The mineral-rich residue historically obtained by burning collected seaweeds and leached as a source of iodine-bearing salts.

Historical pharmaceutical preparation; its composition and clinical use must be identified by edition. ^[ref]

Historical pharmaceutical preparation; its composition and clinical use must be identified by edition. ^[ref]

Historical pharmaceutical preparation; its composition and clinical use must be identified by edition. ^[ref]

For an incompressible phase, this coefficient compares its actual axial momentum flux with the uniform-profile flux at the same intrinsic mean velocity. Cauchy-Schwarz gives $D_1\ge 1$ where the mean is nonzero. A spatially varying profile can have a larger second moment even when its mean speed is unchanged.

With constant $D_1$, expansion of ${\partial }_t(\alpha u_1)+{\partial }_x(\alpha D_1{u}_1^2)$ and use of ${\alpha }_t+(\alpha u_1{)}_x=0$ gives $\alpha (u_1{)}_t+(2D_1-1)\alpha u_1(u_1{)}_x+(D_1-1)u_1^2{\alpha }_x$. Thus a profile correction changes both convection and phase-fraction-gradient coupling in the averaged momentum equation. If $D_1$ varies, an extra $\alpha u_1^2(D_1{)}_x$ remains; a zero mean should be treated through the undivided second moment rather than an undefined coefficient.

The family $x_{n+1}=\mu -x_n^2$ has fixed points $(-1\pm \sqrt{1+4\mu })/2$. Its upper branch attracts linearly for $-1/4<\mu <3/4$ and loses stability in a period-doubling bifurcation. The resulting two-cycle has points $(1\pm \sqrt{4\mu -3})/2$ and multiplier $4(1-\mu )$, attracting for $3/4<\mu <5/4$. At $\mu =2$, the map is conjugate on $[-2,2]$ to the full tent map.

On a rectangle, bilinear interpolation is linear in each coordinate separately and matches the four corner values. With normalized coordinates $s,t\in [0,1]$, it is $(1-s)(1-t)u_{00}+s(1-t)u_{10}+(1-s)t{u}_{01}+st{u}_{11}$. A multigrid method can use it to prolong a coarse correction to finer-grid nodes. It is not generally an affine function of the pair $(s,t)$, since it can contain a mixed $st$ term.

A singular integral operator has a kernel whose ordinary integral can fail at points on the integration domain. A specified limiting prescription, such as a Cauchy principal value, can still define the operator by cancellation. For the Hilbert transform kernel $1/(t-x)$, subtracting $f(x)$ from a locally Hölder-continuous density leaves $[f(t)-f(x)]/(t-x)$ locally integrable; the constant part is interpreted by symmetric deletion. The limiting prescription and the class of allowed densities are part of the operator's definition, not optional details.

Let $f=X^n+a_{n-1}{X}^{n-1}+\cdots +a_0\in \mathbb{Z}[X]$ satisfy Eisenstein's criterion at a prime number $p$, and let $\theta$ be a root. ThenIndeed, ${\theta }^n/p$ is an algebraic integer, whereas for $p\nmid r$ and $0\le i<n$ the field norm of $r{\theta }^i/p$ has $p$-adic valuation $i-n<0$, so that element is not integral. If $\gamma =p^{-1}{\sum }_{i=0}^{n-1}{r}_i{\theta }^i$ is integral, multiply it by ${\theta }^{n-1}$. Every term except $r_0{\theta }^{n-1}/p$ is integral; thus $p\mid r_0$. Subtract $r_0/p$, multiply the remainder by ${\theta }^{n-2}$, and continue to obtain $p\mid r_i$ for every $i$. Hence there is no element of order $p$ in the finite group ${\mathcal{O}}_{\mathbb{Q}(\theta )}/\mathbb{Z}[\theta ]$, proving the claim. The result concerns the Eisenstein prime only: other primes may still divide the index.

For a nonzero squarefree integer $d\equiv 3(\mathrm{mod}4)$, set $\theta =\sqrt[4]{d}$. The power basis $1,\theta ,{\theta }^2,{\theta }^3$ is an integral basis, with discriminant $-256d^3$. The polynomial is Eisenstein at every prime dividing $d$; for $d=-1$, shift $X^4+1$ by one to obtain an Eisenstein polynomial at two. The discriminant-index formula for an integral lattice restricts any power-basis index prime to $2d$. An Eisenstein prime cannot divide the index: the norm of $r{\theta }^i/p$ has valuation $i-4<0$ for $p\nmid r$ and $i<4$, while ${\theta }^4/p$ is integral; multiplying a putative denominator-$p$ integral element successively by powers of $\theta$ forces each numerator coefficient divisible by $p$.

