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The cosmic web is the network of matter sheets, filaments, nodes, and intervening voids produced by anisotropic gravitational collapse.

After photon decoupling in an expanding universe, every frequency scales as $\nu \propto a^{-1}$. Therefore a Planck occupation factor retains its thermal form when its temperature is assigned the same scaling,

Observed time intervals from a comoving source are stretched by the same factor $1+z$ as photon wavelengths.

A mode crossing during matter domination acquires the growth factor $({\tau }_0/{\tau }_H{)}^2$. An initial amplitude proportional to ${\tau }_H^2{k}^{1/2}$ therefore becomes proportional to $k^{1/2}$ today.

With $a(t_0)=1$, $k_0=2\pi /(c{t}_0)$, and $1+z_{\mathrm{eq}}=(t_0/t_{\mathrm{eq}}{)}^{2/3}$,

If primordial horizon-crossing perturbations satisfy $V⟨|{\delta }_k{|}^2⟩=C/k^3$, remain frozen during radiation domination, and grow as $a$ during matter domination, then

In a decelerating expanding universe, $|\Omega -1|$ grows, so its small present value requires extremely fine-tuned early initial data unless an earlier mechanism drives $\Omega$ toward one.

For subgroups $H,P\le G$, the double coset of $x\in G$ isThe double cosets partition $G$. The action of $H$ on the left cosets of $P$ givesby the orbit-stabilizer theorem.

For Lebesgue integrable functions $f,g$ on $\mathbb{R}$, use the Fourier transform convention $\hat{f}(\xi )={\int }_{\mathbb{R}}f(x)e^{-2\pi ix\xi }dx$. ThenThus the integral combining all translated products becomes an ordinary multiplication at each frequency. Different Fourier transform normalizations can place a constant in this identity; fixing the convention prevents that ambiguity. Its use in filtering is illustrated by ^[ref].

For the indicator function $q=1_{[0,1]}$,with $\mathrm{sinc}(0)=1$. Therefore the triangular function $q\ast q$ has Fourier transform $e^{-2\pi i\xi }{\mathrm{sinc}}^2(\xi )$. The extra factor of the sinc function improves the large-frequency bound from a constant times $1/|\xi |$ to one times $1/|\xi {|}^2$, matching the smoother output in the overlap picture. Fourier decay of repeated interval convolutions extends this calculation to further repetitions.

For a proof outline, Fubini's theorem permits exchanging the two integrals because the integral of $|f(y)g(x-y)|$ over $(x,y)$ is $\Vert f{\Vert }_1\Vert g{\Vert }_1<\infty$. Set $u=x-y$ in the inner integral:The factorization of the complex exponential is the mechanism behind the simplification. The discrete convolution version similarly turns coefficient sums into products of generating functions.

For sequences $a,b$ indexed by the integers, discrete convolution adds products whose indices sum to the output index:For finite sequences, treat every unspecified entry as zero; the sum then has finitely many nonzero terms. For infinite sequences, a convergence hypothesis is needed, for example absolute summability of both sequences.

Take $a=(1,2,1)$ at indices $0,1,2$ and $b=(1,1)$ at indices $0,1$. Their convolution is $(1,3,3,1)$ at indices $0,1,2,3$:Equivalently, $c_n=a_n+a_{n-1}$: add an unshifted and a one-step shifted copy of $a$. This explains the output by superposition, with every index included.

The same products appear in polynomial multiplication and generating functions:A term $a_m{z}^m$ times $b_k{z}^k$ contributes to coefficient $m+k$, precisely the index condition in discrete convolution. In probability, $a$ counts the outcomes of two independent fair coin tosses by their number of heads, while $b$ counts the outcomes of one more toss. Dividing the output by eight gives the probability mass function $(1,3,3,1)/8$ of the binomial distribution with $n=3$ and $p=1/2$. No new arithmetic is needed to pass from coefficient multiplication to counting independent sums.

The full finite convolution is different from circular convolution, where indices wrap modulo a chosen length. Wrapping this four-term output to length three adds its last entry to its first and gives $(2,3,3)$. When using a discrete Fourier transform to compute the full convolution, zero padding to length at least the sum of the input lengths minus one prevents this wraparound. ^[ref]

An approximate identity is a family of integrable kernels $K_{\epsilon }$ whose total mass is one and whose mass concentrates near zero as $\epsilon \to 0$. Under standard hypotheses, $f\ast K_{\epsilon }$ converges to $f$ in norm and at almost every Lebesgue point.

A vector is a subgradient when its supporting affine function lies below the convex function.

The perspective of f is g(t,x)=t f(x/t) for positive t and preserves convexity.

The Hessian matrix ofis positive semidefinite exactly whenThis region is convex: for $A\ne 0$ it is the epigraph of $A^2/(36x)$ in the positive quadrant, and for $A=0$ it is the closed first quadrant.

A function $f$ on a convex set is strictly convex whenwhenever $x\ne y$ and $0<t<1$.

A differentiable function is $\alpha$-strongly convex when

The pointwise maximum of finitely many convex functions is convex because

A finite convex function on an open convex subset of a finite-dimensional real vector space is locally Lipschitz, and therefore continuous. In one dimension this follows by bounding its secant slopes between two slightly wider endpoints.

If a convex subset of ${\mathbb{R}}^n$ is invariant under cyclic coordinate permutation, averaging the $n$ cyclic images of $x$ shows thatalso belongs to the set.

Binomial distributions with trial count tending to infinity and success probability tending to zero converge to a Poisson law when their product converges.