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This is the part-preserving homomorphism density of a four-cycle in a bipartite graph with independently uniform labels $x,x^{\prime }$ in the first part and $y,y^{\prime }$ in the second. Repeated labels are included. It equals the fourth power of the four-cycle norm of the adjacency indicator function. It differs at finite sizes from a count requiring four distinct vertices.

For a nonnegative integer $L$ and $|z|<1$, this specialization of the binomial series sums the normalizing coefficients of a negative binomial distribution.

Partition the positive natural numbers into three infinite coordinate sets. Fix the coordinates on the first two sets to any prescribed binary values, and leave those on the third set free. Reading the third coordinate set embeds all infinite binary sequences in the resulting family, so the family is an uncountable set by Cantor's diagonal argument. For example, fixing $x_{3k-2}=0$ and $x_{3k-1}=a_{3k-1}$ while leaving $x_{3k}$ free guarantees infinitely many zeros and infinitely many agreements with any prescribed binary sequence $(a_n)$. Infinite free coordinates remain even after both requirements are enforced.

In a symmetric monoidal category, a convolution-invertible scalar pairing $\gamma :H\otimes H\to I$ on a bimonoid inducing the comodule braiding $u\otimes v\mapsto \sum v_{(0)}\otimes u_{(0)}\gamma (u_{(1)},v_{(1)})$. Its multiplicativity, normalization and commutation axioms express the hexagon, unit and colinearity conditions. The notation abbreviates morphisms and ambient symmetries, not a requirement for elements.

For nonzero finite-dimensional complex vector spaces $U,V$, suppose a bilinear map never vanishes on a pair of nonzero vectors. Equivalently, each map obtained by fixing one nonzero argument is injective. If its associated map on $U\otimes V$ has image dimension $r$, projectivization gives a map to ${\mathbb{CP}}^{r-1}$ whose positive degree-two class pulls back to $x+y$. The nonzero top power $(x+y{)}^{\dim U+\dim V-2}$ in the product cohomology ring of complex projective space proves the bound. Multiplication of polynomials of bounded degree attains equality. Nonzero hypotheses matter: with a zero factor, slice conditions can be vacuous.

With the convention $m\ast \gamma =\gamma \ast n$ and invertible bialgebra scalar cocycle $\gamma$ for $n$, the same coalgebra has $n=\bar{\gamma }\ast m\ast \gamma$, or equivalently $m=\gamma \ast n\ast \bar{\gamma }$. Thus $n(a,b)=\sum \bar{\gamma }(a_{(1)},b_{(1)})m(a_{(2)},b_{(2)})\gamma (a_{(3)},b_{(3)})$. Specifying the equation prevents opposite twist conventions from being conflated.

With fixed inverse-variance weights and weighted mean $\hat{\mu }$, $Q={\sum }_i{w}_i(y_i-\hat{\mu }{)}^2$. Under a common-effect normal model with $m$ independent estimates and known variances it has a chi-squared distribution with $m-1$ degrees of freedom. Estimated variances make this calibration approximate.

For $m$ studies let $w_i=v_i^{-1}$, $W={\sum }_i{w}_i$, $C=W-{\sum }_i{w}_i^2/W$, and let $Q$ be Cochran's Q statistic. The moment estimate is ${\hat{\tau }}^2=\max \{0,[Q-(m-1)]/C\}$ when $C>0$. It estimates the variance of underlying study effects, not their sampling variance.

For $Q>0$ and $m$ studies, $I^2=\max \{0,[Q-(m-1)]/Q\}$. Usually reported as a percentage, it describes excess variation relative to the observed variation on this heterogeneity scale. It is not a proportion of studies with different effects or a proportion of an individual clinical outcome.

For $J\ge 2$ conditionally independent effects ${\beta }_j\sim N(\mu ,{\tau }^2)$, each pair difference has normal distribution $N(0,2{\tau }^2)$. With $m=\left(\genfrac{}{}{0ex}{}{J}{2}\right)$, an upper bound $A=R/[\sqrt{2}{\Phi }^{-1}(1-\epsilon /(2m))]$ on $\tau$ ensures, by the union bound, that $\mathbb{P}({\max }_j{\beta }_j-{\min }_j{\beta }_j>R)\le \epsilon$. Any proper scale prior distribution supported on $(0,A)$ preserves this bound after averaging. This is conservative simultaneous calibration, rather than the weaker statement about one selected pair. For log odds ratios, a bound $R$ corresponds to a ratio-of-odds-ratios bound $e^R$.

