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This cone records first-order limiting velocities of feasible points approaching $x$. The defining limits are required to exist, and the point sequences must be feasible. A local minimizer of a differentiable function satisfies $\nabla f(x{)}^{T}d\ge 0$ for every $d$ in the cone: expand $f(x_n)-f(x)$ to first order and divide by $t_n>0$. A tangent direction need not itself generate an exactly feasible straight segment. Linearizing active constraints always gives a containing linearized tangent cone; equality requires a suitable constraint qualification.

For inequalities $c_j(x)\ge 0$, the active constraints at a feasible point are those holding with equality. Inactive constraints remain strictly satisfied in a sufficiently small neighborhood and do not restrict first-order feasible velocities there. The active gradients determine the linearized tangent cone and appear in the linear independence constraint qualification. At a KKT point, inactive constraints have zero Lagrange multipliers by complementary slackness.

A saturated fatty acid with sixteen carbon atoms.

For $n\ge 1$ and $a,b$ in a commutative ring, multiplying out the displayed polynomial factorization cancels all intermediate terms, leaving $a^n-b^n$. The case $n=2$ is the difference of squares. Unlike division by $a-b$, this identity also applies when $a=b$.

For real $1\le b\le a$, every summand lies between one and $a^{n-1}$. Consequently $n(a-b)\le a^n-b^n\le n(a-b)a^{n-1}$. This turns a one-step discrepancy into a bound after repeated multiplication, for example when estimating the global error of a one-step method on $y^{\prime }=y$.

A mixed gustatory and secretomotor branch that crosses the middle ear and joins the lingual nerve.

An exposed face of a convex set $K$ is the nonempty set on which a supporting real linear functional attains its maximum. It is a face of a convex set: if $x=ty+(1-t)z$ with $0<t<1$ attains the maximum and $y,z\in K$, each of $f(y),f(z)$ is at most that maximum, so both equal it. On a complex vector space, take the real part of a complex linear functional. An exposed point is an exposed face consisting of one point.

Let $F$ be a nonempty norm-closed face of a convex set $[0,1_A]$ in a unital C-star algebra, with $0\notin F$. Every $a\in F$ has norm one: otherwise $a=\Vert a\Vert (a/\Vert a\Vert )+(1-\Vert a\Vert )0$ is a nontrivial convex decomposition forcing zero into the face. If $g:[0,1]\to [0,1]$ is continuous and nondecreasing with $g(0)=0,g(1)=1$, then $g(a)\in F$.

To prove this, for $n\ge 2$ replace $g$ near the endpoints by a function $g_n$ which is zero on $[0,1/(2n)]$, one on $[1-1/(2n),1]$, agrees with $g$ on $[1/n,1-1/n]$, and interpolates linearly in between. Monotonicity and endpoint continuity give $g_n\to g$ uniformly. With $\delta =1/(2n)$, the function $h_n(t)=(t-\delta g_n(t))/(1-\delta )$ takes values in $[0,1]$: check the two constant regions and use $\delta \le t\le 1-\delta$ elsewhere. The continuous functional calculus gives $a=\delta g_n(a)+(1-\delta )h_n(a)$ with both summands positive contractions. The face property gives $g_n(a)\in F$, and closedness gives $g(a)\in F$.

Suppose a topological space admits an unbounded continuous function $f$ to $\mathbb{R}$. Set $u=|f|$ and choose $x_n$ so that $t_n=u(x_n)$ increases to infinity. There is a continuous function $h:[0,\infty )\to \mathbb{R}$ with $h(t_n)=(-1{)}^{n}n$: interpolate linearly between successive prescribed values and extend constantly below $t_1$. Thenis continuous and takes values strictly between $-1$ and $1$, while its values on the even and odd chosen sequences tend to $1$ and $-1$. Thus neither bound is attained. In particular, unbounded continuous witnesses to noncompactness of a real subspace give this stronger failure of the extreme value theorem for every noncompact real subspace.

