A short rigid nonrotating guide constrains two freely sliding beads to one axis $n$. In its parallel-propagated orthonormal frame, the axial geodesic deviation is $\ddot{L}=-R_{n0n0}L$. For a transverse gravitational wave, $R_{n0n0}=-ϵ{\ddot{h}}_{nn}/2$, so a wave incident after the beads are at rest changes the measured separation by $ϵh_{nn}{L}_0/2$ to first order. Transverse reaction forces do not supply an axial restoring force. Alignment with the propagation direction gives no first-order length change.

For coset permutation modules, Mackey restriction formula and Frobenius reciprocity identify the homomorphism space with a free module having one orbit-sum basis vector per double coset. The same basis works over the residue field, so reduction of homomorphisms is surjective. Applied to endomorphisms of a permutation module, idempotent refinement theorem lifts each modular direct summand to a trivial source lattice. Lifting inverse isomorphisms between reductions shows uniqueness among trivial source lattices.

If $D$ is a normal p-subgroup of $N$, any indecomposable projective module of $k(N/D)$, inflated to $N$, is a summand of $k[N/D]={\mathrm{Ind}}_D^{N}k$. It has trivial source and vertex $D$. Its Brauer quotient of a module at $D$ is the entire module, since $D$ acts trivially and all proper-subgroup relative traces multiply by an index divisible by $p$. This prevents its vertex from being smaller than $D$.

A triple-point cell turns phase coexistence into a reproducible reference for thermometer calibration. The NIST water-cell demonstration uses a sealed glass vessel containing water of specified isotopic composition and an open central thermometer well. Cooling this well with solid carbon dioxide and alcohol freezes an ice mantle while leaving liquid water and its vapor. The coolant is outside the sealed sample; the sample's vapor space does not communicate with room air.^[ref]

In the actual NIST cell-comparison experiment, newly frozen mantles were compared with an aged reference mantle in a controlled maintenance bath. A warm aluminium rod melted a thin inner layer around the well, allowing the mantle to rotate freely and establishing an ice–water interface close to the sensor. A standard platinum resistance thermometer and a resistance bridge measured the electrical resistance ratio to a calibrated, temperature-controlled reference resistor. Readings at two currents permitted extrapolation to zero electrical power, correcting self-heating; measured immersion depths supplied hydrostatic pressure corrections. Comparing preparations over time tested stabilization rather than assuming freshly frozen ice was ready.^[ref]

The inference is a reproducible fixed-point response for the thermometer, not a direct reading of three chemical potentials. The reference cell anchors the comparison; a water point alone does not determine a thermometer's response at every other temperature. The ideal $F=0$ argument excludes impurity and isotope variation, finite interface effects and unintended heat transfer; real cells also require adequate immersion and stable electrical references. Repeatability is evidence that these effects are controlled, not a reason to omit their uncertainty.

Since 20 May 2019 the kelvin has been defined through a fixed numerical value of the Boltzmann constant. The thermodynamic temperature of the water triple point is consequently a measured quantity with uncertainty, rather than an exact value imposed by the kelvin definition. Its continued use as an assigned reference on the International Temperature Scale of 1990 is a separate convention.^[ref][ref]

Write triangle-group generators as products $a=s_1{s}_2$, $b=s_2{s}_3$ of adjacent side reflections. Conjugation by $s_2$ sends both to their inverses. A sheet relabelling satisfying the displayed two permutation identities therefore lifts this base reflection to an orientation-reversing isometry between the two hyperbolic covers. Thus nonconjugate subgroups in the orientation-preserving finite group can still give isometric covers.

The full transformation monoid consists of every self-map of $X$, with function composition as its binary operation. For $X=\{0,1\}$, there are four functions: the identity function $i$, the swap $s(0)=1,s(1)=0$, and the constant functions $c_0(x)=0,c_1(x)=1$. Their composition order matters: $c_0\circ s=c_0$, whereas $s\circ c_0=c_1$. Thus associativity does not imply commutativity.

Only $i$ and $s$ are bijections, and their inverse elements are themselves. They form a two-element group within this monoid. The constant functions instead satisfy $c_0\circ c_0=c_0$ and $c_1\circ c_1=c_1$, so they are idempotent elements of a semigroup. Such irreversible transformations explain why the monoid setting is useful even when a group would be too restrictive.

