A monochromatic internal gravity wave in uniform stratification has a spatially linear phase. In the nonrotating inviscid Boussinesq approximation, its dispersion relation is ${\omega }^2=N^2|{\mathbf{k}}_h{|}^2/(|{\mathbf{k}}_h{|}^2+m^2)$. The wave vector is normal to its phase planes, while the group velocity lies along them.

Nonlinear disruption of an internal gravity wave, for example when large displacement gradients overturn the stable density arrangement or strong shear becomes unstable. The resulting turbulence dissipates wave energy and mixes the stratification, limiting ideal geometric focusing.

Specific internal energy $e=U/M$ is internal energy per unit mass, measured in $\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}$. For a perfect gas with constant specific-heat ratio $\gamma$, choosing the constant reference as zero gives $e=c_{v}T=p/[(\gamma -1)\rho ]$, where $c_v$ is the specific heat capacity at constant volume, $T$ the temperature, $p$ the pressure and $\rho$ the mass density.

For a single-component fluid, the local first law of thermodynamics can be writtenThe material derivative follows velocity $\mathbf{u}$; $\mathbf{\tau }$ is the viscous stress tensor, $\mathbf{q}$ the heat flux density, and $r$ an external heating rate per unit volume. Compression supplies pressure work, the stress contraction accounts for viscous dissipation, and convergence of heat flux raises internal energy. Writing only $-p\nabla \cdot \mathbf{u}+Q$ assumes that viscous heating is negligible or included in $Q$. An insulated sheared fluid can warm even when no heat crosses its boundary.^[ref]

Multiplication by a nonzero element $a$ of an integral domain is injective, by multiplicative cancellation. When the domain is finite, this self-map is surjective, so some element $b$ satisfies $ab=1$. Commutativity gives $ba=1$, so every nonzero element has a multiplicative inverse and the domain is a field.

Even when the domain convention does not initially assume an identity element, finiteness produces one. Surjectivity supplies $e$ with $ae=a$; then $a(ex-x)=0$ forces $ex=x$ for every $x$. Commutativity makes $e$ a two-sided identity, after which the inverse argument applies.

The discriminant of an integral full-rank lattice is the field discriminant multiplied by the square of its index in the ring of integers. A squarefree lattice discriminant therefore forces index one.

If $m>1$ is a square-free integer, $3\nmid m$, and $m\not\equiv \pm 1(\mathrm{mod}9)$, then $1,\sqrt[3]{m},\sqrt[3]{m^2}$ is an integral basis of the pure cubic number field, with field discriminant $-27m^2$. The discriminant-index formula for an integral lattice restricts a possible index to primes dividing $3m$. At primes dividing $m$ the different exponent is $2$ by tame ramification. A shifted Eisenstein polynomial gives total wild ramification at $3$, so its different exponent is at least $3$. These exponents exhaust the polynomial discriminant and force index one.

A Stäckel potential allows additive separation of the Hamilton-Jacobi equation in an orthogonal coordinate system. In axisymmetric prolate spheroidal coordinates, the relative-potential form is $\psi =[G(\lambda )-G(\mu )]/(\lambda -\mu )$. It gives energy, axial angular momentum and an independent third separation integral. Here $G$ is a one-variable generating function, not the gravitational constant. Generic bounded orbits are regular invariant-torus motions; resonance and coordinate degeneracy require separate treatment.

For a square-integrable time series, let ${\mathcal{H}}_{t-1}$ be the closed linear span of its past. The innovation ${\epsilon }_t=X_t-{\mathrm{proj}}_{{\mathcal{H}}_{t-1}}{X}_t$ is orthogonal to every past linear observation. A causal invertible autoregressive moving-average model is driven by these innovations, which are weak white noise for a weakly stationary process.

For complex $m\times n$ matrices, the Frobenius inner product isHere $A^{\ast }$ is the conjugate transpose, and this convention is conjugate-linear in the first argument. It is the ordinary Euclidean inner product on all matrix entries, so $⟨A,A{⟩}_F={\sum }_{i,j}|a_{ij}{|}^2$ is positive for every nonzero matrix. Its induced norm is the Frobenius norm.

On real symmetric matrices it becomes $\mathrm{tr}(AB)$. In particular, for a real symmetric matrix $A$ and a real vector $x$,Thus a quadratic form evaluated at $x$ is a linear test of $A$ against a rank-one positive semidefinite matrix. This connects the geometry of matrices with quadratic measurements of vectors.

The Fischer inner product on polynomials is $⟨x^{\alpha },x^{\beta }{⟩}_F=\alpha !{\delta }_{\alpha \beta }$. Multiplication by $x_i$ is adjoint to differentiation by $x_i$. Therefore multiplication by $|x{|}^2$ is adjoint to the ordinary Laplacian, yielding the harmonic decomposition of homogeneous polynomials by finite-dimensional orthogonality.

