The coefficients $C_{ij}{}^k(q,p)$ in $\{{\phi }_i,{\phi }_j\}=C_{ij}{}^k{\phi }_k$ are structure functions. A finite real span of first-class constraints is a Lie algebra when it closes with constant coefficients. General structure functions instead describe closure with phase-space-dependent coefficients, while still permitting canonical gauge transformations.

For a saline Stefan problem with common thermal properties, $ϵ=\sqrt{D/\kappa }\ll 1$, and fixed latent-to-sensible heat ratio, set $T_0=-m{C}_0$, ${\theta }_{-}=T_0-T_{-\infty }$ and ${\theta }_{+}=T_{\infty }-T_0>0$. A finite-$\lambda$ similarity solution has $T_i=(T_{\infty }+T_{-\infty })/2+O(ϵ)$. At the onset of constitutional supercooling, $\lambda =O(ϵ)$ and the error in this mean is $O({ϵ}^2)$. The salt-rejection function for a saline Stefan front then gives $m\lambda C_i=({\theta }_{-}-{\theta }_{+})/(2\sqrt{\pi })+O({ϵ}^2)$, whereas $\sqrt{Dt}{T}_{l,x}(a^{+})=ϵ{\theta }_{+}/\sqrt{\pi }+O({ϵ}^2)$. Comparing the actual temperature gradient to the liquidus gradient gives the displayed onset, with supercooling on the side of greater ${\theta }_{-}$. For a freezing front at $a\ge 0$ and $D<\kappa$, the ratio of these two gradients decreases into the liquid, so their interface comparison also decides whether a supercooled interval occurs anywhere ahead. The formula is an asymptotic criterion; finite-$ϵ$ onset requires the full Stefan condition and salt balance.

The sets $A_i=[m]\setminus \{i\}$ have every $t$-fold intersection of size $m-t$ and every $(t-2)$-fold intersection of size $m-t+2$. For $m\ge t+1$, their common intersection is empty and they attain $m=k+t-2$ in the constant t-wise intersection dichotomy.

In a massless four-dimensional theory, current conservation gives a transverse tensor structure, $\int d^{4}x{e}^{-ikx}⟨J_{\mu }(x)J_{\nu }(0)⟩=(k_{\mu }{k}_{\nu }-{\delta }_{\mu \nu }{k}^2)G(k^2)$. With current mass dimension three, the scalar coefficient has dimension zero.

An integral of the time component of a conserved current is a conserved charge when the spatial boundary flux vanishes. Charges need not come from an immediately apparent elementary symmetry; a field theory can have an infinite higher-derivative current hierarchy. A Noether charge is obtained from the symmetry current supplied by the Noether theorem.

A species freezing out while relativistic retains its thermal comoving number. If it is nonrelativistic today, ${\rho }_x\simeq m_x{n}_x$. Comparing with photons gives ${\Omega }_x/{\Omega }_{\gamma }=m_x(n_x/n_{\gamma })/⟨E_{\gamma }⟩$. An upper density fraction therefore bounds its mass inversely with its relic number density. The relation does not apply unchanged to a species whose abundance is depleted by nonrelativistic annihilation.

In the scalar-field source Legendre transform, differentiating $h=\delta \Gamma /\delta m$ with respect to the source and using $G=\delta m/\delta h$ yields this inverse-kernel relation. For a scalar quartic Landau-Ginzburg theory, ${\Gamma }_{\mathrm{L}}^{(2)}=-{\nabla }^2+r_0+u_0{m}^2/2$, so the connected response is the Green kernel of that operator. A one-momentum representation requires a translationally invariant background.

For a classical spin model with Hamiltonian $H_h=H_0-hM$, $M={\sum }_n{\sigma }_n$, define the static magnetic susceptibility per site by $\chi =N^{-1}{\partial }_h⟨M⟩$ at fixed temperature. Differentiating its partition function givesFor a translation-invariant array of spins, write $G(r)=⟨{\sigma }_0{\sigma }_r⟩-⟨{\sigma }_0⟩⟨{\sigma }_r⟩$. Expanding the variance of the sum then yields $\chi ={\beta }_{\mathrm{th}}{\sum }_{r}G(r)$. The derivative with respect to the dimensionless source ${\beta }_{\mathrm{th}}h$ removes the explicit inverse-temperature factor. Replacing the sum by an integral over physical coordinates requires the site-density factor $a^{-D}$.

The equal-time variance formula also holds in a quantum system when $[H_0,M]=0$. Without that commutation assumption, differentiating a matrix exponential instead givesThis is an integrated imaginary-time-ordered correlation function, not generally ${\beta }_{\mathrm{th}}$ times the equal-time variance. It follows by insertinginto the derivative of the normalized expectation value and using the cyclic matrix trace.

For example, a single qubit with $H_h=\Delta {\sigma }_z-h{\sigma }_x$, $\Delta >0$, has $\chi =\tanh ({\beta }_{\mathrm{th}}\Delta )/\Delta$ at $h=0$, whereas ${\beta }_{\mathrm{th}}\mathrm{Var}({\sigma }_x)={\beta }_{\mathrm{th}}$. Their difference directly exhibits the quantum noncommutation correction. Below an ordered transition, selecting a pure thermodynamic phase also avoids a macroscopic phase-mixture contribution to the classical sum.

