If $X$ has a Pareto distribution with positive scale $x_m$ and positive shape $\alpha$, then for $z\ge 0$,Thus the rescaled logarithm has an exponential distribution with rate $\alpha$. The tail probability calculation also identifies the support as $z\ge 0$.

The complexity class of complements of formal languages in P. A deterministic polynomial-time algorithm deciding a formal language can have its accepting and rejecting outcomes interchanged, so P is closed under complement and equals co-P.

The component of osmotic pressure due to colloidal constituents such as albumin in the circulating fluid.

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.