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A checksum is a value calculated from data to expose some forms of corruption. Internet Protocol and transport checks cover specified header and data fields; changing covered fields requires updating their checksums.

Let a polynomial $q$ of degree at most $n$ have errors $f(x_j)-q(x_j)=(-1{)}^{j}h$ at $n+2$ ordered distinct points. Then $|h|$ bounds below the error $E_n(f)$ of the best uniform approximation. Otherwise an improved polynomial $p$ with error strictly below $|h|$ would make $p-q$ alternate strictly in sign at all those points, giving at least $n+1$ distinct roots of a polynomial by the intermediate value theorem. Its degree is at most $n$, a contradiction. Moreover, if $q$ is not optimal, $|h|<E_n(f)$. In the equality case, an optimal $p$ gives weak alternating signs of $p-q$; its vanishing order-$(n+1)$ divided difference has same-sign summands, forcing every nodal value to vanish and hence $p=q$.

On $[0,1]$, the chord of $\sqrt{1-x}$ is $1-x$. Its nonnegative gap has maximum $1/4$ at $x=3/4$, as differentiation shows. Shifting the chord upward by $1/8$ gives error $-1/8,1/8,-1/8$ at $0,3/4,1$. The Chebyshev alternation theorem therefore proves that $9/8-x$ is the unique best linear uniform approximation, with error $1/8$. Endpoint differentiability is unnecessary.

A homogeneous equilibrium in a spatial evolution equation is a steady state independent of position. Its stability against homogeneous perturbations need not imply stability against spatially varying modes.

For the equal superposition of the first two box states with zero initial relative quantum phase, put $\Omega =(E_2-E_1)/\hslash$. The probability in the left half is $P_{\mathrm{L}}=1/2+4\cos (\Omega t)/(3\pi )$. The continuity equation and zero wall probability current implyDirect substitution of $\psi (L/2,t)=L^{-1/2}{e}^{-i{E}_{1}t/\hslash }$ and ${\partial }_x\psi (L/2,t)=-2\pi L^{-3/2}{e}^{-i{E}_{2}t/\hslash }$ into the probability current verifies the sign. Positive current means probability leaves the left half. The quantum interference controls transport even though each component has stationary probability density.

The sum of relative atomic masses for the simplest integer composition represented by an empirical formula.

For a uniformly elliptic second-order Dirichlet form with bounded coefficients and lower-order terms, there are $a>0$ and $b\ge 0$ giving the displayed lower bound. Cauchy-Schwarz and Young's inequality absorb the drift term into part of the principal gradient energy plus an $L^2$ correction. Adding a sufficiently large nonnegative multiple of the $L^2$ inner product therefore makes the form coercive, permitting Lax-Milgram. The shift restores coercivity even when the unshifted form is not positive.

Conductors with net charges of the same sign can attract because their internal charge distributions respond to one another. A point-charge calculation misses this electrostatic induction contribution. Exact conducting-sphere results are given in Electrostatics of Two Charged Conducting Spheres by John Lekner.^[ref]

Historical terminology for electrical effects supplied by electrochemical cells. The source supplies energy through chemical changes; the resulting electric charge and electric current obey the same electrical laws as charge supplied by other means.

In a thin wall-withdrawal model with large dimensionless surface-tension parameter $\hat{C}$, matching a parabolic static meniscus fixes inner longitudinal scale $ϵ={\hat{C}}^{-1/3}$ and thickness scale $\delta ={ϵ}^2$. The leading inner equation is $v_0{H}^{\prime }+(H^3{H}^{(3)}{)}^{\prime }/3=0$, matching $H\sim {\bar{z}}^2/\sqrt{2}$ below and a bounded film above. Scaling $H=v_0^{2/3}g$, $\bar{z}=v_0^{1/3}\zeta$ preserves the quadratic matching coefficient and gives the universal equation $3g^{\prime }+(g^3{g}^{(3)}{)}^{\prime }=0$. Its selected bounded solution tends to $0.948$, giving the displayed dimensionless law. It applies to positive slowly varying wall speed in the leading matched regime; the numerical constant comes from that universal inner problem, not a dimensional identity.

For $\dot{x}=2x-x^2-y$, $\dot{y}=y-kx$ with $k>2$, take the positive Dulac function in the title. More generally, $\psi =e^{a(x+y)}$ givesThe equal coefficients in the exponential cancel the $y$ term. Setting $a=-2/(k-2)$ also cancels the linear $x$ term, leaving $\psi (3+2x^2/(k-2))>0$. The Bendixson-Dulac theorem excludes periodic orbits in the open positive quadrant. The axes are not invariant: the vector field points out of the quadrant at every nonzero axis point. A closed orbit staying in the nonnegative quadrant therefore cannot touch an axis, and cannot pass through the equilibrium at the origin by uniqueness. This extends the exclusion to the closed quadrant. Choosing an exponential multiplier turns a differential search for a useful Dulac function into cancellation of polynomial coefficients.

The algebraic double dual of a vector space is the dual space of its algebraic dual. An element assigns a scalar linearly to each linear functional on the original space. The bidual evaluation map sends $v$ to the rule $\ell \mapsto \ell (v)$. It is a canonical linear isomorphism in finite dimension, unlike an identification of $V$ with $V^{\ast }$, which generally requires an additional choice. For spaces with topology, using the continuous dual space instead defines the topological bidual; that is a distinct construction in infinite dimension.

If a normed subspace $Z$ is dense in a normed vector space $X$ and carries the inherited norm, restriction gives an onto linear isometry from the dual space of $X$ to that of $Z$. A bounded functional on $Z$ extends uniquely by limits along sequences from $Z$; its norm is unchanged because its inequality passes to the limit. Thus completion does not change the continuous dual. For example, the subspace of sequences with $\sum j|a_j|<\infty$, equipped with the unweighted ${\ell }^1$ norm, has dual ${\ell }^{\infty }$, since it contains the dense subspace of finitely supported sequences. The inherited norm matters; the weighted norm would give a different dual norm.

The capacity of water to dissolve lead from materials it contacts; dissolved release should be distinguished from carriage of particles containing lead.^[ref]