For distinct scalars ${\mu }_j$, the Lagrange interpolation polynomials ${\ell }_j(t)={\prod }_{k\ne j}(t-{\mu }_k)/({\mu }_j-{\mu }_k)$ sum to one. Their values at $T$ have images in the respective eigenspaces, since $(t-{\mu }_j){\ell }_j$ is a multiple of the annihilating polynomial. Hence every vector is the sum of its ${\ell }_j(T)$ images. On the $i$th eigenspace, ${\ell }_j(T)$ is multiplication by ${\delta }_{ij}$, which proves directness. This gives the decomposition without invoking the primary decomposition theorem. Under this decomposition a linear operator commutes with $T$ exactly when it preserves every eigenspace.

An amortized analysis can bound the total time complexity $T(m,n)$ of every allowed sequence of $m$ operations by $ma(n)+I(n)$, where $n$ is a size parameter and $I(n)$ accounts for initialization or the specified initial state. The bound $a(n)$ is then an amortized per-operation time complexity. Dividing the total bound by $m$ explains the averaging; it is a guarantee over every permitted sequence, without an assumed probability distribution on requests. Individual operations can cost more than $a(n)$.

For an initially zero $k$-bit binary counter, suppose changing bit $i$ costs $2^i$. It changes exactly $\lfloor n/2^i\rfloor$ times in $n$ increments. Consequently the total is $\Theta (n\min \{k,1+{\log }_{2}n\})$ for $n,k\ge 1$. The upper bound counts at most $n$ cost at each active position; the lower bound uses $\lfloor x\rfloor \ge x/2$ for $x\ge 1$. This shows why exponential position costs cancel the decreasing carry frequencies that make the ordinary binary counter have constant amortized cost.

Mistura Ammoniaci is an ammoniacum gum-resin mixture. The botanical gum-resin name must not be confused with an ammonium salt.^[ref]

Mistura Amygdalae is an almond mixture. Dispersed seed constituents make it different from either clear almond oil or intact kernels.^[ref]

The one-point compactification $D\cup \{\infty \}$ of an uncountable discrete space is a hereditarily normal space but not a perfectly normal space. A subspace missing $\infty$ is discrete; one containing it is closed, hence compact Hausdorff and normal. However, every neighborhood of $\infty$ misses only finitely many points of $D$. A countable intersection misses at most countably many points, so the closed singleton $\{\infty \}$ is not a G-delta set.

For locally compact noncompact Hausdorff spaces, a continuous proper map $f:X\to Y$ extends continuously to their one-point compactifications by the displayed rule. The inverse image of $\alpha Y\setminus K$ is $\alpha X\setminus f^{-1}(K)$, a neighborhood of infinity because the inverse image of a compact set is compact. The extension itself is proper, since it is continuous from a compact space to a Hausdorff space. A nonproper map can fail to admit any continuous extension: projection ${\mathbb{R}}^2\to \mathbb{R}$ sends two vertical sequences escaping to the same point at infinity to two distinct constant sequences, incompatible with continuity at that point.

A concept satisfiability procedure that expands concept labels into a graph, branches on disjunctions, creates witnesses for existential restrictions, propagates universal restrictions along existing edges, and tests for a tableau clash. General TBoxes require global constraints and a termination device such as direct blocking in description logic.

Consider fluid at rest with mean depth $d$ and a bottom displacement $a\sin (k(x-ct))$. The linearized free-surface boundary conditions at $y=0$ and moving-bottom kinematic boundary condition at $y=-d$ give a periodic particular response $\eta =b\sin (k(x-ct))$ and velocity potentialThe bottom derivative fixes the signed response coefficient to the title formula. Here $k>0$, $c\ne 0$, and the forcing must avoid $k^2{c}^2=gk\tanh (kd)$. This equality is resonance with the natural finite-depth surface gravity wave; nonzero undamped resonant forcing has no bounded periodic solution. The response reverses phase below resonance. General initial data can add free surface gravity waves, which do not decay in this inviscid model. Near resonance the resulting wave steepness and particle excursions, rather than the bottom displacement alone, determine whether the linearization remains valid.

If $f(x)=Ax+b$ and $g(x)=Cx+d$, their function composition is $g(f(x))=CAx+Cb+d$, again an affine map. With column-vector homogeneous coordinates, its matrix is the product of the matrix for $g$ with that for $f$, so the rightmost factor acts first.

An affine dependence of points is a nonzero coefficient family satisfying the displayed identities. Equivalently, the augmented vectors $(x_i,1)$ are linearly dependent. Both identities are independent of the chosen origin. The absence of an affine dependence is affine independence.

