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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.

When each observation counts untagged captures before the $r$th tagged capture with immediate replacement, its negative binomial distribution has success probability $p=k/N$. For $n$ independent runs, the log-likelihood is $nr\log p+({\sum }_i{x}_i)\log (1-p)$ plus a constant. Differentiation gives the displayed estimator and expected Fisher information. For an interior true parameter, its asymptotic variance is $p^2(1-p)/(nr)$. A confidence interval $(L,U)$ for $p$ transforms to $(k/U,k/L)$ for the population size, with endpoint order reversed.

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.