For a permutation $\sigma$ of a finite set, the cycle containing $x$ is its orbit under the iterates of $\sigma$. Its length $k$ is the least positive integer with ${\sigma }^{k}x=x$; finiteness and invertibility guarantee such a return. Two cycles that meet are the same, because any iterate of a common point belongs to both. Thus the cycles partition the set. In cycle notation, the displayed cycle sends each listed element to the next and the last one back to $x$.

For a bijection $f$, composing $f^2=f^{-1}$ with $f$ gives $f^3=\mathrm{id}$, and composing the latter with $f^{-1}$ gives the converse. Every orbit therefore has size one or three: the permutation consists of fixed points and disjoint three-cycles. This applies to an infinite underlying set as well as a finite one, since each individual orbit has at most three elements.

Motor supply to the respiratory diaphragm, arising chiefly from cervical spinal segments C3–C5.

Peripheral neuropathy is dysfunction or damage of peripheral nerve pathways. Sensory, motor and autonomic components can be affected in different combinations. Distribution across nerves and fiber types is therefore part of its description, rather than one symptom defining every case.

For $\dot{r}=r[1-r^2(2-\cos 2\theta )]$, $\dot{\theta }=2$, the substitution $w=r^{-2}$ gives $dw/d\theta +w=2-\cos 2\theta$. Its positive periodic solution yields the displayed orbit, with full planar period $\pi$. Every other solution at the same phase has $w-w_p=C{e}^{-2t}$, proving uniqueness and attraction. The limiting circles $r=1/\sqrt{3}$ and $r=1$ give a trapping annulus, with tangencies at their extremal angles.

A relatively recent inflammatory syndrome involving the pericardium. It can produce pain and a friction rub with or without appreciable pericardial effusion; fluid amount and inflammatory activity are separate features.^[ref]

Accumulation of fluid in the pericardium. Its size and effect on the blood circulation are distinct questions.

An auscultatory sign produced by movement of inflamed pericardial surfaces.

For an equality-constrained optimization problem, an even integer $p\ge 2$ makes this a nonnegative penalty for constraint violation. Its derivative is $\nabla f+{\mu }^{-1}D{c}^T{c}^{p-1}$, with the power applied coordinatewise. The case $p=2$ is the quadratic penalty. Reducing $\mu$ increases the cost of infeasibility; it does not itself ensure that computed stationary points or minimizers stay bounded.

Explicit light-source sampling at a surface interaction: connect to a sampled source and use a shadow ray to test visibility. It avoids relying solely on a randomly continued path to hit an emitter; this is essential for ordinary continuous direction sampling with ideal point lights.

If $f:X\to Y$ is continuous and $X$ is path-connected, then $f(X)$ is path-connected. Choose preimages of any two image points and a continuous path between them in $X$. Its composition with $f$ supplies the required path inside $f(X)$. The conclusion uses the subspace topology on the image, not an assumption that $f$ maps onto the whole codomain.

If $X$ had a separation into two disjoint nonempty open subsets, a continuous path joining points in the two subsets would pull it back to a separation of $[0,1]$. This contradicts connectedness of an interval. Thus every path-connected space is connected. The converse fails, for example for the topologist's sine curve with its limiting vertical segment.

The displayed subspace of the Euclidean plane is path-connected: each vertical rational tooth joins the horizontal line at height one. Delete $(r,1)$ for rational $r$. The residual tooth $V=\{r\}\times [0,1)$ cannot be joined by a path to the top line. Before a path first reaches height one, its first coordinate is a continuous rational-valued function on an interval and must therefore be constant; it would reach the deleted point.

The punctured space remains connected. Its portions $L,R$ on either side of $x=r$ are path-connected, and $V\subseteq \overline{L}\cap \overline{R}$ by the density of the rational numbers. Adjoining limit points to a connected set preserves connectedness: a separation would place the connected subset on one side and prevent the other side from containing its limit points. Hence $L\cup V$ and $R\cup V$ are connected and intersect in $V$. Their union is connected. This example distinguishes connectedness from path-connected using the loss of a single attachment point.

The vectors need not form orthonormal bases. With the inner product linear in its second entry, expand the rank-one operator $|\Psi ⟩⟨\Psi |$ and take the trace on the second factor. This gives the displayed reduced positive operator. For every observable $A$ on the first factor,Positivity also follows directly from $⟨v,S_{1}v⟩=\Vert {\sum }_j⟨v,{\xi }_j⟩{\eta }_j{\Vert }^2$. Thus the physical density operator is $S_1/\Vert \Psi {\Vert }^2$ when the nonzero source vector is not normalized. The Gram matrix of the second-factor vectors supplies the cross coefficients; replacing those coefficients by Kronecker deltas without checking orthonormality changes the subsystem state. This formula is useful when an efficient product expansion uses nonorthogonal vectors, even though the partial trace itself is basis-independent.

A second-order partial differential equation is of mixed type when the type of its principal part changes across its domain. For $A{u}_{xx}+2B{u}_{xy}+C{u}_{yy}=0$, the sign of $B^2-AC$ distinguishes a hyperbolic partial differential equation, parabolic type and an elliptic partial differential equation. For example, $u_{xx}+\mathrm{sign}(y)u_{yy}=0$ is elliptic above $y=0$, hyperbolic below it, and degenerate on the interface. On its hyperbolic side, the change $\xi =x+y$, $\eta =x-y$ reduces it to $u_{\xi \eta }=0$; above it is the Laplace equation. Type classification alone supplies no matching rule for the normal derivative.

A free boundary problem determines both a field and part of its domain boundary. A glacier terminus, for example, separates positive ice thickness from bare bed. Mass conservation, nonnegative thickness, a rule for ice ablation only where ice exists, and suitable interface or entropy solution conditions are needed; a formal zero of one characteristic branch alone need not give the selected physical boundary.

For a differential equation $L[u]=0$ with its comparison order fixed, a supersolution satisfies $L[w]\ge 0$. For example, if $L={\partial }_t-{\partial }_x^2+Q$ with $Q\ge 0$, a sufficiently regular $w$ satisfying $L[w]\ge 0$ and dominating the initial and boundary data lies above a solution by the parabolic comparison principle. An unbounded-domain barrier such as $w=\eta e^{3t}(1+x^2)$ can provide the missing boundary control at large distance. The sign convention belongs to the chosen differential operator and must be stated.

A type scheme describing a family of instances through type variables. In Hindley-Milner type inference, a principal type describes the most general valid instance pattern; concrete types arise by consistently substituting for the quantified variables. Reusing one variable constrains the corresponding positions to the same type, whereas distinct variables can vary independently.

An object method with its own type parameters, instantiated or inferred at each call. In Scala, def identity[T](x: T): T = x is a generic method. A method def accept(x: T) inside class Box[T] instead uses a class type parameter fixed by the receiver's type argument; it is not itself a generic method with a separately chosen T on every call. This distinction matters in method overloading.