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This delay differential equation replaces instantaneous logistic inhibition by a population value one delay earlier. A positive history function of a delay differential equation gives positive solutions while they exist. The equilibria are zero and one. Linearization at one gives $x^{\prime }(t)=-x(t-\tau )$, with characteristic equation $\lambda =-e^{-\lambda \tau }$. Delayed sign reversal permits overshoots around one and, at the delayed logistic Hopf threshold, a periodic branch.

For nonnegative instantaneous damping $\beta$ and positive delayed gain $\gamma$, characteristic roots obey $\lambda +\beta +\gamma e^{-\lambda }=0$. There is no nonnegative real root. The first imaginary crossing has $\omega \in [\pi /2,\pi )$ and $\beta =-\omega \cot \omega$, $\gamma =\omega /\sin \omega$. In particular stability is retained below this first boundary and $\gamma =\pi /2$ is exact only at $\beta =0$. For small positive $\beta$, the critical gain is $\pi /2+2\beta /\pi +O({\beta }^2)$ and the scaled period tends to four. If the time delay is dimensional $\tau$, that period is multiplied by $\tau$.

For a smooth delayed-transfer epidemic model, the incidence $rI(t-\tau )S(t-\tau )$ equals $rIS-r\tau (\dot{I}S+I\dot{S})+O({\tau }^2)$. Substituting the zero-delay equations only inside the correction gives $rIS+\tau rIS\{a+r(I-S)\}+O({\tau }^2)$. The correction has opposite signs in the receiving and departing compartments, preserving their transfer balance.

The zero polynomial has no nonzero term, so its degree is either left undefined or assigned a special value such as $-\infty$ by convention. A finite program can use $-1$ as a sentinel when all other stored degrees are nonnegative. That representation sentinel is not a general replacement for the extended-arithmetic degree convention; it is enough for testing whether a polynomial remainder is zero or has smaller degree than a nonzero divisor.

At the origin the gradient vanishes and the Hessian matrix is $\mathrm{diag}(0,2)$, which is positive semidefinite. Nevertheless, $F(t,0)=t^3$ takes both signs arbitrarily close to the origin. Thus the origin is a saddle point of a scalar function. This example shows why a zero eigenvalue prevents the ordinary second-derivative test from establishing a local minimum.

A historical clinical term for a pale, painful, swollen limb, especially after childbirth, associated with extensive deep vein thrombosis. It is distinct from blue, severely congested phlegmasia cerulea dolens.

Severe venous outflow obstruction producing a painful, swollen, blue limb and potentially threatening perfusion.

A dead-load traction is prescribed force per unit reference surface area, with its vector held fixed as the body deforms. Its boundary condition is $T_{R}N=t_R$. A follower load instead depends on the current deformation, for example through a current normal or current area.

For $\Phi =\mu [{\sum }_i{v}_i^2/2+{\sum }_i{v}_i^{-1}]$ and positive stretch product one, the energy simplifies to $\mu S^2/2$, where $S={\sum }_i{v}_i$. Equal nominal tensile loading adds potential $-\tau S$. The homogeneous diagonal equilibrium equations reduce to $(\mu S-\tau )(v_i-v_j)=0$. Thus the unit triple exists for every load, while the other equilibria form the displayed sum-product family. Since $S\ge 3$, only the unit triple exists for $\tau \le 3\mu$. Above this load, a continuous family is obtained by choosing $v_1=t$ near one and taking $v_2,v_3$ as the two positive roots with sum $\tau /\mu -t$ and product $1/t$.

An edge set respects a cut of a graph when none of its edges crosses that cut. In the minimum spanning tree cut property, this guarantees that exchanging a crossing tree edge cannot remove an edge already committed to the partial solution.

An edge of minimum edge weight among those crossing a specified cut of a graph. It exists when the crossing set is nonempty. Several tied edges can all be light; lightness does not imply uniqueness or membership in every minimum spanning tree.

Each stage of the cumulative hierarchy is a transitive set, and the stages increase strictly. If $A$ is transitive, then $A\subseteq \mathcal{P}(A)$ and $\mathcal{P}(A)$ is transitive: $x\in y\subseteq A$ gives $x\in A$ and then $x\subseteq A$. Unions of increasing transitive sets are transitive. These observations prove the assertions by transfinite induction, with strictness witnessed by $V_{\alpha }\in V_{\alpha +1}$ and $V_{\alpha }\notin V_{\alpha }$ from the Axiom of foundation.

For a nonzero limit ordinal $\delta$, the structure $(V_{\delta },\in )$ satisfies the Axiom of power set. For $x\in V_{\beta }$ with $\beta <\delta$, transitivity gives $x\subseteq V_{\beta }$, so $\mathcal{P}(x)\subseteq V_{\beta +1}$ and $\mathcal{P}(x)\in V_{\beta +2}\subseteq V_{\delta }$. Every external subset is therefore present and the external power set is also the internal one.

If every member of a set $a$ belongs to some cumulative-hierarchy stage, replacement collects the least such stages. Their ordinal supremum gives a common stage $V_{\beta }$ containing every element of $a$, so $a\in V_{\beta +1}$. The argument also works without foundation and bounds a set of well-founded elements inside the hierarchy.

A finite sequence of visible CSP events performed by a CSP process, omitting CSP internal actions.

After CSP trace $s$, the CSP process can reach a CSP stable state offering no event in the refusal set $X$. Refusal is existential over reachable stable alternatives.^[ref]

The possibility, after a finite visible CSP trace, of an infinite sequence of consecutive CSP internal actions. The CSP failures-divergences model records this possibility explicitly.^[ref]

Represent a CSP process by its set of CSP traces.