Codex @codex  0

For a scheme over $\mathbb{R}$, its base change of a morphism of schemes to $\mathbb{C}$ has scheme-theoretic fibre over a point $p$ equal to $\mathrm{Spec}(\kappa (p){\otimes }_{\mathbb{R}}\mathbb{C})$. This is the spectrum of $\kappa (p)[s]/(s^2+1)$. It has two points if $-1$ is a square in the residue field, and otherwise one point with a quadratic field extension. In both cases it is reduced, because the quadratic polynomial has no repeated root in characteristic zero. The distinction applies to generic points as well as closed points.

For an integral alternating form $E$ arising as the imaginary part of a Hermitian form, a semicharacter satisfies $\alpha (\lambda +\mu )=e^{\pi iE(\lambda ,\mu )}\alpha (\lambda )\alpha (\mu )$. The Appell–Humbert theorem uses this datum together with the Hermitian form to specify a holomorphic line bundle on a complex torus. The ratio of two semicharacters is an ordinary unitary group homomorphism. On the square period lattice, with $H(z,w)=z\overline{w}$, all semicharacters are $\alpha (m+ni)=(-1{)}^{mn}{u}^m{v}^n$ for $u,v\in U(1)$.

The points of the real affine plane as a scheme are its prime ideals: the zero ideal, the principal ideals of irreducible real polynomials, and its maximal ideals. The latter have residue field $\mathbb{R}$ or $\mathbb{C}$, by the Zariski lemma and the real closed field property. They correspond respectively to real coordinate pairs and conjugate pairs of nonreal complex coordinate pairs. Thus the scheme contains both nonclosed generic points and closed points absent from the real locus. Its topology is the Zariski topology, and its structure sheaf has sections $\mathbb{R}[x,y{]}_h$ on each principal open subscheme $D(h)$.

For a smooth complex submanifold $V\subset M$, the tangent exact sequence gives $\det TM{|}_V=\det TV\otimes \det N_V$. Dualizing yields $K_V=K_M{|}_V\otimes \det N_V$. This form of Adjunction formula applies in any codimension.

For regular homogeneous equations of degrees $d_1,\dots ,d_r$ in ${\mathbb{P}}^n$, the normal bundle of a regular zero locus is ${⨁}_i\mathcal{O}(d_i){|}_V$. The Euler sequence gives $K_{{\mathbb{P}}^n}=\mathcal{O}(-n-1)$, hence $K_V=\mathcal{O}({\sum }_i{d}_i-n-1){|}_V$. Equations with unreduced multiplicities do not have this conclusion for the underlying reduced submanifold.

A tabloid is fixed by a permutation exactly when each of its cycles lies wholly within one row. Assigning the distinct length-$q$ cycles to rows gives the displayed coefficient. More explicitly, sum ${\prod }_q{m}_q!/{\prod }_j{a}_{qj}!$ over nonnegative integers $a_{qj}$ with ${\sum }_j{a}_{qj}=m_q$ and ${\sum }_{q}q{a}_{qj}={\lambda }_j$. Equal-length rows remain distinguished. Thus the character counts fixed ordered row sets, rather than unordered set partitions.

An urn with $j$ red balls among $n$ is sampled independently $n$ times with replacement, and the next generation's red count is the number of red draws. Its proportion is a bounded discrete-time martingale. Accelerating generation time by $n$ gives a Wright–Fisher diffusion limit: the one-step variance of the proportion is $x(1-x)/n$, and the higher moments make Taylor remainders negligible.

For a dominant integer tuple $\lambda$ and $\delta =(m-1,\dots ,0)$, the rational Schur character is $\det (x_i^{{\lambda }_j+m-j})/\det (x_i^{m-j})$. For partitions it is a symmetric polynomial; for general tuples it is Laurent on the diagonal torus. The formula follows by comparing the trace of a permuted tensor power with the Frobenius alternant character formula and using independence of symmetric-group characters. Apparent poles at coincident nonzero eigenvalues are removable.

Strict convergence combines strong convergence in $L^1$ with convergence of the total variation seminorm. It implies weak-star convergence in BV but does not make jump sets stable: mollifications of an interior step can converge strictly to the step while every approximant has an empty jump set of a bounded-variation function.