For vectors $v_1,\dots ,v_n$ in a real or complex inner product space, averaging over all $2^n$ sign choices givesExpand the squared norm. Each diagonal term is $\Vert v_j{\Vert }^2$ in every summand. Each mixed term contains a factor ${ϵ}_j{ϵ}_k$ with $j\ne k$; reversing ${ϵ}_j$ pairs it with an equal term of opposite sign. Thus all mixed terms cancel. Equivalently, independent symmetric random signs have $\mathbb{E}[{ϵ}_j{ϵ}_k]=0$ for distinct indices.

For $n=2$ this is the parallelogram law $\Vert u+v{\Vert }^2+\Vert u-v{\Vert }^2=2\Vert u{\Vert }^2+2\Vert v{\Vert }^2$. It expresses a special feature of inner product geometry: the average squared length of a random signed sum is just the sum of the individual squared lengths.

For a commutative ring that is a Noetherian ring, injectivity of a module can be tested at all prime localizations. By the Baer criterion, test ${\mathrm{Ext}}_R^1(R/J,M)$ for every ideal $J$. Localization of Ext over a Noetherian ring identifies its localization with the corresponding ideal test over $R_P$. Every ideal of $R_P$ is extended from its contraction, and localization detects zero elements applies to the resulting Ext module even when it is not finitely generated. These observations prove both directions.

Differential entropy describes the spread of a continuous probability distribution relative to the volume scale of its coordinates. For a real random variable $X$ with probability density function $f$, use natural logarithms and writeThe convention is $0\ln 0=0$. For a finite value, require $\int f|\ln f|<\infty$; if both positive and negative contributions are infinite, the expression is undefined. Unlike a probability, a probability density can exceed one. Consequently differential entropy can be negative. Base-two logarithms give $h_2=h_e/\ln 2$, in bits instead of nats.^[ref]

This quantity does not give the exact number of bits needed to store a real number. A recording device supplies a resolution; equal-width bin entropy connects that resolution to discrete information entropy. The useful continuous quantity is what remains after removing the increasing cost of specifying finer bins. Its dependence on coordinates is visible already in differential entropy of a uniform law.

For a finite-valued random variable, an epsilon-sufficient set $S$ retains probability at least $1-ϵ$. This notion is about covering probability mass and is unrelated to a sufficient statistic. For an IID random variable source, any such set of words with fixed $ϵ<1$ requires exponentially many words at a rate approaching the Shannon entropy.

If one probability in a $k$-point distribution equals $t$, normalize the remaining $k-1$ probabilities to $q$. Then $H=h(t)+(1-t)H(q)\le h(t)+(1-t){\log }_2(k-1)$. Equality for $t<1$ means that the remaining probabilities are equal. This combines the binary entropy of the distinguished event with the largest residual Shannon entropy.

The logic $L_{{\omega }_1,\omega }$ extends first-order logic by countable logical conjunctions and logical disjunctions. Each formula has finitely many free variables, and quantifier strings are finite. In a countable first-order language, every countable structure has a Scott sentence in this logic that characterizes its isomorphism class among countable structures.

Write a small planar displacement in arclength coordinates as $X=s+X_2$ and $Y=y$. The inextensibility relation $X_s^2+Y_s^2=1$ gives $X_{2,s}=-y_s^2/2$ to quadratic order. Thus longitudinal material velocity enters one order later than the transverse velocity. For a periodic material stroke with no mean axial drift, its mean velocity vanishes, although its instantaneous contribution is needed for exact local force balance.

For an $H$-representation $V$ and $g\in G$, the conjugate representation ${}^{g}V$ of $gH{g}^{-1}$ is defined by $(gh{g}^{-1})v=hv$. This transports the acting subgroup; it is distinct from conjugating complex matrix entries. It occurs in the Mackey restriction formula.

If $n$ points and $m$ curves have the property that each pair of distinct points lies on at most $\lambda$ curves, then double counting gives ${\sum }_{\gamma }{k}_{\gamma }(k_{\gamma }-1)\le \lambda n(n-1)$, where $k_{\gamma }=|P\cap \gamma |$. The Cauchy-Schwarz inequality implies $I^2\le mI+\lambda mn(n-1)$ and hence $I\le m+\sqrt{\lambda mn(n-1)}$. The statement needs only this combinatorial multiplicity condition, not algebraicity.

An importance sampling proposal $q$ must be positive wherever the target-weighted integrand can contribute. The usual sufficient condition is that the target probability is absolutely continuous with respect to $q$. Otherwise multiplying observed draws by $f/q$ cannot recover a missing part of the integral. A strict rejection envelope supplies this condition for the rejected-proposal distribution.