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The displayed entire function has exactly the periods $2\pi i\mathbb{Z}$: a period $c$ would make $e^z(e^c-1)$ a constant integer multiple of $2\pi i$, forcing $e^c=1$. If a polynomial addition relation existed, specialize its middle argument at a point with $e^c=\alpha >0$ irrational, avoiding the finitely many specializations making the entire polynomial vanish. With $t=e^z$, the relation becomes a nonzero finite sum of exponentials $e^{(m+\alpha n)t}$. These real exponents are distinct, so the largest one dominates as $t\to +\infty$, a contradiction. This demonstrates that an algebraic addition theorem is stronger than simple periodicity.

The Conway polynomial is normalized by ${\nabla }_U=1$ for the unknot, zero for a split unlink with more than one component, and the skein relation ${\nabla }_{L_{+}}-{\nabla }_{L_{-}}=z{\nabla }_{L_0}$. For a knot, the symmetric Alexander polynomial with value one at $t=1$ is ${\Delta }_K(t)={\nabla }_K(t^{1/2}-t^{-1/2})$. This fixes the Laurent-unit ambiguity of the Seifert-matrix presentation of the Alexander module.

The lattice points of a rational polyhedral cone form a finitely generated additive monoid. Choose lattice generators for the cone. Subtracting integer parts of nonnegative coefficients leaves a lattice point in their bounded fundamental parallelepiped; its finitely many lattice points, together with the chosen generators, generate the monoid. This supplies finite generation of the coordinate ring of an affine toric variety.

// Target: geometry-and-topology.bigb

Suppose a scheme is the scheme-theoretic union of affine closed subschemes $Y=\mathrm{Spec}A$ and $Z=\mathrm{Spec}B$, meaning their ideal sheaves have zero intersection. The intersection $W$ is closed in both, hence affine with coordinate ring $C$ and surjections $A\to C$, $B\to C$. Then the whole scheme is $\mathrm{Spec}(A{\times }_{C}B)$. Both projections of this fiber product are surjective; their kernels have zero intersection and product, and their quotient by the sum is $C$. Thus its spectrum has the same closed pieces, intersection and topology. The structure sheaf is the kernel of $i_{\ast }{\mathcal{O}}_Y\oplus j_{\ast }{\mathcal{O}}_Z\to k_{\ast }{\mathcal{O}}_W$ on both spaces, identifying their locally ringed structures. A reduced Noetherian scheme has finitely many irreducible components, so induction proves it is affine if and only if its reduced irreducible components are affine. The intersection $W$ need not be reduced.

For a nonempty topological space $S$, the constant diagram functor on its open-set presheaf category lies in this five-term adjunction chain. The leftmost functor sends $A$ to the categorical presheaf equal to $A$ at the empty open set and empty elsewhere. The rightmost sends $A$ to the presheaf equal to $A$ at $S$ and a singleton elsewhere. A natural transformation from the former is determined by its empty-open component; a transformation into the latter is determined by its whole-space component. The middle adjunctions follow from restriction maps to the empty open set and from the whole space. Neither endpoint extends: the leftmost functor fails to preserve the terminal object, and the rightmost fails to preserve the initial object.

The complex Adjoint representation of SU(3) is the action by conjugation on traceless $3\times 3$ matrices. With $H_1=\mathrm{diag}(1,-1,0)$ and $H_2=\mathrm{diag}(0,1,-1)$, its highest-weight vector is $E_{13}$ of Dynkin labels $(1,1)$. The six off-diagonal matrix units have weights $(2,-1),(-2,1),(-1,2),(1,-2),(1,1),(-1,-1)$, and the two-dimensional Cartan subalgebra has weight zero. In coordinates $(h_1,(h_1+2h_2)/\sqrt{3})$, the weight diagram is a regular hexagon together with a central point of weight multiplicity two. This representation describes both the gluon colour multiplet and the approximate flavour baryon octet, under different physical SU(3) symmetries.

In the effective group form used for recognition problems, a finite group presentation of $K$ and a word $w$ yield, by an algorithm, a finite presentation of a group $K_w$. If $w=1$ in $K$, then $K_w$ is trivial; if $w\ne 1$ in $K$, then the specified generators embed $K$ in $K_w$. One can additionally arrange that the normal closure of the image of $w$ is all of $K_w$. The finite presentation can be written without first deciding which case holds. This is the collapse-versus-embedding construction behind the Adian–Rabin theorem.

