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.

The nerve of a category $\mathcal{C}$ hasso an $n$-simplex is a string of $n$ composable morphisms. Every inner horn in a nerve has a unique filler.

The boundary distance is the intrinsic distance between boundary points. Its infinitesimal limit along boundary curves recovers the induced metric on the boundary tangent bundle. It does not directly fix the normal and mixed metric components, which can be changed by boundary-fixing diffeomorphisms.

For an inward unit boundary vector, the geodesic scattering relation returns the next boundary point and its outgoing unit velocity. On outward vectors it can be extended using the first backward exit, giving an involution on the full boundary unit bundle, with tangential vectors fixed. Exit time is separate lens data; on a simple manifold it equals boundary distance between the endpoint pair.

With a nonzero hidden period, Simon's algorithm samples uniformly from its $(n-1)$-dimensional orthogonal complement over the binary field. At rank $k$, a new sample raises the rank with probability $1-2^{k-(n-1)}$. Summing the geometric waiting times gives an expected $n-1+{\sum }_{j=1}^{n-1}(2^j-1{)}^{-1}<n+1$ queries. After $n-1+c$ samples, the probability of insufficient rank is less than $2^{-c}$, by a union bound over the nonzero linear functionals that could annihilate all samples.

For $f(z)=\mu +A\mathrm{sgn}(z)|z{|}^{\delta }$, a nonzero fixed point satisfiesAssume $A>0$, $\delta >0$ and $\delta \ne 1$. A saddle-node bifurcation has multiplier $f^{\prime }(z)=A\delta |z{|}^{\delta -1}=1$, so it occurs at $z=\pm r$ withAt either nonzero point, $f^{\prime \prime }$ is nonzero and the parameter enters additively, giving the ordinary nondegenerate fold conditions.

When $0<\delta <1$, define $C=(1-\delta )r/\delta$. The scalar extension has three fixed-point branches for $|\mu |<C$: two stable outer branches and an unstable inner branch. For nonzero fixed points, their derivatives are below and above one because $A\delta |z{|}^{\delta -1}$ decreases with $|z|$. At $\mu =0$, the inner branch lies at the singular separatrix value; the signed gluing-map reduction explains why it is excluded as a finite-time flow return.

For fixed $0<A<1$ and $\delta \uparrow 1$ from below, the fold magnitude has asymptotic value $C\sim (1-\delta )A^{1/(1-\delta )}/e$. This is the half-width; the total parameter interval has width $2C$. The exponentially narrow gluing-map cusp gives the exact rescaling. For $\delta >1$ approaching one, the formal fold coordinate instead tends to infinity, outside the local return-map region.

For an integrable function $f$ on $[a,b]$, the total geometric area between its graph and the horizontal axis is ${\int }_a^b|f(x)|dx$. The absolute value makes portions above and below the axis contribute positively, unlike the signed area under a graph ${\int }_a^{b}f(x)dx$.

For $f(x)=x$ on $[-1,1]$, the signed integral is zero, whereas the total geometric area is $2{\int }_0^{1}xdx=1$. To evaluate it by antiderivatives, split at sign changes and reverse the sign on intervals where $f$ is negative.

For $d{r}_t=a_{0}dt+b_{0}d{W}_t$ under a risk-neutral measure, the instantaneous forward rate and zero-coupon bond prices areIndeed, conditional on current information, ${\int }_t^T{r}_{s}ds$ is Gaussian with mean $(T-t)r_t+a_0(T-t{)}^2/2$ and variance $b_0^2(T-t{)}^3/3$. Its negative exponential expectation gives the bond formula, and differentiating its logarithm gives the instantaneous forward rate. Equivalently the affine forward-rate equation has $A^{\prime }=0$, $A(0)=1$, and $B^{\prime }=a_0-b_0^2\theta$, $B(0)=0$. This is the constant-drift instance of the Ho-Lee Gaussian short rate model; the model permits negative short rates.

Under a risk-neutral measure $Q$, let the short rate be $r_t=g(t)+\sigma W_t^Q$, with deterministic locally integrable $g$ and constant $\sigma >0$. For a zero-coupon bond paying one at $T$, the risk-neutral pricing value isIndeed, conditional on ${\mathcal{F}}_t$, the random part of the integrated future short rate is $\sigma {\int }_t^T(T-u)d{W}_u^Q$, which is Gaussian with mean zero and variance ${\sigma }^2(T-t{)}^3/3$. Its exponential moment gives the formula. At points where $g$ is continuous, the instantaneous forward rate isFor merely locally integrable $g$, the same identity holds almost everywhere. The model allows negative short rates and gives explicit bond prices and forward curves.

A Gaussian short rate model with Ornstein-Uhlenbeck process dynamics under the chosen pricing measure. For $a>0$, its unit zero-coupon bond value is $P(t,T)=A(\tau )e^{-B(\tau )r_t}$, where $\tau =T-t$, $B(\tau )=(1-e^{-a\tau })/a$ andSolve the linear stochastic differential equation, integrate over future time, and use the exponential moment of the resulting Gaussian integral to obtain this price. Its Gaussian law permits negative short rates.

A one-factor short-rate model specifies $d{r}_t=\mu (t,r_t)dt+\eta (t,r_t)d{W}_t$ with deterministic coefficient functions. All zero-coupon bond prices are functions of time, maturity and this single scalar state. The physical drift and pricing drift differ by $\eta \lambda$, where $\lambda$ is the market price of risk.

The Gaussian Hull-White model has pricing dynamics $d{r}_t=(\beta (t)-a{r}_t)dt+\eta d{W}_t^Q$. Its deterministic drift function can fit the initial yield curve exactly, while its Ornstein-Uhlenbeck process part controls stochastic variation. Like the Vasicek model, it permits negative rates.