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.

An affine algebraic group is an affine variety $G$ whose multiplication and inversion are regular maps. Equivalently, its coordinate ring $k[G]$ is a finitely generated reduced commutative Hopf algebra, with comultiplication induced by multiplication in $G$.

For finite matrices, differentiate the ordered product to obtain ${\Phi }^{\prime }-(A+B)\Phi =[e^{tA},B]e^{tB}$. Multiplying the error by $e^{-t(A+B)}$ and integrating gives $\Phi (t)-e^{t(A+B)}={\int }_0^t{e}^{(t-s)(A+B)}[e^{sA},B]e^{sB}ds$. This exact variation-of-constants formula identifies noncommutativity as the source of Lie-Trotter splitting error.

A resident trait $x$ occupies fraction $1-m/c(x)$ when its colonization rate exceeds fire extinction. With net competitive replacement given by ability difference $B(y)-B(x)$, a rare mutant has the displayed per-capita growth. The selection gradient is $g(x)=m{c}^{\prime }(x)/c(x)+[1-m/c(x)]B^{\prime }(x)$. A singular trait with negative mutant-fitness curvature is a strict local evolutionarily stable strategy. Global strict uninvadability requires that $mc(y)+[c(x)-m]B(y)$ be smaller than its resident value for every other trait. Monotonicity alone does not specify the shape or existence of an optimum.

For a family of Levi-Civita connections, the difference of affine connections is a tensor. Dividing by the family parameter and taking the limit shows that its variation is tensorial. Vary the metric compatibility equation and retain the symmetry of the two lower indices. Adding the two cyclic identities and subtracting the third gives the displayed result. In a normal coordinate frame it is the direct derivative of the Christoffel formula; its tensorial form then holds in every frame.

Image parity in gravitational lensing is the sign of the determinant of the source-from-image Jacobian matrix $A=\partial \mathbf{\beta }/\partial \mathbf{\theta }$. A positive sign preserves the local handedness of a small source pattern; a negative sign reverses it. Negative signed lensing magnification therefore means a reflected orientation, not negative received light. This is a two-dimensional property: the derivative of a signed radial equation alone does not determine it.

For a point-mass gravitational lens, differentiate $\mathbf{\beta }=(1-{\theta }_E^2/r^2)\mathbf{\theta }$, where $r=|\mathbf{\theta }|$. The radial and tangential eigenvalues of $A$ areConsequently the outer member of the two images of a point-mass lens has positive image parity in gravitational lensing; the inner member has negative image parity in gravitational lensing. At $r={\theta }_E$ the tangential eigenvalue vanishes: this is the tangential critical curve of an axisymmetric lens, and its gravitational-lensing caustic is the aligned source point. The signed lensing magnification is $\mu =1/\det A$, but measured total light uses $|{\mu }_{+}|+|{\mu }_{-}|$.

An optical focus is a point where a specified bundle of light rays converges, or where its backward extensions meet. For a converging thin lens receiving light rays parallel to the optical axis, the rear optical focus is $(f,0)$ when the thin lens is at $x=0$; its front optical focus is $(-f,0)$. An object's image is generally elsewhere: the optical focus describes the parallel incoming bundle, while the thin lens equation locates the image for a finite object distance.

Let $p$ be an odd prime and $a$ an integer with $p\nmid a$. Reduceto their least positive residue classes modulo $p$. If exactly $N$ of these representatives exceed $p/2$, Gauss's lemma statesThe symbol on the left is the Legendre symbol. For example, $p=7$ and $a=3$ give representatives $3,6,2$. Exactly one exceeds $7/2$, so $3$ is a quadratic nonresidue modulo $7$.

For a proof, replace every representative above $p/2$ by its negative modulo $p$, obtaining numbers $s_1,\dots ,s_m$ in $\{1,\dots ,m\}$. These numbers are distinct: equality would give $ai\equiv \pm aj(\mathrm{mod}p)$, forcing $i=j$ because $1\le i,j\le m$. They therefore form a permutation of $1,\dots ,m$. Multiplying the signed representatives givesThe factorial is invertible modulo $p$. Cancelling it and applying the Euler criterion proves the lemma. Counting signs can thus replace a potentially large modular exponentiation.

For a right-exact additive functor out of an abelian category with enough projectives, apply the functor to a projective resolution and take homology objects. The zeroth derived functor is $F$, positive derived functors vanish on projective objects, and the degreewise split horseshoe lemma produces a long exact sequence from every short exact sequence in an abelian category. The target can be any abelian category, including an opposite category of one.

Such a self-map of ${\Sigma }_{g_1}\times {\Sigma }_{g_2}$ exists exactly when one factor is a torus. Necessity follows because its Lefschetz number equals the Euler characteristic $4(1-g_1)(1-g_2)$ and must vanish. Sufficiency is translation by a nonzero point in the torus factor, times the identity on the other factor. The condition does not determine the product $g_1{g}_2$, since the other genus can be arbitrary.