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For a closed oriented smooth four-manifold $X$ with $b_2^{+}(X)>1$, the Seiberg–Witten invariant of a four-manifold associates an integer-valued invariant, with a choice of homology orientation, to a Spin-c structure. It is obtained from the moduli space of solutions of the Seiberg–Witten equations modulo gauge transformations. When the expected dimension is zero, it is a signed count of solutions after a generic perturbation. Positive-dimensional cases require suitable insertions; when $b_2^{+}=1$, chamber data matters. These analytic counts can distinguish smooth structures and, through Taubes nonvanishing theorem, obstruct decompositions of symplectic manifolds of dimension four.^[ref]

Remove tubular neighborhoods of diffeomorphic embedded oriented surfaces with opposite normal Euler numbers, and glue their boundary circle bundles by a fiber-reversing identification. The resulting smooth manifold is a fiber sum. The square-zero case uses trivial normal bundles. Additional symplectic hypotheses give a symplectic fiber sum along a square-zero surface.

If one-dimensional smooth linear characters of a local multiplicative group have conductors tending to infinity, their direct sum can be admissible. Its algebraic dual is the product of inverse linear characters, but a smooth vector in this product has only finitely many nonzero coordinates. A common open stabilizer contains one fixed principal-unit subgroup, on which every sufficiently ramified linear character is nontrivial. Thus the smooth dual is again a direct sum.

Every projective line through $P$ on a smooth cubic surface lies in its projective tangent plane. Each is a distinct linear factor of the cubic plane section. There can therefore be at most three. Equality occurs when the tangent section consists of three concurrent lines.

Write a line as $L=\{x_2=x_3=0\}$ and the surface equation as $F=x_2{Q}_2+x_3{Q}_3$. Smoothness says $A=Q_2{|}_L$, $B=Q_3{|}_L$ have no common zero. Projection $[x_2:x_3]$ extends over $L$ by $[-B:A]$. Its fibres are the residual plane conics in the planes through $L$, since substituting $x_2=sw$, $x_3=tw$ gives $F=w(s{Q}_2+t{Q}_3)$. Different residual conics are disjoint everywhere, including their points on $L$, because they are fibres of one morphism of algebraic varieties. The family is flat: over the smooth base curve its local rings are torsion-free over the corresponding discrete valuation ring; smoothness of the total surface supplies the reduced-fibre conclusions.

For two skew lines $L,M$ on a smooth cubic surface, their joining line meets the surface at a third point generically. In coordinates with $L=\{x_2=x_3=0\}$ and $M=\{x_0=x_1=0\}$, write $F(\alpha A+\beta B)=\alpha \beta (\alpha U+\beta V)$. The third point is $[VA-UB]$ whenever $UV\ne 0$. Neither $U$ nor $V$ is identically zero, because that would make one of the two lines singular on the surface. The inverse records $[x_0:x_1]$ and $[x_2:x_3]$ of a general surface point. Thus the surface is birational to ${\mathbb{P}}^1\times {\mathbb{P}}^1$, hence to ${\mathbb{P}}^2$.

A right-going, boundary-forced internal gravity wave has horizontal momentum per unit inertial density $P=(k/\omega )\overline{E}$. Its envelope-induced Eulerian mean acceleration is ${\overline{u}}_t=P_t$ to leading slow order. In the frame moving at horizontal phase velocity $c=\omega /k$, the change of mean kinetic energy relative to the initial current $-c$ is $-{\rho }_{0}c\int \overline{u}dz$ at quadratic wave order. Its time derivative is minus the boundary work $\overline{W}={\rho }_{0}d\int \overline{E}dz/dt$. The lab-frame mean kinetic energy alone is fourth order in wave amplitude; retaining the moving-frame cross term is essential.

A natural transformation $P\to F$ assigns a set of points over every element $x\in F(c)$. These fibers form a functor on the category of elements of $F$. Conversely, a functor on the category of elements gives $P(c)$ as the disjoint union of its values over all $x\in F(c)$. The two constructions are inverse up to natural isomorphism. Smallness of $\mathcal{C}$ makes the category of elements small and allows this description to transfer Cartesian closure to every slice.

For the Integer Heisenberg group, choose $x=I+E_{12}$, $y=I+E_{23}$ and central $z=I+E_{13}$. Every element has a unique form $y^c{z}^b{x}^a$, and $xy=zyx$. The group algebra is therefore the displayed skew Laurent polynomial ring, with $\sigma (z)=z$. The base is a commutative Laurent polynomial ring and an integral domain; the skew Hilbert basis theorem gives both-sided Noetherianity and the leading coefficient of a product proves that there are no zero divisors.

Let $A$ have no zero divisors and $\sigma$ be an automorphism. For nonzero Laurent polynomials $f=\sum a_i{x}^i$ and $g=\sum b_j{x}^j$ with greatest exponents $m,n$, the coefficient of $x^{m+n}$ in $fg$ is $a_m{\sigma }^m(b_n)$. No other exponent pair contributes to that highest degree. Both factors are nonzero, so their product is nonzero, proving the noncommutative domain property in every characteristic.

On a finite-dimensional positive inner-product space, a skew-adjoint nilpotent operator is zero. In an orthonormal basis, $\Vert A{\Vert }_F^2=\mathrm{tr}(A^{\ast }A)=-\mathrm{tr}(A^2)=0$, since a nilpotent matrix has zero trace and so does its square. The Frobenius norm therefore vanishes. In particular, a nonzero nilpotent adjoint map rules out a compact Lie algebra.

