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For a decreasing probability sequence $p(n)$, if $G(n,p(n))$ contains a Hamilton cycle with high probability, then $G(n,\min \{Cp(n),1\})$ 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 $0<\epsilon <1$, 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

The primordial bispectrum is the Fourier-space connected correlation function of order three for an early-universe perturbation field. For a zero-mean real scalar field $\varphi$, statistical homogeneity impliesA common spatial translation multiplies the left side by $e^{i({\mathbf{k}}_1+{\mathbf{k}}_2+{\mathbf{k}}_3)\cdot \mathbf{a}}$, so translation invariance requires support on closed wavevector triangles. If the field has a nonzero mean, subtract the disconnected mean/two-point products before defining this connected statistic.

Statistical isotropy is the additional assumption that removes orientation dependence, allowing the shorter notation $B(k_1,k_2,k_3)$ with $k_i=|{\mathbf{k}}_i|$. The three positive magnitudes obey the triangle inequality. Statistical homogeneity alone does not justify replacing the vectors by their lengths. The usual primordial convention is illustrated by the Planck definition.^[ref]

A zero-mean Gaussian random field has vanishing three-point function. A measured nonzero connected primordial bispectrum therefore tests primordial non-Gaussianity beyond the power spectrum. The converse fails: a non-Gaussian field whose distribution is invariant under $\varphi \mapsto -\varphi$ has zero odd moments but can have a nonzero connected four-point function. Comparing predictions requires specifying the field, such as a gravitational potential or comoving curvature perturbation, because nonlinear field redefinitions change its primordial bispectrum.

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 relates thermodynamic state variables. The ideal gas law, $pV=N{k}_{B}T$, gives a simple example: specifying temperature, particle number and volume determines the equilibrium pressure within that model. A fluid momentum equation needs such thermodynamic information to connect compression with pressure response.

One pressure relation does not generally determine every thermal property. An ideal gas's internal energy still depends on its available molecular degrees of freedom; the pressure law alone does not supply its heat capacity. Likewise, the Van der Waals equation requires a compatible thermal description when predicting temperature changes or heat transfer.

An equation of state describes allowed equilibrium states, whereas a particular heating or compression protocol specifies a path among them. Keeping those roles separate prevents treating every process relation as a universal material law.

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].

Shear thinning is the decrease of effective viscosity with increasing shear rate.

In classical power counting in quantum field theory, an irrelevant coupling multiplies a local operator whose mass dimension $\Delta$ exceeds the spacetime dimension $d$. Since the action is dimensionless in natural units, $[g]=d-\Delta <0$. The dimensionless interaction strength scales as

For example, ${\varphi }^6$ in four-dimensional scalar field theory has coefficient $g/{\Lambda }^2$, giving suppression $g(E/\Lambda {)}^2$ at low energy. This explains the usefulness of effective field theory below its cutoff. At an interacting renormalization-group fixed point, relevance is determined by the full scaling dimension, including quantum corrections. ^[ref]

A relevant coupling has positive mass dimension and grows under scaling toward long distances.

A marginal coupling is dimensionless under classical power counting.

The marginal interaction coefficient is distinct from the operator it multiplies.

The superficial degree of divergence is the ultraviolet power obtained by scaling all independent loop momenta of a Feynman diagram together. If a scalar graph in $d$ dimensions has $L$ loops, $I$ internal propagators behaving as momentum to the power $-2$, and $N_{\partial }$ powers of loop momentum from derivative vertices or numerators, thenEach loop integration contributes $d$ powers, each propagator removes two, and each numerator power adds one. A scalar theory with no derivative vertices has the familiar specialization $D=dL-2I$.

For a connected graph in four-dimensional quartic scalar field theory, $L=I-V+1$ and $4V=2I+E$, where $V$ is the number of interaction vertices and $E$ the number of external lines. Eliminating $I,V$ givesThus a two-point graph has superficial degree two, while a four-point graph has degree zero, corresponding to possible logarithmic ultraviolet behavior.

The word superficial matters: tensor or gauge cancellations can reduce the actual divergence, and a graph with negative overall $D$ can still contain divergent subgraphs. Exceptional external momenta can introduce a separate infrared problem. Power counting organizes possible counterterms; convergence requires inspecting the relevant subgraphs and kinematic regime as well.