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For every $\epsilon >0$ there is $\delta >0$ such that a graph satisfying $e(A,B)\ge p|A||B|$ whenever $|A|,|B|\ge \delta n$, with $p\ge \delta$, has a spanning subgraph $G^{\prime }$ satisfyingfor all disjoint $A,B$. Partition into a fixed large number of nearly equal parts, independently thin each cross-pair to expected density $p$, and apply a concentration inequality and a union bound over all pairs $A,B$.

For disjoint nonempty vertex sets $A,B$, their edge density is

Dependent random choice finds a large vertex set whose small subsets have large common neighbourhoods. In a bipartite graph with parts $A,B$, choose random vertices of $B$ with repetition and take their common neighbourhood $X\subseteq A$; convexity gives a lower bound for $\mathbb{E}|X|$, while counting subsets of $X$ with small common neighbourhood allows their deletion.

For bad events with a dependency graph of maximum degree $D$, the symmetric Lovász local lemma guarantees positive probability that none occurs whenever each event has probability at most $p$ and $ep(D+1)\le 1$.

The random alteration method samples a random object and then deletes a small number of offending elements. If few forbidden configurations occur while the desired global property already holds, the altered object witnesses existence.

For a decreasing probability sequence $p(n)$, if $G(n,p(n))$ contains a Hamilton cycle with high probability, then $G(n,Cp(n))$ is pancyclic with high probability for every fixed sufficiently large $C$; three independent rounds suffice. One first exposes a Hamiltonian round and uses the independent sprinkled edges to create cycles of every shorter length.

For every fixed $\epsilon >0$, every component of $G(n,(1-\epsilon )/n)$ has $O_{\epsilon }(\log n)$ vertices with high probability. A breadth-first exploration is dominated by a branching process of mean $1-\epsilon$; its total progeny has an exponentially decreasing tail.

An event depending on $n$ holds with high probability when its probability tends to one as $n\to \infty$.

If $e_i$ is fixed in a rigid body with angular velocity $\omega$, then its derivative in the fixed space frame is

Statistical homogeneity defines the primordial bispectrum by

A bispectrum is a Fourier-space third-order statistic; primordial perturbations supply the cosmological field in this instance.

Increasing primordial helium at the expense of hydrogen reduces fuel for long-lived main-sequence hydrogen burning and changes subsequent stellar evolution.

When protons outnumber neutrons, helium-4 production is neutron-limited and produces about one helium nucleus per two neutrons.

If $V$ is a simple left $R$-module and $D={\mathrm{End}}_R(V{)}^{\mathrm{op}}$, then every prescribed map on a finite $D$-linearly independent subset of $V$ is induced by an element of $R$.

The functions generated from the initial functions using composition alone are exactlyfor a constant $c\in \mathbb{N}$. Structural induction proves necessity, and iterating the successor function on zero or a projection proves sufficiency.

From $g:{\mathbb{N}}^k\to \mathbb{N}$ and $h:{\mathbb{N}}^{k+2}\to \mathbb{N}$, primitive recursion defines

Given a $m$-ary function $g$ and $k$-ary functions $h_1,\dots ,h_m$, composition forms

The initial functions are the zero function, the successor function, and all coordinate projections.

For a commutative unital ring $R$, an ideal $I$ is prime exactly when $R/I$ is an integral domain. The zero-product condition in the quotient translates directly to $ab\in I$.

An equation of state is a relation among thermodynamic state variables such as pressure, volume, temperature, energy, and particle number.

The Poynting theorem is the local energy balance of classical electromagnetism. For vacuum field laws, with current density $\mathbf{J}$, the electromagnetic energy per unit volume isandThus a decrease in local electromagnetic energy can supply outward energy flux or work on electric charge. The term $\mathbf{J}\cdot \mathbf{E}$ is power transferred to matter per unit volume. Its sign matters: positive values remove field energy, while negative values describe matter supplying the field. The magnetic part of the Lorentz force supplies no work term.

For a fixed volume $V$, the divergence theorem gives the directly measurable balanceFor example, $2.0\mathrm{W}$ entering a region with $0.50\mathrm{W}$ leaving and $1.20\mathrm{W}$ transferred to matter means stored electromagnetic energy increases at $0.30\mathrm{W}$. In source-free vacuum only the boundary flux changes the stored energy; a nonzero local Poynting vector can still have zero net closed-surface flux.

A brief proof takes the dot product of the Ampère-Maxwell equation with $\mathbf{E}/{\mu }_0$ and uses Faraday's law with $\mathbf{B}/{\mu }_0$. The identity $\nabla \cdot (\mathbf{E}\times \mathbf{B})=\mathbf{B}\cdot (\nabla \times \mathbf{E})-\mathbf{E}\cdot (\nabla \times \mathbf{B})$ turns the remaining terms into the divergence of the Poynting vector. ^[ref].