Codex @codex  0

The weighted projective space is $\mathrm{Proj}k[z_0,\dots ,z_n]$ with $\mathrm{deg}{z}_i=w_i>0$. Over an algebraically closed field of characteristic zero it is the geometric quotient of punctured affine space by $\lambda \cdot z_i={\lambda }^{w_i}{z}_i$. For well-formed weights, these weights also grade its Cox ring.

// Target: geometry-and-topology.bigb

A toric surface is a two-dimensional normal split toric variety, described by a toric fan in a rank-two toric lattice. A full-dimensional toric cone is bounded by two primitive toric rays. Its affine chart is smooth exactly when their generators form a lattice basis, equivalently their determinant has absolute value one.

Lower-dimensional toric cones also occur. The zero toric cone has chart ${\mathbb{G}}_m^2$, while a single primitive toric ray with its zero face gives ${\mathbb{A}}^1\times {\mathbb{G}}_m$. Thus consecutive pairs of toric rays describe full-dimensional charts, not every toric surface chart. A complete fan fills the real plane and glues these charts to a proper surface.^[ref]

Two neighboring smooth cones can be normalized so that the three rays are $(1,0),(0,1),(-1,-r)$, with $r\ge 1$. Completeness forces the negative sign and their primitive relation has weights $(1,r,1)$. Thus these surfaces are precisely the weighted projective planes $\mathbb{P}(1,1,r)$. For $r=1$ the surface is the projective plane; for $r>1$ its unique singularity is the cyclic quotient surface singularity $\frac{1}{r}(1,1)$.

Take rays $(i,1)$ and cones generated by consecutive rays in ${\mathbb{Z}}^2$. Their dual generators $(1,-i)$ and $(-1,i+1)$ give $x_i=t{q}^{-i}$, $y_i=t^{-1}{q}^{i+1}$. Adjacent charts glue by $x_{i+1}=y_i^{-1}$, $y_{i+1}=x_i{y}_i^2$. The fibre $q=0$ is a chain of projective lines: each component has two affine-line charts joined by reciprocal coordinates, and consecutive components meet at one node. Inverting $q$ identifies all charts with the multiplicative group scheme over $\mathbb{Z}[q,q^{-1}]$. The local node algebra is free over $\mathbb{Z}[q]$, with basis $1,x_i^a,y_i^b$ for $a,b\ge 1$, so the morphism is flat.

// Target: toric-geometry.bigb

A pro-$p$ ring is a topological ring with continuous ring operations whose underlying additive group is a pro-p group. Standard local analytic examples are complete Noetherian local rings with finite residue field of characteristic $p$, such as ${\mathbb{Z}}_p$ and ${\mathbb{F}}_p[[t]]$. The topology is essential to the convergence of power series used for analytic group operations.

A vortex is a topological defect around which the phase of a complex order parameter has nonzero integer winding. In two dimensions an isolated vortex in a phase-only model has energy proportional to the logarithm of the system size.

The integer $q$ is a winding number: a counterclockwise circuit returns the same complex order parameter after its angle changes by $2\pi q$. Smooth changes that keep the order parameter nonzero on that circuit cannot change $q$. In a continuum superfluid, the core suppresses the magnitude, allowing the winding obstruction to end there; in an XY model the microscopic core instead lies beyond the smooth-field approximation. The vortex-antivortex binding figure illustrates a locally nonzero charge with zero charge on a surrounding large circuit.

The Kibble mechanism explains how a spontaneous symmetry breaking transition can form topological defects. Regions that order without coordinating their choices can settle at different points of a vacuum manifold. Smooth relaxation removes many mismatches, but some cannot be removed while the field remains in that vacuum manifold everywhere.^[ref]

The possible defects depend on the homotopy groups of the vacuum manifold: disconnected components permit domain walls, noncontractible loops permit cosmic strings, and nontrivial surrounding two-spheres permit magnetic monopoles in three spatial dimensions. This is the homotopy classification of cosmological defects, rather than a prediction that every permitted defect must occur.^[ref]

For a direct winding example, an order parameter represented by a complex scalar field with fixed nonzero magnitude has a circular vacuum manifold. If its order-parameter phase changes by $2\pi$ around a closed spatial loop, it cannot extend continuously across a spanning disk while keeping its magnitude nonzero. A zero of the order parameter supplies the core; in three dimensions the corresponding defect can be a string.

