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If proton–proton chain heating scales as $X_H^2\rho T^{{\nu }_{pp}}$ and CNO cycle heating as $X_H{Z}_{\mathrm{CNO}}\rho T^{{\nu }_{\mathrm{CNO}}}$, their ratio at fixed hydrogen abundance scales as $Z_{\mathrm{CNO}}{T}^{{\nu }_{\mathrm{CNO}}-{\nu }_{pp}}$. Equating the rates proves the displayed crossover-temperature relation. It applies to a local power-law approximation, with unchanged coefficient normalizations apart from the catalyst abundance.

For positive $\alpha ,\beta$, suppose $p^{\alpha }\le |A|\le p^{1-\alpha }$ and $|B|\ge p^{\beta }$. Repeatedly replace a scalar set $S$ by $3SS-3SS$. Quadratic growth of a sixfold product difference set reaches size at least $p/2$ in a number of steps depending only on $\beta$. The closure bounds for small skew-sumset scalars transfer a putative bound $|A+bA|\le K|A|$ to all these scalars with exponent depending only on the number of steps. Averaging collisions over the resulting large scalar set then forces $K^L\ge \frac{1}{4}\min (|A|,p/|A|)\ge p^{\alpha }/4$. This supplies a positive exponent depending only on $\alpha ,\beta$. The finitely many smaller primes are covered by the strict inequality $|A+bA|>|A|$ for every nonzero $b$ when $1<|A|<p$.

For segments with initial direction vectors $U,T$, initial endpoint displacement $D$, and translation velocities $V,W$, contact is feasibility of the displayed linear equation with $s,r\in [0,1]$ and $t\ge 0$. Minimizing $t$ is a small linear programming problem. A nonsingular three-column coefficient matrix gives at most one candidate; singular, parallel and degenerate cases require the constrained feasibility problem rather than division by a vanishing determinant.

A connected $n$-sheeted covering space of a closed orientable surface of genus $g$ has genus $1+n(g-1)$. Lifting a finite cell decomposition gives $n$ cells above every base cell, so its Euler characteristic is $n(2-2g)$. Equating this with $2-2g^{\prime }$ gives the formula. For $g\ge 1$, every positive $n$ occurs: send one handle generator of the fundamental group to $1$ in $\mathbb{Z}/n$ and all remaining generators to zero, then use the classification of connected covering spaces for the kernel of that surjection.

For metric spaces $X,Y$ with $X$ compact, the projection with a compact factor is closed: if $F\subset X\times Y$ is a closed set and $y_n\in {\pi }_Y(F)$ converges to $y$, choose $x_n$ with $(x_n,y_n)\in F$. Sequential compactness supplies $x_{n_k}\to x$, so $(x_{n_k},y_{n_k})\to (x,y)\in F$, proving $y\in {\pi }_Y(F)$. The projection map is also continuous for the sum metric. Without compactness, the closed set $\{(x,y)\in {\mathbb{R}}^2:xy=1\}$ projects to the nonclosed set $\mathbb{R}\setminus \{0\}$.

A densely defined closed operator is sectorial of angle $\theta$ if its spectrum lies in $\{z:|\arg z|\le \theta \}\cup \{0\}$ and, for every larger angle $\varphi <\pi$, its resolvent outside that larger sector satisfies $\Vert (z-A{)}^{-1}\Vert \le C_{\varphi }/|z|$. If $\theta <\pi /2$, then $-A$ generates an analytic semigroup. A real shift of the generator allows exponential growth without changing the short-time smoothing estimates.

For an everywhere-defined linear map $T:E\to F$ between Banach spaces, its separating space measures the possible limiting defect in continuity at zero. It is a closed linear subspace: linear combinations combine witnessing sequences, and diagonal choices with $\Vert x_j\Vert <1/j$ and $\Vert T{x}_j-y_j\Vert <1/j$ handle a convergent sequence $y_j$ of defects. The closed graph theorem gives $T$ continuous exactly when $S(T)=\{0\}$, since a limit $(x_n,T{x}_n)\to (x,y)$ has $y-Tx\in S(T)$.

