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A rate law specifies how a reaction rate depends on concentrations under stated conditions. A common form iswhere $k$ is a rate constant and the exponents are partial orders of reaction. Determine them experimentally, or derive them from a justified reaction mechanism. Do not infer $\alpha =a$ and $\beta =b$ from the net chemical equation: its coefficients describe consumption ratios, whereas the exponents describe sensitivity to concentration. See IUPAC Gold Book: rate law and Flowers et al., Chemistry 2e, §12.3: rate laws.

A rate law becomes a prediction of time evolution when combined with stoichiometry in a rate equation, for example $d[\mathrm{A}]/dt=-av$. Agreement with a measured rate law is useful evidence for a reaction mechanism, but several routes can give the same dependence.

The disappearance rate of a reactant A is its positive net loss of concentration per unit time, $v_{\mathrm{A}}=-d[\mathrm{A}]/dt$. For the fixed-volume, single-chemical reaction setting of reaction rate, consuming $a$ units of A per conversion gives $v_{\mathrm{A}}=av$. The factor belongs to the stoichiometry; stating it is essential when quoting a rate constant. A negative net disappearance rate means that A is accumulating.

An elementary reaction is one microscopic step in a reaction mechanism, with no intervening reaction intermediate needed within that step. Its molecularity counts the participating reacting entities. In an ideal dilute, uniformly mixed model, the law of mass action gives the directional reaction rate from the step's reactant concentrations: $\mathrm{A}⟶\mathrm{P}$ has $r=k[\mathrm{A}]$, while $\mathrm{A}+\mathrm{B}⟶\mathrm{P}$ has $r=k[\mathrm{A}][\mathrm{B}]$. Applying those powers to a net chemical equation requires separate evidence that it is elementary.^[ref]

For a reversible step $\mathrm{A}+\mathrm{B}\rightleftharpoons \mathrm{I}$, apply that rule separately: $r_f=k_f[\mathrm{A}][\mathrm{B}]$ and $r_r=k_r[\mathrm{I}]$, giving net production of I as $r_f-r_r$. Their difference can vanish at chemical equilibrium while both directional reaction rates remain nonzero. For $2\mathrm{A}⟶\mathrm{P}$, the elementary normalized reaction rate is $r=k[\mathrm{A}{]}^2$, whereas A's disappearance rate is $2r$; see second-order reaction for the complete factor-of-two example.

Calling a step slow does not remove the need for its actual reactant concentrations. If a reaction intermediate enters its rate law, determine that concentration from the other rate equations or a justified approximation. In the consecutive first-order reaction mechanism, an intermediate maximum is only one instant at which its production and consumption match. Replacing its entire evolution by that one balance would lose the initial buildup and delayed reaction product formation. A successful overall first-order reaction fit likewise does not establish an elementary event: both worked reaction mechanisms above provide ways for microscopic detail to remain hidden in one reactant trace.

A reaction coordinate parametrizes changes in molecular configuration along a chemical reaction path. It can involve a bond length, a bond angle, or a combination of geometric changes. It is not elapsed time and is not the bulk extent of reaction: one describes configurations, while the other accounts for how much material has converted. The precise choice matters when interpreting an energy profile. See the IUPAC Gold Book: reaction coordinate.

A reaction intermediate is formed during a reaction mechanism and consumed in a subsequent step, cancelling from the net chemical equation. It can accumulate, as I does in the consecutive first-order reaction mechanism. A reaction intermediate can also connect transformations in energy coupling. Unlike a catalyst, it need not be present before conversion begins.^[ref]

A reaction intermediate is distinct from a transition state: the former can persist between steps, whereas the latter describes passage through a barrier. The existing potential energy surface and energy profile provide the configuration-space picture; the time-dependent concentration of an intermediate instead measures an ensemble of reacting entities. Low concentration or short lifetime can make detection difficult, so failure to detect a proposed reaction intermediate is not by itself proof of absence. Conversely, detecting a chemical species does not establish that it carries the proposed route to the final reaction products; it could belong to a competing route.