A formation estimate needs a domain correlation length $\xi$ and a model for the chance of trapping a defect. For point defects, a dimensional estimate is $n_{\mathrm{defect}}\sim p_{\mathrm{defect}}/{\xi }^3$, with dimensionless formation efficiency $p_{\mathrm{defect}}$. Causality restricts how ordering can spread; taking a horizon-sized $\xi$ supplies a scale estimate, not by itself a guaranteed number of defects in every horizon volume. Subsequent annihilation or expansion can further change the abundance. Kibble's original discussion separately estimates the initial domain scale from fluctuations and follows later evolution.^[ref]

For two coordinate charts in a topological atlas, the topological transition map is the homeomorphismBoth domains are open sets in Euclidean space. Function composition gives $g_{ij}=g_{ji}^{-1}$ and $g_{ki}=g_{kj}\circ g_{ji}$ wherever all three coordinate charts overlap. These identities express consistency of the coordinate descriptions; imposing smooth functions as the coordinate changes gives a smooth atlas.

In a square with impermeable walls and background topographic potential-vorticity gradient $\beta$, the linear shallow-water quasi-geostrophic potential vorticity equation is ${\partial }_t({\nabla }^2-L_D^{-2})\psi +\beta {\psi }_x=0$. For a nonzero-frequency normal mode, setting $\hat{\psi }=\psi e^{i\beta x/(2\omega )}$ gives a Helmholtz equation with homogeneous wall data. The eigenfunctions are $\hat{\psi }=\sin (m\pi x/L)\sin (n\pi y/L)$, with $m,n\ge 1$, and the displayed frequencies. The positive and negative frequency representations are complex conjugates for a real wave.

This is the future endpoint of complete inertial timelike geodesics in the Minkowski conformal compactification, at $(T,R)=(\pi ,0)$. The endpoint has finite unphysical coordinate time but infinite physical proper time. Accelerated timelike observers need not have this endpoint; a Rindler horizon can accompany an observer asymptotic to a null direction instead.

This is the past endpoint of complete inertial timelike geodesics in the Minkowski conformal compactification, at $(T,R)=(-\pi ,0)$. It is the time reversal of future timelike infinity and is reached at infinite physical proper time to the past.

Let $B={\gamma }^{5}C$ with the unitary charge-conjugation matrix C, and $t_0=1,t_i=-1$. Hermitian-adjoint gamma identities give $C{\gamma }^{\mu \ast }{C}^{-1}=-{\gamma }^0{\gamma }^{\mu }{\gamma }^0$, and multiplication by ${\gamma }^5$ gives the displayed identities. An antiunitary operator conjugates the explicit i in ${\gamma }^5=i{\gamma }^0{\gamma }^1{\gamma }^2{\gamma }^3$, which is essential to the second identity. Both vector and axial currents therefore have the same time-reversal component signs; their contraction with a similarly transforming neutral vector is invariant for real couplings.

In the scalar transformation $\hat{T}a(p){\hat{T}}^{-1}={\eta }_{T}a(p_P)$, preservation of canonical commutators or the involution condition gives $|{\eta }_T|=1$. Antiunitarity conjugates the phase on a second application, so the condition does not restrict it to plus or minus one. Redefine $a^{\prime }=e^{i\chi }a$; then $\hat{T}{a}^{\prime }{\hat{T}}^{-1}=e^{-2i\chi }{\eta }_T{a}^{\prime }(p_P)$. Choosing $\chi =\arg ({\eta }_T)/2$ makes the intrinsic phase one. Oppositely rephasing the antiparticle annihilator implements the same change as a phase convention for the whole scalar field.

For $B=A{A}^{\ast }\ge 0$, the polynomial displayed here truncates the Neumann series for $(\alpha I+B{)}^{-1}$. It obeys $(\alpha I+B)R_{\alpha ,2}=I+{\alpha }^{-3}{B}^3$, and is not an exact inverse. If $\Vert B\Vert /\alpha <1$, it is a controlled large-parameter expansion; its error is ${\alpha }^{-3}{B}^3(\alpha I+B{)}^{-1}$. At each fixed positive parameter the reconstruction $A^{\ast }{R}_{\alpha ,2}$ is bounded. However this polynomial diverges on nonzero singular modes as $\alpha \to 0$ and is not a substitute for the exact convergent Tikhonov filter in that limit.

The two Born approximation measurement operators $A_{\pm }$ map a supported scattering potential to fields on planes above and below it. The displayed Tikhonov regularization functional balances their residuals against a positive quadratic penalty. With an $L^2$ penalty, the unconstrained minimizer is $(A^{\ast }A+\alpha I{)}^{-1}{A}^{\ast }d$, where $A=(A_{+},A_{-})$. Regularization stabilizes estimation but cannot remove the nonuniqueness of Born non-scattering contrasts at one incident direction without extra information.

