For two unbiased estimators $A,B$ of the same parameter with variances $V_X,V_Y$ and covariance $C$, the fixed-weight estimator $wA+(1-w)B$ is unbiased and has variance $w^2{V}_X+(1-w{)}^2{V}_Y+2w(1-w)C$. Completing the square gives $w_{\ast }=(V_Y-C)/(V_X+V_Y-2C)$ and minimum variance $(V_X{V}_Y-C^2)/(V_X+V_Y-2C)$ when the denominator is positive. These are optimal among fixed linear unbiased combinations. Substituting weights estimated from the same observations need not preserve unbiasedness.

For standard Cauchy distribution probability density function $f$ and $k>0$, this estimates its upper-tail probability without sampling a heavy tail. The function $f(ku)$ decreases on $[0,1]$, while $f(k(1-u))$ increases, so their covariance is nonpositive. The paired estimator is unbiased and has no larger variance than an estimator using the same number $2r$ of independent probability density function evaluations.

For an antisymmetric payoff matrix, such a vector guarantees that an opponent playing rows cannot win positively against it. Antisymmetry also gives $p^{T}A\ge 0$, so the same mixed strategy guarantees nonnegative payoff as a row strategy and proves game value zero. Furthermore $p^{T}Ap=0$ forces $(Ap{)}_i=0$ whenever $p_i>0$. To prove uniqueness, use one certified strategy $q$: any other optimal $p$ satisfies $q^{T}Ap=-(Aq{)}^{T}p$, so a strict negative component of $Aq$ forces the corresponding probability in $p$ to vanish.

A harmonic conjugate $\varphi$ satisfying ${\varphi }_{,1}=w_{,2}$ and ${\varphi }_{,2}=-w_{,1}$ makes $F=\varphi +iw$ a holomorphic function. Its derivative packages the two antiplane shear stresses into the displayed complex field. On multiply connected domains a global potential additionally requires the appropriate periods to vanish; prescribed displacement jumps can instead be represented by branches of the potential.

For subsonic motion $|V|<c$, an antiplane shear displacement $w(x_1-Vt,x_2)$ satisfies ${\beta }^2{w}_{xx}+w_{x_2{x}_2}=0$. The displayed rescaling makes it a harmonic function of $(x,y)$, so locally $w=\mathrm{Re}H(x+iy)$ for a holomorphic function $H$. Set $F=\beta H$. The Cauchy-Riemann equations then give $\mu F^{\prime }=\beta {\sigma }_{13}-i{\sigma }_{23}$. Keeping this potential normalization explicit avoids losing a factor of $\beta$ in the dynamic stress field. At sonic or supersonic speeds the elliptic harmonic reduction does not apply.

A centred fan consists of straight characteristic curves emanating from one point. Stress and antiplane velocity are constant on each ray and depend on the polar angle $\varphi$. The radial tangent condition is $(\cos \varphi ,\sin \varphi )\parallel (-f_b,f_a)$, so $f_a\cos \varphi +f_b\sin \varphi =0$. A boundary ray carrying a traction-free crack-face stress $b=0$ has slope $\tan {\varphi }_b=-f_a/f_b$, interpreted as vertical when $f_b=0$. For the elliptic yield condition $a^2/A^2+b^2/B^2=1$, a forward fan has $a=-A^2\sin \varphi /H$, $b=B^2\cos \varphi /H$, $H=(A^2{\sin }^2\varphi +B^2{\cos }^2\varphi {)}^{1/2}$. The boundary rays are vertical; the adjacent rear regions have constant stresses $(\mp A,0)$. Velocity is a function of angle whose magnitude requires further loading data.

An endomorphism $J$ that is an antilinear map with $J^2=-I$ equips a complex vector space with a quaternionic structure: complex scalar multiplication supplies the quaternion generator $i$ and $J$ supplies $j$, satisfying $Ji=-iJ$. In finite dimensions the complex dimension must be even. Indeed, there is no $J$-invariant complex line: $Jv=cv$ for $v\ne 0$ would imply $-v=|c{|}^{2}v$. Consequently $v,Jv$ span a $J$-invariant complex two-plane; passing to its quotient vector space and repeating proves evenness.

