A tree is splitting if every node has two incompatible extensions. This property is additional to height and level-size bounds in a kappa-tree. A cofinal branch through a splitting tree of regular uncountable height produces an equally large antichain by taking off-branch extensions and proceeding beyond their divergence heights. Thus a splitting ${\omega }_1$-tree with countable antichains has no uncountable chains; the nonsplitting chain ${\omega }_1$ itself is a counterexample to omitting splitting.

For a finitely generated field extension $L/k$ in characteristic zero, a tuple $t_i$ is algebraically independent exactly when its differentials $d{t}_i$ are linearly independent in the Kähler differentials. A relation of minimum total degree gives a nontrivial differential relation; an independent tuple extends to a transcendence basis. In positive characteristic, derivatives of nonconstant relations can vanish identically.

A semisimple Lie algebra over a field of characteristic zero is split if it has a Cartan subalgebra whose adjoint action is simultaneously diagonalizable over that field. Its root-space decomposition then needs no extension of scalars. The sl2R Lie algebra is split, with the real diagonal Cartan subalgebra. The real special unitary Lie algebra $\mathfrak{su}(2)$ is not: its adjoint operators are skew-adjoint for the positive definite inner product obtained by negating its Killing form. Thus a split adjoint operator would have to be zero.

A continuous semimartingale is a continuous adapted process admitting a decomposition $X=X_0+N+V$, with $N$ a zero-starting continuous local martingale and $V$ continuous adapted finite variation. The decomposition is unique because a continuous finite-variation local martingale starting at zero vanishes. Its quadratic variation is $[X]=[N]$.

An element of a semigroup is idempotent when multiplying it by itself leaves it unchanged. Every element of a left-zero or right-zero semigroup is idempotent, whereas a group has only its identity as an idempotent. The Ellis–Numakura lemma guarantees such an element in a nonempty compact one-sided topological semigroup. Additive idempotence on ultrafilters gives the idempotent ultrafilter relation $p+p=p$.

A cone barrier with this scaling law satisfies $⟨\nabla F(s),s⟩=-\nu$ and $\nabla F(ts)=t^{-1}\nabla F(s)$. Therefore a central path with $y=-\mu \nabla F(s)$ has primal-dual gap $\nu \mu$. A product of orthant and Lorentz cone canonical barriers adds their parameters.

A Seifert circle is a component of the crossing-free diagram obtained by oriented smoothing at every crossing. These circles bound the disks used in the Seifert algorithm.

For a quantity on a periodic Kepler orbit, its orbit average is $⟨x⟩=P^{-1}{\int }_0^{P}x(t)dt$. Equal increments of true anomaly are not equal increments of time. Eccentric anomaly gives the weight $dt=(1-e\cos E)dE/n$, which yields $⟨\cos f⟩=-e$. Slow orbital perturbations can be averaged using elements held fixed over a single orbital period.

The zero locus of a holomorphic section of a vector bundle of a rank-$r$ bundle is regular when its derivative has rank $r$ along the locus. Its kernel is then the tangent bundle of the smooth zero set. Smoothness of the underlying set alone is insufficient: the square of a reduced defining function has the same set of zeros but zero derivative there.

For a normalized second-order equation, a point is singular when its coefficient functions are not both analytic there. If multiplying them by the first and second powers of distance to the point makes them analytic, it is a regular singular point. This classification concerns the equation; some individual solutions may nevertheless extend analytically through the point.

The homogeneous equation has linearly independent solutions $e^{rx}$ and $x{e}^{rx}$, with nonzero Wronskian $e^{2rx}$. Setting $y=e^{rx}u$ converts the forced equation to $u^{\prime \prime }=e^{-rx}f$. For resonant forcing $f=e^{rx}$, the particular response is $x^2{e}^{rx}/2$. This substitution explains the extra powers of $x$ required by repeated-root resonance.

The operator Schmidt rank of a bipartite linear operator is its shortest expansion as a sum of product linear operators. Expanding in local orthonormal operator bases gives a coefficient matrix; its matrix rank is the operator Schmidt rank, by a singular value decomposition. It is unchanged by invertible local operator-basis changes. A rank-one projector onto a bipartite pure state of Schmidt rank $d$ has operator Schmidt rank $d^2$: its expansion contains all $d^2$ independent products of local matrix units.

An irreducible scheme is one whose underlying topological space is a nonempty irreducible topological space. It can have nilpotent elements in its structure sheaf; irreducibility concerns only the topology. A reduced scheme is an integral scheme precisely when it is irreducible.

An open subscheme of a scheme $X$ is an open subset $U$ equipped with the restriction of its structure sheaf. Its inclusion is an open immersion. Every scheme is covered by affine open subschemes. Any nonempty open subscheme of an integral scheme is integral and has the same generic point and function field.

For $g=\exp (-i(x{\sigma }_1+y{\sigma }_2)/2)$, write $r=\sqrt{x^2+y^2}$ and $n=(x/r,y/r,0)$. The Pauli matrix multiplication law gives $g=\cos (r/2)I-i\sin (r/2)n\cdot \sigma$. On the circles $r=2\pi k$, the angular derivative is zero, so the two Cartesian right logarithmic derivatives are proportional to the same radial generator. They commute, giving zero gauge curvature for every scaling constant. At the origin their commutator is $-i{\sigma }_3/2$, which need not vanish.

With a mostly-plus metric signature, define $f_d=(1/i)\int d^{d}k/[(2\pi {)}^d(k^2-iϵ)((P-k{)}^2-iϵ)]$. Wick rotation and a Feynman parameter give $f_d=\Gamma (2-d/2)(4\pi {)}^{-d/2}{\int }_0^1[x(1-x)P^2{]}^{d/2-2}dx$. At nonzero Euclidean $P^2$, the Gamma function supplies the pole. Exceptional massless momentum can introduce infrared ambiguity.

For a bounded-space machine, start with a logarithmic work budget and test accepting reachability and reachability of a transition leaving the budget. Accept if acceptance is found, double the budget if an exit is reachable, and otherwise reject. If all paths use at most $Cs(n)$ space, doubling stops at $O(s(n))$. Each budget test uses the Savitch theorem recursion and the storage is reused. The method discovers a sufficient bound without computing $s(n)$.

A rotlet dipole is the derivative of a rotlet along the separation of two opposed torques. A torque-free bacterium can have this far field because its rotating flagella and counterrotating body exert spatially separated internal torque reactions. The displayed axial formula uses the observation-coordinate derivative convention and signed torque moment $D$.

For contact with $R_d/R_{\odot }=M_d/M_{\odot }$ and $R_L/a=f(M_d/M{)}^{1/3}$, Kepler's third law gives $P_0=2\pi \sqrt{R_{\odot }^3/(f^{3}G{M}_{\odot })}$. The total binary mass cancels. The linear period relation depends on that particular equilibrium radius sequence; a donor with a different stellar radius response exponent follows a different mass-period relation.

Differentiate the genus-one Hadamard factorization. Its complex difference terms converge as $O(|\rho {|}^{-2})$, and the individual real sums converge by the absolute convergence of the real xi logarithmic derivative. The remaining constant is $\Re B+\sum \Re (1/\rho )$. On the critical line the xi logarithmic derivative has zero real part, while the kernel terms cancel in pairs reflected across that line. Hence this constant vanishes and gives the displayed formula away from zeros.