For $X\sim N(m,v)$ with $v>0$ and $0<\epsilon <1$, $P(X\ge C)\le \epsilon$ is equivalent to the displayed chance constraint, because the standard normal distribution function is continuous and strictly increasing. The coefficient is a standard normal quantile, which can be negative. If $v=0$, equality is an atom and the exact condition becomes $C>m$ instead.

A deterministic demand has probability one of meeting or exceeding its mean. Thus replacing a positive-variance Gaussian chance constraint by its naive zero-variance limit with a non-strict capacity inequality loses the boundary condition. The example $X=0$, $C=0$, $\epsilon =e^{-1}$ satisfies $m+\varphi \sqrt{v}\le C$ for finite $\varphi$, but violates the chance requirement.

A coreflexive pair $f,g:B\rightrightarrows A$ has a common retraction $r:A\to B$ with $rf=rg=1_B$. It is the dual of a reflexive pair. Both $f$ and $g$ are split monomorphisms.

A sequence of functionals Gamma-converges in a specified topology when every convergent $u_n\to u$ satisfies $F(u)\le {\mathrm{liminf}}_n{F}_n(u_n)$, and each $u$ has a recovery sequence $u_n\to u$ with $F_n(u_n)\to F(u)$. With appropriate compactness of energy sublevels, limits of approximate global minimizers minimize $F$, and the minimum values converge. This is suited to variational approximation such as the Ambrosio–Tortorelli approximation; it does not describe arbitrary local minima.

The worldvolume is the history of a brane. An embedding $X:\Sigma \to \mathcal{M}$ maps its local coordinates into target spacetime. A string worldvolume is a worldsheet.

The brane tension is the energy per unit spatial brane volume and multiplies its geometric action. In units $\hslash =c=1$, a $p$-brane tension has mass dimension $p+1$. The $p=1$ case is the string tension.

For the normalized isotropic cutoff model, collisions redistribute the direction of relative velocity uniformly while keeping its length. With center $C=(v+v_{\ast })/2$ and relative speed $r=|v-v_{\ast }|$, outgoing velocities are $C\pm r\sigma /2$. Momentum and kinetic energy are preserved, and the collision rate has no relative-speed factor. The displayed operator uses unit total angular rate. General Maxwell molecule kernels can have nonconstant angular dependence.

The integral uses $0\log 0=0$ when it is defined. For the Boltzmann equation, the Boltzmann H theorem makes it nonincreasing under suitable regularity and integrability. This functional is distinct from the microstate-counting Boltzmann entropy; the physical kinetic entropy has the opposite sign.

A Gaussian velocity profile at each position, with coefficients encoding local density, mean velocity and temperature. Its logarithm is a linear combination of collision invariants, so its collision term vanishes pointwise in position. Transport need not vanish: the coefficients must also satisfy their transport equations for a local Maxwellian to solve the full Boltzmann equation.

For equal-mass elastic collisions, one reflection parametrization is $v^{\prime }=v-((v-v_{\ast })\cdot n)n$ and $v_{\ast }^{\prime }=v_{\ast }+((v-v_{\ast })\cdot n)n$. The fixed-$n$ transformation preserves pairwise momentum, kinetic energy and velocity-pair measure. A nonnegative kernel invariant under particle interchange and collision reversal yields a gain-loss operator. Its weak Boltzmann collision identity proves conservation and its entropy identity proves the Boltzmann H theorem. Unlike the Maxwell molecule collision operator, this general kernel may depend on relative speed.

When the velocity distribution is independent of position and there are no forces, the free transport equation contributes zero and only collisions evolve $f$. For a bounded cutoff kernel, $\Vert Q(f,g){\Vert }_1\le C\Vert f{\Vert }_1\Vert g{\Vert }_1$ gives local existence and uniqueness by the Banach fixed-point theorem. Positivity and mass conservation bound the norm and permit global continuation in this particular model. Stronger kernels require moment or weighted estimates.

In a scaling ${\partial }_{t}f+v\cdot {\nabla }_{x}f={\epsilon }^{-1}Q(f,f)$ with small Knudsen number, the leading collision balance formally forces a local Maxwellian. The conserved moments then give the Euler equations with a monatomic ideal-gas closure. The next nonequilibrium correction produces viscous and heat-conduction terms. These substitutions are formal: rigorous limits need uniform estimates, initial and boundary conditions, and sufficient convergence.

A weak solution framework in which a suitable nonlinear transform $\beta (f)$ satisfies a transport equation with transformed collision source ${\beta }^{\prime }(f)Q(f,f)$, interpreted with the required integrability and possible framework-dependent defect terms. Renormalization addresses distributions whose raw collision term is not integrable enough to support the ordinary weak formulation. This is different from a classical smooth solution and does not by itself imply smooth global well-posedness.

For a Dirichlet character $\chi$ of conductor $f$, define the generalized Bernoulli numbers byThey express special values of Dirichlet L-functions at nonpositive integers and occur in the interpolation formula for Kubota-Leopoldt p-adic L-functions.

A Bell pair consists of two qubits prepared jointly in a Bell state. It is a maximally entangled two-qubit resource with Schmidt rank two. Two independent Bell pairs have resource Schmidt rank four across the same party partition.

This barotropic pressure potential obeys $\nabla h=\nabla p/\rho$ for a barotropic fluid. It gives the pressure contribution to a steady Bernoulli equation. In isentropic thermodynamics it agrees with specific enthalpy up to a constant; for an externally maintained isothermal ideal gas it is $c_s^2\log \rho$ and is not the constant thermodynamic enthalpy at fixed temperature.

Steady spherical continuity equation gives $r^2\rho u=\text{constant}$. Combining this with Euler equations for an inviscid fluid and $c_s^2=dp/d\rho$ gives the displayed equation. Smooth transonic passage requires $u^2=c_s^2$ and ${\Phi }^{\prime }=2c_s^2/r$ simultaneously. The barotropic enthalpy function gives the integral $u^2/2+h+\Phi =\text{constant}$.

For fixed real $\mu \ne 0$, roots of $\tan x=\mu x$ tending to positive infinity approach tangent poles $a_n=(n+1/2)\pi$. Expanding $-\cot (x-a_n)$ givesThe expansion works for either sign of $\mu$ when $|\mu |a_n\gg 1$, but is nonuniform at $\mu =0$, where the roots lie at $n\pi$ instead. The pole labeling need not equal the ordered root index at small $n$.

In a connected inviscid uniformly rotating barotropic fluid, momentum balance makes the displayed sum constant. The pressure term is the barotropic enthalpy function. Perturbations experience Coriolis acceleration, and their rotating-frame frequency depends on the chosen exponential sign convention.

For an axisymmetric reflection-symmetric thin disk, the Laplace equation in cylindrical coordinates separates into regular radial Bessel functions and vertical decaying exponentials. The divergence theorem imposes ${\Phi }_z^{+}-{\Phi }_z^{-}=4\pi G\Sigma$. Hankel transform inversion gives $S(k)=-2\pi G{\int }_0^{\infty }\Sigma (R)J_0(kR)RdR$. Its radial derivative gives the galaxy rotation curve. Infinite scale-free models may require subtracting an additive divergent constant.