If $ad\equiv 1(\mathrm{mod}N)$, the determinant-one matrix with rows $(a,(ad-1)/N)$ and $(N,d)$ sends $\tau -d/N$ to $a/N-1/(N^2\tau )$. The transformation of the cusp form gives the displayed identity. It turns the small-height portion of an additive-twist Mellin transform into exponentially decaying values at the inverse rational cusp.

For even weight $k$, complete the additive twist of a cusp-form L-function as $M(f,a/N,s)=(N/(2\pi ){)}^s\Gamma (s)L(f,a/N,s)$. Split its Mellin transform at $y=1/N$ and use rational-cusp inversion of a level-one cusp form in the lower integral. The resulting two integrals over $[1/N,\infty )$ converge locally uniformly for every complex $s$, giving an entire continuation and the displayed functional equation. The relation $ad\equiv 1(\mathrm{mod}N)$ identifies the opposite cusp.

For a finite surjective map of normal integral schemes with function fields $K\subseteq L$ in characteristic $p$, the condition $L^p\subseteq K$ lets the Absolute Frobenius morphism of the source factor through the map. On an affine chart $A\subseteq B$, send $b$ to $b^p\in A$: it lies in $K$, is integral over $A$, and $A$ is integrally closed. The condition $K\subseteq L^p$ instead lets the map factor after the source Frobenius. Send $a$ to its unique $p$th root in $L$; that root lies in the normal ring $B$ because it satisfies a monic polynomial over $B$. These rules localize and glue; injectivity of Frobenius in the function field supplies uniqueness.

For an effective divisor on an algebraic curve $D$ on a smooth projective curve, the Zariski tangent space to its symmetric product of a curve is $H^0({\mathcal{O}}_D(D))$. The derivative of the Abelian sum map is the connecting homomorphism of $0\to {\mathcal{O}}_X\to {\mathcal{O}}_X(D)\to {\mathcal{O}}_D(D)\to 0$. Its dual under Serre duality is restriction of holomorphic differential forms to $D$, including all jets at repeated points. Hence its rank is $\mathrm{deg}D+1-h^0({\mathcal{O}}_X(D))$, and its image annihilates $H^0(K_X(-D))$.

For constant kinematic viscosity $\nu$, use the phenomenological viscous torque in an accretion disk, specific angular momentum $h=rc\sqrt{m/(r-3m)}$, and $h_0=\sqrt{12}mc$ at the innermost stable circular orbit. Neglect radial kinetic motion in the clock relation and neglect differential light travel time. The Schwarzschild time to drift formally from $r_1\ge 6m$ to $6m$ is the displayed expression, withIndeed $\mathcal{T}(2)=0$ and ${\mathcal{T}}^{\prime }(X)=6X[X-2\sqrt{X-1}]/(X-1)\ge 0$ for $X\ge 2$. The zero-torque inner boundary condition makes the formal drift speed diverge as $r\downarrow 6m$, so the nearly circular approximation must give way to the plunging region of a black-hole accretion disk sufficiently close to that orbit. This formula describes the stipulated approximation, not an exact relativistic flow solution.

For $F=dA+A\wedge A$, the gauge-theory Bianchi identity is $DF=0$ and the variation is $\delta F=D\delta A$. The graded product rule for the covariant exterior derivative gives the displayed boundary integral. Thus a constant-coefficient Yang-Mills theta term leaves the bulk Yang-Mills equations unchanged, but can change the boundary variational condition. Its four-form density is exactly gauge-invariant by conjugation of $F$ and cyclicity of the matrix trace, even if the gauge parameter is nonzero on the boundary. Gauge changes of a local Chern-Simons 3-form must not be confused with gauge changes of this globally defined curvature density.

