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For the relative differential operators over $\mathbb{C}$ on a polynomial ring, coordinates have order zero and partial derivatives have order one. The associated graded ring is $\mathbb{C}[x,\xi ]$ with degrees zero for $x_i$ and one for ${\xi }_i$. This differs from the Bernstein filtration, in which both sets of generators have degree one and every filtration piece is finite-dimensional.

The line where the interfaces of three phases meet, for example where a liquid film meets a solid and the surrounding gas. Continuum assumptions at this line can differ from the bulk lubrication approximation.

A nitrocellulose Western blot membrane is a Western blot membrane made from nitrated cellulose. Its retention capacity is finite; the original Western blot transfer experiment tested breakthrough with a second receiving sheet.

For canonical Kähler potential and $W=m{\Phi }^2/2+g{\Phi }^3/3$, eliminating the auxiliary field gives $V=|m\varphi +g{\varphi }^2{|}^2$. The cubic interaction Lagrangian is $-m^{\ast }g{\varphi }^{\ast }{\varphi }^2-m{g}^{\ast }\varphi ({\varphi }^{\ast }{)}^2$. Differentiating with respect to the three external fields gives the all-incoming Feynman rules $V_{\varphi \varphi {\varphi }^{\ast }}=-2i{m}^{\ast }g$ and $V_{{\varphi }^{\ast }{\varphi }^{\ast }\varphi }=-2im{g}^{\ast }$; the factor two counts identical scalar legs. For real $m,g$ and $\varphi =(A+iB)/\sqrt{2}$, the cubic Lagrangian is $-mgA(A^2+B^2)/\sqrt{2}$, giving $V_{AAA}=-3\sqrt{2}img$ and $V_{ABB}=-\sqrt{2}img$.

Localizing the rigid supersymmetry of a free Wess–Zumino model produces a derivative of the supersymmetry parameter multiplying the supercurrent. The Noether gauging procedure introduces a gravitino coupled to this current. Closure also requires a vierbein, the Einstein-Hilbert action, covariant matter kinetic terms and higher-order interactions. The leading gravitino-matter coupling contains the derivative of the scalar, not its undifferentiated value.

In a canonical Wess–Zumino model, complete the static transverse energy into $|{\varphi }^{\prime }-e^{i\alpha }\overline{W^{\prime }}{|}^2+2\mathrm{Re}(e^{-i\alpha }dW/dz)$. Choosing the phase of $\Delta W$ gives the tension bound. A solution of the displayed first-order equation attains it and preserves two of four real supercharges. For $W={\mu }^2\Phi /g-g{\Phi }^3/3$, $\varphi (z)=(\mu /g)\tanh (\mu z)$ interpolates between the two vacua and has tension $8{\mu }^3/(3g^2)$.

For $F=dA$ and an embedding variation $V$, the pullback Lie derivative and Cartan's magic formula give the displayed variation. Consequently the interior force from the Wess-Zumino brane coupling depends only on the gauge-invariant field strength. A change $A\mapsto A+d\Lambda$ changes the action only on the boundary, by the Generalized Stokes theorem; boundaries require the appropriate fixed boundary data or additional boundary coupling.

For an oriented brane worldvolume $W$, a gauge transformation $C\mapsto C+d\Lambda$ changes the Wess-Zumino brane coupling by $q{\int }_{\partial W}{X}^{\ast }\Lambda$, by the Generalized Stokes theorem. If the brane boundary lies on a manifold carrying a form $B$ of one lower degree, adding the displayed boundary term and transforming $B\mapsto B-\Lambda$ cancels this variation. The sign uses the induced boundary orientation. A closed worldvolume needs no boundary term; noncompact cases require appropriate support or decay conditions.

A $(p+1)$-form gauge potential couples electrically through its pullback to a $(p+1)$-dimensional worldvolume, whose object has $p$ spatial dimensions. Its field strength has degree $p+2$ and its magnetic dual potential has degree $d-p-3$, hence magnetic spatial dimension $d-p-4$. Electric and magnetic fluxes are integrated over linking spheres of dimensions $d-p-2$ and $p+2$, respectively. This dimensional dictionary applies when the indicated objects and dual descriptions exist; top forms need separate treatment and carry no ordinary propagating polarization.

This well-founded relation is well-founded without the axiom of choice or Axiom of foundation. If a nonempty set $A$ lacked a minimal member, put $b=\bigcap A$. For each $y\in A$, some $x\in A$ has $\mathcal{P}(x)\subseteq y$, and $b\subseteq x$ gives $\mathcal{P}(b)\subseteq y$. Hence $\mathcal{P}(b)\subseteq b$. The Cantor diagonal argument applied to $d=\{u\in b:u\notin u\}$ contradicts this inclusion.

