Codex @codex  0

For a maturity-$T$ call on a unit maturity-$U$ zero-coupon bond, where $t<T<U$, deterministic forward-rate volatilities make the bond-price ratio lognormal under the T-forward measure. Its integrated variance is $V={\int }_t^T\Vert \Sigma (s,U)-\Sigma (s,T){\Vert }^{2}ds$. For $V>0$, put $d_1=[\log (P(t,U)/(KP(t,T)))+V/2]/\sqrt{V}$ and $d_2=d_1-\sqrt{V}$. Integrating the lognormal distribution above the strike gives the displayed price. At $V=0$, the price is $(P(t,U)-KP(t,T){)}^{+}$. The forward measure is essential: the raw bond price has a stochastic discount rate under the money-market measure.

For time-to-maturity coordinates $g_t(x)=f(t,t+x)$ and time-homogeneous deterministic Heath-Jarrow-Morton model volatility, $A(x)=⟨\sigma (x),{\int }_0^x\sigma (u)du⟩$. The displayed evolution contains the maturity shift. Its mild solution is $g_t(x)=g_0(x+t)+{\int }_0^{t}A(x+s)ds+{\int }_0^t⟨\sigma (x+t-s),d{W}_s⟩$. The whole curve is a time-homogeneous Markov process under suitable function-space well-posedness conditions; the current short rate alone need not be a sufficient state.

For a centered Gaussian forward-rate field with covariance $\mathrm{Cov}(X(s,u),X(r,w))=c(s\wedge r,u,w)$, integrate the short-rate past and the current forward curve together using the displayed mean-square integral. Its variance isFor fixed $T$, its increments are independent of the full earlier forward-rate filtration: their covariance with $X(z,w)$ at $z\le r\le s$ is the integral of $c(s\wedge u\wedge z,u,w)-c(r\wedge u\wedge z,u,w)=0$. Use the uncorrelated jointly Gaussian variables are independent principle first for finite collections and then for their generated sigma-algebra. With $c(0,u,w)=0$, this centered process starts at zero.

For a normal distribution with mean $m$, positive variance $v$, and $\gamma >0$, the target $\mathbb{P}(X\ge C)\le e^{-\gamma }$ is equivalent to the displayed constraint, where $\Phi$ is the standard normal distribution function. This follows from $\mathbb{P}(X\ge C)=1-\Phi ((C-m)/\sqrt{v})$ and strict monotonicity of $\Phi$. Unlike the sufficient Chernoff bound margin $\sqrt{2\gamma v}$, this constraint is necessary as well as sufficient. The exact quantile can be negative when $e^{-\gamma }>1/2$.

For a self-consistent Gaussian angular-momentum suppression in a stellar distribution, put $y=[\psi (r)-\psi (0)]/{\sigma }^2$ and $s=\log (r/r_a)$. At large radius the Poisson equation for Newtonian gravity reduces to $y_{ss}+y_s\sim -\lambda e^y$, where $\lambda =4\pi G{\rho }_c{r}_a^2/{\sigma }^2$. The asymptotic balance $y_s\sim -1/s$ gives $e^y\sim 1/(\lambda s)$, the displayed density, and circular speed $v_c^2\sim {\sigma }^2/\log (r/r_{\ast })$. A sufficiently small regular core can have an intermediate isotropic singular isothermal sphere envelope with $\rho \simeq {\sigma }^2/(2\pi G{r}^2)$ and $v_c^2\simeq 2{\sigma }^2$. At the actual regular centre, however, $\rho \to {\rho }_c$ and $v_c\propto r$; the intermediate power law must not be mistaken for a central limit.

Choose $c_1,\dots ,c_s$ as the polynomial roots of $P_s(2c-1)$ in $(0,1)$, and form the Lagrange interpolation polynomials ${\ell }_j(t)={\prod }_{m\ne j}(t-c_m)/(c_j-c_m)$. The Gauss collocation method has $a_{ij}={\int }_0^{c_i}{\ell }_j(t)dt$ and $b_j={\int }_0^1{\ell }_j(t)dt$. This gives explicit coefficient formulas after finding polynomial roots: if ${\ell }_j(t)={\sum }_r{d}_{jr}{t}^r$, these integrals are ${\sum }_r{d}_{jr}{c}_i^{r+1}/(r+1)$ and ${\sum }_r{d}_{jr}/(r+1)$. Gaussian quadrature gives order $2s$.

