Attaching a $q$-cell along $\varphi :S^{q-1}\to X$ forms the quotient of $X\bigsqcup D^q$ identifying each boundary point with its image under $\varphi$. A constant attaching map gives $X\vee S^q$. An attaching map can affect the cohomology ring even when it does not change the additive cohomology groups.

A CW pair consists of a CW complex $X$ and a subcomplex $A$. Its inclusion is a cofibration, so homotopies on $A$ extend across $X$. The relative lifting property of a Serre fibration for such pairs allows specified boundary lifts to be retained while lifting a homotopy.

For an all-integer dictionary row $x_B+{\sum }_j{a}_j{x}_j=b$ with nonnegative variables, this cut follows because $x_B+{\sum }_j\lfloor a_j\rfloor x_j$ is an integer and ${\sum }_j\{a_j\}{x}_j\ge 0$. Consequently that integer is at most $\lfloor b\rfloor$. Negative coefficients use the usual fractional part $u-\lfloor u\rfloor$, so $\{-1/5\}=4/5$. Normalize a resulting inequality sensibly before introducing a new integer-valued slack variable.

At equal argument and large integer order, the Bessel function of the first kind has phase $n(\sin t-t)=-n{t}^3/6+\cdots$ near the endpoint. The cubic stationary endpoint yields the displayed positive constant. It is also the zero-argument value of the Airy function in the uniform Bessel turning-point approximation.

For the Prandtl normal-mode equation with viscosity scaled to one and a profile having a nondegenerate stationary point, $U-c_0\sim (y-y_c{)}^2/2$. A piecewise outer mode proportional to $U-c_0$ has a second-derivative jump. Balancing the advective and third-derivative terms smooths that jump over width ${\alpha }^{-1/4}$, with velocity amplitude ${\alpha }^{-1/2}$ and a phase-speed correction of the same order. Matching selects the eigenvalue through a third-order inner problem.

The random time at which credit default occurs. In an observed default model it is a stopping time for the market filtration.

For an observed transformed time $Y=H(\min (T,C))$ and event indicator $\Delta$, the correction $Y^{\ast }=Y+1-\Delta$ has mean one under independent censoring. Its complement $1-Y^{\ast }=\Delta -Y$ is the corresponding martingale residual.

For a correct fitted survival law, complete Cox–Snell residuals are approximately unit exponential. Residuals computed at observed censored times retain their censoring indicators. Fit a Kaplan–Meier estimator to the transformed times and indicators and compare it with $e^{-r}$, or compare their Nelson–Aalen estimator cumulative hazard with $r$. Treating censored transformed times as complete exponential observations gives a distorted Q–Q check.

At an untied event time, a Schoenfeld residual is the event subject's covariate vector minus its fitted risk-weighted average over the risk set. Under a correctly specified Cox proportional-hazards model, these residuals have no systematic mean trend with event time. They are defined at event times, unlike Cox–Snell residuals which can be evaluated at both events and censoring times.

A proportional hazards assumption test assesses whether fitted covariate coefficients remain constant over time. A common implementation uses trends in scaled Schoenfeld residuals, with separate covariate tests and a joint global test. Failure to reject is compatible with proportional hazards, but does not establish the correct covariate functional form or independent censoring.

At a distinct event time, the Schoenfeld function is the event subject's covariate minus the hazard-weighted covariate mean over the risk set, as a function of a trial coefficient $\beta$. Its evaluation at the fitted coefficient is a Schoenfeld residual. The sum of these functions is the Cox partial likelihood score function, and each is conditionally centered at zero at the true proportional-hazards coefficient.

For a path-connected, locally path-connected, semilocally simply connected topological space $Y$, the universal abelian cover is the connected covering space associated to the commutator subgroup of $G={\pi }_1(Y,y_0)$. It is a regular covering with deck transformation groupEvery connected covering space of $Y$ with Abelian deck transformation group that is also a regular covering is a quotient of this cover: its corresponding normal subgroup contains $[G,G]$. This is universality among such abelian regular coverings, rather than a claim that the total space is simply connected.^[ref]

For a torus, $G\cong {\mathbb{Z}}^2$ is already Abelian, so this cover is the universal cover ${\mathbb{R}}^2$. For a wedge sum of two circles, $G$ is a free group on two generators and $[G,G]$ is nontrivial; the universal abelian cover is therefore not simply connected. If $Y$ is a CW complex, lifting its cells gives a cellular chain complex of free modules over the group ring $\mathbb{Z}[H_1(Y;\mathbb{Z})]$.

Old covariant quantization retains all target-space string oscillators and imposes $(L_0-a)|\psi ⟩=0$ and $L_n|\psi ⟩=0$ for $n>0$. Negative-mode constraints are adjoints, not further annihilation conditions. Quotienting physical null string states removes redundant polarizations.

Time-coordinate oscillators produce negative norms in the covariant indefinite Fock space. A physical negative-norm state would violate probabilistic unitarity. Such ghosts differ from the anticommuting worldsheet ghost fields introduced by gauge fixing.

For stable density jumps in the three-layer stratified piecewise-linear shear flow, the upper interface's intrinsically left-going gravity-vorticity interface wave and the lower interface's intrinsically right-going wave have opposite laboratory phase velocities. They coincide at zero when $1+J=2\alpha$. At large wavenumber, interaction is of order $e^{-2\alpha }$ and the unstable band of a three-layer stratified shear flow has width $4\alpha e^{-2\alpha }+O(\alpha e^{-6\alpha })$ around this resonance.

Only finite subsets are compact in the cocountable topology. An infinite subset contains distinct $c_1,c_2,\dots$. The open sets obtained by removing the tails $\{c_j:j\ge n\}$ form an increasing open cover, but every finite selection misses a tail point. Finite sets are compact in every topology.

For a free electromagnetic field with suitable boundary conditions, combine Coulomb gauge ${\partial }_i{A}_i=0$ with $A_0=0$. Gauss's law makes the second condition consistent in the source-free problem. The remaining field operator is spatially transverse and describes the two radiative photon polarizations. A constrained scalar potential must be restored when charges are present.

Between a resting wall at $y=0$ and a wall moving at speed $U$ at $y=h$, an adverse pressure gradient $p_x=G$ givesZero volume flux requires $G=6\mu U/h^2$, yielding $u/U=3(y/h{)}^2-2y/h$. Reverse flow below $2h/3$ balances the positive flux near the moving wall. The minimum is $-U/3$ at $h/3$.

In hierarchical galaxy formation, low-mass dark-matter structures typically collapse before larger structures because the smoothed linear density variance is larger on smaller mass scales. Accretion and mergers assemble increasingly massive haloes. Cooling, feedback and angular momentum determine how baryons form galaxies within that halo hierarchy; the ordering is statistical rather than an absolute rule for every object.

The squared adiabatic sound speed is the ratio of background pressure and energy-density time derivatives. For a barotropic equation of state it is $dP/d\rho$. The non-adiabatic pressure perturbation is the difference $\delta P-c_a^2\delta \rho$; in a general field or multicomponent model this background quantity need not equal the rest-frame propagation speed.