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If $p\equiv 11(\mathrm{mod}12)$ and $K=\mathbb{Q}(\sqrt{-p})$, then ${\mathcal{O}}_K=\mathbb{Z}[(1+\sqrt{-p})/2]$. The generator's minimal polynomial reduces modulo $3$ to $T(T-1)$, so $(3)$ splits into two distinct prime ideals of norm three.

Ramification specializes to primes as follows: a prime $p$ ramifies in a number field when its prime-ideal factorization contains a factor with exponent greater than one.

Under $H=-\gamma B{S}_z$, a coherent spin direction precesses around the $z$-axis with angular velocity $-\gamma B$.

At azimuth $\phi$, the normalized spin-$3/2$ coherent-state components are proportional to $1,\sqrt{3}{e}^{i\phi },\sqrt{3}{e}^{2i\phi },e^{3i\phi }$.

The Press-Schechter formalism estimates the halo abundance from the Gaussian probability that a smoothed linear overdensity crosses the spherical-collapse threshold, with a factor of two enforcing total mass conservation.

The peak height of a halo of mass $M$ is $\nu ={\delta }_c/\sigma (M)$, where ${\sigma }^2(M)$ is the variance of the linear density field smoothed on the corresponding mass scale.

When viscous drag creates a large axial tension $T$, a clamped filament turns toward its outer direction in a boundary layer of length $\sqrt{A/T}$.

The full spectrum of an element is its spectrum together with every bounded component of its complement. Its complement is the unbounded resolvent component, and the maximum modulus principle controls a holomorphic function on each filled hole by its values on the spectral boundary.

For $x$ in a unital Banach algebra and $f$ holomorphic near $\sigma (x)$, the holomorphic functional calculus defineswhere $\Gamma$ winds once around the spectrum. It is a continuous unital algebra homomorphism and satisfies the spectral mapping theorem $\sigma (f(x))=f(\sigma (x))$.

If a closed unital subalgebra $A\subseteq B$ contains $x$, then ${\sigma }_A(x)$ is obtained from ${\sigma }_B(x)$ by adjoining some bounded connected components of its complement. Membership of $(x-\lambda 1{)}^{-1}$ in $A$ is constant on every connected component of the resolvent set, and it always holds on the unbounded component.

The resolvent is $R(\lambda ,a)=(\lambda 1-a{)}^{-1}$ on the complement of the spectrum of an element. In a Banach algebra it is analytic there.

Ifthen truncating the sum after $N$ terms gives finite-rank Hermitian $T_N$ with

A Matsubara correlator is obtained from the retarded spectral density by a Cauchy transform evaluated at an imaginary frequency.

For $L(y;\lambda )=y^{m+1}$ near a normalized simple null mode $y_0$, put $y=\epsilon y_0+O({\epsilon }^2)$ and $\lambda -{\lambda }_0={\epsilon }^m\mu$. Orthogonality to $y_0$ yields

For $1\le p<\infty$, no bounded map from $L^p({\mathbb{R}}^n)$ to $L^p({\mathbb{R}}^{n-1})$ can agree with restriction on continuous functions. Indeed,has a fixed nonzero trace while $\Vert u_k{\Vert }_p=k^{-1/p}\Vert \varphi {\Vert }_p\Vert \psi {\Vert }_p\to 0$.

For $1\le p<\infty$, boundary restriction on smooth functions extends uniquely to a bounded linear map $T:W^{1,p}({\mathbb{R}}_{+}^n)\to L^p({\mathbb{R}}^{n-1})$. The estimate follows by applying the fundamental theorem of calculus on normal line segments and then integrating over the boundary variables.

A Hölder space consists of functions satisfying a Hölder condition. A bounded function belongs to $C^{0,\alpha }({\mathbb{R}}^n)$ when

Translations of one compactly supported bump have equal Sobolev norms. Widely separated translates have disjoint supports and remain a fixed positive distance apart in $L^2$, obstructing compactness on an unbounded domain.

The Gagliardo-Nirenberg interpolation inequalities bound an intermediate derivative norm by powers of a higher derivative norm and a lower norm. In two dimensions, one useful case is

If a periodic function $u$ has zero mean, its nonzero Fourier frequencies giveThus inversion of the Laplacian on mean-zero periodic functions gains two Sobolev derivatives.