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A superpotential is a holomorphic function of chiral superfields integrated over chiral superspace, $\int d^2\theta W+h.c.$

An auxiliary field has no kinetic term and can be eliminated algebraically. The complex field $F$ in a chiral superfield balances bosonic and fermionic degrees of freedom off shell.

For a reversible chain with spectral gap $\gamma$ and Cheeger constant $\Phi$,

A non-characteristic hypersurface satisfies $p(x,d\varphi (x))\ne 0$. This lets the equation solve for the highest derivative normal to the hypersurface.

Replace the original halt state by a continuation state that tests whether its one-letter output is $0$ or $1$, removes it, appends the other symbol, and enters a new halt state. This computes the complement language without changing termination behavior.

To recognize words ending in a fixed symbol $a$, test the final input letter, choose one of two erasing loops, and append $1$ after the accepting loop or $0$ after the rejecting loop. The empty word follows the rejecting branch.

Two string registers can recognize $\{a^n{b}^n:n>0\}$ by storing one marker per $a$ and removing one per $b$. A machine with only its input register cannot: two different terminal $b$-block lengths eventually reach the same state and remaining prefix, forcing equal outputs for one member and one nonmember.

Gradient blow-up means that a solution stays bounded while the norm of a spatial derivative tends to infinity. In a scalar conservation law this happens as characteristics meet, because the derivative contains the reciprocal of the characteristic-map Jacobian.

For an algebra $A$ of complex-valued functions on a set, evaluation at $x$ is the multiplicative linear functional ${\delta }_x(f)=f(x)$. It is an algebra character precisely when it is nonzero, meaning that some $f\in A$ has $f(x)\ne 0$. This holds automatically if $A$ contains the constant function one.

The nonzero requirement matters for algebras with a common zero: evaluating $\{f\in C[0,1]:f(0)=0\}$ at zero gives the zero linear functional, which is not an algebra character.

The Gelfand topology on ${\Phi }_A$ is the weak-star topology inherited from $A^{\ast }$, equivalently the coarsest topology making every map $\phi \mapsto \phi (a)$ continuous.

This is the topology on the character space used in the Gelfand representation.

Hyperspherical coordinates extend polar coordinates to $n$ dimensions with one radius and $n-1$ angles.

Polar coordinates use $x=r\cos \theta$, $y=r\sin \theta$ and area element $rdrd\theta$.

For twice differentiable functions,

At zero, $f(x)=\sqrt[3]{x}$ is not differentiable and $g(y)=y^3$ is differentiable, while $g\circ f(x)=x$ has derivative one.

Chain maps $f,g:C\to C^{\prime }$ are chain homotopic when there are maps $h_i:C_i\to C_{i+1}^{\prime }$ such thatChain-homotopic maps induce the same map on homology.

A chain map sends cycles to cycles and boundaries to boundaries, and therefore induces $f_{\ast }:H_i(C)\to H_i(C^{\prime })$ by $[x]\mapsto [f_i(x)]$.

The entropy method for certifiable functions turns short certificates into bounds on coordinate sensitivity. Suppose a nonnegative integer-valued certifiable function $f$ changes by at most $L$ when one coordinate changes and has certificates of size at most $g(k)$ at value $k$. Define the coordinate-deleted functionsassuming these functions are measurable. At an input of value $k$, changing a coordinate outside a chosen certificate cannot reduce the value below $k$. Hence $f_i=f$ outside that certificate, while $0\le f-f_i\le L$ on it. ConsequentlyFor independent random variables, the Efron–Stein inequality already gives $\mathrm{Var}(f(X))\le L^2\mathbb{E}g(f(X))$. When $L=1$ and $g(k)=k$, the stronger bound ${\sum }_i(f-f_i)\le f$ makes $f$ a self-bounding function. Tensorization of entropy and a modified logarithmic Sobolev inequality control exponential rather than quadratic variation, producing concentration inequalities. The certificate size alone needs this coordinate-change bound; an arbitrarily large change in one coordinate is not controlled by a short certificate.

For a nonnegative integer-valued certifiable function $Z=f(X)$ of independent random variables, suppose changing one coordinate changes $f$ by at most $L>0$ and a value $f(x)\ge b$ has a certificate involving at most $rb$ coordinates, with $r>0$. Then for $0\le a<b$,For a high-value input $x$, every low-value input must disagree with its certificate on at least $(b-a)/L$ coordinates: replacing its certificate positions by those of $x$ would change the low value by less than $b-a$ but force a value at least $b$. Uniform weights $1/\sqrt{rb}$ on certificate positions therefore give Talagrand convex distance at least $(b-a)/(L\sqrt{rb})$ from the low-value event. Apply Talagrand's convex distance inequality to that event.

The bound is useful when one of the two events has known nonnegligible probability, for example at a median. A certificate alone supplies no coordinate-change constant $L$; both hypotheses are used in the proof. The two-threshold concentration for certifiable functions article develops the resulting mean-median comparison.

The coefficient of an $(n-1)$-cell in the cellular boundary of an $n$-cell is the degree of the attaching map after collapsing the complement of that $(n-1)$-cell to obtain a map $S^{n-1}\to S^{n-1}$.

A linear contrast is a coefficient-weighted sum of cell means whose coefficients total zero, used to test interpretable differences among groups.