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Four positive-radius circles with pairwise disjoint interiors, successive external tangencies, and four distinct contact points have those contact points on a circle. Apply the reciprocal Möbius transformation $w=1/(z-T)$ at one contact $T$. The two circles through $T$ become disjoint parallel lines; the other two disks lie between them and avoid zero. Collinearity for tangent circles between parallel lines places the three finite transformed contacts on a straight line. That straight line misses zero: otherwise its segment across the strip would pass through zero, while that segment is contained in the two transformed closed disks. Its inverse is therefore a genuine circle through $T$, containing all four original contacts. Allowing internal tangency without the pictured disjoint-interior hypotheses admits collinear degeneracies instead.

The pork tapeworm. Human intestinal adults and tissue cysticerci follow different routes of infection.

The beef tapeworm.

An explicit contradiction in a node label, such as both $A$ and $\neg A$, or $\perp$. Opposite universal restrictions alone cause no clash when there are no successors.

A test that stops further witness generation at a node when an appropriate earlier ancestor already represents its required pattern of labels. The exact test depends on the logic and tableau calculus. An isolated root has no ancestor and is not directly blocked.

For a symplectic matrix $M$ of size $2m$ with determinant one, the defining identity $M^{T}JM=J$ gives $M^{-1}=J^{-1}{M}^{T}J$. Since similarity and transposition preserve the characteristic polynomial, ${\chi }_{M^{-1}}={\chi }_M$. Factoring $xI-M=M(x{M}^{-1}-I)$ then proves the displayed reciprocal identity for $x\ne 0$. If ${\chi }_M(z)=(z-\lambda {)}^{d}R(z)$ with $R(\lambda )\ne 0$, this identity gives ${\chi }_M(x)=x^{2m-d}(1-\lambda x{)}^{d}R(1/x)$ near $x={\lambda }^{-1}$. Thus reciprocal eigenvalues have the same algebraic multiplicity. The fixed points $\pm 1$ require the stronger parity argument in even multiplicities at fixed points of reciprocal root pairing.

Suppose distinct Sylow subgroups for $p$ in a finite group have trivial intersections. Conjugation by one of them, $P$, fixes $P$. The stabilizer of any other $Q$ is $H=P\cap N_G(Q)$. Since $H$ normalizes $Q$, the product $HQ$ is a subgroup of $p$-power order; maximality of $Q$ gives $H\le Q$. Thus $H\le P\cap Q=1$. All other orbits have size $|P|$, proving the congruence.

In the alternating group $A_6$, of order $360$, a Sylow subgroup of order five is generated by a five-cycle. There are $6\cdot 4!=144$ five-cycles, four in each such subgroup, giving $36$ subgroups. A Sylow subgroup of order nine is generated by disjoint three-cycles, hence is $C_3\times C_3$. The unordered partitions of six letters into two triples give all such subgroups by Sylow conjugacy, and their single three-cycles recover the partition uniquely. There are $\left(\genfrac{}{}{0ex}{}{6}{3}\right)/2=10$ partitions.

For a bottom-forced gravity wave with signed surface coefficient $b$, evaluate the first-order velocity field at a parcel's mean position $(\bar{x},\bar{y})$ and integrate, with $\Theta =k(\bar{x}-ct)$. The horizontal and vertical displacement coefficients areThus the pathline is an ellipse with horizontal and vertical semiaxes $|A|,|B|$ whenever both are nonzero, and a line segment if one vanishes. The identity $A^2-B^2=b^2[(g/(k{c}^2){)}^2-1]$ determines which axis is longer. Also $B(0)=b$ and $B(-d)=a$, verifying the surface and bottom displacements. Closed first-order orbits do not rule out a higher-order mean drift.

Set $p=(0,0)$, $q=(1,0)$, ${\epsilon }_m=1/(m+2)$, and let $A_m$ be the polygonal arc joining $({\epsilon }_m,0)$, $({\epsilon }_m,m)$, $(1-{\epsilon }_m,m)$, and $(1-{\epsilon }_m,0)$. In the planar subspace $Z=\{p,q\}\cup {\bigcup }_m{A}_m$, each arc is clopen. The component of $p$ is therefore just $\{p\}$. Any clopen neighborhood of $p$ contains the left endpoints of all sufficiently high arcs, hence those entire connected arcs, and by closedness also the limit $q$ of their right endpoints. Thus its quasi-component is $\{p,q\}$. Higher arcs escape to infinity while their endpoints approach the two distinguished points.