In a finite perfect-information game without chance moves, solve terminal decision points first. At each earlier decision point, its player chooses an action maximizing its own payoff among the already solved continuations. Record a choice in every subgame, including those not reached by the root choice. This constructs a subgame perfect equilibrium. Tied choices may yield several equilibrium strategies and several outcomes. The name refers to Ernst Zermelo; the computational procedure is an application of backward induction.

Two players alternate proposals for dividing a pie, with a fixed final round. Agreement ends the game; rejection shrinks the next available pie by a factor $0<\alpha <1$. Final rejection gives both players zero. Offers range continuously over the entire available pie. Backward induction in games determines immediate agreement and the proposer's share through the continuous-offer bargaining acceptance lemma and the finite-horizon bargaining parity effect. Discrete offer grids can change both the boundary behavior and the exact shares.

A leading fixed-discharge, level-surface approximation to a shallow erodible-bed flow gives perturbation equations $C_x=4U-C$ and $U_t+3\beta U_x=-4U+C$. The displayed Fourier growth rate has negative real part for every nonzero wave number, independently of bedload parameter $\beta$. Bedload changes the propagation speed but not this damping. The zero-wave-number mode is neutral, so absence of growing modes should not be confused with strict asymptotic decay of every spatially uniform perturbation.

At an endpoint of a second-derivative variational problem, a free value $y$ gives the first displayed condition, while a free slope $y^{\prime }$ gives the second. Each condition is required only when its corresponding endpoint variation is free; a prescribed value or slope instead forces that variation to vanish. For $F=K(y^{\prime \prime }{)}^2/2+qy$, the two conditions become $y^{\prime \prime \prime }=0$ and $y^{\prime \prime }=0$, respectively. A clamped beam has both value and slope fixed, a pinned beam fixes value but leaves slope free, and a free end leaves both free.

For a smooth functional $\int F(x,y,y^{\prime },y^{\prime \prime })dx$, let $P=F_{y^{\prime }}$ and $Q=F_{y^{\prime \prime }}$. Along an extremal, the Euler-Lagrange equation for a second-derivative functional is $F_y-P^{\prime }+Q^{\prime \prime }=0$. Differentiating $H=y^{\prime \prime }Q+y^{\prime }(P-Q^{\prime })-F$ by the chain rule givesConsequently $H$ is constant whenever $F$ has no explicit dependence on $x$. This extends the Beltrami identity to a second-derivative variational problem. If $F$ is independent of $y^{\prime \prime }$, the expression reduces to $y^{\prime }{F}_{y^{\prime }}-F$.

With values $y(0)=y(\pi /2)=y^{\prime }(\pi /2)=0$ and $y^{\prime }(0)=1$, the higher-order Euler-Lagrange equation is $y^{(4)}-y=0$. Write $c=\cosh (\pi /2)$ and $s=\sinh (\pi /2)$. Solving the four linear boundary equations gives $y=[c(\cos x-\cosh x)+(1-s)\sin x+(1+s)\sinh x]/2$. The added function $x^2$ does not affect the first variation.

A sublattice is a subgroup of a Euclidean lattice. It is a Euclidean lattice in its real linear span, which may have smaller dimension than the original ambient Euclidean space. The equal-parity integer points in ${\mathbb{Z}}^d$ form a full-rank sublattice of index $2^{d-1}$: reduction of all coordinate differences modulo two gives the quotient.

There are infinitely many primes in each of the classes one and three modulo four. For the first, a hypothetical finite list $p_1,\dots ,p_r$ is contradicted by any prime divisor $q$ of $(2p_1\cdots p_r{)}^2+1$: it is odd, not on the list, and admits a square root of minus one, so $q\equiv 1(\mathrm{mod}4)$. For the second, $4p_1\cdots p_r-1$ is three modulo four; its prime factorization must include a prime that is three modulo four, and none of the listed primes divides it. These special congruence classes therefore admit elementary product-plus-or-minus-one arguments without the general theorem on primes in arithmetic progressions.

A cyclic redundancy check uses a polynomial remainder over the field with two elements as checking information. In the basic systematic convention, appending the remainder of $x^{r}M$ modulo the CRC generator polynomial makes the transmitted polynomial divisible by that generator.