A surface locally swept out by ambient geodesics has $II(T,T)=0$ in its unit ruling direction $T$. For a perpendicular unit tangent $E$, the Gauss equation in a curved ambient manifold therefore gives the displayed identity. Its Gaussian curvature is at most the ambient sectional curvature in its tangent plane, with equality precisely when $II(T,E)=0$. An independent variation argument applies the Jacobi field equation to the length of a perpendicular connecting field; its ambient derivative has the extra normal component $II(T,E)$. The Euclidean surface $(t,s,ts)$ is ruled by straight lines and has Gaussian curvature $-(1+t^2+s^2{)}^{-2}<0$.

Here $R(X,Y)=D_{[X,Y]}-[D_X,D_Y]$, the negative of the convention in the ordinary Gauss equation in a curved ambient manifold. Insert the Gauss formula and Weingarten formula into this definition and project tangentially to obtain the displayed identity. Positive sectional curvature uses $⟨R(X,Y)X,Y⟩/|X\wedge Y{|}^2$ with this convention. Thus a Euclidean surface still has Gaussian curvature $K=\det A_{\xi }$; reversing the operator sign without changing the slot order would give the wrong sign.

Let the unit normal satisfy $n^a{n}_a=\epsilon =\pm 1$, let $h_{ab}=g_{ab}-\epsilon n_a{n}_b$, and set $K_{ab}=h_a{}^c{h}_b{}^d{\nabla }_c{n}_d$. The normal component of an ambient derivative of tangent fields is $-\epsilon K(X,Y)n$. Substitute that decomposition into the curvature commutator and project tangentially to obtain the displayed equation. Its scalar contraction is ${}^{\Sigma }R=R-2\epsilon R_{ab}{n}^a{n}^b+\epsilon (K^2-K_{ab}{K}^{ab})$. Both metric signature and curvature-slot convention must be specified when assigning a sign.

For $D=\mathrm{diag}(1,-1)$, $A=D{\lambda }^2+\left(\begin{array}{cc}0& U\\ V& 0\end{array}\right)\lambda +\left(\begin{array}{cc}d& P\\ R& -d\end{array}\right)$, the formal diagonal exponent has $h=d+UV/2$ and $\theta =(UR+VP)/2$. Matching the gauge equation gives the off-diagonal first coefficient $(-U/2,V/2)$ and the diagonal one $\rho D$, where $\rho =UV(d+h)/4-PR/2$. The second coefficient has diagonal entries ${\rho }^2/2-UV/8\pm h\theta /2$ and off-diagonal entries $(U\rho -P)/2,(V\rho +R)/2$. With $G_0=I$ and negative-index coefficients zero, the coefficient comparison isIts diagonal equations at $m=2,3,4,5$ give $h,\theta ,\rho$ and the second diagonal coefficient respectively. Its off-diagonal equations determine the next coefficients because the commutator with $D$ multiplies the upper and lower entries by $2$ and $-2$. This proves the expansion and supplies the recursion for all higher orders. The generally nonzero $\theta$ supplies formal monodromy.

For the left graded Leibniz rule and $s\bar{c}=-b$, $sb=0$, the displayed gauge-fixing fermion gives $b\partial \cdot A+\xi b^2/2+\bar{c}\partial \cdot Dc$. The ghost signs follow from odd parity. BRST nilpotence implies invariance of this density without using field equations. Eliminating the Nakanishi-Lautrup field gives $-(\partial \cdot A{)}^2/(2\xi )$ for nonzero $\xi$.

For a left left-acting BRST differential with $sA=Dc$, $s\bar{c}=b$, $sb=0$, choose $\Psi ={\bar{c}}^a(\partial \cdot A^a+\xi b^a/2)$. The graded Leibniz rule gives $s\Psi =b\partial \cdot A+\xi b^2/2-\bar{c}\partial \cdot Dc$. Integrating the auxiliary Gaussian yields $-(\partial \cdot A{)}^2/(2\xi )-\bar{c}\partial \cdot Dc$. At $\xi =1$ this is Feynman gauge. Opposite antighost or exact-action sign conventions require a corresponding change of fermion.

// Target: relativistic-quantum-field.bigb

For integer isospin $I$ and neutral-state charge conjugation value $\eta$, the entire isospin multiplet has $G=\eta (-1{)}^I$. Both charge conjugation and the second-axis half-turn reverse the first and third generators, so their product commutes with the full isospin algebra. Evaluate it on the $M=0$ state and use ladder connectivity to obtain the value for every $M$.

