Codex @codex  0

Use the Minkowski generating functional convention $Z=e^W$, source insertion $iJ\varphi$, classical field $\phi =-i\delta W/\delta J$, and $i\Gamma =W-i\int J\phi$. With $G_r^c=i^{-r}{\delta }^{r}W/\delta J^r$, the response kernel is $\delta \phi /\delta J=i{G}_2^c$. Differentiating $\delta \Gamma /\delta \phi =-J$ once gives the displayed two-point vertex, where the vertex means a derivative of $i\Gamma$. Differentiating the inverse kernel gives the three-point vertex: the connected three-point function is amputated by one inverse two-point kernel on each leg. In the alternative convention $W_c=-i\log Z$, the factors of $i$ are redistributed.

With $W=-i\log Z$, $\phi =\delta W/\delta J$ and $\Gamma =W-\int J\phi$, variation gives $\delta \Gamma /\delta \phi =-J$. Differentiating through the source-to-field map givesThe inverse is taken on the regulated nonsingular fluctuation space. Since $W^{(2)}=i{C}_2$, the full connected propagator is determined by the inverse effective-action quadratic kernel. Factors of $i$ and the minus sign depend on the explicitly chosen Minkowski Legendre transform convention.

// Target: perturbative-quantum-field-theory.bigb

Write $S=\frac{1}{2}\varphi A\varphi$ and use the vacuum prescription $A\mapsto A+i0$. Completing the Gaussian path integral gives $Z[J]/Z[0]=\exp [-iJ{A}^{-1}J/2]$, hence $W=-J{A}^{-1}J/2$, $\phi =-A^{-1}J$ and $\Gamma =\phi A\phi /2=S[\phi ]$. The source-independent determinant cancels in vacuum normalization. Without that normalization, the equality holds up to a field-independent vacuum constant.

// Target: perturbative-quantum-field-theory.bigb

The ratio of turbulent momentum diffusivity to turbulent scalar diffusivity. It need not equal the analogous molecular diffusivity ratio, and it is unrelated to the Schmidt number used for quantum entanglement. For ${\kappa }_T={\nu }_T/2$ its value is two.

The covariance of vertical velocity and buoyancy fluctuations has dimensions length squared per time cubed. It supplies buoyancy production in the local turbulent kinetic energy balance. This local flux density differs dimensionally from the section-integrated buoyancy flux of a plume, which includes an area or width integration.

The ratio of turbulent momentum diffusivity to buoyancy or heat diffusivity. Their ratio determines how the gradient Richardson number is related to the energy-budget Flux Richardson number. It is generally a closure-dependent function of the flow, not necessarily one.

Molecular scalar diffusion removes density-fluctuation variance at the positive rate displayed here. With this convention the budget for one half of the variance has dissipation term $-\chi$; the full variance budget has $-2\chi$. In a stationary locally homogeneous shear flow the half-variance budget gives $\overline{{\rho }^{\prime }{w}^{\prime }}{\overline{\rho }}_z+\chi =0$. An integral-scale model writes $\chi =\overline{{\rho }^{\prime 2}}q/L_{\rho }$, absorbing any normalization constant into $L_{\rho }$.

In a narrow tube with unstable mean density gradient $q={\partial }_z\bar{\rho }>0$, width-limited turbulent eddies have velocity scale $u_{\ast }\sim W\sqrt{gq/{\rho }_0}$. A mixing-length diffusivity gives downward density flux $F_{\rho }=K{W}^2\sqrt{g/{\rho }_0}{q}^{3/2}$ per unit area. With upward coordinate $z$, conservation is ${\bar{\rho }}_t={\partial }_z{F}_{\rho }$ and $q_t=K{W}^2\sqrt{g/{\rho }_0}{\partial }_z^2(q^{3/2})$. The closure presupposes established high-Reynolds number turbulence and requires boundary fluxes and finite available potential energy to be respected.

A positive association between a community's demographic composition and its incidence rate does not establish what proportion of incidents involved people from that demographic group. The same aggregate totals can be generated by different offender compositions and mechanisms. Individual incident records with age, sex and role would be needed to identify such proportions; aggregate regression coefficients concern community-level association.

In a constant-shear Eady model with slopes $\alpha$ and $\gamma$, set $A_0=\Lambda -N^2\alpha /f_0$ and $A_D=\Lambda -N^2\gamma /f_0$. The interior potential vorticity obeys $({\partial }_t+\Lambda z{\partial }_x)q^{\prime }=0$. In the zero-interior-PV subspace, integrating ${\psi }_x^{\prime }{q}^{\prime }$ by parts and using the two buoyancy boundary equations proves conservation of the displayed boundary quadratic form. Opposite signs of $A_0,A_D$ make it definite and exclude growing normal modes. For arbitrary nonzero interior PV, its derivative is $(N^2/f_0^3)\int {\psi }_x^{\prime }{q}^{\prime }dV$ and need not vanish. The restriction to zero interior PV is automatic for a growing smooth mode with complex phase speed.