A maximal clique is a clique not strictly contained in another clique. This is distinct from a maximum clique, whose size equals the clique number. In a triangulated probabilistic graphical model, maximal cliques are the natural nodes of a junction tree.

Maximality is a condition on set inclusion: no outside vertex is adjacent to every member of the current clique. In the five-vertex example, $\{c,d\}$ is maximal because $a,b$ are not adjacent to $d$, while $e$ is not adjacent to $c$. Yet $\{a,b,c\}$ is larger. Checking every possible one-vertex extension is enough: if a larger containing clique existed, any one of its new vertices could already be added.^[ref]

For fixed $k\ge 2$, a binomial random graph has no $k$-cliques with probability tending to one if ${\mu }_n\to 0$, and has at least one with probability tending to one if ${\mu }_n\to \infty$. The first assertion uses the first moment method. The second uses the overlap formula for the variance of a clique count: overlapping $k$-sets with $l\ge 2$ contribute $O(n^{-l}{p}^{-\left(\genfrac{}{}{0ex}{}{l}{2}\right)})$, and each contribution tends to zero above this threshold. The fixed-size qualification matters.

Put $s=j_1+j_2$, with both $j_i>0$. At magnetic quantum number $-s+1$, let $|a⟩$ raise only the first factor of the lowest product state and $|b⟩$ raise only the second. Applying total $J_{+}$ to the lowest product gives the normalized $J=s$ combination $\sqrt{j_1/s}|a⟩+\sqrt{j_2/s}|b⟩$. Its orthogonal combination is annihilated by total $J_{-}$ and therefore has $J=s-1$. This obtains a useful Clebsch-Gordan coefficients without a table.

Apply a total ladder operator to a coupled state and evaluate against an uncoupled product state. The total operator is the sum of the two factor operators, so its one coupled ladder amplitude times the corresponding one of the Clebsch-Gordan coefficients equals the sum of two factor ladder amplitudes times shifted coefficients. An overall common rescaling of the ladder operators cancels from the recurrence. Nonzero coefficients obey $m_1+m_2=m$, and the positive ladder amplitudes require a consistent phase convention within each irreducible summand.

// Target: lie-theory.bigb

For a path-connected, locally path-connected, semilocally simply connected space $X$, base point $x_0$, and subgroup $H\le {\pi }_1(X,x_0)$, take paths starting at $x_0$ modulo the relation $\alpha \sim \beta$ when their endpoints agree and $[\alpha {\beta }^{-1}]\in H$. Projection sends a class to its endpoint. Over a path-connected neighbourhood whose loops die in $X$, classes $[\alpha \eta ]$, with $\eta$ inside that neighbourhood, form a sheet mapping homeomorphically to it. These sheets define a covering space. A loop lifts closed precisely when its class belongs to $H$, so the injective induced fundamental group map has image $H$.

For a semilinear partial differential equation $u_t=\Delta u+F(u)$ with smooth $F$ and smooth fixed boundary data, local boundedness permits positive-time elliptic regularity and parabolic smoothing. If $u_t$ is continuous with values in a Hölder space, elliptic estimates first give continuous $C^{2,\beta }$ spatial regularity for $u$. Time difference quotients then solve a linear parabolic equation with coefficient converging to $F^{\prime }(u)$; smoothing away from the initial time gives $u_t\in C(C^2)$, hence $u\in C^1(C^2)$. It is essential to gain spatial regularity for the time derivative too; $u\in C(C^2)\cap C^1(C^0)$ alone does not imply $C^1(C^2)$.

An argument has logical validity in classical propositional logic when no Boolean valuation makes all premises $\Gamma$ true and the conclusion $A$ false: $\Gamma \models A$. With no premises, $\models A$ says that $A$ is a tautology. See forall x: Calgary, sections 12.4–12.6.

