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The Kähler potential is a real function integrated over full superspace. The canonical choice $K={\Phi }^{†}\Phi$ gives canonical kinetic terms.

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 of functions on a space, evaluation at $x$ is the character ${\delta }_x(f)=f(x)$.

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 combines a coordinatewise certificate with tensorization of entropy to bound the moment-generating function of a certifiable random quantity and hence its tails.

This is an application of Talagrand's concentration inequality to certifiable functions. Talagrand's product-space inequality gives sub-Gaussian-type concentration for a bounded-differences, certifiable function, with a variance scale controlled by the certificate size.

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 $S^{n-1}\to S^{n-1}$.