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For $L=-d^2/d{x}^2+1$ on $[-1,1]$ with zero endpoint values, the displayed Green kernel is positive and symmetric. Each side solves the homogeneous equation; the kernel is continuous at $x=s$ and its first derivative has jump $-1$. Hence $LG(\cdot ,s)={\delta }_s$. Integrating against a forcing function solves $Ly=f$. For the indicator of the right half-interval, the value at zero is $(\cosh 1-1)/(2\cosh 1)$, obtained by integrating $\sinh 1\sinh (1-s)/\sinh 2$ over $0\le s\le 1$. The derivative-jump sign changes if the operator is written with the opposite overall sign.

If a group has carrier $A$ and a disjoint set $B$ is in bijection with $A$, identify $A\cup B$ with $A\times C_2$, sending the original carrier to $A\times \{0\}$. Transporting the direct-product multiplication gives a larger group operation that agrees exactly with the old one on $A$. A maximal partial group on a fixed carrier universe therefore has complement of cardinality strictly below its own, using cardinal comparability.

For a scalar fixed-endpoint problem on a bounded interval, suppose the continuous integrand $f(x,y,p)$ is convex in $p$ and satisfies $f(x,y,p)\ge \Theta (|p|)-C$, where $\Theta$ is nonnegative convex and $\Theta (r)/r\to \infty$. If there is one admissible absolutely continuous function of finite energy, a global minimizer exists in $W^{1,1}$.

The lower bound gives uniform integrability of derivatives in a minimizing sequence. Fixed endpoint values then give boundedness and equicontinuity of the functions. Extract uniform convergence of the functions and weak $L^1$ convergence of their derivatives; the limit retains the endpoint values. Convexity in the derivative gives weak lower semicontinuity of convex gradient energies, so the limit attains the infimum. Strict convexity of the full integrand on a convex admissible class additionally gives uniqueness.

For $I_{\epsilon }(u)={\int }_0^1[(u^4-x{)}^2+\epsilon |u^{\prime }{|}^{4/3}]dx$ with endpoint values zero and one, the zero-penalty minimizer $x^{1/4}$ has infinite derivative penalty. Use $u=x^{1/4+\delta }$ instead. The mismatch is at most $(32/27){\delta }^2$, while the derivative penalty is $(3\epsilon /(4\delta ))(1/4+\delta {)}^{4/3}$. Balancing ${\delta }^2$ and $\epsilon /\delta$ by $\delta ={\epsilon }^{1/3}$ gives an upper energy bound of order ${\epsilon }^{2/3}$. Nonnegativity then proves convergence of minimum energies to zero. This is a recovery construction; no claim of a matching lower asymptotic constant is needed.

A Lavrentiev gap is a strict change in an infimum when the admissible regularity class is restricted, despite retaining the same endpoint values. For example $I(u)={\int }_0^1(u^3-x{)}^2{u}^{\prime 6}dx$ with $u(0)=0,u(1)=1$ has value zero at the absolutely continuous function $u=x^{1/3}$. Every Lipschitz competitor has a uniform positive lower energy bound. Indeed its ratio to $x^{1/3}$ starts at zero, so let $x_0$ be its first crossing of $u(x)=x^{1/3}/2$. Before that point, $|u^3-x|\ge 7x/8$. Holder inequality givesThus $I(u)\ge (49/4096)(3/5{)}^5/x_0\ge (49/4096)(3/5{)}^5>0$. This proves a gap directly. The example's integrand lacks a uniform superlinear lower bound along its zero set; it illustrates the admissible-class issue rather than by itself furnishing a counterexample under stronger coercivity assumptions.

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.