In complex dimension four the invariant complex two-planes are quaternionic lines, forming the quaternionic projective space ${\mathbb{HP}}^1\cong S^4$. This is the Euclidean reality structure on twistor space used in twistor theory.

On a regular bipartite graph with coordination number $q$, replace each bond product by ${\sigma }_B{\sigma }_W\simeq M_W{\sigma }_B+M_B{\sigma }_W-M_B{M}_W$. For $N$ sites there are $Nq/2$ bonds, so the constant in the effective Hamiltonian is $-NqJ{M}_B{M}_W/2$. The mean-field self-consistency equations are $M_B=\tanh \beta (h+g-qJ{M}_W)$ and $M_W=\tanh \beta (h-g-qJ{M}_B)$. At zero fields the antisymmetric branch has a continuous mean-field Ising critical bifurcation at $k_B{T}_c=qJ$.

The stable bundle consists of the transverse directions exponentially contracted in positive time. The unstable bundle has the corresponding negative-time property. A positive time change of a smooth flow can tilt these bundles in the flow direction, even though it preserves the underlying orbits.

The unstable bundle consists of transverse directions exponentially contracted in negative time. Together with the stable bundle and the flow direction it gives the invariant splitting of an Anosov flow. Under a positive time change its graph coefficient is obtained from the convergent past integral $-\rho (x{)}^{-1}{\int }_{-\infty }^{0}d\rho (D{\varphi }_{r}v)dr$.

The Livšic theorem turns periodic-orbit data into a global derivative along a hyperbolic flow. Let ${\Phi }_t$ be a transitive Anosov flow on a compact smooth manifold, with generating vector field $F$, and let $h$ be a real Hölder continuous observable. There is a Hölder continuous function $u$, differentiable along flow lines, satisfyingif and only if ${\int }_0^{T}h({\Phi }_{t}p)dt=0$ for every periodic orbit $p$ of period $T$. The solution preserves the Hölder exponent.^[ref]

Necessity is immediate by putting ${\Phi }_{T}p=p$ in the identity. Two solutions differ by a continuous invariant function. Transitivity supplies a dense orbit; their difference is constant on that orbit, and continuity makes it constant everywhere. Thus the solution is unique up to an additive constant.

The harder sufficiency argument defines $u({\Phi }_t{x}_0)$ by integrating $h$ along a dense orbit. The Anosov closing lemma converts a sufficiently close return into a nearby periodic trajectory. Its vanishing integral and exponential shadowing bound the discrepancy between the endpoint values in terms of their distance. This gives Hölder control on the dense orbit and permits extension to the whole manifold.

When both the flow and $h$ are smooth, a separate regularity theorem gives smooth $u$. This does not follow merely from the preceding continuity argument.^[ref] The theorem is named for A. N. Livšic.

Invariance gives $dh(v_s)=dh(D{\varphi }_t{v}_s)$ for a stable direction. Boundedness of $dh$ and exponential contraction make this zero; negative time does the same for unstable directions. Also $dh(F)=0$. The Anosov splitting spans the tangent bundle, so $dh=0$ and the function is constant on each connected component. This proof does not need an ergodicity hypothesis.

Let a nonsingular smooth flow on a closed manifold $N$ be generated by $F$. Suppose a continuous nondegenerate indefinite quadratic form $Q$ on the transverse vector bundle $TN/\mathbb{R}F$ has constant signature, is differentiable along the linearized flow, and has continuous positive-definite derivative there. Then the flow is an Anosov flow.^[ref]

Here differentiating the form means differentiating $Q_{{\Phi }_{t}x}(D{\Phi }_{t}v)$. Compactness gives constants $M,c>0$ such that $|Q(v)|\le M\Vert v{\Vert }^2$ and $\dot{Q}(v)\ge c\Vert v{\Vert }^2$. On the positive cone,so the form increases at least exponentially. Negative cones supply the analogous backward-time estimate. Strict cone invariance produces uniformly expanding and contracting transverse bundles; the full splitting also contains the neutral flow direction.