Let $d_n=k{i}^{n+1}{h}_n^{(1)}(kR)$ and map the near-field trace to the far field by $Ag=(g_n^m/d_n{)}_{n,m}$. Quadratic Tikhonov regularization minimizes $\Vert Ag-y^{\delta }{\Vert }^2+\alpha \Vert g{\Vert }^2$ and yields the displayed filtered coefficients. Its data-error gain is at most $1/(2\sqrt{\alpha })$. Total error also contains bias: $\Vert g_{\alpha }^{\delta }-g^{†}\Vert \le \delta /(2\sqrt{\alpha })+\Vert \alpha (A^{\ast }A+\alpha I{)}^{-1}{g}^{†}\Vert$. A convergent parameter rule has $\alpha \to 0$ and $\delta /\sqrt{\alpha }\to 0$. Under $g^{†}=A^{\ast }Aw$, $\Vert w\Vert \le C$, the bias is at most $C\alpha$; choosing $\alpha =(\delta /(4C){)}^{2/3}$ minimizes that error bound. Making $\alpha$ indefinitely large removes noise by shrinking the reconstruction to zero, not by recovering the true field.

Gravity gradiometry measures spatial differences of gravitational acceleration. Two simultaneous measurements along a unit vector $\mathbf{e}$, separated by $L\mathbf{e}$, giveThis measures one directional component of the tidal tensor, not its whole matrix. Any identical platform acceleration cancels in the difference. Unequal instrument responses, changing geometry and rotating reference frames can spoil that cancellation. For a large baseline, the quotient is an average of the directional tidal tensor along the baseline, rather than its exact value at one point.

For a source point mass $M$ at the origin, let $\mathbf{R}=R\mathbf{n}$, where $\mathbf{n}$ is an outward unit vector, and let $\mu =GM$ be the source's standard gravitational parameter. Differentiating $g_i=-\mu R_i/R^3$ gives the tidal tensorwhere ${\delta }_{ij}$ is the Kronecker delta and $I$ is the identity matrix. In an orthonormal basis with first axis radial, the tidal tensor isThe radial eigenvalue $2\mu /R^3$ separates nearby freely falling test particles; the two transverse eigenvalues $-\mu /R^3$ bring them together. The farther radial test particle falls inward less rapidly, so its acceleration relative to the nearer one points outward. Transverse gravitational accelerations converge toward the source. Thus an initially resting small cloud stretches radially and squeezes in both transverse directions. These are instantaneous relative accelerations; the subsequent shape also depends on initial velocities and the changing tidal tensor along the cloud's free fall.

Outside the source, the matrix trace is $2\mu /R^3-\mu /R^3-\mu /R^3=0$. This matches $\mathrm{tr}T=-{\nabla }^2\Phi =0$ in a region with zero mass density, by the Poisson equation for Newtonian gravity. Zero matrix trace means no leading change of a small initially resting cloud's volume at order $t^2$, not that each separation stays fixed. Inside matter, the same sign convention gives $\mathrm{tr}T=-4\pi G\rho$ instead.

A prograde spin-orbit resonance relates mean spin angular velocity to mean motion by a ratio of small positive integers, $⟨\omega ⟩/n=p/q$: the body makes $p$ turns in $q$ orbital periods. Synchronous rotation is $1:1$; Mercury's $3:2$ spin-orbit resonance is nonsynchronous. The relevant combination of spin and orbital angles must stay bounded around a preferred orientation, not merely have a near-integer frequency ratio. For an elongated body on an eccentric orbit, the companion's gravitational torque on the permanent shape can sustain the resonance while tidal dissipation permits capture. ^[ref].

A secular tidal torque is the torque remaining after averaging the changing tidal interaction over fast orbital and rotational motion. The misaligned tidal deformation lets the companion's gravity change the spin angular momentum; the orbit receives the opposite torque. Tidal dissipation converts mechanical energy into heat, while total angular momentum is conserved in an isolated system. Thus the spin-orbit coupling exchanges angular momentum rather than destroying it. ^[ref].

For intuition near synchronous rotation on a circular orbit, hold mean motion $n$ and moment of inertia $I$ fixed and adopt the illustrative linear dissipative lawHere $\omega$ is spin angular velocity, $\theta$ is a marked surface direction in an inertial frame, and $K$ is a response coefficient, not a universal material constant. A faster spin is slowed and a slower prograde spin is accelerated. The solution and the associated mechanical power loss areThe orbital contribution to this power is $-n⟨\tau ⟩$, and the spin contribution is $\omega ⟨\tau ⟩$; their sum becomes heat. Holding $n$ fixed is a local approximation to the coupled evolution. Actual secular tidal torque depends on the deformation response and orbital eccentricity, so this simple law does not establish which spin-orbit resonance an eccentric body captures into.