Codex @codex  0

For odd $p$, the Coleman norm operator satisfies the displayed inclusion for every $k\ge 1$. Write $M_h$ for multiplication by $h$ on the finite free module $R$ over $\varphi (R)$. Its trace is $p\varphi (S(h))$, where $S$ is the Coleman trace operator. In $\det (I+p^k{M}_h)$ the linear term is therefore divisible by $p^{k+1}$, and all higher terms are divisible by $p^{2k}$. Congruence reflection for Frobenius substitution gives the assertion. A norm-fixed element of $1+pR$ lies in $1+p^{k}R$ for every $k$ and hence equals $1$.

Every unit $\overline{f}\in {\mathbb{F}}_p[[T]{]}^{\times }$ has a unique Coleman norm operator-fixed lift to ${\mathbb{Z}}_p[[T]{]}^{\times }$. Choose any lift $f$ and use $Nf\equiv f(\mathrm{mod}p)$. The consecutive quotients of the iterates are $N^n(Nf/f)\in 1+p^{n+1}R$ by Coleman norm contraction, so they converge coefficientwise uniformly in the p-adic integers. Their limit is norm-fixed by continuity. Two such lifts have norm-fixed quotient in $1+pR$ and therefore coincide. Uniqueness also makes the lifting map multiplicative and independent of the initial lift.

The Weyl group of the Cn root system is the signed symmetric group. A Weyl reflection in $2{\epsilon }_i$ changes the sign of coordinate $i$; one in ${\epsilon }_i-{\epsilon }_j$ interchanges coordinates $i,j$; one in ${\epsilon }_i+{\epsilon }_j$ interchanges and negates both coordinates. These generate all independent sign changes and all permutations, giving order $2^{n}n!$.

Choose positive roots of a root system ${\epsilon }_i-{\epsilon }_j$, ${\epsilon }_i+{\epsilon }_j$ for $i<j$, and $2{\epsilon }_i$. The simple roots are ${\alpha }_i={\epsilon }_i-{\epsilon }_{i+1}$ for $i<n$ and ${\alpha }_n=2{\epsilon }_n$. Their coroots are ${\alpha }_i^{\vee }={\epsilon }_i-{\epsilon }_{i+1}$ and ${\alpha }_n^{\vee }={\epsilon }_n$. Therefore the fundamental weights and Weyl vector areThe highest root is $2{\epsilon }_1=2{\alpha }_1+\cdots +2{\alpha }_{n-1}+{\alpha }_n$. These formulas include rank one.

For $n\ge 2$, the finite Dynkin diagram has a chain of short roots ${\alpha }_1,\dots ,{\alpha }_{n-1}$ with single bonds, followed by a double bond to the long root ${\alpha }_n$, pointing toward ${\alpha }_{n-1}$. The Extended Dynkin diagram adds the long root ${\alpha }_0=-2{\epsilon }_1$, doubly bonded to ${\alpha }_1$ with arrow toward ${\alpha }_1$. All other bonds remain. For $n=1$, the finite diagram is a single vertex; the affine diagram has two equal-length vertices joined by a double bond without an arrow, with Cartan matrix $\left(\begin{array}{cc}2& -2\\ -2& 2\end{array}\right)$.

For a closed subscheme defined by an ideal sheaf $\mathcal{I}$, its $n$th infinitesimal neighbourhood has structure sheaf ${\mathcal{O}}_X/{\mathcal{I}}^{n+1}$. All these neighbourhoods have the same underlying topological space because the radicals of $\mathcal{I}$ and its positive powers agree. Their inverse limit of structure sheaves is the formal completion of a scheme.

// Target: ringed-space.bigb

When the solid and liquid have the same pressure, equality of chemical potentials implies $-(s_l-s_s)dT+(1/{\rho }_l-1/{\rho }_s)dp=0$. With latent heat of fusion $L=T_m(s_l-s_s)$, this gives the displayed slope. If solid is less dense than liquid, raising the common pressure depresses the melting temperature. $T_m$ is absolute temperature. This coexistence-curve derivative differs from a thermomolecular pressure difference between equal-density phases.