For a geodesic flow on a compact negatively curved surface, a transverse Jacobi field has scalar coordinates satisfying $\dot{y}=z$ and $\dot{z}=-K(t)y$, with Gaussian curvature $K\le -\kappa <0$. The indefinite form $Q(y,z)=yz$ obeysfor every nonzero transverse vector. This directly verifies the criterion. The quotient by $F$ is essential: $D{\Phi }_{t}F=F\circ {\Phi }_t$ gives the neutral direction that cannot satisfy this strict transverse positivity. The estimates concern the form; constructing the invariant bundles is the additional cone argument, rather than an assertion that every positive-cone vector has monotonically growing norm.

The right annihilator of a subset $X$ of a ring is the right ideal of elements killed on the left by every member of $X$. Its sidedness matters for a noncommutative ring.

For a Gaussian kernel $K{q}_{\parallel }^2+L{q}_{\perp }^4+t$, keeping both gradient coefficients fixed gives $q_{\parallel }^{\prime }=b{q}_{\parallel }$ and $q_{\perp }^{\prime }=b^{1/2}{q}_{\perp }$. The volume rescales by $b^{(d+1)/2}$. Fourier-field normalization is $m^{\prime }(q^{\prime })=m(q)/b^{(d+5)/4}$; the real-space normalization instead divides by $b^{(3-d)/4}$. The thermal coefficient scales by $b^2$ and the uniform source by $b^{(d+5)/4}$. Thus ${\xi }_{\parallel }\sim t^{-1/2}$, ${\xi }_{\perp }\sim t^{-1/4}$ and the formal Gaussian free-energy powers are $\alpha =(7-d)/4$, $\Delta =(d+5)/8$.

For a flat universe with a single constant equation-of-state parameter $w>-1$, $H(z)=H_0(1+z{)}^{\alpha }$. Its angular diameter distance is $H_0^{-1}[(1+z{)}^{-\alpha }-(1+z{)}^{-1}]/(1-\alpha )$, with logarithmic limit $H_0^{-1}\log (1+z)/(1+z)$ at $\alpha =1$. Differentiation gives the displayed unique maximum, interpreted as $z_m=e-1$ at $\alpha =1$. The distance vanishes at both endpoints $z=0$ and $z\to \infty$. As $w\downarrow -1$, the maximum moves to infinite cosmological redshift.

Every algebra character of the analytic Wiener algebra is evaluation at a unique point $z$ of the closed unit disk. Indeed, for $u(w)=w$, continuity and multiplicativity give $z=\varphi (u)$, $|z|\le \Vert u\Vert =1$ and $\varphi (f)={\sum }_{n\ge 0}{a}_n{z}^n$. Conversely, this formula defines a continuous algebra character for every such $z$. The evaluation correspondence is a homeomorphism for the Gelfand topology, since the power series are uniformly convergent on the closed unit disk.

A boundary of a domain of a holomorphic function is natural if the function has no analytic continuation through any of its points. For $f(z)={\sum }_{n\ge 1}{z}^{2^n}$, each root of unity of order a power of two is a singular boundary point: along $z=r\zeta$ all sufficiently late terms equal $r^{2^n}$ and their sum tends to infinity as $r\uparrow 1$. These roots are dense on the unit circle, so no boundary arc admits analytic continuation.

For a non-dividend-paying stock and nonnegative deterministic interest, the European call lower bound is $C_E\ge S-XB$, where $B\le 1$ is the zero-coupon bond price. This is at least the in-the-money exercise payoff $S-X$; the call also preserves the possibility of avoiding payment of the strike. Positive interest makes the comparison strict when exercise would have positive payoff. At zero interest there can be indifference, and with negative interest the statement can fail: a deterministic stock $S_u=S_0{e}^{ru}$ with $r<0$ can give a larger payoff by immediate exercise.

For a single known cash dividend at $t_1$, nonnegative interest excludes exercise strictly before the last cum-dividend opportunity. Immediately before the dividend, continuation has lower bound $S_{t_1-}-D-XB(t_1,T)$, whereas positive immediate exercise pays $S_{t_1-}-X$. The displayed inequality makes continuation strictly better. With no further dividends, it therefore excludes all early exercise. If the inequality fails, exercise just before the dividend may be worthwhile, but is not automatically optimal because remaining option time value also matters.