In a unit-spacing periodic spinless chain, self-paired momenta are $k=0$ and, when available, $k=\pi$. An anomalous pair of the same creation operator vanishes by the canonical anticommutation relations. Such a mode is a single normal number level, filled or empty according to its energy sign, rather than a two-distinct-mode Bogoliubov transformation. This excludes the paired-angle construction at those special momenta.

For an affine variety with coordinate ring $A$ acted on by a finite group $G$, the affine quotient is $\mathrm{Spec}{A}^G$. Elements of $A$ satisfy monic orbit polynomials over $A^G$, so the quotient map is a finite morphism. Over an algebraically closed field, invariant functions separate finite orbits; in characteristic not dividing $|G|$, interpolate and average a function separating two orbits. The quotient fibres are consequently the orbits.

The Engquist-Osher method is the finite volume method using the Engquist-Osher flux. Under $(k/d)\sup |f\prime |\le 1$ on the invariant state interval, its update is monotone. It preserves the state range, is contractive for summable perturbations in $L^1$, and decreases discrete total variation.

Let $T$ be a monotone conservative scalar update on a periodic grid. Since $T(U\vee V)\ge TU,TV$, one has $(TU-TV{)}_{+}\le T(U\vee V)-TV$. Conservation makes the sum of the right side equal to $\sum (U-V{)}_{+}$. Reverse $U,V$ and add to obtain $\Vert TU-TV{\Vert }_1\le \Vert U-V{\Vert }_1$. On an infinite grid the same proof holds for summable differences with a Lipschitz flux on the state range.

A total variation diminishing scheme is a numerical update $T$ that satisfiesThe total variation measures accumulated changes across adjacent cells. The displayed property is the definition; monotonicity is one sufficient condition, rather than an equivalent requirement.

For a translation-invariant monotone conservative scheme, let $S$ be the one-cell shift. Since $TS=ST$, L1 contraction of a monotone conservative scheme applied to $U,SU$ givesThis proof applies on a periodic grid, or with the appropriate summability conditions on an infinite grid. It explains why that class of finite volume methods cannot increase total variation. The property controls oscillation, but a numerical shock wave can still be spread over several cells.

For $V=C_p(\Phi /M_{\mathrm{Pl}}{)}^p$ and the leading endpoint estimate ${ϵ}_V=1$, the slow-roll approximation gives ${\Phi }_N^2/M_{\mathrm{Pl}}^2=2pN+p^2/2$ and ${ϵ}_V=p/(4N+p)$. Thus ${\mathcal{P}}_{\mathcal{R}}=C_p(2pN+p^2/2{)}^{p/2}(4N+p)/(24{\pi }^2{M}_{\mathrm{Pl}}^{4}p)$ and the scalar spectral index obeys $n_s-1=-2(p+2)/(4N+p)$. The coefficient sets the amplitude, whereas the monomial power and exit e-fold count set the leading tilt. Changing the power without specifying the coefficient leaves the overall normalization undetermined.

For any commutative ring $A$ and a monic polynomial $F$ of positive degree $D$, monic polynomial division over a ring gives unique representatives of degree less than $D$. Thus the classes of $1,T,\dots ,T^{D-1}$ form a basis of a module over $A$. This is a finite free algebra, hence flat, and the map $A\to A[T]/(F)$ is injective. The monic hypothesis makes leading-degree arguments valid even when $A$ has zero divisors.

Fix a monad on $\mathcal{C}$. Objects are adjunctions with left-hand category $\mathcal{C}$ whose induced monads are identified with the fixed monad. A morphism between two such adjunctions is a functor between their right-hand categories commuting with both adjoints and respecting the specified units and counits. Strict identifications give a category; coherent identifications give the analogous universal property up to compatible natural isomorphism.

Let $X_1,X_2,\dots$ be independent and identically distributed nonnegative random variables. Fix $t>0$ with $\mathbb{E}{e}^{t{X}_1}<\infty$, and defineThen the empirical moment-generating function is uniformly strongly consistent on this interval:The dominated convergence theorem, with envelope $e^{t{X}_1}$, makes $m$ continuous on $[0,t]$. Both $m$ and every ${\hat{m}}_n$ are nondecreasing because the observations are nonnegative. Given $\epsilon >0$, choose a finite grid $0=s_0<\cdots <s_k=t$ whose successive $m$ increments are at most $\epsilon$. The strong law of large numbers holds simultaneously at its finitely many endpoints. Squeezing between adjacent endpoints givesUse a countable sequence $\epsilon \downarrow 0$ to conclude. continuity of the sample and population functions also makes the supremum measurable by reducing it to rational parameter values. This is the finite-bracketing mechanism behind the uniform strong law from finite L1 bracketing. The iid sampling assumption cannot simply be omitted: repeatedly observing the same nonconstant random variable would not create averaging.

Suppose a positive function $M$ satisfies the displayed fourth-power identity and $\log M(h)=h^2/2+o(h^2)$ at zero. Iteration gives $\log M(t)=4^m\log M(t/2^m)$, which tends to $t^2/2$. A related first-power symmetry step is: if $\psi (t)=\psi (t/2{)}^2$ and $\psi (h)=1+o(h^2)$, then $\log \psi (t)=2^m\log \psi (t/2^m)\to 0$. The powers $4^m$ and $2^m$ must not be interchanged in the Gaussian characterization by independent sum and difference.