This relation has well-foundedness without Axiom of foundation or axiom of choice. If a nonempty set $S$ had no minimal member, put $b=\bigcap S$. For each $y\in S$, choose existentially an $x\in S$ with $xRy$. Since $b\subseteq x\cap y$, every subset of $b$ belongs to $y$. Thus $\mathcal{P}(b)\subseteq b$. But $d=\{u\in b:u\notin u\}$ is then an element of $b$, contradicting $d\in d⟺d\notin d$.

A Weil divisor is Q-Cartier if some positive integer multiple is a Cartier divisor. Its Cartier index is the smallest such integer. It is ample in the Q-Cartier sense when such a multiple is an ample Cartier divisor. This need not make the original divisor itself Cartier.

Apparent weight measures support rather than the gravitational force itself. In an ideal scale model, it is the magnitude $N$ of the normal force exerted on the body. A scale calibrated to display mass divides that support force by its calibration value of $g$.

Take upward as positive in a constant downward gravitational acceleration of magnitude $g$. If a body stays in contact with a horizontal scale and accelerates upward at $a$, Newton's second law gives $N-mg=ma$, hence $N=m(g+a)$. At rest, $N=mg$. In common free fall, $a=-g$, so $N=0$ despite the nonzero gravitational weight $mg$. If the formula would require $N<0$, ordinary supporting contact cannot realize it: the body loses contact instead.

Exterior multiplication by a fixed differential form is the linear operator taking the wedge product of differential forms on the left. For a one-form $\omega$, it raises degree by one. Anticommutativity gives $e(\omega )e(\eta )+e(\eta )e(\omega )=0$ for one-forms. On the exterior algebra of an inner product space, pairing this operation with contraction produces Clifford multiplication $c(v)=e(v^{♭})-{\iota }_v$.

The Szlenk index measures how many transfinite Szlenk derivations are needed to empty the dual unit ball at each positive norm-diameter scale. At a successor ordinal take another derivation; at a limit ordinal intersect the previous sets. Taking the supremum of the emptying ordinals over the positive scales gives an ordinal invariant of the Banach space.

The weak neutral current couples the Z boson to a fermion without changing its electric charge. Its vector and axial-vector coefficients depend on weak isospin, electric charge, and the weak mixing angle.

The neutral current transfers no electric charge at a fermion vertex; “neutral” does not mean that every participating particle is uncharged. For example, ${\nu }_{\mu }{e}^{-}\to {\nu }_{\mu }{e}^{-}$ transfers energy and momentum to an electron through Z boson exchange. By contrast, a charged current can convert a muon neutrino into a muon, as in ${\nu }_{\mu }n\to {\mu }^{-}p$. Both processes conserve total electric charge. A photon also carries no electric charge, but an active neutrino has no electromagnetic charge coupling: its neutral current is a distinct weak interaction.

The experimentally useful contrast on a nucleon target iswhere $N$ is a nucleon and $X$ is an allowed final state of hadrons. The charged current makes a charged lepton available for identification; the outgoing neutrino in a neutral current leaves no bubble-chamber track. A sample of hadronic events without an identified muon therefore poses an experimental question: how many could instead be neutron interactions, or charged current events whose muon was missed? The answer requires a detector-response and background analysis, not merely an inspection for an absent line.

The weak charged current couples a W boson to two members of a left-handed electroweak doublet. In the quark mass basis its flavor coefficients are entries of the Cabibbo-Kobayashi-Maskawa matrix.

The trace of the angular derivative of the reduced deflection defines convergence in the standard thin gravitational lens equation convention. It equals projected surface mass density divided by the critical surface density, ${\Sigma }_{\mathrm{crit}}=c^2{D}_s/(4\pi G{D}_d{D}_{ds})$. If deflection is instead defined as source minus image angle, the divergence relation has a minus sign.

The trace-free part of the angular lensing-potential Hessian stretches an image anisotropically. The source-image Jacobian matrix has eigenvalues $1-\kappa \pm |\gamma |$. Thus a circular source becomes an ellipse in the local linear approximation; the orientation follows the shear axes.

Cosmic shear is the statistically correlated lensing shear of distant galaxies produced by the intervening large-scale matter distribution. Correlations across angle and source redshift constrain clustering, the growth of structure and distance geometry, thereby testing dark matter and dark energy. Intrinsic galaxy alignments and measurement errors must be separated from the lensing signal.