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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.

When an energy Hamiltonian contains $-h{\sum }_n{\sigma }_n$, differentiating its partition function gives $\chi =({\beta }_{\mathrm{th}}/N)\mathrm{Var}({\sum }_n{\sigma }_n)$. Translation invariance converts this to the connected-correlation sum. A dimensionless source ${\beta }_{\mathrm{th}}h$ removes the explicit inverse-temperature factor. In continuum physical coordinates the lattice sum includes the site-density factor $a^{-D}$. A selected pure thermodynamic phase is needed below an ordered transition to avoid a macroscopic mixture contribution.

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 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 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.

This nondispersive cubic complex Ginzburg–Landau equation balances linear growth against nonlinear damping. Its real-wavenumber Ginzburg-Landau plane wave has amplitude $\sqrt{\mu -k^2}$ when $\mu >k^2$. Relaxation within that single-mode amplitude family does not by itself establish stability against general perturbations.

Assign a primitive proposition true exactly when it belongs to a maximal consistent set. An induction on formulas proves that truth agrees with membership: maximal consistency gives the negation dichotomy, and deductive closure gives the implication truth table. This builds a model of every consistent propositional theory.

In finite dimensions, a completely positive map has the form $\mathcal{N}(X)={\mathrm{Tr}}_E(VX{V}^{†})$ for $V:{\mathcal{H}}_Q\to {\mathcal{H}}_{Q\prime }\otimes {\mathcal{H}}_E$. From a Kraus representation take $V={\sum }_j{K}_j\otimes |j⟩$. The observable-picture adjoint is ${\mathcal{N}}^{†}(Y)=V^{†}(Y\otimes I_E)V$. A general completely positive map need not have isometric $V$; it is a linear isometry of Hilbert spaces precisely when the map is trace preserving.