Two disjoint ordinary circle boundaries can be taken to distinct concentric circles by a Möbius transformation. After their centres are put at $0,d$ with radii $a,b$, the map $(z-s)/(z-t)$ works when $st=a^2$ and $s+t=(d^2+a^2-b^2)/d$. Disjointness makes the quadratic have distinct real roots. Already concentric circles require no transformation.

The special linear group $S{L}_2(\mathbb{R})$ acting by Möbius transformations has exactly three group orbits on the Riemann sphere: the extended real line and the two open half-planes. The identity $\mathrm{Im}((az+b)/(cz+d))=\mathrm{Im}z/|cz+d{|}^2$ preserves these sets. Upper triangular determinant-one matrices send $i$ to any prescribed point of the complex upper half-plane, and similarly send $-i$ through the lower half-plane. The stabilizer subgroups of $i$ and $-i$ are both $SO(2)$.

A finite mixture distribution of exponential distributions with rates $r_i>0$ and nonnegative weights summing to one. Its expected value is ${\sum }_i{w}_i/r_i$ and its moment-generating function is ${\sum }_i{w}_i{r}_i/(r_i-t)$ on the common finite domain. The mixture chooses one component for each observation; it is not a sum of independent exponential variables.

Let $\psi$ be the second Chebyshev function. SupposeThen the Mertens function satisfies $M(x)\ll x/\log x$ for $x\ge 2$.

First, the Möbius divisor-sum identity gives ${\sum }_{d\le x}\mu (d)\lfloor x/d\rfloor =1$. Replacing each floor function by its input with error at most one shows ${\sum }_{d\le x}\mu (d)/d=O(1)$. The exact Dirichlet convolution identity $\mu \log =-\mu \ast \Lambda$, with $\Lambda$ the Von Mangoldt function, now givesIndeed, the main term is $-x{\sum }_{d\le x}\mu (d)/d=O(x)$. The error is at most a constant times $x{\sum }_{d\le x}|E(x/d)|/d$. On each interval where $2^j\le x/d<2^{j+1}$, the harmonic sum of $1/d$ is bounded by an absolute constant, so the assumed convergence of the series bounds the total error by $O(x)$.

Finally,because ${\sum }_{n\le x}\log (x/n)=O(x)$ by integral comparison. The classical Prime number theorem error $E(u)=e^{-c\sqrt{\log u}}$ satisfies the dyadic condition. Thus the Mertens function estimate follows directly from the second Chebyshev function error, without a separate bound for the reciprocal Riemann zeta function.

For a finite first absolute moment, the minimizers of expected absolute loss are the medians. With a density, differentiating the loss gives $2F(a)-1$, so every point where the cumulative distribution function equals one half minimizes it. A density gap may give a whole minimizing interval. In contrast, finite squared loss is uniquely minimized by the expectation.

The spherical derivative compares the mass of shrinking centered open balls with their Lebesgue measure, when the limit exists. This is a density of one measure relative to Euclidean volume, rather than the relative density of a set within one fixed measure. A singular finite Borel measure has spherical derivative zero Lebesgue almost everywhere.

A measure on a topological space is outer regular on its measurable sets when the displayed open-cover identity holds. Lebesgue measure has this property. It allows a null set to be placed in open neighbourhoods of arbitrarily small measure. For Lebesgue measure in Euclidean space, outer regularity also lets a compact set of small measure receive a smooth cutoff with a comparably small integral.

For a finite positive measure on the real line, this convention for its Stieltjes transform is analytic off the real line and has positive imaginary part in the upper half-plane if the measure is nonzero. Some sources use $(z-x{)}^{-1}$ instead, changing the sign. The transform of an empirical spectral measure equals the normalized trace of the Stieltjes matrix resolvent.

For $a<b$ and real-valued functions $f,g$ that are continuous on the closed interval $[a,b]$ and differentiable on the open interval $(a,b)$, there exists $\xi \in (a,b)$ withApply Rolle's theorem to the linear combination $h(x)=[f(b)-f(a)]g(x)-[g(b)-g(a)]f(x)$, which has equal endpoint values. The displayed product identity permits zero derivatives. For the ratio form, additionally assume $g^{\prime }(x)\ne 0$ for every $x\in (a,b)$. Then Rolle's theorem ensures $g(b)\ne g(a)$, so both denominators below are nonzero:Neither derivative needs to be continuous, and $f^{\prime }$ may vanish. Choosing $g(x)=x$ recovers the mean value theorem; the ratio form also gives the zero-over-zero endpoint case of L'Hôpital's rule. ^[ref]

A shifted exponential distribution has location estimate equal to the sample minimum and rate estimate given above, for a nondegenerate sample of at least two observations. The minimum exceeds the true location on average by $1/(n\lambda )$. The residual sum above the minimum is gamma with shape $n-1$, so the rate estimate has expectation $n\lambda /(n-2)$ for $n>2$ and infinite expectation at $n=2$. The estimated location plus reciprocal rate is the unbiased sample mean.