A solid resists sustained changes of shape on the timescale being considered. It can support a static shear stress, whereas an ordinary liquid continues to deform under sustained shear. Small loads often produce recoverable elastic deformation; larger loads can cause plastic deformation or fracture.

A solid need not be a crystal. A crystalline specimen has long-range structural order, while an amorphous solid such as ordinary glass lacks the same repeating lattice arrangement. The mechanical classification and the structural classification answer different questions. Applying a small lateral load to the top of a block while holding its base fixed tests shear resistance; a finite shear displacement at static balance is a solid-like response. At long times some solids also creep, so the observation interval and applied stress matter when comparing a material's response with an ideal elastic model.

A protostar is a forming star whose energy comes chiefly from gravitational contraction and accretion, before sustained core hydrogen burning supplies its luminosity. A fixed-mass contraction model approximates the stage after most accretion has ended; an actively accreting protostar need not have fixed mass or the same Hayashi track.

In a normal linear model with residual degrees of freedom $\nu \ge 2$, an internally studentized residual satisfies $|{\eta }_i|\le \sqrt{\nu }$ and ${\eta }_i^2/\nu \sim \mathrm{Beta}(1/2,(\nu -1)/2)$, provided its leverage is less than one. Choose an orthonormal basis of the residual space with its first direction proportional to the projection of the $i$th coordinate vector. The Gaussian coordinates in this basis are independent standard normal random variables $Z_1,\dots ,Z_{\nu }$, and ${\eta }_i=\sqrt{\nu }{Z}_1/({\sum }_j{Z}_j^2{)}^{1/2}$. The beta distribution follows from the independent chi-squared distributions of $Z_1^2$ and ${\sum }_{j=2}^{\nu }{Z}_j^2$. The bounded residual is not exactly distributed according to Student's t-distribution; it approaches the standard normal distribution as $\nu \to \infty$.

An upstream-facing probe that senses stagnation pressure. Determining speed also requires a separate static pressure and mass density.

A sequential model of a topological spectrum consists of based spaces $E_j$ and structure maps $S^1\wedge E_j\to E_{j+1}$. Indexed models use finite-dimensional subspaces of a universe for spectra and structure maps $S^{W\ominus V}\wedge E(V)\to E(W)$. Different standard models present the same stable homotopy category.

The stable homotopy category is obtained from topological spectra by inverting stable equivalences of spectra. Its morphisms are abelian groups; for a cellular source and fibrant target they are spectrum homotopy classes of point-set maps of topological spectra.

The derived smash product of spectra is the symmetric monoidal product on the stable homotopy category. It has unit sphere spectrum, preserves cofiber sequences of spectra and homotopy colimits of spectra in each variable, and satisfies ${\Sigma }^{\infty }X\wedge {\Sigma }^{\infty }Y\simeq {\Sigma }^{\infty }(X\wedge Y)$.

The homotopy cofiber of spectra of $u:A\to B$ is the derived pushout $B{\cup }_{A}CA$. It fits into a cofiber sequence of spectra.

A homotopy colimit of spectra is the derived colimit of a diagram of topological spectra. For a sequential diagram $T_0\to T_1\to \cdots$, it is the cofiber of spectra of $1-s:{\bigvee }_j{T}_j\to {\bigvee }_j{T}_j$, where $s$ uses the bonding maps into the next summands.

A universe for spectra is a countably infinite-dimensional real inner product space $U$, used to index finite-dimensional suspension directions. A standard model is the algebraic direct sum $U={⨁}_{i\ge 1}\mathbb{R}{e}_i$ with an orthonormal basis, carrying its direct-limit topology. Its vectors have finite support; this universe is not the completed Hilbert space ${\ell }^2$. Indexed topological spectra assign spaces to finite-dimensional vector subspaces $V\subseteq U$ and use the corresponding orthogonal differences for structure maps.^[ref]

A family of indexing vector subspaces is cofinal when every finite-dimensional $W\subseteq U$ is contained in some member of the family. The flag $V_n=\mathrm{span}(e_1,\dots ,e_n)$ is cofinal: a finite basis of $W$ uses only finitely many of the $e_i$. In contrast, $W_n=\mathrm{span}(e_2,e_4,\dots ,e_{2n})$ has unbounded dimension but never contains $e_1$, so it is not cofinal. Cofinality ensures that restricting the index system does not discard any finite collection of suspension directions.

For binary linear codes $C_2\subseteq C_1\subseteq {\mathbb{F}}_2^N$, the CSS code has basis $|x+C_2⟩=|C_2{|}^{-1/2}{\sum }_{y\in C_2}|x+y⟩$ for $x\in C_1/C_2$. Its X stabilizers come from $C_2$ and its Z stabilizers from the dual code $C_1^{\perp }$; their commutation follows from orthogonality. For positive logical dimension, $d_X=\min \{\mathrm{wt}(x):x\in C_1\setminus C_2\}$ and $d_Z=\min \{\mathrm{wt}(z):z\in C_2^{\perp }\setminus C_1^{\perp }\}$, with quantum code distance $d=\min (d_X,d_Z)$. Thus arbitrary errors on at most $\lfloor (d-1)/2\rfloor$ qubits are correctable. Stabilizer errors are harmless even if their weight is small; the distances exclude them.

