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The topological realization of a simplicial complex with infinitely many vertices is not compact. Its vertex open star in a simplicial complex form an open cover, but a vertex belongs only to its own star. Thus every finite collection of vertex stars omits all the other vertices. The argument uses the weak topology of a simplicial realization, in which a vertex star is open because its complement is the realization of the subcomplex of simplices missing that vertex. No local finiteness assumption is required.

Let $M$ be a countable omega-saturated model and $D\subseteq M$ finite. Then $D$ is a parameter-free definable set exactly when every structure automorphism preserves it setwise. One direction is automorphism invariance of parameter-free definability. For the converse, saturation implies homogeneity and the back-and-forth method show that equal empty-set types are in the same automorphism orbit. For $a\in D$, if no formula in $\mathrm{tp}(a)$ has its realization set contained in $D$, the partial type $\mathrm{tp}(a)\cup \{x\ne d:d\in D\}$ is finitely satisfiable. Its parameters form a finite set, so saturation realizes it outside $D$, contradicting invariance. Choose a defining neighbourhood for each $a\in D$ and take their finite disjunction. The empty set is defined by $x\ne x$. Setwise invariance permits automorphisms to permute points within $D$.

An NP decision problem is strongly NP-complete when its NP-hardness persists with numerical parameters written in the unary numeral system. An important proof method bounds every relevant numerical value by a polynomial in the remaining explicit input size. For the clique decision problem, values $k>|V(G)|$ can be mapped to one fixed no-instance; otherwise writing $1^k$ takes only polynomial time. The distinction is representation-sensitive, so an explicit graph encoding must be specified.

For a divergence-form differential operator $-\nabla \cdot K\nabla$, its conormal derivative is the flux factor appearing in integration by parts. It generally differs from the ordinary normal derivative ${\partial }_{n}u=\mathbf{n}\cdot \nabla u$ when $K$ is anisotropic. Decomposing the gradient along a smooth boundary into its normal and tangential components gives ${\partial }_{n,K}u=(\mathbf{n}\cdot K\mathbf{n}){\partial }_{n}u+(\mathbf{n}\cdot K\mathbf{t}){\partial }_{\mathbf{t}}u$ in two dimensions. Thus zero normal derivative does not in general remove the boundary term in the usual symmetric weak formulation. A zero conormal derivative does remove it and produces the natural homogeneous Neumann boundary condition for that energy.

A planar node whose Jacobian matrix has a repeated nonzero real eigenvalue and only one independent eigenvector is an improper node. In suitable real coordinates its linearization of a dynamical system has the displayed form, with solutionsIt is a stable node for $\lambda <0$ and an unstable node for $\lambda >0$. The factor $t$ makes generic trajectories tangent to the unique eigendirection as they approach the equilibrium in the attracting time direction. For a nonlinear autonomous system with this hyperbolic equilibrium, the sign of $\lambda$ still determines local stability; the local phase portrait has the same topological type as its linearization.

For independent $N(0,{\sigma }^2)$ observations, testing a fixed variance against larger variances uses the sum of squares $S={\sum }_i{X}_i^2$. The likelihood ratio for any simple larger variance is strictly increasing in $S$. The Neyman-Pearson lemma therefore gives the displayed rejection rule, where $F_n$ is the chi-squared distribution function with $n$ degrees of freedom. Its threshold is independent of the selected larger variance, so it is a uniformly most powerful test for the one-sided composite alternative. Its power function is $1-F_n(({\sigma }_0^2/{\sigma }^2)F_n^{-1}(1-\alpha ))$, increasing continuously from zero to one and equaling $\alpha$ at the null. A known nonzero mean is handled by replacing each observation by its difference from that mean.

For independent Weibull distribution observations of known shape two, the likelihood ratio between any scale ${\lambda }_1>{\lambda }_0$ and the null scale ${\lambda }_0$ increases with ${\sum }_i{X}_i^2$. Under the null, $2{\sum }_i{X}_i^2/{\lambda }_0^2$ has a chi-squared distribution with $2n$ degrees of freedom. The same upper-tail critical region is therefore most powerful for every larger scale, giving a uniformly most powerful size-$\alpha$ test.

A symmetric uniformly elliptic divergence-form energy with zeroth-order coefficient at least $\mu >0$ controls the full $H^1$ norm, including constants. Its quadratic energy minus a bounded load has a unique minimizer by the direct method, and vanishing first variation gives the natural Neumann weak equation. Unlike a pure Neumann Laplacian, this problem needs no zero-mean compatibility condition. The exact energy difference from the weak solution is half the bilinear energy of the difference.

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.