Port forwarding configures a mapping from an external transport port to an internal service, allowing selected inbound traffic through a translator.

Consider an incompressible neo-Hookean prism with energy $\alpha (|F{|}^2-3)$, zero nominal end traction normal to $e_3$, and equal nominal lateral tractions $TN$ on side faces whose normals enforce both in-plane principal stresses. Within positive diagonal homogeneous deformations the first Piola-Kirchhoff stress tensor has entries $t_i=2\alpha v_i-p/v_i$. Equilibrium makes $p$ constant; end freedom gives $p=2\alpha v_3^2>0$.

The function $2\alpha v-p/v$ is strictly increasing on positive $v$, so equal lateral stresses require $v_1=v_2$. The determinant constraint then gives $v_3=v^{-2}$ and the displayed load relation. Its derivative is $2\alpha (1+5v^{-6})>0$, with limits minus infinity and infinity at the two ends of the positive axis. Thus there is exactly one positive diagonal equilibrium for every signed load. This uniqueness is confined to the homogeneous diagonal class and does not rule out other modes or nonhomogeneous equilibria under a general dead-loading problem.

Suppose a population has $N$ individuals, $k$ tagged, and captures are made independently with immediate replacement. Stop each run at the $r$th tagged capture, with known positive integer $r$, and let $X$ count untagged captures before stopping. Then $X$ has a failure-count negative binomial distribution with tagged-capture probability $p=k/N$ and mass $\left(\genfrac{}{}{0ex}{}{x+r-1}{x}\right)p^r(1-p{)}^x$.

For $n$ independent runs and $s={\sum }_i{x}_i$, the log-likelihood function apart from constants is $nr\log p+s\log (1-p)$. Treating $p$ as continuous in $(0,1)$ givesIndeed, $\mathbb{E}X=r(1-p)/p$ yields the second-derivative Fisher information. For an interior true $p$, the usual continuous-parameter asymptotic variance is $p^2(1-p)/(nr)$. If $s=0$, the likelihood is maximized at $p=1$ on the closed parameter space.

For actual unknown population size $N$, the constraint $p=k/N$ restricts $p$ to a discrete set. The relaxed estimate $\hat{N}=k/\hat{p}$ need not be an integer; evaluate the likelihood at the admissible integers on either side of it rather than claiming the continuous optimum is an attained integer-population maximum-likelihood estimator. A confidence interval $(L,U)$ for $p$, with $0<L<U$, transforms to $(k/U,k/L)$ and then is intersected with the allowed integer population sizes. Immediate replacement is essential; sampling without replacement gives a different likelihood.

The long ciliary nerves are branches associated with the nasociliary nerve reaching the eye without passing through the ciliary ganglion. They carry sensory pathways and sympathetic fibers. Their route differs from that of the short ciliary nerves.^[ref]

The anterior ethmoidal nerve is a branch of the nasociliary nerve supplying parts of the nasal cavity and external nose. Its course passes between orbital, cranial and nasal regions, illustrating why the apparent surface territory does not describe the whole pathway.

The posterior ethmoidal nerve is a branch of the nasociliary nerve supplying sensory pathways from posterior ethmoidal and nearby sinus regions. Its distribution differs from the more anterior territory of the anterior ethmoidal nerve.

The perpendicular distance from a muscle force's line of action to the relevant axis of an anatomical joint. It can change with posture; multiplying it by the muscle force gives the corresponding torque.

For independent standard normal random variables $X,Y$, put $X=R\cos \Theta$, $Y=R\sin \Theta$, with $R\ge 0$ and $0\le \Theta <2\pi$. The polar coordinates Jacobian determinant is $r$, so the joint probability density is $r{e}^{-r^2/2}/(2\pi )$ on $r>0$. It factorizes into a Rayleigh distribution density for $R$ and a continuous uniform distribution density for $\Theta$. Consequently the radius and angle are independent. The change of variables for probability densities $S=R^2$ gives $f_S(s)=e^{-s/2}/2$ for $s>0$, the exponential distribution of rate $1/2$, also the chi-squared distribution with two degrees of freedom. Measurable transformation preserves independence, giving $\Theta \perp S$. The angle at $R=0$ is arbitrary on an event of probability zero.

Thus a sector with angular length $\ell$ has probability $\ell /(2\pi )$, independently of any radial restriction. For instance $Y>|X|$ occupies a sector of angle $\pi /2$ and has probability $1/4$.

The function mapping each possible element to its multiset multiplicity. Finite multisets correspond to nonnegative functions with finite support of a multiset. A functional representation can answer queries without supporting enumeration of all elements.