For a nonzero real-on-the-circle trigonometric polynomial of actual order $d$, the polynomial $P(z)=z^{d}p(z)$ satisfies $P(z)=z^{2d}\overline{P(1/\overline{z})}$. Its constant term is the nonzero conjugate of its leading coefficient. Hence its nonzero roots of a polynomial occur in pairs $\zeta ,1/\overline{\zeta }$ with the same multiplicity of a root. The coefficient reversal and complex conjugation preserve the exponents of the paired factors. Fixed roots of a polynomial on the unit circle require nonnegativity, rather than this symmetry alone, to have even multiplicity.

The residual of an approximate solution $x$ of $Ax=b$ is $r=b-Ax$. It is the unbalanced right-hand side, while, for an invertible matrix $A$, the solution error is $x_{\ast }-x=A^{-1}r$. The conjugate gradient method makes residuals orthogonal to successive Krylov subspaces.

A finite sum ${\sum }_j{\lambda }_j{s}_j$ with ${\lambda }_j\ge 0$. Unlike a convex combination, the coefficients need not sum to one. Zero coefficients and the empty sum are allowed.

Expose the independent edge indicators of a binomial random graph in a fixed order, and set $M_i=\mathbb{E}[Z\mid \text{the first }i\text{ indicators}]$ for an integrable statistic $Z$. This is a martingale from $\mathbb{E}Z$ to $Z$. If changing one edge changes $Z$ by at most $c$, coupling the unexposed indicators gives $|M_i-M_{i-1}|\le c$, permitting the Azuma-Hoeffding inequality.

Disintegrate over a factor map and take the $L^2$ norm using the conditional probability on each fiber. Its square integrates to the global squared norm. The uniform conditional L2 norm takes the essential supremum of these fiber norms.

If $Y$ is a real square-integrable random variable and $X=\mathbb{E}[Y\mid \mathcal{G}]$, then $\mathbb{E}[XY]=\mathbb{E}{X}^2$ and the displayed identity follows by expansion. Thus equality of the second moments forces $X=Y$ as an almost sure equality. This is the equality case of the conditional Jensen inequality for the strictly convex function $z\mapsto z^2$.

For an event $A$ in a product of finitely many probability spaces with its product measure, take the closed convex hull of the mismatch vectors $(1_{x_i\ne y_i}{)}_i$, $y\in A$. The distance of this hull from zero is $d_T(x,A)$. Equivalently it is the supremum, over nonnegative coordinate weights of squared sum at most one, of their minimum mismatch sum. Averaging mismatch patterns is the feature that distinguishes this quantity from ordinary Hamming distance.

Suppose $f(0)=0$, $f^{\prime }(0)=c$, $h=(f^{\prime }{)}^{-1}$ is differentiable, and $k_{-}<h^{\prime }/2<k_{+}<0$. Then $h(c)=g(c)=0$ for the concave Legendre dual $g$, andIntegrate the derivative bounds from $c$ to $z$, then integrate $g^{\prime }=h$. Multiplying the first inequality by a negative $z-c$ reverses both comparisons. A bound valid without cases is $|h(z)|\le -2k_{-}|z-c|$.

A linear map is injective on the class of sparse vectors of order $s$ exactly when its null space contains no nonzero vector with at most $2s$ nonzero entries. Differences of two sparse vectors have at most $2s$ active coordinates. Conversely, splitting the support of a vector of a $2s$-sparse null vector into two parts constructs two distinct $s$-sparse vectors with the same image. The strict null space property implies sparse injectivity by applying it to both parts of such a split.

A composition factor is an irreducible successive quotient in a composition series of a module. For a finite-dimensional complex Lie algebra representation and a central Casimir operator, its scalar on a composition factor is an eigenvalue of the operator on the original Lie algebra representation.

For $[{\nabla }^2+\xi (1-|\psi {|}^2)]\psi =0$ and a regular unit-charge core $\psi =f(r)e^{i[\theta +\chi (r)]}$, write current velocity $u_r=2{\chi }^{\prime }$ when the time-dependent kinetic operator is $-{\nabla }^2$. Separating the imaginary part givesWith $f=ar+O(r^3)$ and no singular core flux, integration gives $u_r=-\mathrm{Im}\xi r/2+O(r^3)$. If $\xi =g{n}_{\infty }-i\alpha$, the slope is $\alpha /2$. The phase-gradient convention $k={\chi }^{\prime }$ has half this slope, $\alpha /4$. These are local regularity results, not a global vortex-existence assertion.

A normal mode $e^{i(kx-\omega t)}$ has dispersion relation $\omega =Uk+(c_d-i)k^2+i\mu$. The absolute wavenumber is the saddle where $d\omega /dk=0$. The Green function of the linear complex Ginzburg-Landau equation distinguishes convective hydrodynamic instability from absolute hydrodynamic instability.