A stabilizer group with $r$ independent generators maps injectively to an r-dimensional isotropic subspace of the binary symplectic space of Pauli labels. For generator rows $(A\mid B)$, commutation is $A{B}^T+B{A}^T=0$ over ${\mathbb{F}}_2$. Its stabilizer subspace has dimension $2^{n-r}$. The labels of the centralizer of a stabilizer group form $L^{\perp }$; labels in $L^{\perp }\setminus L$ act as nontrivial logical Pauli operators.

A different interpretation can treat cyclic spreadsheet formulas as simultaneous equations and seek a fixed point by repeated numerical updates. This requires an explicit update rule, initial values and stopping condition. It is not automatically the meaning of a spreadsheet formula. In particular, a directed cycle alone tells us neither whether a fixed point exists nor whether a chosen update rule approaches it.

Two original toy examples make the distinction exact. The cyclic pair X1=Y1+1, Y1=X1+1 has no solution over the real numbers: substitution gives $X1=X1+2$. For X1=Y1, Y1=X1, every equal pair is a fixed point, so there is no unique answer. Simultaneously updating both spreadsheet cells from the previous pair gives $(x_{n+1},y_{n+1})=(y_n,x_n)$. Starting at $(0,1)$ produces the periodic sequence $(0,1),(1,0),(0,1),\dots$, which does not approach a fixed point. Updating X1 first and immediately using its new value for Y1 instead gives $(1,1)$ in one sweep. That difference follows from the specified rules, not from a claim about any current spreadsheet product. Numerical iteration can be useful for a deliberately chosen model with a justification such as the contraction mapping theorem; ordinary acyclic spreadsheet dependency evaluation already has its answer from prerequisite order.

A linear spline is a continuous piecewise linear function on a specified spline knot sequence. For uniform integer spline knots, the hat basis in the displayed representation interpolates the data $s(i)=c_i$. The derivative is constant on each open span and can jump at a knot. Coinciding neighboring slopes remove a particular jump, but generic data give only $C^0$ regularity.

For $|s⟩=(|01⟩-|10⟩)/\sqrt{2}$ and any two-by-two unitary matrix, expand the two columns of $U$. The equal-index coefficients cancel, and the coefficient of $|01⟩-|10⟩$ is the determinant. Its modulus is one, so a common change of qubit basis preserves the spin singlet state up to a global phase. The density projector is exactly invariant.

Larmor precession is the precession of a magnetic moment about an applied magnetic field. For a spin with gyromagnetic ratio $\gamma$ and Hamiltonian $H=-\gamma \mathbf{B}\cdot \mathbf{S}$, its expectation value obeys $d⟨\mathbf{S}⟩/dt=\gamma ⟨\mathbf{S}⟩\times \mathbf{B}$. A spin coherent state retains its coherent form during this rotation.

A Stern-Gerlach experiment separates atomic beams according to a component of their magnetic moment in a nonuniform magnetic field. For an appropriate two-state spin system, calibrated outputs distinguish the two spin eigenvalues; Stern-Gerlach expectation estimation explains how repeated fresh preparations turn channel counts into an expectation value.

The decisive contrast is between two separated components of an atomic beam and the continuous spread predicted for randomly oriented classical magnetic moments. Neither component need be an infinitely narrow spot: velocity spread and the resolving power of the scientific instrument affect the spatial image. The apparatus converts an internal magnetic moment into a displacement large enough to inspect, connecting angular momentum eigenstates to macroscopic evidence.

For a correctly specified conditional rate, transformed interspike intervals have independent unit-exponential distributions, apart from observation-boundary censoring. Their transforms $1-e^{-z_i}$ should be uniform. Distributional checks and serial-correlation checks assess a fitted spike model on held-out data. A history-dependent model must include past spikes in its conditional rate; stimulus-only Poisson spike-train likelihood fitting cannot automatically capture refractoriness.

The Rayleigh-Plesset equation balances the radial inertia of an incompressible liquid around a spherical bubble against internal and ambient pressure, surface tension and viscous stress. Removing the last two effects and setting the bubble pressure to zero gives the following vacuum-collapse case.

In radial potential flow outside a spherical bubble, incompressibility and the moving-boundary condition give $u_r=R^2\dot{R}/r^2$ and $\varphi =-R^2\dot{R}/r$. The Unsteady Bernoulli equation, with its gauge fixed at infinity, yields the displayed radius equation when surface tension and viscosity are neglected. Pressure continuity at the interface identifies the liquid pressure with the gas pressure $p_g$. The vacuum special case is Rayleigh collapse of a spherical cavity.