The ratio of thermal diffusivity $\kappa$ to molecular or solutal diffusivity $D$. In coupled heat equation and solute diffusion equation similarity problems, the thermal coordinate $z/(2\sqrt{\kappa t})$ becomes $\sqrt{Le}$ times larger in the solute complementary error function profile. This follows by substituting the thermal coordinate into $C_t=D{C}_{zz}$, giving $C^{\prime \prime }+2Le\eta C^{\prime }=0$. The definition must specify the direction of this ratio because reciprocal conventions occasionally occur.

A unary description interpreted as a subset of the domain. Atomic names can be combined using the constructors of the chosen description logic.

A binary relation on individuals used by description logic restrictions such as $\exists R.C$ and $\forall R.C$.

An axiom requiring every instance of the description logic concept $C$ to be an instance of $D$. It holds throughout the domain, not just at named individuals.

A collection of terminological axioms, including general concept inclusions, constraining the meanings of description logic concepts and description logic roles.

A collection of assertions about individuals, such as $A(a)$ and $R(a,b)$, interpreted together with a TBox.

The description logic with atomic concepts, top and bottom, Boolean concept constructors, and existential and universal restrictions on description logic roles.

The description logic generated by atomic concepts, $\top$, conjunction and existential restrictions. Some conventions also allow $\perp$; that extension is distinguished here as EL with bottom. Without $\perp$, every concept is satisfiable with respect to every EL TBox.

A rule-oriented description logic fragment designed for translation to function-free Horn rules, allowing constraints but excluding existential witness generation in rule conclusions. In an equivalence, each direction must meet the relevant restrictions. ^[ref]

A description logic concept is satisfiable with respect to a TBox if some model of that TBox interprets the concept as a nonempty set.

The assumption that distinct individual names denote distinct domain elements. Standard ALC semantics does not impose it; Herbrand interpretations do treat distinct ground terms as distinct elements.

For a nonzero gradient, this unit direction minimizes the directional derivative among Euclidean unit directions. The Cauchy-Schwarz inequality gives $\nabla f(x{)}^{T}p\ge -\Vert \nabla f(x){\Vert }_2$, with equality for the displayed direction. The steepest descent method usually uses the equivalent unnormalized direction $-\nabla f(x)$ and chooses a separate step length. A different choice of norm can change the steepest direction.

Visit all child subtrees before visiting their parent. A depth-first traversal of a tree produces this order by processing a node on return from its recursive calls. It lets dynamic programming use the already computed answers of every child subtree.

Overcompensation occurs when a larger present population produces a smaller total population in the next generation over part of the relevant range. It is stronger than a decline of per-capita growth alone. For $F(u)=\alpha u/(\beta +u^{\gamma })$ with $\gamma >1$, the large-abundance response $F(u)\sim \alpha u^{1-\gamma }$ decreases with abundance. Stronger feedback can drive a fixed-point multiplier through minus one, but that local threshold alone does not establish chaotic behaviour.

There are $2^{\mathfrak{c}}$ dense subsets of $\mathbb{R}$, where $\mathfrak{c}=|\mathbb{R}|$. For a fixed Cantor set $C$ of size $\mathfrak{c}$, its complement $D$ is dense, and $A\mapsto D\cup A$ injects $\mathcal{P}(C)$ into the family of dense subsets. The reverse injection is the inclusion into $\mathcal{P}(\mathbb{R})$. The Cantor-Schröder-Bernstein theorem gives equality.

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 $\gamma$, roots obtained by substituting $q(T)=e^{\lambda T}$ 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 $\tau$.

Consider a smooth delayed compartmental model with susceptible and infected populations $S,I$, incidence $rI(t-\tau )S(t-\tau )$, and removal rate $aI(t)$, where $r,a>0$. The zero-delay equations are $S^{\prime }=-rIS$ and $I^{\prime }=rIS-aI$. A Taylor expansion of the incidence givesSubstitute the zero-delay equations only in the first-order correction to obtainThe remainder is controlled on a compact time interval when the required second derivatives are bounded. The same approximated transfer enters the susceptible and infected equations with opposite signs, preserving their compartmental model transfer balance. Defining the zero-delay dynamics explicitly is essential: different population equations would give a different correction.

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.