For the positively oriented boundary of a spherical triangle on the unit sphere, parallel transport is rotation by its angle excess modulo $2\pi$. The tangent is parallel along each geodesic side and turns by $\pi -\alpha$ at a corner of angle $\alpha$. The three corner turns sum to $3\pi -\alpha -\beta -\gamma$, and combining them with parallel transport returns the initial tangent. Hence the transport angle is the negative of that sum modulo $2\pi$, giving the formula. The spherical excess formula identifies the angle with enclosed area; reversing traversal changes its sign.

A vector field along a smooth curve is parallel if its covariant derivative along that curve is zero. Its value is the parallel transport of its initial value. A Levi-Civita connection preserves inner products under this transport, so the length of a parallel field is constant. Along a unit-speed curvature-one geodesic, the normal Jacobi field with zero initial value and initial derivative $a$ is $\sin t$ times the parallel transport of $a$.

In flat three-dimensional spacetime with polar spatial orthonormal coframe $(dT,d\rho ,\rho d\varphi )$, the spin connection has ${\omega }_{12}=-d\varphi$. Thus every parallel spinor has the displayed components. Since $({\gamma }^1{\gamma }^2{)}^2=-1$, these components change sign under $\varphi \mapsto \varphi +2\pi$. This is the lift of a full rotation to the spin structure extending over the disk; the polar frame itself does not extend over the origin. In a Cartesian frame the same parallel spinors have constant components.

Consider an approximate conditional likelihood in which a paired total $t_i$ leads to a structural zero with probability $\theta$, and otherwise a binomial distribution with success probability $p$. For an observed zero, the posterior cure weight is $q_i=\theta /[\theta +(1-\theta )(1-p{)}^{t_i}]$; for a positive count it is zero. Differentiating the log-likelihood gives the displayed equations at an interior maximum-likelihood estimator, with the weights evaluated at that estimator. Recomputing the weights and then updating the parameters is an expectation-maximization algorithm. Boundary fits must also be considered. These equations describe a conditional mixture: if cure is instead specified before two Poisson distributions are conditioned on their total, its conditional probability generally depends on the total and on the baseline nuisance rate. A zero observed in a short interval is not by itself proof of permanent biological cure.

Fix an ancillary statistic $a$ and a parameter value at which the score function is evaluated. Express that score as a function $u(v,a)$ of the varying maximum-likelihood estimator coordinate $v$. The ordinary density change-of-variables formula applied to the P-star approximation gives the displayed density; several monotone branches require a sum. Its derivative is a sample-coordinate derivative, not generally the parameter observed information.

For a size-$n$ sample from an exponential distribution of mean $\theta$, the likelihood in the maximum-likelihood estimator coordinate $v=\bar{Y}$ has difference $\ell (\theta ;v)-\ell (v;v)=n\log (v/\theta )-nv/\theta +n$, and the fitted observed information is $n/v^2$. Their P-star approximation is the displayed gamma density shape. Its normalization gives exactly the sampling density $n^n{v}^{n-1}{e}^{-nv/\theta }/(\Gamma (n){\theta }^n)$. This illustrates an exact transformation-model case rather than a general claim of exactness.

For independent samples of size $n$ from exponential distributions with rates $\lambda$ and $\psi \lambda$, the maximum-likelihood estimators are $\hat{\lambda }=n/S$ and $\hat{\psi }=S/T$, where $S$ and $T$ are the respective sample sums. Their normalized P-star approximation is exact:Indeed, $S$ and $T$ have independent gamma distributions with shapes $n$ and respective rates $\lambda ,\psi \lambda$. The Jacobian determinant of their inverse transformation is $n^2/({\hat{\psi }}^2{\hat{\lambda }}^3)$, giving the displayed density. Alternatively the square root of the determinant of the fitted observed information is $n/(\hat{\psi }\hat{\lambda })$; multiplying by the exponential of the fitted log-likelihood difference gives the same shape. Its normalizing constant in the P-star approximation is $c_n=n^{2n-1}{e}^{-2n}/\Gamma (n{)}^2$. Integrating over $\hat{\lambda }$ givesso $\hat{\psi }/\psi$ has the F-distribution with degrees of freedom $(2n,2n)$.

The total space of the orientation line bundle is orientable even when its base is not: an orientation reversal in a base coordinate transition is matched by a reversal in the line coordinate, so their product has positive sign. Fiberwise multiplication by $1-t$ gives a deformation retraction onto the zero section. For the real projective plane this total space is $(S^2\times \mathbb{R})/((v,t)\sim (-v,-t))$, an oriented noncompact three-manifold homotopy equivalent to the plane.

A faithful transitive order-primitive lattice permutation group is either a regular Archimedean translation action, an order two-transitive action, or a periodic action. In the periodic case a fixed-point-free order automorphism of the Dedekind completion commutes with the action and has cofinal and coinitial iterates; the point stabilizers act order two-transitively within period intervals. Only the regular case satisfies $|f||g|\le |g{|}^2|f{|}^2$ for every pair. In either other case, order two-transitivity on the whole chain or on a period interval supplies positive $a,b$ and $x<y<z<t$ with $xa=y$, $ya=z$, $xb=x$, $yb=t$, giving $xab=t>z=x{b}^2{a}^2$. The classification itself is a theorem, not a consequence of that witness calculation.

Suppose $h_{ij}$ sends the $i$th and $j$th nodes of an increasing input tuple to the corresponding output nodes. At the $i$th input, every $h_{ij}$ agrees, so their pointwise minimum takes the required value. At every other input it is bounded above by the required value, using the corresponding $j$ term. The pointwise maximum over $i$ therefore interpolates the entire tuple. The construction uses closure under finite join and meet, without assuming arbitrary restrictions of group elements belong to the group.

