Let $f$ be order-preserving and inflationary on a directed-complete partial order. Define $xRy$ when every set containing $x$ and closed under $f$ and directed joins contains $y$. This is a partial order refining $\le$. The order-preserving maps $h$ satisfying $xRh(x)$ form a composition-closed family $H$. The values $Hx$ are directed, since $h(k(x))$ bounds $h(x)$ and $k(x)$. Their least upper bound $h_0(x)$ lies in every such closed set containing $x$, so $h_0\in H$. Then $f\circ h_0\in H$ gives $f(h_0(x))\le h_0(x)$, while inflationarity gives the reverse inequality. If $p=f(p)$ and $x\le p$, the lower set $\{y:y\le p\}$ is closed, proving $h_0(x)\le p$. Thus $h_0(x)$ is the least fixed point of $f$ above $x$.

A directed-complete partial order with a least element. Here directed suprema concern nonempty directed sets. Morphisms are Scott continuous maps and need not preserve the least element. Requiring strictness would change the category and its cartesian-closure argument.

With the fixed algebra-filtration convention in dimension of a filtered module, the multiplicity is factorial times the common leading coefficient of the cumulative Hilbert function for $M\ne 0$, and is zero for $M=0$. It is a positive integer with a standard graded algebra normalization, and can be a positive rational number for a weighted grading. Fixed shifts between good filtrations of a module preserve its leading Hilbert coefficient. Rescaling the algebra filtration itself can change multiplicity, so its normalization must be specified.

A commutative differential graded algebra has multiplication $ab=(-1{)}^{|a||b|}ba$ and a degree-one differential with $d^2=0$ and $d(ab)=d(a)b+(-1{)}^{|a|}ad(b)$. Over $\mathbb{Q}$, its free algebra on graded generators is polynomial on even-degree generators and exterior on odd-degree generators. Rational rational polynomial differential forms on a space supply such an algebra, modelled up to quasi-isomorphism by a Sullivan minimal model.

A reduction relation has the Church-Rosser property when its equivalence relation generated by forward and backward steps satisfies $b{\leftrightarrow }^{\ast }c\Rightarrow \exists d(b{\to }^{\ast }d{\leftarrow }^{\ast }c)$. It is equivalent to confluence of ${\to }^{\ast }$. The Church-Rosser theorem establishes it for beta reduction.

Here $N_r$ counts $k$-subsets $i_1<\cdots <i_k$ with ${\sum }_a(i_a-a)=r$, equivalently partitions of an integer $r$ contained in a $k$ by $(n-k)$ rectangle. The diagonal trace Morse function on a complex Grassmannian has only even Morse indices. Its Morse chain complex therefore has zero differentials, proving the displayed torsion-free integral homology. For $\mathrm{Gr}(2,4)$ the even-degree ranks are $(1,1,2,1,1)$ in degrees $0,2,4,6,8$.

For constant buoyancy frequency, density forcing $r_0{e}^{-\mu z}\cos kx\cos {\omega }_{0}t$ gives two frequencies $\omega =\pm {\omega }_0$. The quasi-geostrophic streamfunction satisfies ${\lambda }_{\omega }^2=(N^2/f_0^2)(k^2+\beta k/\omega )$ and $C_{\omega }=ig\mu r_0/(2{\rho }_0{f}_0\omega )$. With zero streamfunction at the bottom, a positive ${\lambda }_{\omega }^2$ selects the decaying solution $C_{\omega }(e^{-\mu z}-e^{-{\lambda }_{\omega }z})/({\mu }^2-{\lambda }_{\omega }^2)$. For $\beta ,k>0$, the westward branch propagates vertically when ${\omega }_0<\beta /k$; a radiation condition selects positive vertical group velocity, not merely bounded amplitude. Coincident exponential exponents are handled by the finite limit $-C_{\omega }z{e}^{-\mu z}/(2\mu )$.

For a proper map of locally compact spaces and a constructible complex $E$, the derived stalk satisfies $(R{f}_{\ast }E{)}_y\cong R\Gamma (f^{-1}(y),E{|}_{f^{-1}(y)})$. In particular $(R^i{f}_{\ast }\mathbb{Q}{)}_y\cong H^i(f^{-1}(y);\mathbb{Q})$. Properness is what prevents fibre cohomology from missing nearby sections.

