Two paths satisfy the synchronous fellow-traveller property with constant $K$ if $d(p(t),q(t))\le K$ for all equal times, with completed paths held at their endpoints. Checking integer times suffices up to increasing $K$ by at most two for unit-speed edge paths. A synchronous combing of a group requires a uniform constant for all adjacent endpoints.

A finitely generated group is synchronously combable if it admits a combing of a group with the synchronous fellow-traveller property. Without a regular-language assumption, this is more general than an automatic group. A bounded-rung strip argument gives a finite group presentation, and finite-state row shortening gives a recursive exponential filling bound and a soluble word problem for a group.

The Kirby color is the weighted sum of annular closures of the surviving Jones-Wenzl idempotents at a root of unity. Its weights are their closed bracket traces. The fusion of Jones-Wenzl colors gives $a\Omega =d_a\Omega$ in the annular fusion ring. Inserting the corresponding fusion resolutions along a band gives the Kirby color handle-slide identity, the central ingredient of surgery invariance.

The coronal approximation treats an optically thin, low-density plasma with electron-impact ionization balanced by radiative and dielectronic recombination. Excited-level populations are determined mainly by collisional excitation and radiative decay, not by a Boltzmann distribution over levels. Ion fractions can depend primarily on electron temperature while metastable populations retain density dependence; this distinction enables density diagnostics.

The coefficient of restitution compares normal separation and approach speeds at the actual contact points of a collision. Choose a unit normal vector $\mathbf{n}$ from the first body toward the second, and let ${\mathbf{u}}_{-}$ and ${\mathbf{u}}_{+}$ be the second contact point's velocity relative to the first before and after impact. For approach ${\mathbf{u}}_{-}\cdot \mathbf{n}<0$,For a rigid body, the contact velocity includes the rotational contribution $\mathbf{\omega }\times \mathbf{r}$ about its centre of mass. Substituting only centre velocities can therefore give the wrong coefficient.

For two smooth point masses with unchanged tangential relative motion, the loss of kinetic energy is $\frac{1}{2}\mu (1-e^2)({\mathbf{u}}_{-}\cdot \mathbf{n}{)}^2$, where $\mu$ is the reduced mass. In that model $e=1$ gives an elastic collision. The value $e=0$ only removes immediate normal separation; a tangential relative velocity can remain, so it does not imply a perfectly inelastic collision with complete sticking. A vertical bounce on a fixed plate gives the measurable relation $e=\sqrt{h_{\mathrm{rebound}}/h_{\mathrm{drop}}}$ when constant gravity and negligible air resistance connect each height to its impact speed.

An experiment by S. F. Foerster, M. Y. Louge, H. Chang (collision mechanics) and K. Allia released small spheres without initial rotation and inferred incoming and outgoing velocities from photographs made with timed flashes. They measured normal restitution together with friction and tangential rebound properties, illustrating why one normal coefficient cannot describe all contact dynamics.^[ref]

// Target: classical-mechanics.bigb

The pressure-area law of a collapsible tube specifies transverse mechanical equilibrium of a deformable tube wall. With physical internal and external pressures $P,P_e$ and cross-sectional area $A$, write $P-P_e=\mathcal{P}(A)$. For constant fluid mass density $\rho$,is the squared tube wave speed about an undisturbed area $A_0$.

The kinematic pressure convention used in the parent model instead writes $p=P/\rho$, $p_e=P_e/\rho$ and $p-p_e=\tilde{P}(\alpha )$. It gives the equivalent expression $c^2={\alpha }_0{\tilde{P}}^{\prime }({\alpha }_0)$, whether $\alpha$ denotes area or area divided by a fixed reference area. Linearized area and momentum conservation yield this coefficient of the resulting wave equation. A stable local wall branch has positive derivative; prescribed external pressure is held fixed when taking it. This distinguishes a wall stiffness from a change in applied external loading.

Let $R(\alpha )\propto {\alpha }^{-n}$, with $n>0$, and use disturbances $e^{i(kx-\omega t)}$ about uniform velocity $U$. The intrinsic frequency is $\nu =\omega -kU$. Linearized mass conservation and the decrease in friction as area grows give the displayed dispersion relation. For nonzero $k$ and positive flow, one branch grows exactly when $nU>c_0$. At $nU=c_0$, one branch is neutral, $\nu =k{c}_0$, while the other remains damped, $\nu =-k{c}_0-i{R}_0$. The maintained external-pressure gradient supplies the energy; positive friction alone does not imply linear stability of a driven flow.

For $p=P_0+\rho c_0^2{\alpha }^2/2$ and constant flux $c_0{A}_{0}q$, a downhill collapsible tube flow obeys the displayed steady equation. Uniform area satisfies $qR({\alpha }_1)=\beta$. Since $R^{\prime }<0$, linearizing about this root gives downstream exponential decay precisely when ${\alpha }_1^2<q$. This is spatial stability of a steady inlet-value problem, not a guarantee of temporal stability of the driven flow. The wave speed is $c_0\alpha$, so the stable spatial branch is supercritical.

