A transition state theory expression relating a rate constant to the standard free energy of activation. For a bimolecular reaction expressed in concentration units, $k_2=(k_{\mathrm{B}}T/h)(c^{\circ }{)}^{-1}\exp [-\Delta G^{‡\circ }/(RT)]$ in the unit-transmission approximation.

The standard enthalpy difference between the activated-complex reference state and the reactants at the same temperature and pressure. Under the usual Eyring assumptions it controls the slope of $\ln (k/T)$ against $1/T$.

The standard entropy difference between the activated-complex reference state and the reactants. Its numerical value depends on standard-state conventions; association, molecular reorientation, internal motions and solvent reorganization can all contribute.

The pressure derivative of the standard Gibbs energy of activation: $\Delta V^{‡\circ }=(\partial \Delta G^{‡\circ }/\partial p{)}_T$. In the corresponding rate convention, $\Delta V^{‡\circ }=-RT(\partial \ln k/\partial p{)}_T$. A negative value makes increasing pressure accelerate the reaction. Solvent reorganization can contribute as well as the intrinsic reacting geometry.

A constant tensile force changes a one-coordinate free-energy landscape from $G_0(x)$ to $G_0(x)-Fx$. To first order, an escape barrier decreases by $F\Delta x$, where $\Delta x$ is the saddle-to-initial-minimum displacement at zero force. An approximately fixed attempt prefactor gives the displayed rate law. Its characteristic force is $F_0=k_{B}T/\Delta x$. Large changes in saddle location or competing pathways invalidate a constant activation distance. The barrier is measured relative to the initial state; an absolute saddle energy is not enough to determine an escape rate.

For exchange of closely related ester and alcohol groups, an excess alcohol reservoir and dilution of the outgoing alcohol can favour conversion through mixture chemical potentials. Equal molecular stoichiometry does not itself imply a positive standard entropy change; a reaction quotient is needed.

A nondecreasing list of nonnegative integers $s_1\le \cdots \le s_n$ is the list of outdegrees of a tournament exactly whenNecessity follows because any $k$ vertices have $\left(\genfrac{}{}{0ex}{}{k}{2}\right)$ internal edges, each contributing one to their total outdegree. Sufficiency is the substantive part of Landau's theorem on tournament scores.

The list $(1,1,1)$ is realized by a directed triangle; $(0,1,2)$ is realized by a transitive tournament. The list $(0,0,3,3)$ has the correct total but fails the inequality at $k=2$, so it cannot occur. Total score alone therefore does not characterize a tournament.

A transitive tournament is a tournament whose vertices admit an ordering $v_1,\dots ,v_n$ with every edge directed from $v_i$ to $v_j$ when $i<j$. Equivalently, it has no directed cycle, or no directed triangle. The implication from no triangle follows by shortening any shortest directed cycle of length greater than three using its chord.

Its outdegrees are $n-1,n-2,\dots ,0$, so the ordering records a consistent ranking with no cyclic preference. Every three-vertex tournament is either transitive or a directed triangle. This makes transitive tournaments the simplest model of a complete pairwise comparison that agrees with a total order.

For principal inertias $I_{\perp },I_{\perp },I_{\parallel }$, the transverse angular-velocity components rotate in body axes at signed rate $(I_{\parallel }-I_{\perp }){\omega }_3/I_{\perp }$. The symmetry axis precesses about the inertially fixed angular momentum at rate $L/I_{\perp }$. These are different frame-dependent descriptions of the same motion.

A defect localized at a boundary of an ordered medium. A spherical inclusion with degenerate planar anchoring in a nematic liquid crystal can have a bipolar texture with two surface boojums, unlike a bulk point-defect pair.

A three-dimensional point defect characterized by how the surrounding nematic director covers orientation space. Its hedgehog charge is not the same quantity as a planar disclination winding strength.

An extended order-parameter winding associated with a nontrivial third homotopy group of a vacuum manifold in three spatial dimensions. A texture can collapse and unwind when its field leaves the vacuum manifold; it differs from a permanent monopole or string core.

Disconnected vacuum components support walls, noncontractible vacuum loops support line defects, and nontrivial vacuum two-spheres support monopoles in three spatial dimensions. Allowed topology does not alone predict the number formed or their subsequent survival.

A transform-limited pulse needs coherent phases across its contributing frequency range. A laser's usable gain bandwidth therefore sets a duration scale through the time-bandwidth product, as displayed. The wavelength interval must first be converted to frequency; the simple conversion $c\Delta \lambda /{\lambda }^2$ is only a narrow-band approximation. Numerical constants depend on pulse shape and width convention: a Gaussian intensity envelope has $\Delta t_{\mathrm{FWHM}}\Delta {\nu }_{\mathrm{FWHM}}=2\ln 2/\pi \simeq 0.441$. A hard gain interval is not automatically a spectral FWHM. Mode locking can arrange the longitudinal modes' phases, while group velocity dispersion must be controlled to approach the transform limit. The cavity round-trip time sets pulse spacing, rather than this bandwidth-limited width.

For $\Phi =-GM/r$, the acceleration tidal tensor isIts radial eigenvalue is $2GM/r^3$ and its two transverse eigenvalues are $-GM/r^3$. This stretches a small freely falling cloud radially and compresses it transversely. The zero trace expresses the vacuum Poisson equation outside the source; within matter the trace is $-4\pi G\rho$.

A point-mass host creates radial differential acceleration $2GMx/D^3$, while a uniform satellite restores with $4\pi G\rho x/3$. Equality gives $D=[3M/(2\pi \rho ){]}^{1/3}$. The displacement cancels; the robust scaling is $(M/\rho {)}^{1/3}$. A rotating circular-orbit Jacobi calculation changes the radial coefficient from two to three.

For a converging thin lens forming a real image of a real object, let $u,v$ be positive distances from the lens and $f$ the focal length. Set $u^{\prime }=u-f$ and $v^{\prime }=v-f$, measuring from the neighboring focal points. Substituting into $1/u+1/v=1/f$ gives $(f+u^{\prime })(f+v^{\prime })=f(2f+u^{\prime }+v^{\prime })$, so $u^{\prime }{v}^{\prime }=f^2$. Other object-image geometries require an explicitly consistent signed-distance convention.

For an optical element, a virtual object is a point toward which incident rays already converge, so they would meet beyond the element if it were removed. It is distinct from a virtual image, which is an apparent origin of outgoing rays. In the real-is-positive thin-lens convention, a virtual object beyond the lens has $u<0$, and $1/u+1/v=1/f$. Thus $u\to -\infty$ gives a real image in the rear focal plane of a converging lens.

An optical coating designed to reduce reflection through refractive-index choice and thin-film interference. A quarter-wave design is a single-wavelength approximation, not broadband perfection.