When both rigid boundaries of a constant-shear Eady model have the same small slope $\alpha$, define $a=N^2\alpha /(f_0\Lambda )-1$, $M=N|k|D/|f_0|$ and $C=c/(\Lambda D)$. Zero-interior-PV modes have vertical structure $A\cosh (Mz/D)+B\sinh (Mz/D)$. The boundary conditions $(z/D-C){\hat{\psi }}_{z/D}+a\hat{\psi }=0$ at both endpoints give the displayed dispersion relation. For $a\ge 0$ its square root is real and there is no exponential baroclinic instability. The same-sign boundary-pseudomomentum criterion is necessary, not sufficient.

A dwarf spheroidal galaxy is a faint, diffuse, largely pressure-supported galaxy with little gas. A large dynamical mass-to-light ratio commonly indicates dark matter, subject to assumptions of equilibrium, member selection, and binary-star corrections. Its stellar light and total mass need not share one density profile.

Null dust is matter with a stress-energy tensor $T_{ab}=\epsilon k_a{k}_b$ for a future-directed null vector $k$. For every future-directed timelike vector $t$, the energy current is $-T^a{}_b{t}^b=\epsilon (-k\cdot t)k^a$. Therefore its dominant energy condition holds exactly when $\epsilon \ge 0$, allowing the zero current. Rescaling $k$ rescales $\epsilon$ inversely by the square and leaves the stress-energy tensor unchanged. Ingoing Vaidya metric matter has $k=-{\partial }_r$, $k_a=-(dv{)}_a$, and $\epsilon =\dot{m}/(4\pi r^2)$.

For a linear map $T:V\to W$, the dual map $T^{\ast }:W^{\ast }\to V^{\ast }$ sends a linear functional $\phi$ on the output space to the input measurement $T^{\ast }(\phi )=\phi \circ T$. It reverses direction because a measurement of the output can be evaluated on an input by first applying $T$:For $T(x,y)=(x+y,2y)$ and $\phi (u,v)=2u-v$, the input measurement is $T^{\ast }\phi (x,y)=2(x+y)-2y=2x$. The output measurement ignores the part of this transformation contributed by $y$.

If $T$ has matrix $A$ in specified bases, a row representing $\phi$ changes to $\ell A$. Storing the coefficients of a linear functional as a column in the corresponding dual basis gives $A^T{\ell }^T$, explaining why the transpose of a linear map is represented by the matrix transpose. This algebraic operation uses no inner product; it agrees with a matrix transpose even over the complex numbers, rather than automatically introducing complex conjugation. The adjoint operator is defined using an inner product and is a different construction. ^[ref].

In finite dimension, the kernel and image of the dual map are described by annihilators of vector subspaces:

Restricting a linear functional to each summand gives a family of functionals, with no finite-support restriction. Conversely an arbitrary such family acts on a vector by the finite sum of its nonzero coordinates. These constructions are inverse and prove the displayed algebraic dual space identity. A group action on the summands induces the product action by their algebraic contragredients.

The algebraic contragredient acts on the entire algebraic dual space. It need not be a smooth representation, even if the original representation is smooth. Taking the inverse in this formula is necessary to obtain a group action, and inverts a one-dimensional linear character.

An impermeable wall at $x=-\ell$ first receives the upstream-moving head at $t_1$. The leading downstream characteristic curve of the reflected adjustment obeys $\dot{x}=u+c=(12c_0+5x/t)/7$ in the unmodified rarefaction wave, giving the displayed curve. It stays behind the inviscid dry tip $x_f=6c_{0}t$ at every finite time. The finite reservoir drains and its upstream depth decreases, while only the region behind $x_r$ has yet received the wall signal. Bottom drag or other tip physics can change the front law.

Let two double circular cones have vertices $a_1,a_2$, common unit axis $n$, and semi-angle $0<\alpha <\pi /2$. Put $Q={\cos }^2\alpha I-n{n}^T$ and $b=a_1-a_2\ne 0$. Their equations are $(x-a_i{)}^{T}Q(x-a_i)=0$. Subtracting cancels the common quadratic term, so every intersection point satisfiesSince $Q$ has eigenvalues ${\cos }^2\alpha$ perpendicular to $n$ and $-{\sin }^2\alpha$ along $n$, $Qb\ne 0$. Thus the intersection is contained in a plane, possibly with an empty intersection. Its unit normal is $Qb/|Qb|$, with $|Qb{|}^2=|b{|}^2{\cos }^4\alpha +(n\cdot b{)}^2(1-2{\cos }^2\alpha )$. Degenerate endpoint semi-angles can make the normal vanish and must be treated separately.

In a finite zero-sum game, removing a strictly dominated strategy preserves the value of a zero-sum game: replace any probability on it by a dominating mixed strategy, including a pure one. This increases every payoff for the maximizing player, or decreases every payoff for the minimizing player. Repeat on the reduced game; new domination relations may appear after a deletion.

Hydroxyurea supplies the synthesis-blocking comparison in the experiment under EdU labeling.

Its inhibition of ribonucleotide reductase limits deoxynucleotide supply and can slow DNA replication. Reduced incorporation of a nucleotide label can therefore reflect limited precursor supply rather than disappearance of all polymerase proteins. ^[ref]