For centered square-integrable random variables $X_1,\dots ,X_T$, their finite-past innovations are the successive errors of best linear prediction from a finite past:Here the inner product is $⟨X,Y⟩=\mathbb{E}[XY]$ in the Hilbert space of square-integrable random variables. By the defining property of orthogonal projection, $U_t$ is orthogonal to every earlier observation, hence to every earlier innovation. Recursively, each $X_t$ is $U_t$ plus a linear combination of earlier innovations. ThusThis equality remains true when some innovations vanish. Only the nonzero innovations form an orthogonal basis: if $X_2=X_1$, then $U_2=0$, and a formula dividing by $\mathrm{var}(U_2)$ is invalid. When all innovation variances are positive, the construction is the Gram-Schmidt process applied to the observations. Finite-past innovations depend on the starting time; unlike innovations from best linear prediction from an infinite past, they do not use observations before initialization. Additional stability assumptions are needed to assert that this initialization effect disappears.

For a strictly decreasing positive sequence ${ϵ}_n\to 0$, put $a_0={ϵ}_0-{ϵ}_1$ and $a_k={ϵ}_{3^{k-1}}-{ϵ}_{3^k}$ for $k\ge 1$. These are positive with sum ${ϵ}_0$. The positive lacunary Chebyshev series $f_{ϵ}={\sum }_k{a}_k{T}_{3^k}$ has error ${ϵ}_0$ at degree zero and error ${ϵ}_{3^{K-1}}\ge {ϵ}_n$ whenever $3^{K-1}\le n<3^K$. This proves the lower-bound form of Bernstein's lethargy theorem with an explicit continuous function.

For outcomes in $\{0,\dots ,d-1\}$, let $[x{]}_d$ be their nonnegative modular representative. A deterministic local hidden-variable model satisfiesThe unreduced chain telescopes to $-1$, so its reduced nonnegative sum is at least $d-1$ for each assignment. For binary outcomes and two settings, it is a form of the CHSH inequality: $I_{2,2}=2-(E_{11}+E_{21}+E_{22}-E_{12})/2$.

The Bell states are joint eigenstates of commuting $Z_A{Z}_B$ and $X_A{X}_B$. Two shared Bell state meter pairs realize their two entanglement-assisted nondemolition parity measurements using spacelike local circuits. Their projector products give the four rank-one Bell state outcomes and preserve each input Bell state. The globally identified outcome requires later local operations and classical communication.

At a regular sonic point, differentiating the vanishing coefficient and numerator yields $(2+C/c_s^2)(u^{\prime }{)}^2+4C{u}^{\prime }/(ru)+(4C+2c_s^2)/r^2+{\Phi }^{\prime \prime }=0$. A real slope compatible with the desired branch is needed in addition to the sonic equalities. For constant sound speed this reduces to $(u^{\prime }{)}^2=-c_s^2/r^2-{\Phi }^{\prime \prime }/2$.

For the nearest integer $a$ to $q\sqrt{2}$, the nonzero integer $2q^2-a^2$ has absolute value at least one. Dividing by $q\sqrt{2}+a\le 4q$ proves the bound. Thus the points $0,\sqrt{2},\dots ,W\sqrt{2}$ modulo one are separated by at least $1/(4W)$. Ordering their distances from zero gives ${\sum }_{h\le W}\Vert h\sqrt{2}{\Vert }^{-1}\ll W\log (2W)$.

The axiom of countable choice allows one to choose an element from each member of a countable family of nonempty sets. It is a restricted form of the axiom of choice. In the usual proof that a countable union of countable sets is countable, it suffices to choose one enumerating injection for each set; if those enumerations are already supplied, that proof requires no further choice.

An invertible time-series representation reconstructs the driving white noise from present and past observations. The stable convention requires absolute summability of the inverse coefficients. A convergent bilateral inverse using future observations does not establish this one-sided property.

For a minimal ARMA representation $\varphi (B)X=\theta (B)ϵ$, causality requires every zero of $\varphi$ to lie strictly outside the unit disk, and invertibility requires the same of $\theta$. This makes the transfer function and its reciprocal analytic on a disk larger than the unit disk, giving geometrically decreasing one-sided filter coefficients. Cancel common factors before applying the criterion to the noise-driven representation.

Reflecting zeros across the unit circle can convert a noncausal or noninvertible ARMA representation to a causal invertible one with the same time-series spectral density. Scale the driving white noise variance according to the modulus factors. Define the new noise as a filter of the actual process; equality of two spectra alone proves equality of second-order structure, not equality of non-Gaussian laws. The resulting noise is weak white noise and need not be independent across time.