For $d$ free target coordinates, normal ordering of $L_m={\sum }_n:{\alpha }_{m-n}\cdot {\alpha }_n:/2$ gives a matter central charge $d$. On the formal zero-momentum vacuum, the double contractions yield $⟨L_m{L}_{-m}⟩=(d/2){\sum }_{r=1}^{m-1}r(m-r)=d(m^3-m)/12$. Two target-metric contractions give ${\eta }_{ab}{\eta }^{ab}=d$, so timelike target coordinates do not subtract from this central charge. Transverse-only and covariant generators must not be confused.

A method-of-moments variance component estimate equates residual mean squares in ANOVA to their model expectations and solves for the variance components. An unconstrained estimate can be negative even when the model component must be nonnegative; boundary treatment then requires a stated estimation convention. The method gives point estimates, not certainty about future experiments.

The unit-width centred box kernel is $K(u)=1_{[-1/2,1/2]}(u)$. It has integral one and squared $L^2$ norm one. Its rescaling $K_h(u)=h^{-1}K(u/h)$ averages a function on intervals of length $h$. It is obtained from the uniform smoothing kernel with support $[-1,1]$ by rescaling, and is useful in both kernel density estimation and local polynomial regression.

With $\omega =-d\alpha$ and ${\omega }_{\sigma }=\omega +{\pi }^{\ast }\sigma$, the one-form graph of $\theta$ is a Lagrangian submanifold exactly when $d\theta =\sigma$. Indeed its pulled-back symplectic form is $\sigma -d\theta$ and its dimension is half that of the total space. This extends graph of a closed one-form is Lagrangian.

If $\sigma =d\beta$, cotangent fiber translation by $+\beta$ gives a symplectomorphism from $(T^{\ast }X,-d\alpha )$ to $(T^{\ast }X,-d\alpha +{\pi }^{\ast }\sigma )$. Its pullback changes $\alpha$ by ${\pi }^{\ast }\beta$, so the extra two-form cancels with $-{\pi }^{\ast }d\beta$. The opposite translation gives the inverse direction, globally without compactness.

If $\ell$ is a prime different from the characteristic, multiplication by $\ell$ on an elliptic curve has degree ${\ell }^2$ and nonzero differential. It is a separable isogeny, so its geometric kernel has ${\ell }^2$ points, all killed by $\ell$. This gives the displayed group over an algebraic closure; the rational subgroup over the base field may be smaller.

If ${\nabla }_X{e}_i={\sum }_j{\omega }_{ij}(X)e_j$ and ${\varphi }_i$ is the dual coframe, the first index of $\omega$ is the input frame vector. Expand the connection derivative of $Y={\sum }_i{\varphi }_i(Y)e_i$ and antisymmetrize in $X,Y$. The formula for the exterior derivative of a one-form evaluated on vector fields gives ${\tau }_j=d{\varphi }_j-{\sum }_i{\varphi }_i\wedge {\omega }_{ij}$. This is the first structure equation with the indicated index convention. Reversing the wedge order changes the sign.

A smooth time-dependent vector field on a compact manifold without boundary has flow maps throughout each compact time interval on which the field is defined. Finite-time escape is impossible on the compact state space. Uniqueness and the backward-time equation supply smooth inverses, so the flow is a smooth isotopy.

At exactly thirty degrees the quadratic curvature vanishes, but the specified straight-line point-mass effective potential is downhill for small positive displacement at cubic order. Thus the strict negative-curvature criterion does not exclude every nonlinear one-sided marginal launch. An exactly stationary bead remains at equilibrium until displaced.

This tensor records the ordered kinetic energy of the streaming velocity $u_i=⟨v_i⟩$. It excludes random motion; its vanishing is compatible with a nonzero stellar pressure-energy tensor.

This tensor integrates the local pressure tensor, where ${\sigma }_{ij}=⟨(v_i-u_i)(v_j-u_j)⟩$ is the velocity covariance. Half its trace is the random kinetic energy. It enters the tensor virial theorem with coefficient one.