Conditional iid accepted values have the ordinary target-sample CLT at scale $\sqrt{N}$. Since $N/n\to 1/M$, the same estimator has asymptotic variance $M{\mathrm{Var}}_f\varphi$ at scale $\sqrt{n}$. The centered count has limiting variance $(1/M)(1-1/M)$. Any fixed convention on an empty output is asymptotically negligible.

An attractive inverse-square force together with radial drag $-k\dot{r}/r^2$ preserves angular momentum $h=r^2\dot{\theta }$ but modifies the Binet equation for $u=1/r$ by a first derivative. For $k/|h|<2$ and $h\ne 0$, the reciprocal radius is a decaying oscillation about $p/h^2$. Forward angular distance $\mathrm{sgn}(h)(\theta -{\theta }_0)$ makes its damping sign explicit for either orbital orientation.

Conjugation by a Controlled-Z gate leaves the $a$ bits unchanged and updates the displayed $b$ bits and phase, using the old bits. The phase occurs when the extra $Z$ from one line is moved past $X$ on the other line. These constant-size updates support Heisenberg propagation of a Pauli observable through a Clifford circuit.

For $d(a,b)=a\log (a/b)+(1-a)\log ((1-a)/(1-b))$, the second derivative with respect to $a$ is $1/[a(1-a)]\ge 4$. Its value and first derivative at $a=b$ vanish, so $d(a,b)\ge 2(a-b{)}^2$. Boundary values follow by continuity or the infinite-divergence convention. Together with the binary partition bound for relative entropy, this gives Pinsker's inequality with the natural-logarithm constant.

An initial segment of the binary order on the discrete cube contains the vertices whose integer encodings are $0,\dots ,m-1$. It is a down-set, so its internal edges and two-dimensional cube faces can be counted at their unique upper vertices by ${\sum }_{j<m}\mathrm{wt}(j)$ and ${\sum }_{j<m}\left(\genfrac{}{}{0ex}{}{\mathrm{wt}(j)}{2}\right)$.

Fix a coordinate of the Boolean hypercube, delete it in each of the two sections, and replace each section by a binary initial segment of the same size. This preserves the total size. Mathematical induction on dimension proves that it does not decrease contained edge counts; the crossing contribution is the size of the section intersection, maximized by nested initial segments.

Incompressibility and periodicity imply that the spatial mean of the normal velocity is constant with height. Zero normal flow at infinity therefore requires zero mean normal boundary velocity. In the bounded biharmonic stream function for planar Stokes flow solution, the mean streamfunction is a constant plus a term linear in height.

A bounded linear operator between Banach spaces is weakly compact exactly when its Banach-space adjoint is weakly compact. Equivalently, that adjoint is continuous from its domain's weak-star topology to its range's weak topology. For the reverse direction, the bidual condition applied to $T^{\ast }$ forces $T^{\ast \ast \ast }$ to annihilate every functional on $Y^{\ast \ast }$ vanishing on $J_{Y}Y$; the Hahn-Banach theorem then places $T^{\ast \ast }(X^{\ast \ast })$ inside $J_{Y}Y$.

The tensor of left modules over a bialgebra has action $h(u\otimes v)=\sum (h_{(1)}u)\otimes (h_{(2)}v)$. The unit module $k$ has action through the counit.

A normalized scalar map $\gamma :H\otimes H\to k$ satisfying $\sum \gamma (a_{(1)},b_{(1)})\gamma (a_{(2)}{b}_{(2)},c)=\sum \gamma (b_{(1)},c_{(1)})\gamma (a,b_{(2)}{c}_{(2)})$, and $\gamma (1,a)=\epsilon (a)=\gamma (a,1)$. Some conventions require convolution invertibility; that extra hypothesis distinguishes strong from lax tensor transformations.

For a Boolean quantum oracle whose function is $g(x,y)=a_y\cdot x$, place its target in $|-⟩$. Conjugating the oracle by a Walsh-Hadamard transform on the $x$ register translates that register by $a_y$. This is Bernstein-Vazirani phase kickback without measuring its output, so the index $y$ may remain in an arbitrary quantum superposition. A second application undoes the translation. A phase evaluation between the two calls therefore transfers a hidden-label-dependent phase to the index register while uncomputation returns the decoding register to zero.

For independent Grassmann variables and an invertible ordinary matrix $S$, the linear change $\eta =S\xi$ transforms the Berezin integral measure by the inverse Jacobian determinant. Indeed, the highest-degree monomial transforms by $\det S$, and coefficient extraction must undo that factor. For paired variables $\eta =S\xi$ and $\bar{\eta }=\bar{\xi }{S}^{-1}$ the two factors cancel. The order of odd variables and differentials fixes the overall sign and must be kept consistent.

An attractive interaction $V_{k,k^{\prime }}=-V/\mathcal{V}$ within $|{\xi }_k|,|{\xi }_{k^{\prime }}|<{\Omega }_D$ gives a constant gap inside that energy shell and zero outside. With approximately constant single-spin density of states ${\nu }_s$ per volume,The weak coupling limit is $\Delta \simeq 2{\Omega }_D{e}^{-1/({\nu }_{s}V)}$. A density counting both spin species is twice ${\nu }_s$ and must be divided by two. The shell cutoff is a relative-energy condition, even when every Cooper pair has zero total momentum.

The distribution of a new or missing quantity integrates its model over the Bayesian posterior: $p(y^{\ast }\mid y)=\int p(y^{\ast }\mid \varphi ,y)p(\varphi \mid y)d\varphi$. Both parameter and residual uncertainty should enter predictive draws used for multiple imputation.

A posterior variance is the variance of a parameter under its Bayesian posterior, namely $\mathbb{E}[(\theta -\mathbb{E}[\theta \mid y]{)}^2\mid y]$ when finite. It describes remaining uncertainty after observing data and is distinct from the repeated-sampling variance of a fitted estimator.

If a hidden coin is selected once with prior probability $\pi$ of coin $B$, repeated tosses are independent conditional on that coin. Bayes' theorem weights the prior by each all-heads likelihood. They need not be independent without conditioning on the chosen coin.

Every linearly independent subset of a finite-dimensional vector space extends to a basis. Starting with its vectors, add a vector outside their current span until the span is the whole space. Each addition preserves linear independence, and the process stops because the dimension is finite. This also lets a linear functional on a vector subspace extend to the ambient space by assigning arbitrary values on the additional basis vectors.

For a basis $e_1,\dots ,e_n$ with cyclic indexing, the vectors $e_j+e_{j+1}$ have coefficient matrix $I+S$, where $S$ cyclically shifts coordinates. Its determinant is $1-(-1{)}^n$. They therefore form a basis exactly when $n$ is odd over a field of characteristic different from two. For even $n$, alternating coefficients produce a nontrivial linear dependence.

For $\dot{z}=Az+\Vert z{\Vert }^{2}z$, suppose a positive-definite matrix $P$ solves $A^{T}P+PA=-2I$. The quadratic function $V=z^{T}Pz$ then obeys $\dot{V}=2\Vert z{\Vert }^2(V-1)$. Its unit ellipsoid separates decay toward the origin from escape. In a planar system with nonzero angular speed it is an invariant periodic orbit.

Subglacial till is a mixture of rock fragments and fine sediment beneath ice. A wet deformable layer can lubricate basal sliding. Its constitutive equation must be specified; Newtonian till lubrication is one idealization, rather than a claim that all subglacial till is a Newtonian fluid.