For fixed $r$ and $f\in C^r[0,1]$, uniform convergence of normalized finite differences to $f^{(r)}$, uniform continuity of $f^{(r)}$, and $(n{)}_r/n^r\to 1$ imply $g_{n,r}\to f^{(r)}$ uniformly. Positivity and partition of unity make Bernstein polynomials contractions in the supremum norm. Consequently $\Vert B_{n-r}{g}_{n,r}-f^{(r)}{\Vert }_{\infty }\le \Vert g_{n,r}-f^{(r)}{\Vert }_{\infty }+\Vert B_{n-r}{f}^{(r)}-f^{(r)}{\Vert }_{\infty }\to 0$. The differentiability assumption is essential; continuity alone does not define the target derivative.

Smooth convergence on a compact interval requires uniform convergence of the functions and of their derivatives of every order separately, as measured by the derivative seminorms. The need for several scales is visible inFor $n\ge 2$, $p_0(f_n)=1/n$ because $t=\pi /(2n)$ lies in $[0,1]$. The derivative has supremum norm one for every $n$, already at $t=0$. Thus $f_n$ converges uniformly to zero, but fails the test for smooth convergence to zero. Andreas Buttenschoen, Applied Functional Analysis, spaces of continuous and differentiable functions discusses this example and the family of smoothness measurements.

More generally, for any fixed $K\ge 0$, $h_n(t)=\sin (nt)/n^{K+1}$ has $p_K(h_n)\le 1/n$ but $p_{K+1}(h_n)=1$ for $n\ge 2$. If a norm $N$ induced the smooth topology, its continuity would give $N(f)\le C{p}_K(f)$ for some finite $K$, while continuity of $p_{K+1}$ for $N$ would give $p_{K+1}(f)\le C^{\prime }N(f)$. Substituting $h_n$ contradicts these two bounds. Smooth convergence therefore cannot be reduced to one norm. Its beauty is that every finite order can be controlled without fixing a final order in advance.

For bound Kepler orbits with gravitational parameter $\mu$, integrating the Delaunay variables over $0<G<L$, $-G<H<G$ and their three full angle periods gives the displayed population. A globally defined thermal eccentricity distribution requires this integral to converge, or an explicit finite action range. Merely having finite $F(E)$ at every energy does not suffice: $F=1$ yields an infinite integral at large $L$. At each fixed energy, or on every finite normalized action band, the cumulative eccentricity fraction remains $e^2$.

In the completed natural Brownian filtration, suppose $d{S}_t/S_t={\mu }_{t}dt+{\sigma }_{t}d{W}_t$ with ${\sigma }_t\ne 0$, and let $d{\zeta }_t=-{\zeta }_t(r_{t}dt+{\kappa }_{t}d{W}_t)$, ${\kappa }_t=({\mu }_t-r_t)/{\sigma }_t$. For an affordable nonnegative terminal claim $X$, represent the conditional-expectation martingale $N_t=\mathbb{E}[{\zeta }_{T}X\mid {\mathcal{F}}_t]$ as $d{N}_t=h_{t}d{W}_t$. Then $w_t=N_t/{\zeta }_t$ and the displayed dollar amount in the stock replicate $X$. The Itô product rule gives $d(\zeta w)=\zeta (\sigma \theta -\kappa w)dW$, proving the formula; share holdings are ${\theta }_t/S_t$.

In a complete decomposing sphere system for a knot, a disk inside a prime-factor ball that meets the knotted arc once cuts its factor into a connected sum of knots. One side must be an unknot. Replacing the ball by the side carrying the prime factor therefore preserves the unordered factor list. A disk missing the arc cuts off an empty ball and gives the same conclusion. After a small transverse perturbation, such a replacement removes an innermost intersection circle with another sphere system. If a chosen innermost disk contains both marked points, another innermost disk, or its complementary disk when it is the only intersection, contains neither. Iterating produces a prime-factor ball disjoint from both systems. The replacement changes only the location of a joining arc in the intervening punctured ball; capping the other boundaries still recovers the same oriented factors.

Let a uniformly random bit $A$ be copied into a three-state classical register $B$ with probability $\epsilon$, and otherwise replaced in $B$ by a flagged erasure independent of $A$. Then $I(A:B)=\epsilon$ bits. Projecting $B$ onto its two nonerased states and conditioning on success gives a perfectly correlated bit pair with mutual information one. The postselection branch is completely positive and trace decreasing, but renormalization after conditioning is not a deterministic quantum channel. This does not contradict data processing for quantum mutual information, which applies to trace-preserving operations or an unconditioned instrument including its outcome flag.

