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For a clamped elastic filament of filament bending modulus $A$, a transverse tip force $f$ gives $h(x)=f{x}^2(3L-x)/(6A)$ in the small-slope model. Thus $h(L)/f=L^3/(3A)$. This differs from the effective tip stiffness under a distributed load. The zero-load Gaussian response reproduces the thermal bending fluctuations of a clamped filament, $⟨h(L{)}^2⟩=k_{B}T{L}^3/(3A)$ for one transverse coordinate.

A smooth phase gives ${\partial }_{X_i}{k}_j={\partial }_{X_j}{k}_i$ and ${\mathbf{k}}_T=-\nabla \omega$. These mixed-derivative identities allow the spatially differentiated dispersion relation to be rewritten as Hamiltonian ray equations for a local dispersion relation. Away from phase singularities they express the curl-free character of a local wavevector.

The first-order imaginary balance of the wave equation in geometrical optics gives ${\partial }_T(\omega A^2)+\nabla \cdot (c^2\mathbf{k}{A}^2)=0$ for a real leading amplitude. The action density $\omega A^2$ travels with the group velocity $c^2\mathbf{k}/\omega$. For a complex amplitude replace $A^2$ by $|A{|}^2$.

For a finite relational first-order language, with finitely many constants allowed, the Duplicator wins the $n$-round game exactly when the structures are rank-n equivalent. Game-winning implies rank equivalence in arbitrary languages; the converse needs a finiteness condition and can fail even for an infinite unary vocabulary. Over a finite relational vocabulary, winning every announced finite horizon is equivalent to elementary equivalence.

With $k$ moves remaining, preserve corresponding gap sizes exactly when smaller than $2^k-1$, and otherwise require both gaps to have at least that many points. One move splits a gap; $2^k-1=2(2^{k-1}-1)+1$ permits matching both smaller gaps. This strategy proves that $(\mathbb{N},<)$ and $(\mathbb{N}+\mathbb{Z},<)$ agree at every finite quantifier rank despite the second order containing points with infinitely many predecessors.

For relativistic species at a common temperature, the mass-equivalent energy density is $\rho ={\pi }^2{g}_{\ast }(k_{B}T{)}^4/(30{\hslash }^3{c}^5)$. Using mass rather than energy density accounts for the powers of $c$ in the Friedmann equation.

A uniformly stratified quasi-geostrophic approximation fluid with constant vertical shear occupies $z>0$, has a flat rigid lower wall, and has decaying perturbations aloft. On an f-plane, the basic interior potential vorticity is constant. Its smooth edge-wave modes have zero interior anomaly and propagate along the lower-boundary buoyancy gradient. The absence of an upper wall removes the pair of interacting edge waves that makes the finite-depth Eady model unstable.

For constant shear $U=\Lambda z$ between impermeable horizontal walls at $0,D$, with constant buoyancy frequency and no interior potential-vorticity gradient, define $m=NK/|f_0|$. The boundary conditions yieldThe bracket is negative for $0<mD<2.399\dots$, giving baroclinic instability for nonzero zonal wavenumber. For large separation the boundary waves decouple; the lower-wave speed tends to $\Lambda /m$, matching the dispersion relation of a semi-infinite Eady edge wave.

A strategy is weakly dominated if another gives at least as much payoff against every opponent strategy and strictly more against at least one. Unlike a strictly dominated strategy, it may remain a best response to opponents who never choose a strategy where the comparison is strict. Thus weak domination does not exclude equilibrium strategies.

For a scalar field with $X=-q^2/2$, Lagrangian $F(X)$ and stress-energy tensor $T_{ab}=F\prime q_a{q}_b+F{g}_{ab}$, the inequalities $A=F\prime \ge 0$ and $Y=XF\prime -F\ge 0$ imply the dominant energy condition for any gradient type. In a local Lorentz frame, $\rho =A(q_0^2+{\sum }_i{q}_i^2)/2+Y\ge 0$ and ${\rho }^2-{\sum }_i{T}_{0i}^2=A^2{X}^2+A(q_0^2+{\sum }_i{q}_i^2)Y+Y^2\ge 0$.

If $P({\lambda }_0,0)=0$ and $P_{\lambda }({\lambda }_0,0)\ne 0$, the implicit function theorem gives a local root branch. For analytic coefficients it has an ordinary power series in the perturbation parameter. Its corrections follow by substituting that series into the polynomial and comparing coefficients. A vanishing leading root may require its first nonzero correction to capture a slow exponential mode.

A polynomial whose degree decreases when its parameter vanishes may have roots escaping to infinity. Such a singular perturbation root is found by balancing powers after rescaling $\lambda ={\epsilon }^{-p}\mu$. If the root enters an exponential, even a lower-order additive root correction can matter on a fixed interval when its leading exponential grows. This phenomenon is different from a repeated root of a fixed polynomial.

A linear system of divisors is a projective family of effective Weil divisors linearly equivalent to a fixed divisor. A complete linear system of a divisor contains every effective representative of that class; a projective subspace gives a subsystem. For a Cartier divisor, the family is obtained from global sections of its associated line bundle. The algebraic curve case recovers the usual moving family of effective divisors.

The divisor function satisfies ${\sum }_{n\le X}d(n{)}^2\ll X{\log }^3(2X)$. Partial summation consequently gives ${\sum }_{n\le X}d(n{)}^2/n\ll {\log }^4(2X)$.

A squarefree divisor of an integer $n$ is a divisor $d\mid n$ that is a squarefree integer. It is specified by choosing a subset of the distinct prime factors of $n$.

A quaternion is $a+bi+cj+dk$ with real coefficients, where $i^2=j^2=k^2=ijk=-1$. They form a noncommutative real division algebra. The norm squared is $a^2+b^2+c^2+d^2$, and every nonzero quaternion has inverse its conjugate divided by that squared norm. The unit quaternions form the Lie group $S^3$.

Let $D$ be a finite-dimensional division algebra over an algebraically closed field $k$. Left multiplication by $a\in D$ has an eigenvalue $\lambda \in k$ and a nonzero eigenvector $v$. Thus $(a-\lambda )v=0$; multiplying by $v^{-1}$ gives $a=\lambda$. Every element is scalar, so $D=k$.

A locally integrable function $f$ defines a distribution by $⟨u_f,\phi ⟩=\int f\phi$. The pairing is bilinear if distributions are taken complex-linear. This identifies ordinary functions with a subset of distributions without claiming that every distribution is a function.

For distributions $u$ on ${\mathbb{R}}^m$ and $v$ on ${\mathbb{R}}^n$, their tensor product acts on a joint test function by $⟨u\otimes v,\Phi ⟩=⟨u_x,⟨v_y,\Phi (x,y)⟩⟩$. The inner function is smooth with compact support in the projection of $\mathrm{supp}\Phi$, and finite order of a distribution estimates prove continuity. Reversing the pairings gives the same value: finite sums of product kernels are dense in the required smooth seminorms, and the two orders agree on product kernels. Its support is contained in the Cartesian product of the two supports. This provides a rigorous replacement for formal integration in separate distribution variables.

For every positive distance, draw equal-radius circles centred at the points of $P$. Their incidences between points and curves count the ordered pairs at that distance. The Szemerédi–Trotter theorem for unit circles bounds each distance class by $O(|P{|}^{4/3})$. Summing over all classes accounts for $|P|(|P|-1)$ pairs, proving $|\Delta (P)|\gtrsim |P{|}^{2/3}$.