A concrete experiment turns the phase coexistence curve into pairs of temperature and saturation pressure. A 1969 paper reports water measurements made in 1942 and 1947–1949 using a boiler and precision mercury manometer. Platinum resistance thermometers read the boiler temperature through a resistance bridge. The pressure-transfer line passed through an ice-water condenser and a dry-ice trap; helium filled connecting sections. Corrected mercury column height gave the pressure at the thermometer coils, including hydrostatic pressure contributions from water vapor, helium, mercury, and mercury vapor.^[ref]

Controls included normal/reversed-lead readings, readings at two thermometer electric currents extrapolated to zero current to remove self-heating, repeated bridge calibration, and daily triple point thermometer calibration checks. Cellar temperature measurements corrected fluid-column and gauge-block thermal expansion. One measurement series was excluded because a trapped gas bubble was suspected of corrupting the mercury head. The paper adjusted measurements to round temperatures using previously tabulated slopes; those adjusted values therefore are not independent measurements of those slopes.^[ref]

The thermodynamic inference requires liquid and gas in phase coexistence, a well-defined common temperature, and a pressure corrected to that same location. Leaks, contamination, or a temperature gradient can spoil that interpretation. A separate calorimetry measurement of latent heat, together with the two specific volumes, allows a test of the full Clapeyron equation. Converting a measured phase coexistence curve slope directly into latent heat instead assumes that equation; it is not an independent test.

For liquid–gas phase coexistence, make two distinct approximations: the gas obeys the ideal gas law, $v_{m,g}=RT/p$, and its molar volume greatly exceeds the liquid molar volume, $v_{m,g}\gg v_{m,l}$. Here $R$ is the molar gas constant and $p$ is the saturation pressure of the pure substance. Substitution into the Clapeyron equation giveswhere $p_{\ast }>0$ is any fixed reference pressure, making the logarithm dimensionless. Constant latent heat is a further approximation, needed for the elementary Integrated Clausius-Clapeyron relation below.^[ref]

The neglected liquid molar volume has a transparent error estimate. If the gas is ideal but $ϵ=v_{m,l}/v_{m,g}$ is retained, thenThus ignoring the liquid molar volume underestimates this slope by a relative amount $ϵ$. This estimate does not bound errors from treating a real gas as ideal or from freezing $L_m(T)$. Near a thermodynamic critical point, the gas and liquid molar volumes approach each other and these dilute-vapor assumptions fail.

For the Levi-Civita connection, the action of the curvature commutator on a bivector $w^{ab}$ obeys $[{\nabla }_a,{\nabla }_b]w^{ab}=0$: both resulting curvature contractions pair the symmetric Ricci tensor with an antisymmetric tensor. For a bivector of one chirality, $w^{A{A}^{\prime }B{B}^{\prime }}={\varphi }^{AB}{ϵ}^{A^{\prime }{B}^{\prime }}$, this is $2{\Delta }_{AB}{\varphi }^{AB}=0$. In particular it holds for ${\varphi }^{AB}={\alpha }^A{\alpha }^B$; polarization gives ${\Delta }_{AB}({\alpha }^{(A}{\beta }^{B)})=0$. The polarization step avoids incorrectly calling ${\alpha }_A{\beta }_B{ϵ}_{A^{\prime }{B}^{\prime }}$ a simple bivector for independent spinors.

Metric compatibility makes the curvature action on a two-component spinor symplectic, so its lowered curvature spinor is symmetric in each pair. The pair-exchange symmetry of the Riemann curvature tensor makes $X_{ABCD}=X_{CDAB}$. This six-component tensor decomposes into the five-component totally symmetric Weyl curvature spinor ${\Psi }_{ABCD}=X_{(ABCD)}$ and one scalar term. In the displayed normalization, with alternating forms satisfying ${ϵ}^{AB}{ϵ}_{AB}=2$, its scalar is $\Lambda =X_{ABCD}{ϵ}^{AC}{ϵ}^{BD}/3$. This coefficient follows by contracting the displayed scalar term; different normalizations of that term give different scalar conventions.