On a finite-dimensional real inner product space, a skew-adjoint operator has even rank. In an orthonormal basis its matrix is skew-symmetric, so an invertible such matrix must have even dimension by $\det A=\det (-A)=(-1{)}^d\det A$. Moreover $\ker A=(\mathrm{im}A{)}^{\perp }$, so the image and kernel intersect trivially. The restriction of $A$ to its image is an invertible skew-adjoint map, and its dimension, the rank, is therefore even.

The self-energy bubble diagram at loop order one in six-dimensional cubic scalar field theory has Feynman-diagram symmetry factor two. With positive Euclidean scalar propagators, its amputated Green's function is the displayed integral. Its superficial degree of divergence is two. Expanding at large internal momentum and taking the angular average shows that its divergent part is a constant plus a multiple of $p^2$, precisely the mass counterterm and wave-function renormalization structures.

For a spherical ultraviolet cutoff $|k|\le \Lambda$ and $m^2>0$, its value at $p=0$ isThe divergent $p^2$ coefficient is $-g^2\log ({\Lambda }^2/m^2)/(768{\pi }^3)$. Finite terms depend on the regularization convention. Tadpole diagram insertions are excluded from the one-particle-irreducible Feynman diagram, and are removed from the connected two-point function by imposing a vanishing one-point function.

A unibranch singular point of an algebraic variety on a plane algebraic curve. The ordinary cusp has local equation $y^2=x^3$ and parametrization $(x,y)=(t^2,t^3)$; its branch has a repeated tangent and a vanishing first derivative at the singular point. Higher-order cusps may have different local equations. A cuspidal cubic gives the basic projective example.

For a continuous map $f:X\to Y$, precomposition with the induced map on singular chains sends a singular cochain on $Y$ to one on $X$. It commutes with the coboundary and induces the induced map on cohomology.

In the single-excitation subspace, take a connected real nearest-neighbor hopping matrix $A$ with every coupling nonzero and endpoint control $B=\mathrm{diag}(-1,1,\dots ,1)$. Endpoint commutators isolate the first transition, then propagate along the chain to generate both off-diagonal skew-Hermitian directions and adjacent diagonal differences. These generate $\mathfrak{su}(N)$. For $N\ge 3$, $\mathrm{Tr}B=N-2\ne 0$ supplies the central direction, giving $\mathfrak{u}(N)$. For $N=2$ the algebra is $\mathfrak{su}(2)$, which still gives full state controllability but omits an arbitrary global phase. A broken coupling separates an unreachable chain component.

The isotropic XY exchange interaction between two qubits is $E=(X\otimes X+Y\otimes Y)/2$. Its action isIndeed, $X|0⟩=|1⟩$, $X|1⟩=|0⟩$, $Y|0⟩=i|1⟩$, and $Y|1⟩=-i|0⟩$. The two terms cancel on equal bits and add on unequal bits. Thus a network Hamiltonian operator $H={\sum }_{a,b}{K}_{ab}{E}_{ab}$ preserves excitation number; in the single-excitation subspace, it is the weighted adjacency matrix, moving an excitation from $a$ to $b$ with hopping coefficient $K_{ab}$.

The identically conserved topological current $j$ couples to an external electromagnetic gauge potential through $-A_{\mu }{j}^{\mu }=-A_0{\theta }_x+A_1{\theta }_t$. A gauge transformation changes this coupling only by a divergence. The Euler-Lagrange field equation becomes ${\theta }_{tt}-{\theta }_{xx}+\sin \theta +E=0$, with $E={\partial }_t{A}_1-{\partial }_x{A}_0$. A unit Sine-Gordon kink has unnormalized charge $Q=2\pi$ and mass $8$. For spatially uniform forcing, its field momentum balance is $\dot{P}=QE$ when the boundary stress contributions cancel.

For $\tau =x+t$, $\rho =x-t$, the unit-mass Sine-Gordon equation is ${\varphi }_{\tau \rho }=\frac{1}{4}\sin \varphi$. An auto-Bäcklund transformation with these coordinates isDifferentiate each relation in the other coordinate and add or subtract; the sine addition formula gives ${\varphi }_{j,\tau \rho }=\frac{1}{4}\sin {\varphi }_j$ for $j=0,1$. Doubling both right-hand sides instead gives ${\varphi }_{j,\tau \rho }=\sin {\varphi }_j$, hence a sine-Gordon equation with mass squared four. Those doubled coefficients are appropriate for the half-scaled coordinates $(x+t)/2,(x-t)/2$.

If $A_r\to 0$, then for nonzero $k$ the weighted series is absolutely convergent and its tail after $n$ is bounded by $2{\sup }_{r>n}|A_r|/(n{k}^2)$. This follows from $|\sin (rk)|\le 1$ and the reciprocal-square tail bound. For $N=\lfloor 1/|k|\rfloor$, the prefix multiplied by $|k|$ tends to zero by splitting at a fixed index where $|A_r|$ becomes small. The multiplied tail is bounded by $2{\sup }_{r>N}|A_r|/(N|k|)$, which tends to zero. Thus the whole multiplied series tends to zero as $k\to 0$. Omitting $k^{-2}$ from the unmultiplied tail bound is invalid.