }_n(X)$. In the homological Atiyah-Hirzebruch spectral sequence, the only nonzero term in total degree $n$ is ${\tilde{H}}_n(X;{\pi }_{0}E)$. Incoming differentials have negative coefficient degree, and outgoing differentials have spatial degree below $n$. The Hurewicz theorem identifies this term with ${\pi }_{n}X$.

The covariant exterior derivative extends a connection to bundle-valued forms byLocally it is $d_A=d+A\wedge$.

A smooth self-adjoint projection $p:M\to M_N(\mathbb{C})$ of locally constant rank defines the Hermitian vector bundle $\mathrm{im}p$ inside the trivial bundle. Projecting the flat derivative gives the unitary connection $\nabla s=pds$ for $ps=s$. Differentiating $p^2=p$ gives $pdp=dp(1-p)$ and $pdpp=0$. For a section $s$, ${\nabla }^{2}s=pdp\wedge ds=p(dp{)}^{2}s$, proving the curvature formula. Relative to $\mathrm{im}p\oplus \ker p$, $dp$ is off-diagonal and $(dp{)}^2$ is diagonal. Therefore $d\mathrm{tr}(p(dp{)}^{2k})=\mathrm{tr}(dp{)}^{2k+1}=0$. These are explicit closed representatives of the Chern character.

A vector field is parallel for a specified connection on a vector bundle on the tangent bundle when its covariant derivative in every direction vanishes. Its values along every smooth curve are related by parallel transport. For a Levi-Civita connection, its length is constant on each connected component, since the connection preserves the Riemannian metric.

A U(1) connection is a principal connection on a principal bundle with circle group $U(1)$. In a local trivialization it has a real one-form potential $A$, with convention $A\mapsto A+d\chi$ under a gauge transformation $e^{i\chi }$. Its curvature form $F=dA$ is globally defined, although $A$ need not be a global one-form.

For a closed curve $\gamma$, choose a local trivialization of the pulled-back circle bundle along $\gamma$. In unit-charge normalization the holonomy is $\exp (i{\oint }_{\gamma }A)$. Changing that local trivialization changes the integral by $2\pi$ times an integer, leaving the exponential unchanged. When computing with several local potentials, transition phases must be included at chart changes.

On the trivial circle bundle over a circle, $A=\alpha d\theta$ has $F=0$ but holonomy $e^{2\pi i\alpha }$. Thus zero curvature form need not imply trivial holonomy. On any closed oriented surface, the integral $\frac{1}{2\pi }\int F$ is the integral First Chern class, relating the curvature form to the topology of the circle bundle.

For a smooth strictly convex rigid body with a smooth unique contact point, a configuration consists of its orientation $R\in SO(3)$ and horizontal position $c_{\parallel }\in {\mathbb{R}}^2$; its height is fixed by contact. Translation makes this a principal bundle over $SO(3)$ with group ${\mathbb{R}}^2$. If $r(R)$ points from its reference point to contact and $\Omega$ is its spatial angular velocity, rolling without slipping requires $\dot{c}+\Omega \times r=0$. The displayed translation-valued principal connection vanishes on precisely these velocities. It is translation-invariant and equals the identity on vertical translation vectors. A prescribed orientation curve consequently has a unique horizontal lift from a given initial position. No-slip alone leaves spin free and does not specify a connection over the plane with orientation as its fiber; an additional no-twist constraint is needed for that familiar spherical model.