The number of times a value occurs in observed data. Keeping 26 indexed counters for lowercase alphabet characters gives constant-time frequency queries independent of document length.

The morphism ${\mathbb{A}}^1\to V(x^4-y^3)$ given by $t\mapsto (t^3,t^4)$ is bijective on points over an algebraically closed field, but is not an isomorphism. At a nonzero curve point, $t=y/x$ is the unique inverse parameter, while the origin has unique preimage zero. The induced map on coordinate rings has image $k[t^3,t^4]\subset k[t]$, which lacks the monomial $t$: all its monomial exponents have the form $3a+4b$, with $a,b\ge 0$. Nor can $t$ be regular at the cusp, since an equality $t=P(t^3,t^4)/Q(t^3,t^4)$ with $Q(0)\ne 0$ would produce a nonzero degree-one term in $tQ$ and none in $P$. Thus pointwise bijectivity does not guarantee an algebraic inverse, even though the inverse is regular away from one singular point.

For incompressible simple shear $F=I+\gamma {\mathbf{e}}_1\otimes {\mathbf{e}}_2$ with Mooney-Rivlin energy $\alpha (I_1-3)+\beta (I_2-3)$, zero traction on the third-coordinate faces fixes the pressure reaction. The remaining Cauchy stress has $T_{11}=2\alpha {\gamma }^2$, $T_{22}=-2\beta {\gamma }^2$ and $T_{12}=2(\alpha +\beta )\gamma$. The material face $x_1=\text{constant}$ has current unit normal $\mathbf{n}=(1,-\gamma ,0)/\sqrt{1+{\gamma }^2}$ and tangent $\mathbf{m}=(\gamma ,1,0)/\sqrt{1+{\gamma }^2}$. Resolving $T\mathbf{n}$ gives $N=-2{\gamma }^2(\alpha +2\beta +\beta {\gamma }^2)/(1+{\gamma }^2)$ and $S=2\gamma (\alpha +\beta )/(1+{\gamma }^2)$. Since $N<0$, the traction-normal angle satisfies $\tan (\theta -\pi /2)=-N/|S|$, proving the displayed formula. Consequently its two-sided small-shear slope is $(\alpha +2\beta )/(\alpha +\beta )$. The reference normal cannot replace the transformed normal in this calculation.

For a left module $M$ over a unital associative algebra $A$, right multiplication in $A$ gives a right module structure on ${\mathrm{Hom}}_A(M,A)$ by $(f\cdot a)(m)=f(m)a$. This remains a module homomorphism because $f(bm)a=bf(m)a$; associativity and the unit verify the action axioms. When $A=M_d(\mathbb{C})$ and $M={\mathbb{C}}^d$ is its simple module of columns, every such map is $m\mapsto mr$ for a unique row $r$, making the Hom space the simple row right module. This illustrates how left and right simple modules correspond for a semisimple algebra.

The itch mite. Human-infecting forms burrow in the superficial skin.

A genus of mites associated with hair follicles and sebaceous structures.

This functional measures the surface area of the graph of a function. On a bounded domain it is finite for $u\in H^1$. Its directional derivative is ${\partial }_{v}F(u)=\int \nabla u\cdot \nabla v/\sqrt{1+|\nabla u{|}^2}$, by the dominated convergence theorem, since the pointwise difference quotients are bounded by $|\nabla v|$. The resulting dual-valued operator is monotone: the scalar integrand $p\mapsto \sqrt{1+|p{|}^2}$ is convex and its gradient is $p/\sqrt{1+|p{|}^2}$. It is not a nonlinear coercive operator on the full $H^1$ space, since it vanishes on constant functions whose $H^1$ norms can tend to infinity.

In the metric realization of a simplicial complex, the point on arm $n$ at parameter $t$ is $(1-t)e_v+t{e}_{u_n}$. Its distance from the common vertex is $\sqrt{2}t$. Consequently the vertex has a neighbourhood basis consisting of one uniformly short initial interval on every arm. Away from that vertex, each arm is locally isolated from the other arms.

The map $(n,t)\mapsto t{e}^{i/n}$, with all $t=0$ identified, is a homeomorphism ontoIt is a bijection, preserves the one-dimensional topology on each arm away from zero, and matches the uniform-radius neighbourhood bases at zero. The positive real limiting ray is omitted. The outer endpoints form an infinite closed discrete subset in the original metric realization, so the image is not compact. This planar fan and the weak realization of a countable star have different topologies at their common vertex.