Consider $q$ identical length-$n$ arms joined to one hub, with XY exchange interactions of strength $J_0/\sqrt{q}$ from the hub to each first site and strength $J_k$ between sites $k,k+1$ on each arm. Let $|s_0⟩$ be the hub excitation and $|s_k⟩=q^{-1/2}{\sum }_{a=1}^q|a,k⟩$. This orthonormal symmetric single-excitation subspace is invariant, and its restricted Hamiltonian operator iswith endpoint terms omitted. At the first link the $q$ hub contributions add to $J_0$; all other links act identically on the arms. Thus the linear isometry $F|k⟩=|s_k⟩$ intertwines $H$ with the path Hamiltonian operator $H_T={\sum }_{k=0}^{n-1}{J}_k(|k⟩⟨k+1|+|k+1⟩⟨k|)$: $HF=F{H}_T$. Expanding the matrix exponential gives $e^{-iHt}F=F{e}^{-i{H}_{T}t}$. Perfect quantum state transfer from zero to $n$ on the path therefore produces an equal excitation superposition at the $q$ arm tips. Since all other qubits are zero, the tip state is a pure W state, up to the transfer's common phase.

For $y\prime \prime +py\prime +qy=0$, the Wronskian satisfies $W\prime =-pW$.

To first order in ${\alpha }^{\prime }$, the conventional Euclidean dilaton coupling gives ${\overline{\beta }}_{\mu \nu }^G={\alpha }^{\prime }[R_{\mu \nu }-H_{\mu \rho \sigma }{H}_{\nu }{}^{\rho \sigma }/4+2{\nabla }_{\mu }{\nabla }_{\nu }\Phi ]$ and ${\overline{\beta }}_{\mu \nu }^B={\alpha }^{\prime }[-{\nabla }^{\rho }{H}_{\rho \mu \nu }/2+({\nabla }^{\rho }\Phi )H_{\rho \mu \nu }]$. The scalar coefficient is $(d-26)/6+{\alpha }^{\prime }[(\nabla \Phi {)}^2-{\nabla }^2\Phi /2-H^2/24]$. Here $H$ is the Kalb-Ramond field strength. Vanishing of these coefficients, including higher-order corrections, is the background-field condition for quantum Weyl transformations to remain a gauge symmetry. The scalar coefficient includes the central charge deficit and is not implied by the metric equation alone.

Use the half-normalized operator $(P_{1}v{)}_{ab}=({\nabla }_a{v}_b+{\nabla }_b{v}_a-h_{ab}{\nabla }_c{v}^c)/2$. On a closed worldsheet, a symmetric trace-free tensor obeysThis integration by parts identifies the displayed formal adjoint operator. A convention with twice $P_1$ has twice this adjoint, with identical kernels. The worldsheet conformal Killing group corresponds to the kernel of $P_1$, while infinitesimal worldsheet moduli correspond to the trace-free kernel of its adjoint.

A compact stratified space with boundary is a Witt space with collared boundary when its interior and boundary are Witt spaces and a stratified neighbourhood of the boundary is $\partial X\times [0,\epsilon )$, with product stratification. Orientations are carried by the regular strata and induce the usual boundary orientation.

Fix $m>0$, $k=(m,0)$, and a rotationless Lorentz boost $L(p)$ taking $k$ to $p$ on the positive-energy mass shell. The little group on the Lorentz spinor double cover is SU(2). Its spin-$s$ irreducible representation induces a unitary representation on $L^2(d^{3}p/(2p^0))\otimes {\mathbb{C}}^{2s+1}$. Here $W(\Lambda ,p)=L(\Lambda p{)}^{-1}\Lambda L(p)$ is the Wigner rotation, lifted to SU(2) when necessary, and the translation convention is $U(I,a)|p⟩=e^{ip\cdot a}|p⟩$. The little-group cocycle proves the group law, and Wigner's classification gives irreducibility. Every nonnegative integer or half-integer spin is allowed on the double cover; descent to the ordinary proper orthochronous Poincaré group requires integer spin, because its $2\pi$ rotation must act as the identity.

The scalar coefficient multiplying an unknown half-line transform in a Wiener-Hopf equation is its kernel. This is a transform-domain function, distinct from the null space of a linear map. Its analytic zeros, poles and winding number govern Wiener-Hopf factorization. For mixed scalar Helmholtz half-plane diffraction a typical kernel is the outgoing vertical wavenumber $\sqrt{k^2-{\alpha }^2}$.