When the singular free-energy density has $F_s=A_f|a_{t}t{|}^{2-\alpha }{\Phi }_{\pm }(0)$, the same thermal and free-energy metric factors multiply the singular heat capacity above and below the critical point. Their ratio cancels these factors and depends only on the normalized scaling functions of the universality class. This statement concerns singular amplitudes, not regular heat-capacity backgrounds, and needs separate treatment at logarithmic resonances or zero amplitudes. Branch matching constants belong to the fixed-point scaling functions and cannot be chosen as independent nonuniversal parameters.

The pairing map to a Cartesian product is injective exactly when its coordinate functions are jointly injective in the displayed sense. Either coordinate being injective is sufficient, but neither need be injective individually. For example, pairing three points with $(0,0),(0,1),(1,0)$ distinguishes all points even though each coordinate separately identifies two of them.

An intersection-free uniform set family is a uniform set family $\mathcal{F}$ with no three distinct members $A,B,C$ satisfying $A\cap B\subseteq C$. For distinct members of a rank-$r$ family, $|A\cap B|<r$, so using strict containment gives the same restriction. Fixing one member turns the other members' intersections with it into distinct antichain traces, giving the antichain trace bound for intersection-free families.

A Cauchy sequence in the uniform metric has a pointwise limit in the complete target, and the common Cauchy bound makes convergence uniform. The triangle inequality with one fixed continuous approximant shows that the limit is continuous. Thus the space of all continuous maps into a bounded complete metric space is complete, with no compactness assumption on the domain.

For each fixed finite uniform hypergraph $F$, having few copies of $F$ forces small edge-edit distance from an $F$-free hypergraph. More precisely, for every $\eta >0$ there is $\rho >0$ such that fewer than $\rho v^{|V(F)|}$ copies can be eliminated by deleting fewer than $\eta v^r$ edges in an $r$-uniform hypergraph. Strong regularity for three-uniform hypergraphs and a relative counting lemma prove the case $r=3$.

Strong regularity refines both the vertex classes and the pair graphs supporting triple hyperedges. The resulting triads in a hypergraph regularity partition have nearly constant relative triple density outside a small exceptional weight. Pair and triple errors must be chosen in a hierarchy strong enough for a relative tetrahedron counting lemma; ordinary graph regularity on the vertex partition alone is insufficient. Refinement raises bounded conditional-density energies.

Let $f_n$ be continuously differentiable functions on an interval $I$. If $f_n^{\prime }\to g$ by uniform convergence on $I$ and $f_n(c)\to A$ at one fixed $c\in I$, then $f_n$ converge locally uniformly to a continuously differentiable function $f$ with $f^{\prime }=g$. If $I$ is bounded, the convergence is uniform on all of $I$.^[ref]

The anchor fixes the additive constant that a derivative cannot see. By the fundamental theorem of calculus, defineThe uniform limit theorem makes $g$ a continuous function, so the fundamental theorem of calculus gives $f^{\prime }=g$. The displayed estimate proves uniform convergence wherever $|x-c|$ has a common finite bound, and therefore proves locally uniform convergence on any interval.

For example, on $[0,2]$ take $f_n(x)=3+1/n+x^2/2+x/n$. Its derivative is $x+1/n$, with supremum error $1/n$ from $g(x)=x$, and $f_n(0)\to 3$. The reconstructed uniform limit is $f(x)=3+x^2/2$; the exact supremum error is $3/n$, equal to the bound above. Without an anchor, the constant functions $f_n(x)=n$ have zero derivatives but no finite pointwise limit. On the real line, $f_n(x)=x/n$ has convergent anchored values and uniformly convergent derivatives, but its supremum error from zero is infinite. The bounded-distance condition has a real role.

If $g(t)\to g_{\ast }$, $\gamma (g(t))\to {\gamma }_{\ast }$ and the reference correlator has a finite nonzero fixed-point limit, the characteristic solution of the multiplicative Callan-Symanzik equation gives $\log f(t)=2{\gamma }_{\ast }t+o(t)$. A simple attractive fixed point and smooth anomalous dimension give a finite prefactor multiplying $e^{2{\gamma }_{\ast }t}$. A nonsimple fixed point can retain slower corrections.

For inviscid unforced two-layer QG dynamics, $H_1{F}_1=H_2{F}_2=f_0^2/g\prime$ makes the coupling symmetric in the layer-depth inner product. Multiplication by $H_i{\psi }_i$ gives the positive energy $E=\frac{1}{2}[{\sum }_i{H}_i|\nabla {\psi }_i{|}^2+(f_0^2/g\prime )({\psi }_1-{\psi }_2{)}^2]$. It is conserved after integration when the lateral energy flux vanishes. The interface term is available potential energy; the gradient terms are kinetic energy.

