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For an axisymmetric normal mode in a rigid rotating sphere, substitute $W=z(A{R}^2+B{z}^2+C)$ into the pressure equation and its impermeability condition. With $\beta =1-4{\Omega }^2/{\sigma }^2$, the interior equation gives $2A+3\beta B=0$. The boundary polynomial gives $3\beta B=(2+\beta )A$ and $\beta C=-(2+\beta )A{r}_0^2$. A nontrivial oscillatory solution has $\beta =-4$, hence the displayed frequency and $B=A/6$, $C=-A{r}_0^2/2$. It satisfies both equations identically.

In the large-diffusivity regime, equate diffusion rate $\alpha /r^2$ with inertial eddy rate ${ϵ}^{1/3}{r}^{-2/3}$. This gives the Corrsin scalar microscale, which exceeds the Kolmogorov length scale when $\alpha \gg \nu$.

The congruence axiom says that an affordable alternative indirectly revealed at least as desirable as a chosen alternative must also be chosen. For a chosen bundle $x$ and an affordable unchosen bundle $y$, the axiom rules out $yRx$; because $xRy$, every strict-part-preserving completion ranks $x$ strictly above $y$.

In the genuine indexed point-set model, spectrification $L$ is the left adjoint to the forgetful functor from indexed spectra to indexed prespectra. An indexed spectrum has adjoint structure maps $E(V)\to {\Omega }^{W\ominus V}E(W)$ that are homeomorphisms. Applying $L$ to a cellular indexed prespectrum supplies the associated genuine topological spectrum.

For compact $X$ over the space of linear isometries $U\to U\prime$, choose a subordinate flag for a family of linear isometries $V_i,W_i$. The vector bundle ${\xi }_i$ has fiber $W_i\ominus f_x(V_i)$. Form $P(W_i)=\mathrm{Th}({\xi }_i)\wedge E(V_i)$. Its structure maps use the orthogonal decompositionand the structure map of $E$. The twisted half-smash product is the spectrification $LP$, after extending from the cofinal flag. Unlike an arbitrary replacement by $X_{+}\wedge E$, this definition retains the specified isometry family at the point-set level.

A rotation through $\pi$ about the third coordinate axis preserves the strain components $e_{11},e_{22},e_{33},e_{12}$ and reverses $e_{13},e_{23}$. A symmetry-invariant elastic stiffness tensor cannot couple the two parity spaces. Its independent symmetric blocks therefore contain $4\cdot 5/2+2\cdot 3/2=13$ coefficients. A twofold axis alone does not impose invariance under every rotation about that axis.

Draw two different labels uniformly from $\{1,2,3,4\}$ using sampling without replacement. Each position has a uniform distribution on a finite set, but $\mathbb{P}(X_1=X_2)=0$. Two independent random variables with those same marginal distributions would match with probability $1/4$. Equal marginal laws therefore do not make this sample an IID sample. Mark some of those four labels as successes. Their count in the two draws has a hypergeometric distribution; independent Bernoulli trials with a common success probability give a binomial distribution.

The IID sample property places no integrability condition on its common probability distribution. An IID sample with the standard Cauchy distribution has no finite expected value or variance. The Cauchy sample mean stability result shows that its averages keep that same Cauchy distribution for every sample size. Thus increasing $n$ need not concentrate an IID sample around a population mean that does not exist. The finite-variance bound below applies only when its hypotheses hold.

In the copied-coin model, $X_i=Z$ for every $i$, so the sample mean is exactly $Z$, however large $n$ is. Its expected value is $1/2$ and its variance is $1/4$, matching each observation's marginal variance. The off-diagonal covariances in the variance of a sum explain the difference:For an IID sample, those off-diagonal terms are zero. Repeating the same random input therefore does not create the reduction in fluctuations obtained from fresh independent draws. The general common-plus-individual structure is described by compound-symmetry covariance.

For $n\ge 1$ independent and identically distributed Bernoulli random variables with success probability $p$, the sum $K={\sum }_i{X}_i$ has a binomial distribution. Every length-$n$ pattern with $k$ successes has probability $p^k(1-p{)}^{n-k}$, and the binomial coefficient counts its $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$ possible success positions. ThereforeThe usual endpoint convention includes $p=0$ and $p=1$ as deterministic cases. For $p=1/2$ and $n=4$, the five probability masses are $1,4,6,4,1$ divided by $16$. The sample mean takes values $0,1/4,1/2,3/4,1$, with variance $1/16$ and standard deviation $1/4$. At $n=64$, its variance is $1/256$ and its standard deviation is $1/16$.

