For a phantom map of spectra $p:D\to E$ and a finite CW complex $K$, Spanier-Whitehead duality identifies $D_k(K)$ with maps from the finite spectrum ${\Sigma }^k({\Sigma }^{\infty }K{)}^{\vee }$. Hence $p_{\ast }=0$ there. An arbitrary CW complex is the filtered union of its finite subcomplexes, and represented homology theory commutes with that filtered union because spheres are compact. Thus $p$ induces zero on the entire represented homology theory.

Put $c_1^2=\gamma p_1/{\rho }_1$ and $v_{A1}^2=B_1^2/({\mu }_0{\rho }_1)$. The mass, momentum and energy jump conditions for a perpendicular magnetohydrodynamic shock factor into the unchanged state $x=1$ andFor the compressive physical branch $x>1$, positive downstream pressure implies $x<(\gamma +1)/(\gamma -1)$ when $\gamma >1$ and the upstream sound speed or Alfvén speed is nonzero. Indeed the polynomial impliesIt then gives $u_1^2>c_1^2+v_{A1}^2$: the upstream flow exceeds the perpendicular fast magnetosonic wave speed. The exactly cold unmagnetized strong-shock limit reaches the upper compression bound.

With $B_x=0$, a fast simple wave in magnetohydrodynamics has constant $u_y$ and $B_y/\rho =K$. Its longitudinal velocity satisfies $d{u}_x/d\rho =\pm c_f/\rho$, and ${\rho }_t+(u_x\pm c_f){\rho }_x=0$. Along this state curve the speed derivative is $\pm [c_f/\rho +K^2/(2{\mu }_0{c}_f)]$, which is nonzero. Compressive profiles therefore exhibit shock formation by characteristic intersection, unlike a linearly degenerate characteristic field.

Average neighboring periodogram ordinates with nonnegative weights summing to one. Let the number $L_T$ of averaged ordinates tend to infinity while their frequency span, of order $L_T/T$, tends to zero. For a smooth spectral density and suitable short-memory or fourth-order dependence conditions, the bias tends to zero and variance is of order $1/L_T$. Smooth second-order spectral density alone does not ensure this: a time-constant random scale multiplying independent white noise has a smooth unconditional spectrum but a nonergodic random limiting local power.

For a centered stationary real Gaussian sequence, let $A_T$ and $B_T$ be its cosine and sine coefficients with normalization $1/\sqrt{\pi T}$ at a nonzero Fourier frequency. Grouping the double covariance sum by lag and extending each lag sum to a whole record leaves at most $2k$ omitted bounded products. The full cosine-sine sum is zero, giving the bound. If $\sum k|{\gamma }_k|<\infty$, the two coefficients become asymptotically uncorrelated, and their limiting joint normal distribution consequently has independent components.

At an interior frequency, the real and imaginary Fourier coefficients of a short-memory Gaussian time series converge jointly to independent centered normal variables, each with variance $s(\omega )/2$, where $s$ is the one-sided spectral density. Their squared sum therefore converges to $s(\omega ){\chi }_2^2/2$. Its variance tends to $s(\omega {)}^2$, so an unsmoothed periodogram is not consistent when $s(\omega )>0$. Averaging nearby ordinates can reduce this persistent variance.

A state can have no positive-probability return at any positive time, for example when its transition-matrix column is zero. Its return-time set is empty. The convention $\gcd \varnothing =0$ assigns period zero; conventions defining period only for a nonempty return set instead leave it undefined. Such a state must not be called aperiodic merely because its communicating class is a singleton.

The compact dual is the complex projective flag variety imposing prescribed filtration dimensions and $Q(F^p,F^{w-p+1})=0$, but neither real opposition nor positivity. In weight one it is the complex Lagrangian Grassmannian. In polarized K3 weight two it is an isotropic-line quadric.

Positive complex Lagrangian subspaces for the integral symplectic form $\left(\begin{array}{cc}0& -I\\ I& 0\end{array}\right)$ are graphs $H_Z=\{(Zu,u):u\in {\mathbb{C}}^r\}$. Isotropy is the symmetry of $Z$, and $iQ(v,\bar{v})=2u^{\mathsf{T}}(\mathrm{Im}Z)\bar{u}$ gives positivity. For $g=\left(\begin{array}{cc}A& B\\ C& D\end{array}\right)$ in the symplectic group, the action is $Z\mapsto (AZ+B)(CZ+D{)}^{-1}$.

On the K3 intersection lattice, this open quadric has complex dimension twenty and records the period line $H^{2,0}$. Fixing a positive class $h$ of type $(1,1)$ adds $q(h,\omega )=0$ and yields a nineteen-dimensional domain for primitive polarized cohomology of signature $(2,19)$. The distinction is necessary: the unrestricted full lattice has signature $(3,19)$ and does not satisfy polarization positivity on all of $H^{1,1}$.

For a two-by-two contingency table, subtracting one half from each absolute observed-minus-expected count difference, with truncation at zero, gives the displayed corrected Pearson chi-squared statistic. Expected counts use the independence model $E_{ij}=O_{i+}{O}_{+j}/N$. The subtraction compensates approximately for using a continuous chi-squared distribution for discrete counts. It can make the test conservative. The corrected statistic should be distinguished from the ordinary uncorrected Pearson chi-squared statistic.

For $N$ independent groups each containing $m$ independent Bernoulli trials with common probability $\theta$, the group count has a binomial distribution. If $O_j$ groups have count $j$, the maximum-likelihood estimate is $\hat{\theta }={\sum }_{j}j{O}_j/(mN)$. Form the Pearson chi-squared statistic using $E_j=N\left(\genfrac{}{}{0ex}{}{m}{j}\right){\hat{\theta }}^j(1-\hat{\theta }{)}^{m-j}$. With all $m+1$ categories retained, an interior true parameter, and sufficiently large expected counts, its null limit is a chi-squared distribution with $m-1$ statistical degrees of freedom: one degree is removed by the fixed total and another by fitting $\theta$. Small expected counts require care with this approximation. If categories are pooled, the fitted parameter and calibration should correspond to the pooled statistical model; simply pooling a statistic after fitting the unpooled model does not automatically give the usual chi-squared limit.

For a massive positive-energy unitary irreducible representation of the Poincaré group with signature $(+---)$, choose a rest frame. The Pauli-Lubanski pseudovector has zero time component and spatial components $m{J}_i$, up to a common sign convention. Its square is therefore $-m^2{\mathbf{J}}^2$. The SU(2) little-group representation has spin $s$ and ${\mathbf{J}}^2=s(s+1)$, proving the invariant value. The mass and spin label the massive induced representation of the Poincare double cover.

In $W^{\mu }=-{ϵ}^{\mu \nu \rho \sigma }{P}_{\nu }{M}_{\rho \sigma }/2$, contracting with $P_{\mu }$ gives zero because the commuting momentum products are symmetric in $\mu ,\nu$ while the alternating tensor is antisymmetric. Reordering $P$ and $M$ makes no difference because the resulting metric contractions with the alternating tensor vanish.

// Target: special-relativity.bigb