There are incomparable Turing degrees strictly between the computable degree and the degree of the diagonal halting set. An oracle-assisted finite-extension construction alternately defeats each Turing functional in the two directions. Since the sets are computable in the halting oracle, their degrees are at most $0^{\prime }$; incomparability makes them nonzero and strictly below $0^{\prime }$. This theorem does not require the constructed sets to be computably enumerable. The Friedberg–Muchnik theorem supplies that stronger conclusion.

A clique-free graph near the Turan theorem edge bound is close to a complete multipartite graph. Choose a maximum-degree vertex, split its neighbourhood from the remaining vertices, and apply induction inside the neighbourhood. The degree bound implies that missing cross edges are at least twice the number of edges inside the remaining class. This inequality controls all three types of edits and gives the constant three.

For the sextic Landau free energy $V=r{M}^2/2+v{M}^6/6-hM$, $v>0$, with $r$ linear in reduced temperature, the equation of state is $h=rM+v{M}^5$. Its ordered zero-field minimum satisfies $M^4=-r/v$, the inverse magnetic susceptibility is $4|r|$ below the transition and $r$ above, and the critical isotherm satisfies $h=v{M}^5$. The minimized singular free-energy density is $-|r{|}^{3/2}/(3\sqrt{v})$ below and zero above. These facts give the displayed mean-field critical exponents, obeying both the Rushbrooke scaling relation and Widom scaling relation. Both the quadratic and quartic coefficients must be tuned; a generic path with a positive quartic coefficient instead has ordinary critical behavior. Fluctuations change the result below the tricritical upper critical dimension three, and marginal corrections can add logarithms at three.

Approximating perturbation temperatures by harmonic fields decaying as $e^{\pm \alpha \xi }$, while retaining displacement through the base gradient, produces this reduced dispersion relation. The Gibbs--Thomson relation supplies $\Gamma =\gamma T_m/(\rho L)$. This short-wave reduction suppresses perturbation advection and time dependence; its constant term and a threshold at $\alpha \kappa /V=O(1)$ are not controlled next-order results of the full diffusion problem.

Retaining perturbation thermal diffusion givesThe displaced interfacial temperature and Stefan condition give the displayed implicit dispersion relation. In the capillarity-free limit its growing branch is exactly $\sigma =V\alpha$, which checks the limits of the small-wavelength thermal approximation for a solidification front.

For a paired non-self-conjugate momentum, rotate the Ising-chain Nambu spinor by an angle with $\cos \theta =(g-\cos k)/r_k$, $\sin \theta =-\sin k/r_k$ and $r_k=\sqrt{1+g^2-2g\cos k}$. Opposite angles at $k,-k$ preserve the canonical anticommutation relations. The result is $H={\sum }_k{ϵ}_k({\gamma }_k^{†}{\gamma }_k-1/2)$. Self-paired momenta have no pairing and are handled directly as number levels. Periodic spin-chain sectors additionally constrain fermion parity and allowed mode quantization.

For positive ferromagnetic exchange and nonnegative transverse field, the Ising-chain quasiparticle dispersion closes its gap at $g=1$, with ${ϵ}_k\sim 2J|k|$. This is the quantum phase transition between the model's ordered and field-polarized regimes. The exact dispersion identifies the gapless scale; computing its order parameter additionally requires ground-state correlations.

With translation of a distribution defined by $⟨{\tau }_{h}u,\phi ⟩=⟨u,\phi (\cdot +h)⟩$, differentiation of the test function gives $\frac{d}{dh}{\tau }_{h}u=-{\tau }_h{u}^{\prime }$ in weak convergence of distributions. The Taylor theorem shows convergence in the space of test functions, including a common compact support. Thus $({\tau }_{-h}u-u)/h\to u^{\prime }$; fixing the translation convention prevents a sign error.

Adjoin a new point $\omega$ fixed by a transitive group $G$. A one-point extension is a transitive group on the enlarged set whose stabilizer of $\omega$ is exactly $G$, acting on the remaining points in the prescribed way. Its order is $(n+1)|G|$ in the finite degree-$n$ case. The double-coset criterion for a one-point extension gives a concrete generator test.

An action is sharply $t$-transitive if each ordered $t$-tuple of distinct points is carried to any other by exactly one group element. For finite degree $n\ge t$, its order is $n(n-1)\cdots (n-t+1)$ and each ordered-tuple stabilizer is trivial. Sharp one-transitivity is a regular action. Finite sharp two-transitivity gives a regular kernel of a finite sharply two-transitive group.

A regular permutation subgroup acts transitively with trivial point stabilizers. Each ordered pair of points is related by exactly one subgroup element. A finite regular subgroup on $n$ points has order $n$. Normality and characteristicity are additional properties; they do not follow from regularity alone.