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A sequence of continuous periodic functions satisfying the displayed conditions, with normalized Haar measure $m$ and every $\delta >0$, is a summability kernel. For $f$ in a homogeneous Banach space on the circle, subtract $f\int k_{n}dm$ from its convolution with $k_n$. Translation continuity controls the resulting integral near zero, while boundedness by $2\Vert f{\Vert }_B$ and the vanishing tails control its complement. Thus $\Vert f\ast k_n-f{\Vert }_B\to 0$. The nonnegative Fejér kernel is an example with $L^1$ norm one.

On an antiplane shear $F=I+e_3\otimes (p,q,0)$, the first strain invariant satisfy $I_1=I_2=3+p^2+q^2$, $I_3=1$. Restricting an isotropic objective strain energy density to this family determines only derivatives tangent to the family. It fixes $S_{31}=2w^{\prime }(p^2+q^2)p$ and $S_{32}=2w^{\prime }(p^2+q^2)q$, but does not specify the whole Cauchy stress tensor. For example the neo-Hookean solid energy $W=\mu (|F{|}^2-3)/2-\mu \log \det F$ gives $S=\mu (B-I)$ on this family. Even though the reference state is stress free, this is not generally a scalar multiple of $B$.

For a vector subspace $U$ of a finite-dimensional vector space $V$, its annihilator of a vector subspace $U^0\subseteq V^{\ast }$ determines $U$ exactly: $U={\bigcap }_{f\in U^0}\ker f$. Indeed, a basis of $U$ can be extended to one of $V$. If $v\notin U$, include $v$ among the added basis vectors and define a linear functional taking value one at $v$ and zero on all the other basis vectors. This functional vanishes on $U$ but not at $v$, excluding every outside vector from the intersection.

It suffices to intersect the kernels of any basis $g_1,\dots ,g_k$ of $U^0$: every other annihilating functional is their linear combination. The dual basis for a basis of $V$ extending a basis of $U$ shows $k=\dim V-\dim U$. If $U=V$, the empty intersection is $V$. This translates a subspace description by spanning vectors into an exact description by independent linear equations.

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.

Historical formulation name; composition is identified by pharmacopoeial edition.

Historical formulation name; composition is identified by pharmacopoeial edition.

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.