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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.

An inguinal hernia entering through the deep inguinal ring, lateral to the inferior epigastric vessels. It can follow the inguinal canal toward the scrotum.

For a finite probability distribution with largest atom $\Pi$, its base-two Shannon entropy satisfies $H\ge 2(1-\Pi )$. If $\Pi \ge 1/2$, coarsen the distribution to the indicator of a largest atom: Shannon entropy cannot increase under a deterministic function, so $H\ge h_2(\Pi )$. The concavity of binary entropy places it above the chord from $(1/2,1)$ to $(1,0)$, giving $h_2(\Pi )\ge 2(1-\Pi )$. If $\Pi \le 1/2$, every nonzero probability $p$ has $-{\log }_{2}p\ge -{\log }_2\Pi$, so $H\ge -{\log }_2\Pi$. The function $-{\log }_2\Pi -2(1-\Pi )$ vanishes at $1/2$ and has negative derivative on $(0,1/2]$, proving the same bound. Equality in the combined bound occurs for a point mass or a distribution uniform on two points.

For a random finite binary string $M$ of length $L$, conditioning on $L=\ell$ leaves at most $2^{\ell }$ possible strings. Thus its conditional entropy is at most $\ell$. Since the length is a function of the string, the chain rule for information entropy givesThe term $H(L)$ cannot generally be removed. For concatenations of $m$ words from a fixed finite decipherable code with maximum length $d$, it is at most ${\log }_2(md+1)$. Decipherability preserves the Shannon entropy of the source block. For an independent source block, dividing the resulting inequality by $m$ and taking $m\to \infty$ proves the expected codeword length is at least the one-symbol Shannon entropy.

An inflammatory exudate rich in fibrin.

An inode-based file system uses direct pointers for small files and successive levels of indirect block pointers for larger ones. If an address block contains $q$ pointers, a depth-$k$ region addresses $q^k$ file data blocks. With the inode resident, an uncached byte access in that region requires $k$ address-block reads and one file data block read.