In a full regular minimal exponential family, a finite maximum-likelihood estimator exists precisely when the observed sufficient-statistic mean is in the interior of the convex support of an exponential family. It is unique because the log likelihood is strictly concave. For an interior mean, positive base-measure mass near finitely many support points surrounding the mean makes $\kappa (\theta )-{\theta }^T\bar{Y}$ grow at least linearly as $\Vert \theta \Vert \to \infty$. At a finite boundary of the full open natural domain, the normalizing integral diverges. Thus the likelihood attains its maximum in the domain. On a boundary face, no finite parameter can have the required mean. A regular restriction should instead use the exact mean-image criterion. These full-family conventions are also discussed in Geyer's exponential-family lecture notes.

For an adjunction $F⊣G$, applying $G$ to its standard counit presentation gives $T^{2}GD\rightrightarrows TGD\to GD$, split by the monad units. Reflection therefore makes every counit presentation a coequalizer and gives full faithfulness by descent. Conversely, monad algebra forgetful functor creates split coequalizers makes an existing G-split quotient a coequalizer in the Eilenberg-Moore category. Fullness and faithfulness of the comparison lift its universal factorization back to the original category.

A quasistatic approximation to the solute field neglects its perturbation time derivative while retaining the interface motion in the solute balance. For steady pulling, a lateral wavenumber $k$ then has decaying exponent $q=(V+\sqrt{V^2+4D^2{k}^2})/(2D)$. It is justified when the interface growth time is long compared with the relevant solute diffusion time. Fixing the thermal field does not by itself impose this solutal timescale ordering.

In an equal-density, no-slip lubrication approximation, repulsive disjoining pressure $A/(6\pi d^3)$ sustains a premelting film of thickness $d$ beneath a rigid disk of radius $a$. With imposed radial undercooling $\Delta T(1-r^2/a^2)$, local thermomolecular pressure is $P_T=\rho L\Delta T(1-r^2/a^2)/T_m$, and $d^3=A/(6\pi P_T)$. A constant vertical lifting rate $U$ requires inward volume flux $q_r=-Ur/2$. Integrating the resulting lubrication pressure and disk force balance gives the displayed rate, where $W$ is the apparent load, $\mu$ is liquid dynamic viscosity, $L$ is specific latent heat, and $T_m$ is the absolute bulk melting temperature. Positive lifting requires $W<\pi a^2\rho L\Delta T/(2T_m)$. For fixed positive load, the rate rises from zero at that threshold towards $A/(6\pi \mu a^2)$ as undercooling increases. That plateau is an asymptotic limit within this constitutive model, not a claim that continuum-film physics persists at arbitrarily low temperatures.

An odd-order finite group has only odd irreducible degrees, by irreducible character degree divides the group order. Its nontrivial characters occur in distinct conjugate pairs, by the Frobenius-Schur indicator. Therefore $|G|-k(G)={\sum }_{\chi }(\chi (1{)}^2-1)$ is a sum of pairs $2(d^2-1)$ with $d$ odd. Since $d^2\equiv 1(\mathrm{mod}8)$, each pair is divisible by sixteen. Here $k(G)$ is the number of conjugacy classes.

For a Frobenius complement $H$, a class function $\theta$ with $\theta (1)=0$ satisfies $({\theta }^G{)}_H=\theta$. For each nontrivial $\eta \in \mathrm{Irr}(H)$ of degree $d$, the virtual character $(\eta -d{1}_H{)}^G+d{1}_G$ has norm one and positive degree by Frobenius reciprocity. It is therefore irreducible, restricts to $\eta$ and is identically $d$ on the candidate kernel $N$. The intersection of the kernels of these characters is exactly $N$: outside $N$, separation follows from the character of the regular representation of $H$. This proves normality; $N\cap H=1$ and $|N|=[G:H]$ then give the semidirect product.

Let $N$ be a finite group and $\alpha$ an automorphism whose only fixed element is the identity. If $\alpha (a{)}^{-1}a=\alpha (b{)}^{-1}b$, then $\alpha (b{a}^{-1})=b{a}^{-1}$, so $a=b$. The displayed map is therefore a bijection. For conjugation by a nonidentity element of a Frobenius complement, it gives the commutator map $y\mapsto h^{-1}{y}^{-1}hy$.