For a nontrivial finite-dimensional anti-Hermitian Lie algebra representation of a compact Lie algebra that is simple, its Trace form of a Lie algebra representation is a positive multiple of the Killing form. Define $T$ by $H(X,Y)=\kappa (TX,Y)$. Invariance makes $T$ commute with the Adjoint representation of a Lie algebra. Symmetry makes $T$ self-adjoint for $-\kappa$, so its eigenspaces are ideals. Simplicity implies $T=\mu I$. The kernel of the representation is an ideal and hence vanishes; therefore $H(X,X)=-\mathrm{Tr}(d(X{)}^{†}d(X))<0$ for nonzero $X$, proving $\mu >0$.

// Target: lie-theory.bigb

The cubic trace tensor of a Lie algebra representation is cyclic by the cyclic property of the trace. It need not be fully antisymmetric. Its antisymmetric part in the first two arguments is fixed by the Trace form of a Lie algebra representation:

// Target: lie-theory.bigb

For a finite-dimensional complex representation of a solvable Lie algebra, the Lie theorem gives a basis in which all representing matrices are upper triangular. Commutators are strictly upper triangular, and their products with upper triangular matrices have zero trace. Thus the derived algebra lies in the radical of a bilinear form. In particular, a nonabelian solvable Lie algebra can carry an invariant bilinear form on a Lie algebra that is nondegenerate without that form coming from a finite-dimensional representation trace.

// Target: semisimple-lie-algebra.bigb

If a càdlàg function has finite variation on compact intervals, its total-variation process is finite, nondecreasing, and càdlàg. Left limits follow from monotonicity. For right continuity, approximate the variation on $[t,s]$ by one finite partition. Changing its first endpoint from $t$ to $u\downarrow t$ changes the sum by at most $|X_u-X_t|$, which tends to zero. Additivity of variation then bounds $V(u)-V(t)$ by an arbitrarily small error. Its jump is the absolute jump of $X$. Without locally finite variation, right continuity can fail even as an extended-valued assertion: $X_t=t\sin (1/t)$ with $X_0=0$ has infinite variation on every interval starting at zero.

For a rotating conducting sphere, a geostrophic azimuthal velocity $U=rV$ couples to the cylinder-integrated magnetic stress $T=\int B_r{B}_{\varphi }dz$. Put $H=\int B_r^{2}dz$ and let $2z_b$ be the cylinder height. With an insulating boundary and no flow through it, cylindrical averaging removes the Coriolis force and gives $U_t=(2z_b{\mu }_0\rho r^2{)}^{-1}{\partial }_r(r^{2}T)$, $T_t=Hr{\partial }_r(U/r)$. The local squared wave speed is $H/(2z_b{\mu }_0\rho )$, the cylinder average of $B_r^2/({\mu }_0\rho )$.

For uniform-density incompressible flow with $u_{\varphi }=f(R,z+vt)+g(R,z-vt)$, $B_{\varphi }/\sqrt{{\mu }_0\rho }=f-g$ and $B_z/\sqrt{{\mu }_0\rho }=v(R)$, the azimuthal magnetohydrodynamic momentum equation and ideal magnetohydrodynamic induction equation hold automatically. Let $\Pi =p/\rho +\Phi +[(f-g{)}^2+v^2]/2$. The remaining equations require ${\Pi }_z=0$ and ${\Pi }_R=4fg/R$, so ${\partial }_z(fg)=0$ is necessary and locally sufficient. A single traveling profile is unrestricted; two nonzero profiles on an unrestricted axial domain must be opposite exponentials or independent of their traveling arguments. Bounded profiles of both signs therefore reduce to steady solutions.

For an elliptic curve $E:y^2=x^3+Ax+B$ in characteristic different from two and three, division polynomials are elements of its coordinate ring encoding multiplication and torsion points of an elliptic curve. The initial terms includeThe recurrences for higher $n$ yieldFor odd $n$, ${\psi }_n$ is a polynomial in $x$; for even $n$, it has a factor $2y$, and ${\psi }_n/(2y)$ is a polynomial in $x$. Thus the phrase “roots of the division polynomial” must specify whether it means points on the curve or roots in the $x$-line. When $n$ is prime to the characteristic, its zero points are exactly the nonidentity $n$-torsion points of an elliptic curve; for odd $n$, the two opposite points share an $x$-root.^[ref]