Adding independent white noise of variance $w$ to a process with transfer function $(1+\theta z)/(1-\varphi z)$ and driving variance $v$ gives numerator $A+2C\cos \omega$ in its rational time-series spectral density. Put $P=A+2C$ and $Q=A-2C$. If both are positive, the invertible moving-average factor has coefficient $\alpha =(\sqrt{P}-\sqrt{Q})/(\sqrt{P}+\sqrt{Q})$ and driving variance $\lambda =(\sqrt{P}+\sqrt{Q}{)}^2/4$. Filtering the actual sum by $(1+\alpha B{)}^{-1}(1-\varphi B)$ produces its weak white noise driver.

With the convention $\Phi (B)X_t=\Theta (B){\epsilon }_t$, the one-sided spectral density is ${\sigma }^2|\Theta (e^{i\omega }){|}^2/(\pi |\Phi (e^{i\omega }){|}^2)$ whenever the stationary filter is well defined. Stable autoregressive roots give the causal representation; moving-average invertibility is a separate property.

With positive moving-average sign and $|\varphi |<1$, the noise coefficients are ${\psi }_0=1$ and ${\psi }_j=(\varphi +\theta ){\varphi }^{j-1}$ for $j\ge 1$. Summing their overlap gives ${\gamma }_0={\sigma }^2(1+{\theta }^2+2\varphi \theta )/(1-{\varphi }^2)$ and ${\gamma }_k={\sigma }^2(\varphi +\theta )(1+\varphi \theta ){\varphi }^{k-1}/(1-{\varphi }^2)$ for $k\ge 1$. Negative lags follow by symmetry.

Every nonzero commutative Artinian ring is a finite product of Artinian local rings. The primary components of zero have distinct maximal radicals and are pairwise comaximal ideals, so the Chinese remainder theorem gives the product. Each local factor has nilpotent maximal ideal and finite composition length. These factors and the components of zero are unique.

For $H=J{\sum }_{⟨i,j⟩}{\mathbf{S}}_i\cdot {\mathbf{S}}_j$, with $J>0$, dimensionless spin length $S$ and lattice spacing $a$ on a $d$-dimensional hypercubic lattice, the linear spin-wave approximation givesThe angular frequency is ${\omega }_{\mathbf{k}}=E_{\mathbf{k}}/\hslash$. Near a gapless point, $E_{\mathbf{k}}\simeq 2JSa\sqrt{d}|\mathbf{k}|$, so the physical spin-wave velocity is $2JSa\sqrt{d}/\hslash$. Writing the energy formula directly as an angular frequency assumes $\hslash =1$.

The zero-temperature spin-boson occupation is $[1/\sqrt{1-{\gamma }_{\mathbf{k}}^2}-1]/2$, producing quantum depletion of Néel order. Exact zero modes need an infrared cutoff or separate global-rotation treatment. The energy, angular frequency and spin normalization must remain consistent when comparing different conventions.

Let $|I,I⟩$ be a normalized highest-weight vector in an irreducible spin representation, with the dimensional angular momentum lowering operator $J_{-}=J_x-i{J}_y$. ThenThe factor ${\hslash }^{-(I-m)}$ is necessary because each application of $J_{-}$ carries one unit of angular momentum. Equivalently, define the dimensionless lowering operator $j_{-}=J_{-}/\hslash$ and use $j_{-}^{I-m}$.

To verify the normalization, set $n=I-m$. Successive lowering multiplies the squared norm byThe displayed prefactor and power of $\hslash$ cancel this product exactly. All factorial arguments are nonnegative integers even when $I$ is half-integral.

For $I=1$, $m=0$, the result is $|1,0⟩=J_{-}|1,1⟩/(\sqrt{2}\hslash )$. Choosing positive lowering coefficients fixes a consistent phase convention used in Clebsch-Gordan coefficients and isospin rotations.

Any at-most-$\kappa$ subfamily of an almost disjoint family on a regular cardinal becomes pairwise disjoint after deleting fewer than $\kappa$ points from each member. Enumerate it in length at most $\kappa$ and apply the displayed trimming. Each removed part is a union of fewer than $\kappa$ small intersections, so regular cardinal arithmetic bounds it below $\kappa$.

A quasi-projective algebraic set is a locally closed subset of projective space in the Zariski topology: it is an open subset of a closed algebraic subset. Equivalently it is the intersection of a Zariski-open subset and a Zariski-closed subset of the same projective space. Reducibility is allowed; an irreducible such space is a quasi-projective variety.

Every projective variety is quasi-projective, by taking the whole closed subset as the open part. Every affine variety is also quasi-projective through its usual affine chart in projective space. Quasi-projective does not mean affine: the projective line itself is quasi-projective but not affine, since all its global regular functions are constant whereas it has more than one point. Its complement of one point is an affine line.

A transcendence basis of a field extension $L/k$ is an algebraically independent set $T$ such that $L/k(T)$ is algebraic. Its cardinality is the transcendence degree. For finitely generated extensions it is finite, and the remaining algebraic extension is finite.

The probability distribution of the nonnegative integer number of claims over a specified accounting period. Its probability generating function determines the count probabilities. A Poisson distribution gives equal count expected value and variance; a Poisson mixture allows heterogeneity in the underlying risk intensity.