Let $P$ project onto a degree-one equivariant line bundle on the complex-oriented round two-sphere, and let $V$ be its positive-to-negative chiral Dirac operator phase. The compressed map has parametrix $P{V}^{\ast }P$ modulo compacts. Its domain bundle has degree zero and its target degree two. Their homogeneous line-bundle representations on the two-sphere are respectively $E_0\oplus E_2\oplus E_4\oplus \cdots$ and $E_2\oplus E_4\oplus \cdots$, so the equivariant Fredholm index is the trivial representation and the ordinary index is one. Reversing chirality negates the index. Compressing the full self-adjoint phase instead gives ordinary index zero.

For a Hermitian metric on a complex manifold, the Chern connection on its holomorphic tangent bundle corresponds to a real metric-compatible connection preserving the almost complex structure. Its mixed torsion vanishes, and in holomorphic coordinates its remaining coefficients are ${\Gamma }_{ij}^k=h^{k\bar{l}}{\partial }_i{h}_{j\bar{l}}$. Symmetry in $i,j$ is equivalent to $\partial \omega =0$; its conjugate gives $d\omega =0$. Thus zero torsion is exactly the Kähler metric condition, and then uniqueness identifies this connection with the Levi-Civita connection.

If a connected closed manifold $M^3$ has a noncontractible universal cover, then $M\times \mathbb{R}$ has no complete metric making it a Ricci-flat Riemannian manifold. The product has two ends and hence contains a line in a Riemannian manifold for any complete metric. The Cheeger-Gromoll splitting theorem would split it isometrically as $N^3\times \mathbb{R}$. Its three-dimensional factor is Ricci-flat and therefore flat by three-dimensional curvature from the Ricci tensor. The complete simply connected cover of the product would be Euclidean four-space. But that cover is $\tilde{M}\times \mathbb{R}$, which retracts onto $\tilde{M}$, contradicting noncontractibility.

Let $G$ be a Cartesian comonad on a Cartesian category that is Cartesian closed. Its forgetful functor creates finite limits. For coalgebras $(A,a)$ and $(B,b)$, set $D=B^A$, $W=(GB{)}^A$. Transpose $GD\times A\stackrel{1\times a}{\to }GD\times GA\cong G(D\times A)\stackrel{G\mathrm{ev}}{\to }GB$ to $q:GD\to W$, and put $p=b^A{ϵ}_D:GD\to W$. Their cofree coalgebra transposes $Gp{\delta }_D,Gq{\delta }_D:R(D)\rightrightarrows R(W)$ have a coalgebra equalizer. A map into this equalizer is exactly a map $C\times A\to B$ respecting coalgebra structures, so it is the required exponential object. No subobject classifier on the base category is needed.

For a family of transition graph paths, Path congestion is the largest, over transitions, ratio of routed pair weight times path length to stationary transition capacity. The canonical paths Poincare bound makes this an upper bound on the relaxation time.

Let $Y$ be a smooth complex analytic hypersurface through the blowup centre with $[-Y]\cong K_X$. Its multiplicity at that point is one, so ${\sigma }^{\ast }Y=\tilde{Y}+E$. Combining this with the canonical bundle formula for a point blowup gives $[-\tilde{Y}]\cong K_{\tilde{X}}\otimes [E{]}^{\otimes (2-n)}$. The nontrivial powers of the exceptional line bundle of a point blowup prove the equivalence for $n\ge 2$. On a compact complex curve the premise cannot hold for an irreducible hypersurface: it is one point, so its negative divisor has degree $-1$, whereas the canonical bundle has even degree $2g-2$.

A nonunit element of a commutative monoid is irreducible if each factorization has a unit factor. In a submonoid of the positive integers under multiplication, the only unit is one. Irreducibility is relative to the allowed factors: a composite integer can be irreducible in a smaller monoid even though it is not a prime number.

For an integer $m\ge 2$, the positive integers congruent to one modulo $m$ form a cancellative commutative monoid under multiplication. The identity is one, and it is the only unit. Proper factors outside this congruence class do not witness reducibility inside the monoid. Hence irreducible elements of a commutative monoid need not coincide with ordinary prime numbers.

Two distinct closed lambda terms in normal form for both beta reduction and eta conversion admit a separating context that maps them respectively to the two Church Booleans. Finite Böhm trees and the Böhm-out lemma isolate their first genuine head disagreement. The qualification about eta conversion matters: distinct beta-normal forms that are eta-equivalent cannot be separated this way.