A cost-effectiveness acceptability curve displays the probability of positive incremental net monetary benefit as a function of the willingness-to-pay threshold. Under a posterior distribution this is a posterior probability. A frequentist paired bootstrap version displays the fraction of replicates for which the net benefit is positive, a resampling measure of uncertainty rather than literally a probability that a fixed true parameter is positive. Using the sign of net benefit handles all quadrants of the cost-effectiveness plane; counting only replicates whose ratio is below the threshold does not.

The acoustic tensor determines the plane waves of a locally uniform incremental elastic medium. For a unit material direction $\nu$, its eigenvalues are ${\rho }_0{c}^2$ and its eigenvectors are displacement polarizations. Major symmetry of the incremental elastic moduli makes $Q$ a real symmetric matrix. Positivity on every rank-one matrix $m\otimes \nu$ makes all its wave speeds real and nonzero.

Strong ellipticity requires the displayed inequality for every nonzero displacement vector $m$ and material direction $\nu$. Equivalently, every acoustic tensor is positive-definite. It is weaker than positivity of the moduli on all matrices. A quadratic minor null Lagrangian vanishes on rank-one matrices, so adding such a term does not change the acoustic tensor or this condition.

For a static base state with zero incremental body force, homogeneous incremental tractions or homogeneous Dirichlet boundary conditions eliminate the boundary term in integration by parts. The displayed Rayleigh quotient then gives real squared frequencies because of major symmetry. Positive moduli give strictly positive squared frequencies for modes with nonzero displacement gradient. Pure dead loading leaves constant translations at zero; fixed displacement conditions remove those modes.

A compression disk for a two-sided embedded surface $F$ in a three-manifold is an embedded disk meeting $F$ exactly in its boundary, with boundary essential in $F$. Replacing an annular neighborhood of this boundary by two parallel copies of the disk reduces the surface's genus or separates it into simpler pieces.

A hierarchy is a finite sequence of cuts along properly embedded two-sided incompressible surfaces, without spherical components, ending in disjoint three-balls. Disks are allowed. For an ambient manifold which is not orientable the two-sided cutting surfaces need not themselves be orientable. The existence of a hierarchy does not imply that every choice of incompressible cuts terminates.

Write the Cauchy stress tensor in the local fibre frame as $\sigma =se\otimes e+h(e\otimes n+n\otimes e)+tn\otimes n$. The compatible geometry has $\nabla \theta =\kappa e$, $\nabla \cdot e=0$ and $\nabla \cdot n=-\kappa$. With zero body force, force balance becomes $e\cdot \nabla s=2\kappa h-n\cdot \nabla h$ and $n\cdot \nabla t-\kappa t=-e\cdot \nabla h-\kappa s$. For specified shear stress $h$, the first equation determines $s$ along fibres, then the second determines $t$ along transverse curves. Pressure and fibre tension supply these two normal constraint reactions. This is a local stress construction; boundary tractions must be chosen consistently with its integration data.

Take an initially isolated virialized self-gravitating stellar population with kinetic energy $T=|W|/2$ and Newtonian gravitational potential energy $W<0$. Instantaneous spatially uniform removal of mass, without kicks to surviving material, leaves fraction $f$ with $T^{\prime }=fT$ and $W^{\prime }=f^{2}W$. Hence $E^{\prime }=|W|f(1/2-f)$: the homologous whole-system energy is negative only for $f>1/2$. Positive whole-system energy does not forbid a bound core after selective escape, and slow mass loss follows the adiabatic mass-loss expansion law instead.

The Haldane prior has formal beta distribution parameters $(0,0)$. It is an improper prior, with infinite mass at both endpoints of $(0,1)$. For a binomial distribution likelihood with $s$ successes and $f$ failures, its posterior is $\mathrm{Beta}(s,f)$ only when $s,f>0$. A density flat in the log odds induces this kernel by the Jacobian determinant $d\log [p/(1-p)]/dp=1/[p(1-p)]$.

For $C>0$, nonzero equilibrium points satisfy $\eta =g(a)=a-C{a}^2+a^4$ and are stable exactly when $a{g}^{\prime }(a)>0$. The origin is stable for $\eta <0$. The quadratic term removes the odd symmetry and changes the local branch crossing into a transcritical bifurcation, with $a=\eta +O({\eta }^2)$. The negative quintic term bounds trajectories. There is one negative-amplitude fold, supplying a stable finite-amplitude branch and hysteresis with the stable origin. Positive-amplitude folds occur precisely when $C>3/2$: $g^{\prime }(a)=1-2Ca+4a^3$ has two positive roots then, a double positive root at $C=3/2$, and none below it. All branches and stability assignments therefore depend on $C$; a single symmetric pitchfork sketch cannot describe this family.