For $u\in C^2(D)\cap C(\overline{D})$ on a bounded domain in ${\mathbb{R}}^n$ with $\Delta u\le 0$, the minimum lies on the boundary. Define $u_{\epsilon }=u-\epsilon |x{|}^2$, so $\Delta u_{\epsilon }\le -2n\epsilon <0$. An interior minimum would have positive semidefinite Hessian matrix and nonnegative trace, a contradiction. Thus $u_{\epsilon }$ attains its minimum on the boundary. Boundedness controls $|u-u_{\epsilon }|$ uniformly by a constant times $\epsilon$; letting $\epsilon \downarrow 0$ proves the weak minimum principle for the original superharmonic function. The perturbation handles the non-strict inequality, including constant functions.

On three players, let $v(S)=1$ exactly when $S$ contains player 1 and at least one of players 2 and 3, and zero otherwise. This simple cooperative game is superadditive: disjoint coalitions cannot both contain player 1, so at most one can have value one. Its core of a cooperative game is exactly $\{(1,0,0)\}$, since singleton nonnegativity and the two winning-pair constraints force the other two coordinates to zero. In a random player ordering, player 1 is pivotal unless it comes first, whereas each other player is pivotal in just one of the six orders. Thus its Shapley value is $(2/3,1/6,1/6)$, outside the core: coalition $\{1,2\}$ receives only $5/6<1$. Convex cooperative games have the stronger property that their Shapley value belongs to the core of a convex game.

For vector subspaces $U\subseteq W\subseteq V$ of a finite-dimensional vector space, such a $W^{\prime }$ always exists. Extend a basis of $U$ to one of $W$, then extend again to one of $V$. Let $W^{\prime }$ be spanned by the original basis of $U$ together with the vectors added in the second extension. Uniqueness of coordinates gives the displayed intersection and sum. Equivalently, choose a direct sum complement $C$ with $V=W\oplus C$ and put $W^{\prime }=U\oplus C$. Necessarily $\dim W^{\prime }=\dim V-\dim W+\dim U$. In $M_2(\mathbb{R})$, the skew-symmetric subspace $U$ inside the traceless subspace $W$ admits $W^{\prime }=\mathrm{span}\{I,\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)\}$.

Let $A=D+L+L^T$ be a real symmetric positive-definite matrix, with $D$ diagonal and $L$ strictly lower triangular. Put $B=D/\omega +L$ and $C_{\omega }=I-B^{-1}A$ for $\omega \ne 0$. The matrix norm induced by an energy norm reduces the question to $T=A^{1/2}{C}_{\omega }{A}^{-1/2}$. Direct multiplication and $B+B^T-A=(2/\omega -1)D$ giveThe left side is positive semidefinite. The matrix following the scalar coefficient on the right is positive-definite, since $D$ has positive diagonal and $B^{-1}{A}^{1/2}$ is an invertible matrix. For $0<\omega <2$, every eigenvalue of $T^{T}T$ therefore lies in $[0,1)$. Hence its spectral radius is below one and $\Vert C_{\omega }{\Vert }_A=\Vert T{\Vert }_2<1$. Iteration errors consequently decay geometrically in the energy norm. The argument proves convergence from every initial vector, without requiring the original iteration matrix to be symmetric.

The values of all closed terms in a first-order structure form a substructure when the first-order language has at least one constant. The constant ensures nonemptiness, and applying a function symbol to closed-term values again gives a closed-term value. Relations are restricted from the ambient first-order structure. Every universal sentence true in the ambient first-order structure remains true in this substructure.

Replacing one designated subformula occurrence in a propositional formula by another propositional formula preserves well-formedness. Induct on the context surrounding the occurrence: at the root the result is the replacement formula; under a connective apply the induction hypothesis to the affected child and rebuild the parent using its formation rule.

A subformula occurrence specifies a particular position in the syntax tree of a formula. Two occurrences can contain the same logical formula while being different nodes. Replacement of one occurrence does not necessarily replace the others.

A structure isomorphism is a bijective structure embedding: it preserves constants and functions, and both preserves and reflects relations. Its inverse has the same properties. It therefore preserves truth of every first-order formula, by structural induction, so it preserves realization and omission of parameter-free types.