A tautology is a propositional formula true under every Boolean valuation in classical propositional logic, such as the law of excluded middle $p\vee \neg p$. A primitive proposition $p$ is not a tautology, even if it represents a true statement. See forall x: Calgary, section 12.1.

The zero-field Hubbard–Stratonovich transformation of an exponentially coupled Ising model produces $U(m)=A{m}^2-\log [2\cosh (2m)]$, where $A=\tanh (\kappa /2)/J>0$. Since $U^{\prime \prime }(0)=2A-4$ and $U$ tends to positive infinity, a symmetric double well exists exactly when $A<2$. Its nonzero stationary points solve $Am=\tanh (2m)$; the ratio $\tanh (2m)/m$ decreases from 2 to 0 for $m>0$, proving that there is exactly one positive minimum. At $A=2$ the minimum is a single quartic minimum. A small conjugate field adds $-h\tanh (2m)+O(h^2)$ and tilts the wells, while a sufficiently large field can remove the metastable well. A finite barrier permits quantum tunnelling; two classical minima alone do not establish spontaneous symmetry breaking.

For a nearest-neighbor auxiliary-field representation of an Ising model, sharing each local potential between adjacent bonds gives the displayed symmetric integral operator. Its periodic partition function is the trace of its $N$th power, up to Gaussian normalization constants. In the slowly varying continuum limit, a slice of length $ϵ$ has kernel $[M/(2\pi ϵ){]}^{1/2}\exp [-M(m^{\prime }-m{)}^2/(2ϵ)-ϵ\{U(m^{\prime })+U(m)\}/2]$. Expanding its action on a smooth function gives $1-ϵ\hat{H}+O({ϵ}^2)$, with $\hat{H}=-(2M{)}^{-1}{\partial }_m^2+U$. Thus fixed endpoints give an imaginary-time transition kernel and periodic endpoints give a Euclidean path integral trace. The finite-spacing integral operator and the continuum differential operator need not have identical high-energy spectra.

The thermal de Broglie wavelength is a length scale from quantum mechanics entering the classical ideal-gas partition function for nonrelativistic particles of mass $m$ at temperature $T$:

For one free particle in a large three-dimensional volume $V$, the Gaussian integral over momentum givesThe factor $h^{-3}$ counts phase space cells; it is essential for a dimensionless partition function. The thermal wavelength represents a thermal momentum scale rather than a wavelength shared by every particle. ^[ref]

For a single internal state, $n{\lambda }^3\ll 1$ is the dilute classical regime. When $n{\lambda }^3$ becomes comparable to one, Bose-Einstein statistics or Fermi-Dirac statistics matters. Cooling increases $\lambda$ as $T^{-1/2}$, bringing quantum degeneracy within reach even at fixed number density $n$. ^[ref]

For the potential $U=r^{2n}/V^{2n/3}$, rescaling $r=V^{1/3}(kT{)}^{1/(2n)}u$ in the classical one-particle partition function gives the formula above. The parameter-conjugate force is $P=NkT/V$, the internal energy is $(3/2+3/(2n))NkT$, and $C_P-C_V=Nk$. The kinetic term contributes $3NkT/2$, while the potential term contributes $3NkT/(2n)$. For indistinguishable particles the factorial $N!$ changes free energy and entropy but not these force and heat-capacity formulas.

For continuous real coefficients with $p>0$ and $q,r\ge 0$ on a bounded interval, use the clamped second-order Sobolev space and the displayed bilinear form. A forcing in $L^2$ defines $\ell (v)=\int fv$. The weak solution satisfies $a(u,v)=\ell (v)$ for every clamped test vector. Compactly supported test functions give $(p{u}^{\prime \prime }{)}^{\prime \prime }-(q{u}^{\prime }{)}^{\prime }+ru=f$ as a distribution; density extends that equation to the energy space. Continuous coefficients alone do not justify differentiating them classically. The clamped interval derivative norm bound gives coercivity, so Lax-Milgram theorem supplies a unique solution and the quadratic energy has that unique minimum.