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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.

On the new subspace of cusp forms, a common eigenspace for the Hecke operators away from the level has dimension one. Distinct newforms, even at different primitive levels, have different systems of these eigenvalues. Consequently the commuting bad-prime operators also act by scalars on these lines. This is the multiplicity-one part of Atkin-Lehner-Li theory, not a consequence of normal operator diagonalization alone. See Stein's statement of the theorem.

For a newform of weight $k\ge 2$, level $M$ and trivial Dirichlet character, the bad-prime Hecke operator $U_p$ has the displayed possibilities. The local trace relation on the new subspace of cusp forms gives $U_p=-p^{k/2-1}{W}_p$ when $p\parallel M$, where the normalized local Atkin-Lehner operator squares to one; if $p^2\mid M$, the new-vector relation gives $U_p=0$. In particular $|a_p|<p^{k/2}$, a strict inequality useful for proving irreducibility of old blocks.

Here ${\log }^{+}x=\max (0,\log x)$. The function $m(r,f)$ measures logarithmic proximity of $f$ to infinity. For a finite target $a$, proximity is $m(r,1/(f-a))$. Logarithmic singularities on a circle are integrable.

The number $n(t,f)$ counts poles in $|z|<t$ with multiplicity, and $n(0,f)$ is the pole order at zero. Thus $N(r,f)=n(0,f)\log r+{\sum }_{0<|p|<r}\log (r/|p|)$. Applying this to $1/(f-a)$ counts occurrences of the value $a$.

The characteristic of a meromorphic function is the sum of its Nevanlinna proximity function and Nevanlinna integrated counting function. It treats large values and actual poles together. Replacing $f$ by $f-a$ changes the characteristic by at most $\log (1+|a|)$.

For a fixed finite target $a$, the displayed identity compares growth with proximity to and occurrences of $a$. If $g(z)=c{z}^{\nu }(1+O(z))$ near zero, where $\nu$ is its signed zero order, the exact reciprocal identity is $T(r,g)-T(r,1/g)=\log |c|$. It follows by factoring out zeros and poles on a disc, applying the mean value property for harmonic functions to the remaining nonvanishing factor, and using ${\log }^{+}u-{\log }^{+}(1/u)=\log u$. Translation changes $T$ by a bounded amount, proving the general assertion.

For a nonconstant meromorphic function, its deficiency at a point of the Riemann sphere is the displayed asymptotic proportion missing from the full Nevanlinna integrated counting function. The Nevanlinna first main theorem bounds $N(r;a)$ by $T_f(r)+O(1)$ and identifies deficiency with the lower limiting normalized proximity. An omitted value has deficiency one.