Draw $P\sim \mathrm{Beta}(\alpha ,\beta )$, $X\mid P\sim \mathrm{Binomial}(n,P)$ and $Q\mid X\sim \mathrm{Beta}(\alpha +X,\beta +n-X)$, with $\alpha ,\beta >0$ and integer $n\ge 0$. The joint density of $(P,X,Q)$ is symmetric in $P,Q$, making them exchangeable random variables with identical beta distribution marginal distributions. The posterior beta normalization constant depends on $X$ and must be retained. The correlation coefficient is $n/(n+\alpha +\beta )$: this constructs tunable dependence without altering either marginal.

For a bounded function on an interval with finitely many jumps and piecewise $C^{\alpha }$ bounds, localized orthonormal wavelets with sufficient vanishing moments give squared best N-term approximation error $O(N^{-2\alpha })$. There are $O(2^j)$ smooth-region coefficients of size $O(2^{-j(\alpha +1/2)})$, and only $O(K)$ jump-crossing coefficients of size $O(2^{-j/2})$ per level. Retain all coefficients through level $J$ and the jump-crossing coefficients through $L=\lceil 2\alpha J\rceil$ when $\alpha >1/2$. This costs $O(2^J+KJ)$ and leaves squared tails $O(2^{-2\alpha J}+2^{-L})$. Choosing $2^J$ comparable to $N$ proves the rate by the Parseval identity.

The worldsheet ghost fields form a bc system with conformal weights $2$ and $-1$. Setting $J=2$ in $c=-2(6J^2-6J+1)$ gives central charge $-26$ in each chirality. The worldsheet Weyl anomaly of $D$ free embedding bosons cancels at $D=26$.

A Gaussian practical-null mixture compares a narrow centered normal distribution prior with a wider centered normal distribution prior. Neither component is a point mass. With observed mean $y\mid \beta \sim N(\beta ,1/n)$ and prior precisions $q_0>q_1>0$, its model predictive variances are $V_i=1/n+1/q_i$, and the Bayes factor isThe overall posterior density is a Bayesian model averaging mixture with component means $ny/(n+q_i)$. A wide-prior penalty can favor the practical null near zero; its dominance depends quantitatively on both prior scales and the observation, not merely on the observation being of order $n^{-1/2}$.

A very ample linear system defines an embedding into projective space. On a smooth projective curve this means it has no base points and separates both distinct points and tangent directions; the complete system of a very ample divisor is an example.

For a nonzero dimension vector, quotient ${\prod }_i{\mathrm{GL}}_{n_i}(k)$ by the common scalar subgroup $(\lambda I_{n_i}{)}_i$. It acts trivially, so the quotient acts algebraically on the quiver representation space. This is one scalar quotient, rather than a product of the individual projective general linear groups. At dimension vector zero the action is trivial.

Choose integer vertex weights increasing along every arrow of a finite acyclic quiver. Acting at vertex $i$ by $t^{w_i}I$ multiplies each arrow by a positive power of $t$. At $t=0$ all arrows vanish. Thus every orbit closure contains the origin of the quiver representation space.

The infinitesimal base change action on quiver representations is ${\xi }_x(u{)}_{\rho }=u_{t(\rho )}{x}_{\rho }-x_{\rho }{u}_{s(\rho )}$. Its image is the Zariski tangent space to the orbit, because the stabilizer is a smooth open subset of the endomorphism ring. The extension complex of quiver representations identifies the quotient of the ambient tangent space by this image with ${\mathrm{Ext}}_Q^1(X,X)$. Consequently rigid quiver representations have open orbits.

The stellar-to-halo mass ratio compares a galaxy's galactic stellar mass with its host's total halo mass. In a simple baryon-conversion model it is ${ϵ}_{\ast }=f_b{f}_{\ast }$. Observational comparisons must distinguish this total-mass ratio from the fraction of baryons converted, and must specify whether central, satellite or all stellar populations are included.

A monotonic luminosity assignment to a counted halo population converts its differential abundance per mass into a luminosity function by a change of variable. If $n_h\propto M_h^{-a}$ and $L\propto M_h^p$, then $\Phi \propto L^{-1-(a-1)/p}$. This simple mapping assumes one counted galaxy per halo, no scatter and a fixed selection. A mass-dependent baryon conversion efficiency of a halo, satellites and stellar-population differences modify it.