For the cyclic quotient surface singularity $\frac{1}{r}(1,q)$, expand $r/q=[b_1,\dots ,b_s{]}^{-}$ with $b_i\ge 2$. In the quotient lattice, start with $v_0=(0,1)$, $v_1=(1,q)/r$, and set $v_{i+1}=b_i{v}_i-v_{i-1}$. The last vector is $(1,0)$, each consecutive pair is a lattice basis, and subdivision by these rays is a resolution. Its exceptional chain has self-intersections $-b_i$. None is a minus-one curve, so the resolution is minimal. All coefficients are two exactly for $q=r-1$.

// Target: geometry-and-topology.bigb

First make a toric fan simplicial by compatible rational subdivisions. In a nonregular simplicial cone, choose a nonregular face minimal under inclusion. Its half-open fundamental parallelepiped contains a nonzero point of the toric lattice $w={\sum }_i{\alpha }_i{v}_i$, with every $0<{\alpha }_i<1$: a zero coefficient would put it in a regular proper face and force all coefficients to be integers. A star subdivision at the primitive point on this ray replaces a generator by $w$ and multiplies the affected lattice determinant by a number strictly between zero and one. The maximum multiplicity and the number of maximal cones realizing it decrease lexicographically, giving a finite toric resolution of singularities.

For this toric cone, one smooth fan subdivision has maximal cones $abc,acd$ and no exceptional divisors. Another has $abd,bce,cde,dbe$, with $e=a+c=(b+c+d)/2$. Its only exceptional divisor is a projective plane, since the quotient fan at $e$ has the three rays $\bar{b},\bar{c},\bar{d}$ with sum zero and unimodular neighboring cones. Both subdivisions admit no proper smooth coarsening. Their common smooth refinement is obtained from the first by adding $e$ in $ac$ and then $w=a+e$ in $ae$, and from the second by adding $w=b+d$ in $bd$. Each step is a toric blowup along an orbit closure.

For an ample invariant divisor $D$ and the blowup $\pi$ of a torus-fixed smooth point in dimension at least two, $c{\pi }^{\ast }D-E$ is ample for sufficiently large integers $c$. In the subdivided cone, its local characters are $c{m}_{\sigma }+u_i$, where $u_i$ is dual to the omitted ray generator. The new strict inequalities have gap one on the omitted ray and grow linearly with $c$ elsewhere, proving the claim by the toric ampleness criterion.

On the Hardy space of the circle, a continuous symbol gives a Fredholm Toeplitz operator exactly when it never vanishes. Nonvanishing gives a parametrix $T_{1/f}$ modulo compact semicommutators. Write $f(z)=z^n{e}^{h(z)}$, where $n$ is its counterclockwise winding number and $h$ is continuous. Homotopy through nonvanishing symbols and local constancy of the Fredholm index reduce the index to that of the shift $T_{z^n}$, namely $-n$. If $f$ vanishes at a point, normalized Hardy reproducing kernels concentrating there are weakly null unit vectors on which $T_f$ tends to zero, contradicting a Fredholm parametrix.

Split a smooth symbol into its strictly negative and strictly positive Fourier modes. Their Toeplitz operators $A,B$ exponentiate exactly to the Toeplitz operators of the corresponding exponentials, and $T(e^f)T(e^{-f})=e^A{e}^B{e}^{-A}{e}^{-B}$. The trace of $[A,B]$ is ${\sum }_{n>0}n{f}_n{f}_{-n}$ because the shift relation gives $\mathrm{Tr}[(S^{\ast }{)}^m,S^n]=n{\delta }_{mn}$. Smoothness makes the series converge in trace norm, so the Fredholm determinant of an exponential commutator applies.

In an $L$-round iterated prisoner's dilemma, Tit for tat against always defect gets $S$ once and $P$ thereafter; the defector gets $T$ once and $P$ thereafter. Their self-payoffs are $R,P$. At introduced frequency $ϵ$, subtracting mixed-population payoffs gives $ϵ[(R-P)-(T+S-2P)/L]-(P-S)/L$. The displayed positive threshold applies when its denominator is positive and the threshold is less than one. Taking $L\to \infty$ first removes this finite entry cost, so the horizon and rare-mutant limits cannot be casually interchanged.