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Antiplane shear has displacement normal to a plane and independent of that normal coordinate. In an isotropic linear elastic body, ${\sigma }_{13}=\mu w_{,1}$ and ${\sigma }_{23}=\mu w_{,2}$, together with their symmetric partners, are the only nonzero stress tensor entries. Static balance without body force reduces to the Laplace equation for $w$.

The bulk modulus relates hydrostatic compressive pressure to fractional volume compression. In isotropic three-dimensional linear elasticity, the spherical part of the stress tensor is $\kappa \mathrm{tr}(\epsilon )I$, so uniform dilation has strain energy density $\kappa (\mathrm{tr}\epsilon {)}^2/2$. Positive bulk modulus and positive shear modulus make the isotropic elastic stiffness tensor positive on nonzero symmetric strains.

For a homogeneous comparison elastic stiffness tensor $C^0$, the elastic strain Green operator maps an elastic polarization to minus the induced infinitesimal strain tensor. In a bounded domain the induced displacement field has zero boundary data; in an infinite medium it is the decaying perturbation. The defining static elastic equilibrium equation is $\nabla \cdot (C^0\mathrm{sym}\nabla u+\tau )=0$. The bounded-domain operator is self-adjoint, annihilates constant elastic polarizations, has zero mean output, and satisfies $\Gamma C^0\Gamma =\Gamma$.

Apply the divergence theorem to ${\partial }_k(x_i{\sigma }_{jk})$ and use momentum balance ${\sigma }_{jk,k}=\rho (a_j-b_j)$ and stress symmetry to obtain the displayed mean stress identity. Applying the same theorem to the infinitesimal strain tensor gives $V{\overline{e}}_{ij}=\frac{1}{2}{\int }_S(u_i{n}_j+u_j{n}_i)dS$. The acceleration term cannot be dropped during a transient.

For a nonrelativistic polytropic equation of state $P=K{\rho }^{4/3}$ during matter domination, $c_s^2\propto a^{-1}$ and the comoving Jeans wavenumber is constant. The leading density equation becomes $\ddot{\delta }+4\dot{\delta }/(3t)+[2(q-1)/(3t^2)]\delta =0$, giving the displayed powers. A growing mode exists for $q<1$; at $q=1$ one mode is constant. At $q=25/24$ the repeated root has the second solution $t^{-1/6}\ln t$. For $q>25/24$ both real solutions oscillate in $\ln t$ with envelope $t^{-1/6}$. Thus the instability threshold and the onset of oscillatory solutions are distinct in an expanding background.

For a flat Newtonian expanding background $R\propto t^{2/3}$ with polytropic equation of state $p=K{\rho }^{\gamma }$, a comoving density contrast satisfies $\ddot{\delta }+4\dot{\delta }/(3t)+[{\Lambda }^2{t}^{-2\nu -2}-2/(3t^2)]\delta =0$. Here $\Lambda =kc(t_0)t_0^{\gamma -1/3}$ and $\nu =\gamma -4/3$. For $\nu >0$, substitution $z=\Lambda /(\nu t^{\nu })$ and $\delta =t^{-1/6}y(z)$ gives the Bessel equation of order $\lambda$. A universally valid basis uses the Bessel function of the first kind and Bessel function of the second kind, $t^{-1/6}{J}_{\lambda }(z)$ and $t^{-1/6}{Y}_{\lambda }(z)$. When $\lambda$ is not an integer, $J_{-\lambda }$ can replace $Y_{\lambda }$. For $4/3<\gamma <5/3$, the early oscillatory envelope is $t^{(3\gamma -5)/6}$; late modes have leading powers $t^{2/3}$ and $t^{-1}$. Late growth requires a nonzero coefficient of the growing branch.

A linear code over the two-element finite field. Its dimension $r$ gives $2^r$ codewords. Its weight enumerator in the convention $W_C(s,t)={\sum }_{c\in C}{s}^{w(c)}{t}^{n-w(c)}$ is symmetric in $s,t$ exactly when the all-ones word belongs to $C$: adding that word complements every bit and replaces weight $w$ by $n-w$. Conversely symmetry makes the coefficient of $s^n$ equal the coefficient of $t^n$, which is one.

For a linear code of length $n$ and dimension $k$ over ${\mathbb{F}}_q$, the code rate is $R=k/n$. There are $q^k$ possible messages represented by $q^k$ codewords of length $n$. More generally a block code $C\subseteq {\mathbb{F}}_q^n$ has rate ${\log }_q|C|/n$. This is a dimensionless redundancy measure, distinct from information transmitted per unit physical time.