The double integrator is the linear time-invariant system in which one control input is integrated into velocity, and velocity into position. It is a mathematical idealization of one-dimensional motion under commanded acceleration; for a constant mass $m$ and net force $F$, Newton's second law gives $u=F/m$. The worked examples here use normalized dimensionless quantities and assume no control disturbances, rather than describing a laboratory record. ^[ref]

Writing $x=(p,v{)}^T$ givesThe Kalman rank criterion therefore gives controllability in every $T>0$ for unrestricted real control inputs. More concretely, choose constants $a,b$ on the first and second halves. Starting at $x_0=(p_0,v_0{)}^T$, direct integration givesThe endpoint map has determinant $T^3/8\ne 0$, so these two amplitudes can supply any desired endpoint displacement. For a rest-to-rest move of distance $d$, take $a=4d/T^2$, $b=-4d/T^2$. At $T=2$ and $d=1$ these are the two commands in the first figure. No impulse is needed; shorter deadlines require larger finite commands.

The reachable set ${\mathcal{R}}_T(x_0)$ collects all endpoints attainable from $x_0$ at exactly $T$ by admissible control inputs. If arrival any time up to $T$ is allowed, use ${\mathcal{R}}_{\le T}(x_0)={\bigcup }_{0\le t\le T}{\mathcal{R}}_t(x_0)$, with ${\mathcal{R}}_0(x_0)=\{x_0\}$. These definitions differ: a general dynamical system may drift away after early arrival, so waiting need not preserve the target. The reachable set is always relative to a declared input class and any constraints on the entire path.

For the double integrator starting from $(0,0)$, now require measurable functions as control inputs with $|u(t)|\le 1$ almost everywhere. Write $(p,v)$ for its endpoint at $T>0$. Two integrals determine the endpoint:Our mathematical example has the exact reachable setTo see the upper boundary, fix $v$: put $u=1$ first and $u=-1$ last, switching at $s=(T+v)/2$. This maximizes the weighted integral for position because early commands have the largest weight $T-t$; the required net velocity fixes $s$. Formally, $(T-t)-(T-s)=s-t$ has the same sign as this maximizing command, so changing to any other admissible command with the same integral cannot increase $p$. Reversing the order gives the lower boundary. Pointwise convex combinations of these two commands preserve $v$ and attain every intermediate $p$, so the formula describes the whole region, not merely a bound. The extreme commands are bang-bang control.

In particular, the bounded double integrator can move from rest to position $d$ and rest again exactly when $|d|\le T^2/4$. Its minimum duration is therefore $2\sqrt{|d|}$ in these normalized units. With freely chosen time every rest-to-rest distance is attainable; with a fixed deadline many are excluded. Changing the amplitude bound to $|u|\le a$ with $a>0$ scales this condition to $|d|\le a{T}^2/4$. This shows why specifying the controls matters even when the Kalman rank criterion gives full rank of a matrix.

In closed quantum control, a density operator reachable set lies inside its unitary orbit of a density operator, since coherent evolution preserves the spectrum. That orbit need not be fully reachable under the available control inputs.

Consider uniform fields $B_{-}\hat{\mathbf{y}}$ below and $B_{+}\hat{\mathbf{x}}$ above an interface, with local constant densities ${\rho }_{-},{\rho }_{+}$. For ideal magnetohydrodynamics and local incompressible potential flow, the interface displacement $h{e}^{st+ikx+imy}$ has displacement potentials ${\chi }_{-}=h{e}^{Kz}/K$ and ${\chi }_{+}=-h{e}^{-Kz}/K$, $K=\sqrt{k^2+m^2}$. Magnetic flux freezing gives ${\mathbf{b}}_i=({\mathbf{B}}_i\cdot \nabla )\nabla {\chi }_i$. The linear magnetohydrodynamic total pressure is $-[{\rho }_i{s}^2+({\mathbf{k}}_h\cdot {\mathbf{B}}_i{)}^2/{\mu }_0]{\chi }_i$. Matching Lagrangian total pressure gives the displayed dispersion relation. For ${\rho }_{+}>{\rho }_{-}$ the most unstable orientation has its wavevector along the weaker field.

// Target: continuum-mechanics.bigb

A fraction $a/b$ represents the quotient of its numerator $a$ by its nonzero denominator $b$. For integers $a,b$ this is a rational number. Different fractions can represent the same number: $252/198=14/11$. Dividing both entries by their greatest common divisor gives a reduced fraction, with the sign convention that the denominator is positive.