In a one-period utility problem with gross riskless return $R$, the marginal utility price is the infinitesimal claim price leaving optimized expected utility unchanged. If optimal terminal wealth is $W^{\ast }$, the pricing density is $Z=U^{\prime }(W^{\ast })/(R\mathbb{E}{U}^{\prime }(W^{\ast }))$, and $p_H=\mathbb{E}[ZH]$. The denominator is the marginal utility of initial wealth. Under the standard first-order optimality conditions this density prices the traded assets and gives the state-price density associated with the investor.

In the optimal initial state for triangular stable shear, expansion of $e^{t/R}$ gives the displayed leading parameter with relative error $O(t/R)$. Keeping both terms distinguishes a large-$R$ shear regime from the limit $t\to 0$ at fixed $R$. The latter direction is the largest-eigenvalue direction of the Hermitian part of $M$; the former can instead have large absolute time and substantial shear amplification even though $t/R$ is small.

At the equilibrium $v_i/\sqrt{\alpha }$, linearization along orthonormal eigenvectors of $C$ gives the displayed rates, with the first equation applying to $j\ne i$. A simple positive largest eigenvalue makes its two normalized eigenvectors locally asymptotically stable; smaller eigenvalues have an unstable direction. Repeated largest eigenvalues give a manifold of equilibria with neutral directions. Generic convergence also needs a nonzero initial component in the largest eigenspace. The squared norm satisfies $\tau d\Vert w{\Vert }^2/dt=2(w^{T}Cw)(1-\alpha \Vert w{\Vert }^2)$.

A Noetherian ring is normal exactly when it satisfies $(R_1)$, regularity in codimension one, and $(S_2)$, depth at least $\min (2,\dim )$ at every localization.

For an integral scheme, take on each affine chart the integral closure of its coordinate ring in its function field. Integral closure commutes with localization, so these rings glue to a normal scheme with an integral birational morphism to the original scheme. This is the normalization. It need not be finite without additional hypotheses; for an algebraic curve of finite type over a field it is finite. It is an isomorphism exactly when the original scheme is normal.

For a finite unramified extension of non-Archimedean local fields, the field norm on residue units is surjective: on a cyclic group of order $q^n-1$ it is exponentiation by $(q^n-1)/(q-1)$ and has image of order $q-1$. For a common uniformizer $\pi$ and $r\ge 1$, expansion of the conjugate product givesThe field trace of a finite finite field extension is onto: its polynomial $X+X^q+\cdots +X^{q^{n-1}}$ is nonzero of degree below $q^n$, so defines a nonzero linear map to the base field. Lift a residue norm and correct successively with elements of $1+{\pi }^r{\mathcal{O}}_L$. The infinite product converges by completeness, and its norm is the prescribed unit. This argument also works in positive characteristic and when $p$ divides $n$.

In homogeneous ideal magnetohydrodynamics, a constant longitudinal field $B_0>0$ and transverse field $a{B}_0(\cos \zeta ,\sin \zeta )$, $\zeta =k(z-v_{a}t)$, give an exact traveling wave with constant mass density and pressure, $u_z=0$, $v_a=B_0/\sqrt{{\mu }_0{\rho }_0}$ and the displayed transverse velocity. Induction and transverse momentum reduce to the same wave relation. Since $|{\mathbf{B}}_{\perp }{|}^2=a^2{B}_0^2$ is constant, there is no longitudinal magnetic pressure gradient. All nonlinear terms are compatible with the constant-density solution, even in a compressible fluid.

A finite sinusoidal transverse field $B_x=a{B}_0\cos [k(z-v_{a}t)]$, $B_y=0$, has varying magnetic pressure. In a purely transverse traveling Alfvén wave, longitudinal momentum requires ${\partial }_z[p+B_x^2/(2{\mu }_0)]=0$, but the adiabatic pressure equation with $u_z=0$ requires ${\partial }_{t}p=0$. A traveling compensating gas pressure cannot satisfy both conditions. The uncompensated force is quadratic in amplitude and drives a compressive response. A regular periodic one-dimensional traveling solution cannot evade this by varying mass density: its mass, transverse momentum and induction integrals force constant mass density at the zeros of the transverse field and hence throughout the nontrivial wave.

In a supersymmetry-preserving local Wilsonian effective action, the gauge kinetic function receives only a one-loop perturbative correction. For one simple gauge factor, $\mu df/d\mu =b_0/(8{\pi }^2)$ with $b_0=3C_2(G)-{\sum }_{i}T(R_i)$. A perturbative shift of the topological angle leaves the running coefficient invariant; the Cauchy-Riemann equations then make its holomorphic beta function independent of the gauge coupling. Holomorphic regularity and spurionic R-charges exclude higher-loop Yukawa combinations, which would require antichiral couplings. The constant is fixed by one-loop matching. This does not assert one-loop exactness of a canonically normalized physical coupling, or forbid nonperturbative terms.

// Target: quantum-field-theory.bigb

The momentum constraint relates the incoming scalar mode $a^{-1}\sqrt{\hslash /(2k)}{e}^{-ik\tau }$ to the Newtonian-potential Hankel modes in stiff contraction. Large-argument Hankel asymptotics give $|N_k{|}^2=3\pi \hslash /(32k^2)$. The small-argument behavior $H_1^{(1)}(z)\sim -2i/(\pi z)$ then gives the displayed spectrum. Its dimensional power is $k^{-4}$ and its power per logarithmic interval is proportional to $k^{-1}$. These are Newtonian-potential spectra; a curvature-perturbation spectrum need not have the same exponent.