For canonical Kähler potential, set $a=|{\varphi }_0{|}^2$, $b=|{\varphi }_{+}{|}^2$, $c=|{\varphi }_{-}{|}^2$. The supplied supergravity F-term potential is $|\lambda {|}^2{e}^{{\kappa }^2(a+b+c)}[ab+ac+bc+3{\kappa }^{2}abc+{\kappa }^{4}abc(a+b+c)]$. Adding $g(b-c-\zeta {)}^2$ with $g>0$, $\lambda \ne 0$ and $\zeta >0$ gives a supersymmetric vacuum at $a=c=0$, $b=\zeta$. The branch $b=c=0$ has positive energy $g{\zeta }^2$, D-term breaking and a flat direction of a scalar potential in ${\varphi }_0$. Its charged squared masses are $|\lambda {|}^{2}a{e}^{{\kappa }^{2}a}\mp 2g\zeta$, so it is a local minimum valley only when both are positive. For negative $\zeta$, exchange the charged fields. These are branches rather than two isolated minima; a conventional constant Fayet–Iliopoulos term additionally needs a consistent supergravity gauge completion.

For one charged chiral superfield, canonical kinetic terms, no nonconstant gauge-invariant superpotential, and $g>0$, the sign of $q\xi$ selects the vacuum. If $q\xi >0$, the minimum has $\varphi =0$, positive vacuum energy, unbroken internal gauge symmetry and a gaugino Goldstino; the scalar squared mass is $gq\xi$ while its chiral fermion is massless. If $q\xi <0$, a scalar condensate cancels the auxiliary field, preserving supersymmetry while Higgsing the gauge group. These are classical statements; a consistent quantum charged spectrum must cancel its gauge anomalies.

Fix a finite excluded set $S$ containing two, and let $W_S$ contain the roots of unity of orders coprime to every prime in $S$. A root-of-unity Euler system is a nonzero algebraic-valued function $\Phi$ on $W_S$ that is even and Galois equivariant, satisfies ${\prod }_{{\rho }^q=1}\Phi (\rho \zeta )=\Phi ({\zeta }^q)$ for $q\notin S$, and satisfies $\Phi (\rho \zeta )\equiv \Phi (\zeta )(\mathrm{mod}\mathfrak{q})$ when the order of $\zeta$ is prime to $q$, for every prime $\mathfrak{q}$ over $q$. Values are required to be locally integral units where the congruence is used. These distribution and congruence properties provide auxiliary cyclotomic extensions, beyond a single norm-compatible p-power sequence.

The cyclic quotient surface singularity $\frac{1}{r}(q,1)$ is the only singular point of a complete four-ray toric surface exactly under the displayed condition. Normalize the three smooth cones to the consecutive rays $(a,-1),(1,0),(0,1),(-1,b)$. The singular cone has determinant $1-ab=r$, so $ab=1-r$. In the basis $(-1,b),(0,-1)$, its other generator is $-a(-1,b)+r(0,-1)$, giving quotient parameter $a\mathrm{mod}r$. Thus $a=\pm s$ for a positive divisor $s$ of $r-1$. Conversely these rays construct the surface. Exchanging the two axes replaces $q$ by $q^{-1}$; replacing a factor by its complementary factor preserves the criterion under this equivalence.

Left multiplication of a permutation by a transposition $(ab)$ swaps the two target labels $a,b$. If they occur in different cycles, $(a{x}_1\dots x_p)(b{y}_1\dots y_q)$ becomes the single cycle $(a{x}_1\dots x_{p}b{y}_1\dots y_q)$. If they occur in the same cycle, that formula reverses and splits the cycle in two. All other cycles are unchanged, so the total number of cycles, including fixed points, decreases or increases by exactly one. Empty intermediate lists yield one-cycles and require no exceptional case.

Squaring an odd-length permutation cycle retains its length, whereas an even-length cycle splits into two cycles of half its length. Thus if any even cycle is present, the number of cycles strictly increases, so the cycle type changes. Since conjugate permutations in a symmetric group have the same cycle type, a permutation is conjugate to its square precisely when all its cycle lengths are odd, equivalently when its order of a group element is odd.