For independent geometric observations with mass $p^{r-1}(1-p)$ on $r=1,2,\dots$, where $0\le p<1$, the total $T$ is sufficient. Conditional on $T=t$, positive-integer compositions are uniformly distributed. The indicator $1_{\{X_1>1\}}$ is unbiased for $p$, and conditioning it on $T$ gives $\left(\genfrac{}{}{0ex}{}{t-2}{n-1}\right)/\left(\genfrac{}{}{0ex}{}{t-1}{n-1}\right)=(t-n)/(t-1)$. For one observation, use $1_{\{T>1\}}$ instead, including its zero value at $T=1$.

For independent uniform random variables on $(\theta ,3\theta )$ with $\theta >0$, the sample minimum and maximum form a sufficient statistic. Conditional on these endpoints, exchangeability and the uniform law of the remaining observations give $E[X_1\mid X_{(1)},X_{(n)}]=(X_{(1)}+X_{(n)})/2$. Applying the Rao-Blackwell theorem to $X_1/2$ gives the displayed unbiased estimator. Its variance is ${\theta }^2/[2(n+1)(n+2)]$, compared with ${\theta }^2/12$ for $X_1/2$.

In a one-brane warped background, the proper extra dimension can be infinite while the four-dimensional graviton zero mode has finite norm. The Einstein-Hilbert action weighs its four-dimensional curvature term by the warp factor integral shown above, finite for $k>0$. Thus the proper length is not the effective gravitational volume. This differs from an unwarped noncompact direction, whose volume integral diverges. It also differs from the two-brane hierarchy mechanism, where finite brane separation redshifts masses on the distant brane. The noncompact construction is established in the original single-brane warped-gravity paper.

For a spatially flat, tuned one-brane background without energy transfer to the bulk, the Israel junction conditions produce the displayed modified Friedmann equation. Positive brane tension $\lambda$ gives ordinary expansion when $\rho \ll \lambda$, while the quadratic density correction enhances expansion at high energy. Big Bang nucleosynthesis, early acoustic evolution and primordial spectra can constrain this change. Spatial curvature, a residual cosmological constant and bulk energy exchange require their additional terms rather than this special formula.

Let $R_k(3)$ be the least order of a complete graph whose every $k$-edge-colouring has a monochromatic triangle in a graph. At a vertex of a graph of order $k(R_{k-1}(3)-1)+2$, some colour joins at least $R_{k-1}(3)$ neighbours. An edge of that colour among these neighbours completes a triangle; otherwise the neighbours use at most $k-1$ colours and contain a triangle by induction. This proves the recurrence and the weaker explicit bound $R_k(3)\le 3k!$.

Every finite colouring of the positive integers contains a pair with nonzero square difference. One proof takes limits of the colour autocorrelations and represents the resulting positive-definite sequence by a positive spectral measure. Its atom at zero has mass at least the sum of squared colour densities. Averaging along squares kills irrational frequencies; a common multiple makes finitely many rational atomic frequencies equal to one. The remaining rational atomic mass can be made arbitrarily small, leaving a positive average correlation.

Radius stabilization fixes the size of a compactification dynamically rather than choosing it as an undetermined parameter. If the four-dimensional low-energy effective action contains a radion $R$ with positive kinetic coefficient $Z(R)$ and scalar potential $V(R)$, a stable radius $R_0$ requires $V^{\prime }(R_0)=0$ and $m_R^2=V^{\prime \prime }(R_0)/Z(R_0)>0$. This matters for the hierarchy problem: the volume or warp factor explains a hierarchy only if a consistent dynamics selects and maintains the required geometry.

