If $g$ is smooth and $\nabla g\ne 0$ on $S=\{g=0\}$, transverse local coordinates give $⟨\delta (g),\phi ⟩={\int }_S\phi /|\nabla g|dS$. Thus ${\delta }_S=|\nabla g|\delta (g)$ as distributions. In particular ${\delta }_{S^1}=2\delta (|x{|}^2-1)$. Omitting this Jacobian confuses the delta of a defining function with a geometric surface delta distribution.

For the geometric surface delta distribution on the radius-$a$ sphere in ${\mathbb{R}}^3$, $⟨u_a,\varphi ⟩=a^2{\int }_{S^2}\varphi (a\omega )d\sigma (\omega )$. Angular integration gives the Fourier transform ${\hat{u}}_a(\xi )=4\pi a\sin (a|\xi |)/|\xi |$, with removable value $4\pi a^2$ at zero. The convolution of distributions with a compactly supported factor gives the regular distributionThis density has total mass $16{\pi }^2{a}^2{b}^2$. Values on endpoint spheres do not affect the distribution. When $a=b$, the inverse-distance singularity is locally integrable in three dimensions and is not an additional point mass. The annulus expresses the triangle inequality for the sum of two vectors of fixed lengths.

An effort support is the support of a probability distribution used to randomize effort in a contest. The all-pay indifference equation with random entry determines this support together with its distribution function.

A four-dimensional massive $\mathcal{N}=1$ vector multiplet contains one spin-one particle, two spin-one-half particles and one real spin-zero particle. Its on-shell state counts are $3+1=4$ bosonic and $2+2=4$ fermionic. These masses agree in an unbroken supersymmetric vacuum. The supersymmetric Higgs mechanism assembles it from a massless vector multiplet and one chiral superfield per broken gauge generator.

For an impulsive supernova kick in a binary star initially in a circular Kepler orbit, let $\beta$ be the remaining total mass fraction and $\alpha$ the kick speed divided by the original relative speed. The displayed post-kick specific orbital energy is negative for a bound pair and positive for escape. Over all directions, the velocity factor ranges from $(1-\alpha {)}^2$ to $(1+\alpha {)}^2$. Some direction leaves a bound pair exactly when $2\beta >(1-\alpha {)}^2$; some direction permits positive-energy escape exactly when $2\beta <(1+\alpha {)}^2$. Equalities are marginal parabolic Kepler orbits. The circular initial orbit and impulsive approximation are essential.

For a positive stock satisfying $dS=S\sigma dW$, the Itô formula gives $d\sqrt{S}=\frac{1}{2}\sqrt{S}\sigma dW-\frac{1}{8}\sqrt{S}{\sigma }^{2}dt$. Localization and the conditional Fatou lemma make this nonnegative local supermartingale a true supermartingale.

With left Grassmann derivatives, let $Q_{\alpha }={\partial }_{\alpha }-i{\sigma }_{\alpha \dot{\beta }}^{\mu }{\bar{\theta }}^{\dot{\beta }}{\partial }_{\mu }$ and ${\bar{Q}}_{\dot{\alpha }}={\bar{\partial }}_{\dot{\alpha }}-i{\theta }^{\beta }{\sigma }_{\beta \dot{\alpha }}^{\mu }{\partial }_{\mu }$. Left-placed odd parameters generate $\delta \theta =ϵ$, $\delta \bar{\theta }=\overline{ϵ}$ and $\delta x^{\mu }=i\theta {\sigma }^{\mu }\overline{ϵ}-iϵ{\sigma }^{\mu }\bar{\theta }$. A scalar superfield transforms by pullback under this superspace translation. The parameter-weighted variation is an even derivation, although each supercharge is odd.

At fixed scale factor and unchanged relativistic counts, instantaneous conversion of a relic into thermal radiation gives the displayed entropy-density enhancement. It exceeds one because the gas receives energy from outside its prior equilibrium sector. This does not contradict cosmological entropy conservation, whose source-free reversible assumptions fail during decay and thermalization.

A category is well-copowered if, for each object $C$, the epimorphisms $C\to Q$ have only a set of equivalence classes, where two are equivalent through an isomorphism of their codomains compatible with the quotient maps. Equivalently, its opposite category is well-powered.

In a complete category, the intersection of a set-indexed family of subobjects of $C$ is their wide pullback in a category over $C$. The resulting morphism into $C$ is a monomorphism, and it factors through every member. In a well-powered category, even the intersection of all subobjects satisfying a specified property is small.

An irradiated grain loses solid material by sublimation. Its lifetime depends strongly on temperature and composition. In a planetary dust tail, a short lifetime can truncate the distribution well before a relative orbital wrap.

A full subcategory contains all ambient morphisms between its chosen objects. Its inclusion is a fully faithful functor.

A replete subcategory contains every ambient object isomorphic to any of its objects. This closure is at the object level; fullness is a separate property.

For this quotient, homotopy groups through degree six and in degree eight vanish. The degree-seven and degree-nine groups are $\mathbb{Z}/2$. A map to $K(\mathbb{Z}/2,7)$ induces an isomorphism on degree-nine integral homology because the relevant square of the bottom mod-two class is nonzero. Two relative Hurewicz steps then compute degrees eight and nine.

Anchor a countable dense subset of a closed c-cyclically monotone support. Take the infimum of accumulated differences $c(x_{j+1},y_j)-c(x_j,y_j)$ along chains ending at a variable point. Cyclical monotonicity bounds this potential below on the support projection. Its cost transform gives the other potential, with equality on the support. Continuity makes both potentials upper semicontinuous and thus Borel.

For a nonnegative cost, simultaneous symmetric clipping preserves feasibility of a pair of Kantorovich potentials. On their equality set the clipped sums are nonnegative and increase to the cost. Their bounded integrals are fixed by the marginals, so monotone convergence theorem proves optimality without subtracting undefined infinite marginal integrals.

The leading massless-sector spacetime action is proportional to $\int \sqrt{-G}{e}^{-2\Phi }[R+4(\nabla \Phi {)}^2-H^2/12]$, with $H=dB$. The backgrounds are the string-frame metric, dilaton, and Kalb–Ramond field. Massive string exchange generates higher-derivative corrections ordered by ${\alpha }^{\prime }$. This perturbative bosonic massless-sector truncation does not remove the tachyon instability.

The canonical commutation relations imply $[N,{\alpha }_{-k}^i]=k{\alpha }_{-k}^i$ for $N={\sum }_{k>0}{\alpha }_{-k}\cdot {\alpha }_k$. With zero level assigned to the oscillator vacuum, a Fock space basis state containing $r_{ki}$ creation operators of each mode has level ${\sum }_{k,i}k{r}_{ki}$. The level is therefore a nonnegative integer; it counts oscillator excitation weighted by frequency, not simply the number of creation operators.

A stationary path law is ergodic when every time-shift-invariant event has probability zero or one. The Birkhoff ergodic theorem then identifies integrable time averages with deterministic ensemble expectations. A shared random scale multiplying iid noise gives a stationary counterexample: it is uncorrelated across distinct times, but the path retains information about the random scale.

Let ${\mathcal{F}}_{-\infty }^0$ and ${\mathcal{F}}_n^{\infty }$ be the sigma-algebras generated by the past and future observations. The mixing coefficient is $\alpha (n)={\sup }_{A\in {\mathcal{F}}_{-\infty }^0,B\in {\mathcal{F}}_n^{\infty }}|P(A\cap B)-P(A)P(B)|$. Strong mixing means $\alpha (n)\to 0$. Appropriate quantitative mixing and moment conditions can imply a central limit theorem; absolute summability of autocovariance alone does not.