For $y>0$, the displayed function, expressed using the complementary error function, satisfies $\Delta u=u_x$, with $u=1$ on the positive boundary ray and $u_y=0$ on the negative boundary ray. Put $x=a^2-b^2$, $y=2ab$, so $u=\mathrm{erfc}b$. Then $\Delta u=u_{bb}/[4(a^2+b^2)]$ and $u_x=-b{u}_b/[2(a^2+b^2)]$; the derivative identity $u_{bb}=-2b{u}_b$ verifies the equation. Its boundary traces are $u(x,0)=\mathrm{erfc}\sqrt{-x}$ for $x<0$ and $u_y(x,0)=-1/\sqrt{\pi x}$ for $x>0$. At fixed nonzero polar angle it decays at infinity, but along fixed $y>0$ and $x\to +\infty$ it approaches one. The downstream thermal layer has width $O(\sqrt{x})$, so uniform radial decay would be an incompatible far-field condition.

For a finite graph, let $\lambda (D)$ count the vertices on a longest directed path in an acyclic orientation $D$. Its minimum over those orientations is the chromatic number. A proper $k$-colouring gives an orientation from smaller to larger colours, so every directed path has strictly increasing colours and at most $k$ vertices. Conversely, assign to each vertex the number of vertices in a longest directed path ending there. Along an edge $u\to v$ this number increases: an ending path at $u$ cannot contain $v$ without forming a directed cycle, so it can be extended. The resulting labels give a proper colouring using at most $\lambda (D)$ colours. These two constructions prove equality. Counting edges instead of vertices in path length subtracts one from the displayed minimum.

Every subgroup of an Abelian group generated by $n$ elements has at most $n$ generators. Lift it under a surjection ${\mathbb{Z}}^n\to A$. A subgroup $L\le {\mathbb{Z}}^n$ has at most $n$ generators by induction: its first-coordinate image is either zero or $d\mathbb{Z}$. In the latter case choose a lift of $d$, and subtract its multiples to reduce to the kernel in ${\mathbb{Z}}^{n-1}$. The images of these generators generate the original subgroup. The Abelian hypothesis matters: $S_{2n+2}$ is generated by a transposition and a full cycle but contains the elementary abelian group generated by $n+1$ disjoint transpositions, requiring $n+1$ generators by its vector-space dimension over ${\mathbb{F}}_2$.

Assign each item to neither bag or exactly one of two bags, respecting both capacities and maximizing total value. With nonnegative integer capacities $W_1,W_2$, a dynamic programming state indexed by the item prefix and both capacities gives $O(n(W_1+1)(W_2+1))$ time complexity. Every transition uses the preceding item layer to prevent reuse of an item.

The cumulative hierarchy is a transitive class even without the ambient Axiom of foundation. If $a$ is a nonempty member, choose the least ordinal $\alpha$ with $a\cap V_{\alpha }\ne \varnothing$. It is a successor $\beta +1$. An element $x$ of that intersection is a subset of $V_{\beta }$, while $a$ has no element in $V_{\beta }$, so $a\cap x=\varnothing$. This proves internal Foundation. If the hierarchy also satisfies the other ZF axioms, restricting every quantifier of a finite ZF proof to this definable class translates it into a proof from ZF without foundation. Thus adjoining Foundation preserves consistency relative to the remaining axioms.

In a normal Hausdorff space, strong zero-dimensionality means that any disjoint closed sets can be separated by a clopen set, equivalently by disjoint clopen neighborhoods. For general Tychonoff spaces, the usual definition asks this for completely separated sets. In the normal setting Urysohn's lemma makes the formulations agree. The condition is equivalent to zero-dimensionality of the Stone-Čech compactification.

The characteristic functions of a clopen topological basis give a topological embedding of a zero-dimensional $T_0$ topological space into a product of discrete two-point topological spaces. The separation axiom makes those functions distinguish points, and their coordinate preimages recover the original topological basis.

The conjugate-power constrained product maximum proves Young's inequality for products by rescaling. For nonnegative $x,y$ and conjugate exponents $p,q>1$, let $A=x^p/p+y^q/q$. If $A=0$, both variables vanish. Otherwise the displayed normalized variables satisfy $X^p/p+Y^q/q=1$, hence $XY\le 1$. Since $1/p+1/q=1$, this means $xy/A\le 1$, proving $xy\le x^p/p+y^q/q$. Equality in the constrained maximum requires $X=Y=1$, equivalent to $x^p=y^q$; the zero case gives the same equality criterion. This approach simultaneously gives the sharp constant and every equality case.

The inner, generally nonconducting portion of a woody stem, distinguished from the more recently formed sapwood.