An elastic jump is a short transition from a fast, narrow supercritical collapsible tube flow to a slower, wider subcritical flow. Neglecting body force and distributed friction across the jump conserves flux and the momentum flux shown above. Distinct upstream and downstream areas therefore obey ${\alpha }_2^3+{\alpha }_1{\alpha }_2^2+{\alpha }_1^2{\alpha }_2=3q^2/{\alpha }_1$. The flux function has its unique minimum at $\alpha =\sqrt{q}$, so an upstream state below this value has one conjugate state above it. The jump dissipates mechanical energy; it does not conserve a Bernoulli constant.

A positive collapsible tube flow is supercritical when both characteristic speeds $u\pm c$ are positive relative to the wall. Information then propagates downstream on both branches. For a quadratic pressure-area law and fixed dimensionless flux $q$, the condition is $A/A_0<\sqrt{q}$.

A positive collapsible tube flow is subcritical when its characteristic speeds $u+c$ and $u-c$ have opposite signs. A downstream boundary can then send information upstream. For a quadratic pressure-area law and fixed dimensionless flux $q$, this corresponds to $A/A_0>\sqrt{q}$.

For steady uniform-density inviscid flow with pressure law $p=p_0+k(r-R)$ and a prescribed thickness $h$, mass conservation and the Bernoulli equation give $h/R=1-s^{-1/2}+\lambda (1-s^2)/4$, where $s=u/V$ and $\lambda =2\rho V^2/(kR)$. The function has a unique maximum at $s={\lambda }^{-2/5}$, producing the displayed choking bound. Below it there are two positive roots; continuity from the upstream state selects the lower-speed root for $\lambda <1$ and higher-speed root for $\lambda >1$. At $\lambda =1$, the maximum is zero and any positive prescribed thickness is incompatible with this model.

For a matrix $A$ over the complex numbers with the standard Hermitian inner product, the linear system $Ax=b$ is solvable exactly whenThe star is the conjugate transpose; for matrices over the real numbers, use the transpose. Necessity follows from $y^{\ast }Ax=(A^{\ast }y{)}^{\ast }x=0$. For sufficiency, $(\mathrm{im}A{)}^{\perp }=\ker A^{\ast }$, and taking the orthogonal complement again recovers $\mathrm{im}A$ in finite dimension. This is the cokernel obstruction expressed as scalar equations. ^[ref]

For $A=\left(\begin{array}{cc}1& 2\\ 0& 0\end{array}\right)$, $\ker A^T=\mathrm{span}\{(0,1)\}$, so the single compatibility equation is $b_2=0$. If it holds, the kernel supplies the remaining input freedom; if it fails, changing the input cannot remove the obstruction.

The torus and the wedge sum $S^1\vee S^1\vee S^2$ both have integral cohomology $\mathbb{Z},{\mathbb{Z}}^2,\mathbb{Z}$ in degrees zero, one and two. Their fundamental groups differ: the torus has ${\mathbb{Z}}^2$, while the wedge has the nonabelian free group on two generators. Thus even compact finite cell complexes can have the same additive cohomology without being homotopy equivalent.

A generalized cohomology theory assigns graded groups contravariantly to suitable topological pairs, with homotopy invariance, excision and a natural long exact sequence for each pair. It omits the dimension axiom of ordinary cohomology, so its coefficient groups at a point can occur in degrees other than zero. A multiplicative theory also has compatible external and relative cup products.

For an oriented proper smooth map $f:M^m\to N^n$ of oriented manifolds, the Gysin homomorphism is a degree-shifting pushforward $f_{!}:H^k(M)\to H^{k+n-m}(N)$, defined using Poincare duality and the appropriate pushforward on homology. For an oriented sphere bundle it is integration along the fiber and participates in the Gysin sequence of a sphere bundle. Oriented embeddings give the corresponding Thom-class construction.

For $\dot{a}=-\gamma a/2-\sqrt{\gamma }{b}_0$ and $b_1=\sqrt{\gamma }a+b_0$, zero-initial-data Laplace transforms give the displayed transfer function. For $\gamma >0$ its pole is in the open left half-plane, while $|G(i\omega )|=1$ and its zero lies in the right half-plane. Passing through $-1$ in this plant's frequency plot is not evidence of an unstable open cavity: a Nyquist stability criterion needs an explicitly specified feedback return ratio.

For a broadband bosonic input with $[b(t),b^{†}(s)]=\delta (t-s)$, disjoint field increments commute and $[d{B}_t,d{B}_t^{†}]=dt$. An adapted system operator commutes with the next incoming increment. Noise increments scale as $\sqrt{dt}$, so quadratic products contribute to first-order stochastic evolution.

The rotating-wave approximation to a weak quadrature coupling of a cavity mode to a broadband field retains photon-exchange terms and removes counter-rotating pair creation and annihilation. Flat coupling around resonance produces the displayed interaction in units $\hslash =1$. Its short-time propagator is generated by $\sqrt{\gamma }(ad{B}^{†}-a^{†}dB)$; the Gaussian quantum noise Ito table supplies the quadratic drift correction. The resulting passive cavity has the passive cavity input-output transfer function.

In a coherent category, maps $A\to Y$ and $B\to Y$ agreeing on the intersection define a relation from $A\cup B$ to $Y$ by taking the union of their graphs. It is total because the two subobjects cover their union, and single-valued because the maps agree on overlap. Total functional relations define morphisms, giving a unique glued map. This is the pushout universal property without assuming a general supply of pushouts.