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.

Let $\chi$ be a primitive Dirichlet character of conductor of a Dirichlet character $f$. Its generalized Bernoulli numbers are the coefficients of the exponential generating functionComparing this with the Bernoulli polynomial generating function gives the finite formulaThe formula makes the definition computable using finitely many ordinary Bernoulli polynomials.^[ref]

The convention at the trivial character deserves attention. For conductor $f=1$, the generating function is $t{e}^t/(e^t-1)$, so $B_{1,1}=+1/2$, whereas the ordinary Bernoulli number convention in this article is $B_1=-1/2$. All other coefficients agree. For the nontrivial character modulo $4$, with $\chi (1)=1$, $\chi (3)=-1$ and zero values at even inputs, the finite formula gives $B_{1,\chi }=B_1(1/4)-B_1(3/4)=-1/2$.^[ref]

These character-weighted coefficients encode special values of Dirichlet L-functions at nonpositive integers and occur in Kubota-Leopoldt p-adic L-functions. They let the arithmetic carried by a Dirichlet character enter the same generating function framework as ordinary Bernoulli numbers.

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.

If all residues of a modular arithmetic progression have integer representatives in an interval of diameter less than half the modulus, the representatives form an integer arithmetic progression. Consecutive differences lie in an interval shorter than the modulus, so the common modular difference has a unique possible integer representative.

A sum of character values over a set. Arithmetic cancellation, rather than the termwise absolute-value bound, is the useful feature. A Gauss sum of a Dirichlet character is a weighted example, and the Pólya–Vinogradov inequality bounds interval sums.

There is an absolute finite exponent $L$ bounding the least prime in any reduced arithmetic progression by a constant times $q^L$. The multiplicative large sieve inequality is one of the tools in proofs of this theorem.

An L-function is an arithmetic function given initially by a Dirichlet series and often an Euler product, with analytic continuation and a functional equation in important cases. Dirichlet L-functions and L-functions of cusp forms are basic examples; p-adic L-functions interpolate suitably modified special values.

Iwasawa theory studies arithmetic objects in infinite towers of number fields, often Zp-extensions. Passing to inverse limits turns growing ideal class groups, unit groups and Galois groups into modules over an Iwasawa algebra. Their algebraic ranks and characteristic ideals organize the growth and relate it to p-adic L-functions.

The Leopoldt conjecture says that the diagonal image of the global units of a number field in its local $p$-adic unit groups has ${\mathbb{Z}}_p$-rank $r_1+r_2-1$. A defect would measure a loss of rank after $p$-adic completion. It is a theorem for abelian extensions of $\mathbb{Q}$.

Birational geometry studies algebraic varieties up to isomorphism of dense Zariski-open subsets. For irreducible varieties, this is equivalent to isomorphism of their function fields over the base field. For example, $t\mapsto (t,t^2)$ identifies the affine line with the parabola $y=x^2$. The rational parametrization $t\mapsto ((1-t^2)/(1+t^2),2t/(1+t^2))$ identifies suitable open subsets of the affine conic $x^2+y^2=1$ and the affine line over a field of characteristic different from two. A blowup changes a variety along a centre while preserving a dense open subset, so it is a central birational operation.

For every $\delta >0$, every sufficiently long integer interval has a nonconstant three-term arithmetic progression in each subset of subset density at least $\delta$. This is a theorem about additive combinatorics, distinct from the Roth theorem on approximation of algebraic irrational numbers. The Roth density-increment step proves it by repeatedly increasing subset density on a shorter arithmetic progression.

Use four parts indexed by $0,1,2,3$ in a cyclic group $G$, and include the triple missing part $i$ when $L_i\in A$. A transversal three-uniform tetrahedron then has the four values $S_1-i{S}_0$, where $S_0={\sum }_j{x}_j$ and $S_1={\sum }_{j}j{x}_j$, so it encodes a four-term arithmetic progression. Constant progressions give $|A||G{|}^2$ edge-disjoint tetrahedra. This is the bridge from the tetrahedron removal lemma to the Szemerédi theorem.

A frequency function built from an affine variable and its fractional part. Irrational $\gamma$ and $\beta$ can give many exact additive relations because each pair sum has only two possible floor carries. For example $\beta =\sqrt{3}$, $\gamma =\sqrt{2}$ gives $\gg N^3$ additive quadruples on $[-N,N]$, but agreement with every ordinary affine function occurs on only $o(N)$ shifts. Integer floor points underlying affine agreement must be collinear.