Let $1\le k\le n\le q$ and let $a_1,\dots ,a_n$ be distinct elements of a finite field ${\mathbb{F}}_q$. The Reed-Solomon code consists of evaluation vectors $(f(a_1),\dots ,f(a_n))$ for polynomials $f$ of polynomial degree less than $k$. It is a linear code with parameters $[n,k,n-k+1]$ and therefore a maximum distance separable code.

Indeed, evaluation is an injective linear map: a nonzero polynomial of polynomial degree at most $k-1$ cannot vanish at all $n\ge k$ evaluation points. The same polynomial root bound shows that every nonzero codeword has at least $n-k+1$ nonzero coordinates. Equality is attained by a polynomial vanishing at any chosen $k-1$ evaluation points. Thus the minimum Hamming distance is exactly $n-k+1$, allowing correction of up to $\lfloor (n-k)/2\rfloor$ unknown symbol errors.

Primitive cyclic Reed-Solomon codes use all nonzero field elements, ordered as powers of a primitive root of unity. An arbitrary evaluation set need not give a cyclic code.

A nonzero length-$n$, dimension-$k$ linear code is maximum distance separable if its minimum Hamming distance equals $n-k+1$, attaining the Singleton bound. Reed-Solomon codes and generalized Reed-Solomon codes provide examples because a nonzero polynomial of degree at most $k-1$ has at most $k-1$ distinct polynomial roots.

Two linear codes are monomially equivalent if a permutation of coordinates followed by multiplication of individual coordinates by nonzero field elements takes one to the other. This preserves dimension and Hamming distance. Choosing different representatives for projective parity-check columns produces this equivalence.

For line segments $P+sd$ and $Q+te$, $0\le s,t\le 1$, the squared Euclidean distance is a convex quadratic function on a square. Its minimum is either an admissible stationary point or a minimum on one of the four edges. Each edge minimum is a point-to-segment orthogonal projection, restricted to the interval. Testing these five candidates gives a complete algorithm; when the directions are parallel, the edge candidates suffice. Independently restricting both coordinates of the unconstrained stationary point is generally incorrect because the variables are coupled by $d\cdot e$.

The midpoint of the line segment joining two points with position vectors $a,b$ is the point $m=(a+b)/2$. Its displacements from the endpoints are $(b-a)/2$ and $(a-b)/2$, so it lies on the segment and is equidistant from its endpoints. It is also the centroid of the two endpoints.

For independent exponential observations of rate $\theta$, stopping after observation $x$ with known probability $\alpha (x)\in (0,1)$ gives a complete-data likelihood proportional to ${\theta }^N{e}^{-\theta S}$. Its minimal sufficient statistic is $(N,S)$ by the parameter-free likelihood-ratio criterion. If only the last value $Y$ is recorded, its density instead is $\alpha (y)e^{-\theta y}/{\int }_0^{\infty }\alpha (x)e^{-\theta x}dx$. The stopping rule therefore changes the coarsened-data inference even though it adds no complete-data likelihood factor involving $\theta$.

A deforming free swimmer must satisfy total linear and angular momentum balances. The unknown lateral translation and yaw are its recoil correction. If the reference translation is the true centre of mass, the prescribed deformation must first have its mass-weighted mean removed. Otherwise the reference translation differs from actual centre of mass displacement. The body mass and its hydrodynamic added mass contribute with the same sign to its effective inertial coefficient.

An active swimming beam has a total internal bending moment $G$ containing passive elasticity and muscular actuation. Neglecting cross-sectional rotary inertia, slice angular momentum balance identifies the internal transverse force with $G_x$. Slice linear momentum balance then gives $G_{xx}=m_b{h}_{tt}-F_{\mathrm{body}}$. In Lighthill elongated-body theory, $F_{\mathrm{body}}=-\mathcal{D}(m\mathcal{D}h)$, where $\mathcal{D}={\partial }_t+U{\partial }_x$. The body and its hydrodynamic added mass therefore contribute with the same sign to the acceleration coefficient. The positive quantity $\mathcal{D}(m\mathcal{D}h)$ is the reaction on the fluid.

For compact Lighthill stress tensor and applied-force distributions, define $S_{ij}=\int T_{ij}{d}^{3}y$ and $F_i=\int f_i{d}^{3}y$. Differentiating the retarded acoustic Green function at far distance gives the displayed leading radiation field, evaluated at $t-r/c_0$. Source compactness requires its size to be small compared with the acoustic wavelength, not merely with the observer distance.