It remains to exclude two. A putative integral element $(a+b\theta +c{\theta }^2+e{\theta }^3)/2$ has relative norm to $\mathbb{Q}(\sqrt{d})$ with coefficientsThe quadratic field has ring of integers $\mathbb{Z}[\sqrt{d}]$, so both coefficients are integers. The first congruence forces $a,c$ to have the same parity. If they are odd, the second requires $b^2+3e^2\equiv 2(\mathrm{mod}4)$, impossible. If they are even, the second forces $b,e$ to have the same parity and the first forces $be$ even, so both are even. Thus no nontrivial denominator-two element exists, proving index one.

The power basis $1,\theta ,{\theta }^2$ is an integral basis. The defining polynomial is irreducible by Eisenstein's criterion at two and its discriminant is $-140$. The discriminant-index formula for an integral lattice restricts its index in the ring of integers to one or two. An index of two would supply an algebraic integer $\eta =(a+b\theta +c{\theta }^2)/2$ with $a,b,c\in \{0,1\}$ not all zero. Its field trace $(3a-4c)/2$ forces $a=0$. If $b=0$ this gives ${\theta }^2/2$ directly; if $b=1$, the identity $\theta \eta ={\theta }^2/2-c\theta -c$ gives the same integrality. But ${\theta }^2/2=-1-{\theta }^{-1}$, while $N({\theta }^{-1})=-1/2$ excludes integrality by the field norm. Thus the index is one, proving the claim.

An integer partition with no repeated part. For example, the partitions into distinct parts of $5$ are $5$, $4+1$, and $3+2$. If $d(i,n)$ counts those with all parts at most $i$, separating those that omit $i$ from those that use it exactly once givesfor $i,n\ge 1$, with $d(i,0)=1$, $d(0,n)=0$ for $n>0$, and negative remaining sums counted as zero. Removing the unique part $i$ gives a bijection proving the second term; retaining the same maximum $i$ there would instead allow repetitions.

Insulin secretion releases stored insulin from pancreatic beta cells through regulated exocytosis. Glucose metabolism can increase the ATP-to-ADP balance, reduce activity of an ATP-sensitive potassium channel, promote depolarization and stimulate calcium-dependent release. Additional inputs modulate the response. Secretion is distinct from insulin synthesis and from peripheral insulin action, so a higher circulating concentration alone does not locate the underlying change. ^[ref]

A beetle is an insect whose adult forewings are modified into protective covers over the hindwings and abdomen. Its larva and adult can occupy different ecological roles, linking metamorphosis to changes in feeding and habitat.

A parasitic insect, distinct from a mite or tick.

At distinct real nodes $x_0,\dots ,x_n$ with positive weights $w_j$, the displayed expression is an inner product on the real vector space of polynomials of degree at most $n$. Symmetry and bilinearity follow from finite sums. If its squared norm vanishes, positivity of every weight forces $p(x_j)=0$ at every node. The root bound for a nonzero polynomial then forces $p=0$, proving positive definiteness.

More generally, on any vector subspace of real functions, the evaluation form is an inner product exactly when the evaluation map $p\mapsto (\sqrt{w_j}p(x_j){)}_j$ is injective. On polynomials of degree at most $n+1$ it fails: the nonzero node polynomial ${\prod }_j(x-x_j)$ vanishes at all nodes. This gives a positive semidefinite form with a nontrivial kernel. On the smaller polynomial space, the Lagrange interpolation polynomial identifies evaluation coordinates with ${\mathbb{R}}^{n+1}$, and Cauchy-Schwarz inequality follows either from the Euclidean inequality in those coordinates or from minimizing $\Vert p-tq{\Vert }^2$ over real $t$.

On the real plane, take the initial topology of all separately continuous real-valued functions. Its coordinate lines have their usual induced topology, and the nonzero rational grid is dense, but its diagonal is uncountable and discrete. Thus the space is separable and nonmetrizable. Rectangular bump functions show that no sequence from the nonzero rational grid converges to the origin, so the space is not first countable either.

On the unit square, these data determine only the zero solution of the heat equation $u_y=u_{xx}$. The squared $L^2$ norm decreases because its derivative is $-2\int u_x^{2}dx$, and its initial value is zero. For Laplace's equation, $\sin (\pi x/2)\sinh (\pi y/2)$ is a nonzero solution with the same three data; adding zero Dirichlet data on the top edge restores uniqueness by Green's first identity. For the wave equation $u_{yy}=u_{xx}$, $\sin (\pi x/2)\sin (\pi y/2)$ is a counterexample; the missing datum is initial velocity. Adding $u_y(x,0)=0$ restores uniqueness by conservation of $\frac{1}{2}\int (u_x^2+u_y^2)dx$. Thus the type of equation determines what information is needed on an unprescribed edge.

An inguinal hernia extending into the scrotum. The external location describes its extent, while the route and condition of its contents require separate anatomical assessment.

An inguinal hernia protruding through the posterior wall of the inguinal canal, medial to the inferior epigastric vessels. It differs in route from an indirect hernia.