An electrical substitution radiometer measures radiant flux by replacing absorbed optical power with calibrated Joule heating. At fixed temperature, feedback control holds the receiver equally warm with the beam blocked or admitted: after corrections, absorbed optical power equals the reduction in supplied electrical heating. Incident radiant flux additionally requires radiometric absorptance and entrance-path corrections. This connects radiometry to electrical standards.^[ref]

Picture a small absorbing cavity connected to a colder reference block by a radiometer thermal link. A heater supplies Joule heating, a thermometer senses the receiver, and feedback control adjusts the heater. When light supplies some of the heating, the controller supplies less. The readout is the change in heater power, rather than the beam's brightness or its number of photons. NIST's institutional description gives a concrete receiver with a conical interior, absorbing coating and weak Kapton support, operated with a receiver heater and thermometer.^[ref]

Irradiance measures incident radiant flux per actual receiving area. Its units $\mathrm{W}{\mathrm{m}}^{-2}$ match radiant exitance, but it counts the incoming side of the radiation exchange. Measure incoming angles from the unit normal pointing toward the incident hemisphere. ThenFor $2\mathrm{W}$ distributed uniformly over $0.010{\mathrm{m}}^2$, $E=200\mathrm{W}{\mathrm{m}}^{-2}$. If the receiving plane is tilted $60^{\circ }$ in the same broad uniform parallel beam, the incident power per actual area is halved to $100\mathrm{W}{\mathrm{m}}^{-2}$. This receiving-surface cosine factor is geometric and does not assume Lambertian reflectance. Together, irradiance and radiant exitance describe how radiation supplies and removes energy at a surface, while radiance preserves the directional information.

Radiant exitance is total outward emitted radiant flux per actual emitting area, measured in $\mathrm{W}{\mathrm{m}}^{-2}$. Integrate radiance over the outward hemisphere, keeping the projected area factor:If the radiance is direction-independent, the spherical coordinates element is $d\Omega =\sin \theta d\theta d\phi$, soThe weighted solid angle is $\pi \mathrm{sr}$, although the whole hemisphere covers $2\pi \mathrm{sr}$. Directions near grazing contribute less because of the projected area. This geometric factor is the bridge between Lambert's cosine law and the total output of a diffuse patch. ^[ref].

For the uniform patch with $L=10\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}$, radiant exitance is $M=10\pi \mathrm{W}{\mathrm{m}}^{-2}$. Its area $0.020{\mathrm{m}}^2$ gives total radiant flux $\Phi =MA=0.20\pi \mathrm{W}\approx 0.628319\mathrm{W}$ over the outward hemisphere. This total differs from its radiant intensity $I(0)=0.20\mathrm{W}\mathrm{s}{\mathrm{r}}^{-1}$, which describes only one direction.

For an ideal blackbody, integrate Planck's law over frequency as well: $M=M_e=j^{\ast }=\sigma T^4$ by the Stefan–Boltzmann law. An ideal $6000\mathrm{K}$ emitting patch of area $1\mathrm{m}{\mathrm{m}}^2$ emits about $73.4881\mathrm{W}$ outward. Irradiance counts incoming radiation separately; for purely emitted output the net outward exchange per area is $M-E$. Reflected output must also be included if the surface reflects incident radiation.

Radiance resolves both position and direction: it is radiant flux per projected area and per solid angle. For a surface element of area $dA$, and a small outgoing cone $d\Omega$ at angle $0\le \theta <\pi /2$ to its outward unit normal,Here $\mathbf{x}$ locates the patch and $\mathbf{s}$ is a unit vector along the outgoing direction. The factor $dA\cos \theta$ is the patch's apparent area viewed along that direction. The same local definition applies to transmitted or received radiation, choosing the positive projected area for the crossing direction. ^[ref].

For a planar patch of area $A$ with spatially uniform, direction-independent radiance $L$, integrating over the patch givesTake $L=10\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}$ and $A=0.020{\mathrm{m}}^2$. The radiant intensities are $I(0)=0.20\mathrm{W}\mathrm{s}{\mathrm{r}}^{-1}$ and $I(60^{\circ })=0.10\mathrm{W}\mathrm{s}{\mathrm{r}}^{-1}$. A sufficiently narrow $0.010\mathrm{sr}$ cone therefore carries approximately $2.0\mathrm{mW}$ or $1.0\mathrm{mW}$, respectively. Both views have the same radiance $L$: the oblique view presents half the projected area. This is Lambert's cosine law, not a loss of radiance with viewing angle.

Radiance is useful because it separates a source's local directional output from the geometry of an observation. Integrating over outgoing directions gives radiant exitance; integrating incoming radiance at a receiver gives irradiance. In the stationary vacuum setting of vacuum conservation of specific intensity, a resolved uniform source retains its radiance along each light ray, while the radiant flux received from the whole source can decrease as its apparent solid angle shrinks.