If the continuity equation has mass source $M$ and the conservative momentum equation has momentum source $S_i$, elimination of momentum gives the displayed wave equation. With no imposed momentum source, its extra term is ${\partial }_{t}M$. A compact source of total mass rate $\mathcal{M}=\int MdV$ radiates the acoustic monopole ${\rho }_M^{\prime }=\dot{\mathcal{M}}(t-r/c_0)/(4\pi c_0^{2}r)$. The momentum source supplies an additional acoustic dipole; specifying mass injection alone does not fix that dipole.

Dynamic pressure is the bulk kinetic energy per unit volume associated with fluid motion at speed $U$ in a specified reference frame:Here $\rho =m/V$ is mass density. The word "pressure" identifies its units and its use as a pressure scale; $q$ is not an extra isotropic stress to add to the local static pressure in every surface traction. A stationary fluid can have large static pressure with $q=0$. Conversely, knowing $U$ and $\rho$ fixes $q$ without fixing static pressure. The definition remains useful in viscous fluid flow and compressible flow, even when the simplest Bernoulli equation is inapplicable. ^[ref]

For stipulated mass density ${\rho }_{\infty }=1.20\mathrm{kg}{\mathrm{m}}^{-3}$ and reference speed $U_{\infty }=20\mathrm{m}{\mathrm{s}}^{-1}$, $q_{\infty }=240\mathrm{Pa}$. At $40\mathrm{m}{\mathrm{s}}^{-1}$ with the same mass density, $q_{\infty }=960\mathrm{Pa}$. This quadratic scaling makes dynamic pressure a useful normalization for the lift coefficient, drag coefficient and pressure coefficient; it predicts four times a particular force only if its corresponding coefficient remains unchanged.

The connection with stagnation pressure requires hypotheses. For steady flow, constant mass density, inviscid flow, and gravity as the only body force, the Bernoulli equation gives, along one streamline,Here $p$ is static pressure, $g$ is gravitational acceleration, and $z$ is height. Bring that flow to rest without losses at the same height and the limiting static pressure is the stagnation pressure $p_0$:There must be no intervening pump supplying work or other energy input. Constancy along one streamline does not assert equal $B$ on all streamlines; irrotational flow in a connected region supplies that additional conclusion. Changing height requires retaining $\rho gz$. In compressible flow the stopping pressure difference generally differs from $\rho U^2/2$, and viscous dissipation can reduce recovered stagnation pressure. ^[ref]

A Pitot-static tube turns this relation into a measurement: its upstream-facing opening senses stagnation pressure, while separate side openings sample static pressure. Connect these to the positive and negative ports of a differential pressure transducer or manometer. Under the hypotheses above, the measured difference $\Delta p$ givesThus the stipulated $240\mathrm{Pa}$ and $1.20\mathrm{kg}{\mathrm{m}}^{-3}$ imply $20\mathrm{m}{\mathrm{s}}^{-1}$, not an independently measured speed. Align the probe with the incoming flow and keep its passages clear: blockage and pinched tubing invalidate the comparison. At very low speed, the difference can be smaller than instrument error; at high Mach number, compressibility and possibly a shock wave require a different reduction. ^[ref]

A useful control is differential-pressure calibration. For a digital manometer, zeroing establishes the reading at equal port pressures; it does not establish the gain away from zero. The manufacturer's Series 475 Handheld Digital Manometer: Specifications - Installation and Operating Instructions by Dwyer Instruments, Inc. directs the user to vent both ports to the atmosphere before zeroing. Its stated accuracy is a fraction of full scale, not of the reading, and depends on instrument temperature. Select a suitable range and record the instrument variant and units; the manual covers several ranges.^[ref]

An independent physical check can use a liquid-column manometer: apply a stable pressure difference to both instruments and compare the digital reading with the hydrostatic pressure inferred from the vertical level difference. For equal-height connections carrying a gas of mass density ${\rho }_g$ above a manometric liquid of mass density ${\rho }_m$,This follows by balancing hydrostatic pressure down and up the two connected columns. Use vertical height, even if a tube is inclined. Known mass densities, measured height and gravitational acceleration set the comparison scale. Check several differences across the working range and reverse the port connections to check the sign. This is a proposed independent check, not a claim that the cited teaching experiment performed a full instrument calibration. Recheck zero after reconnecting the sealed tubing; a zero adjustment alone cannot detect a blocked passage or an incorrect scale.

In characteristic zero, a finite-dimensional Lie algebra is a semidirect sum of its solvable radical and a semisimple Lie subalgebra. The group analogue splits appropriate groups as a semidirect product with a Levi subgroup.