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For a coherent sheaf $\mathcal{F}$ of support dimension $d$ on a projective scheme and a fixed Cartier divisor $D$, $h^0(X,\mathcal{F}(mD))=O(m^d)$. Choose a sufficiently high ample divisor $A$ whose section avoids the associated points of $\mathcal{F}$ and for which $D+A$ is ample. Multiplication by the $m$th power of that section injects $\mathcal{F}(mD)$ into $\mathcal{F}(m(D+A))$. The ample Hilbert polynomial, together with Serre vanishing, bounds the latter section space by $O(m^d)$.

The splitting field has Galois group $S_3$ and quadratic subfield $\mathbb{Q}(\sqrt{-59})$. The extension over that subfield is abelian of degree three and unramified everywhere. The only possible rational ramified prime is $59$; at that prime a simple residue root lifts, so the local splitting field is quadratic and coincides with the quadratic subfield's completion. It follows from the Hilbert class field degree that $3\mid h_{\mathbb{Q}(\sqrt{-59})}$.

If $M/L$ is an extension of number fields, $L$ has no real places, and one finite prime of $L$ is totally ramified in $M$, then $h_L\mid h_M$. The Hilbert class field $H_L$ has trivial intersection with $M$: any common intermediate extension would be both unramified and totally ramified at that prime. Thus $M{H}_L/M$ is an unramified abelian extension of degree $h_L$, and is contained in $H_M$.

For ${\mu }_h^2>0$ and $\lambda >0$, the minimum has ${\varphi }^{†}\varphi =v^2/2$ with $v^2={\mu }_h^2/\lambda$. In unitary gauge, $\varphi =(0,v+H{)}^T/\sqrt{2}$ gives $V=\text{constant}+\lambda v^2{H}^2+\lambda v{H}^3+\lambda H^4/4$. The canonically normalized Higgs boson therefore has $m_H^2=2\lambda v^2$. The angular modes produce the Goldstone bosons absorbed in the Higgs mechanism.

Set $r=\sqrt{2{\varphi }^{†}\varphi }$. The same Higgs potential becomesThe origin has negative radial curvature, while $r=v$ minimizes the scalar potential. Changing the field's angular coordinates at fixed $r$ costs no potential energy; changing its radial magnitude does. This geometric distinction explains why the surviving physical scalar field has a mass even though the angular directions are flat in the scalar potential. Local angular variations still enter the gauge-covariant kinetic term.

For a chosen model, not fitted physical constants, take $v=240\mathrm{GeV}$ and $\lambda =1/8$ in units $\hslash =c=1$. Then ${\mu }_h^2=7200{\mathrm{GeV}}^2$ and $m_H=120\mathrm{GeV}$. Direct expansion gives a quadratic coefficient $7200{\mathrm{GeV}}^2$, cubic coefficient $30\mathrm{GeV}$ and quartic coefficient $1/32$. The quadratic coefficient is $m_H^2/2$, not $m_H^2$. Keeping the same named $\lambda$ in the Higgs radial mass for a half-normalized quartic potential would instead give $m_H^2=7200{\mathrm{GeV}}^2$; these are different potentials. The shifted quartic Higgs potential normalization agrees with this section when $\kappa =\lambda$.

An observed astronomical colour is the sum of a normal intrinsic colour, a nonnegative dust contribution with an exponential distribution, and normal measurement noise. With latent $C_s\sim N(\mu ,v)$, $E_s\sim \mathrm{Exp}(\text{mean }\tau )$ and $O_s\mid C_s,E_s\sim N(C_s+E_s,r_s)$, the Gibbs sampler conditionals for $C_s$ are normal and those for $E_s$ are truncated normal distributions. A flat prior on $\mu$ and inverse-gamma priors on $v,\tau$ give inverse-gamma conditional updates for the two scales. Log-flat priors on both scales instead make the joint posterior improper when all $r_s>0$, despite formally proper full conditionals at generic latent states: the marginal observed likelihood has a positive limit at either zero-scale boundary. Proper positive-scale priors restore a genuine joint posterior.

An indefinite Hermitian form takes both positive and negative values on $⟨v,v⟩$ for nonzero vectors. For example, $-|v_0{|}^2+|v_1{|}^2+|v_2{|}^2+|v_3{|}^2$ is indefinite. It is not a positive inner product defining a Hilbert space, and a nonzero vector can have zero norm without being orthogonal to all vectors. A positive semidefinite restricted form becomes a positive inner product only after quotienting its radical of a Hermitian form.

A Hermitian form is positive semidefinite when $h(v,v)\ge 0$ for every vector. It need not define an inner product, because a nonzero vector may have zero squared norm. The Cauchy-Schwarz inequality still holds: applying nonnegativity to $v+zw$ and minimizing the quadratic expression in $z$ gives $|h(v,w){|}^2\le h(v,v)h(w,w)$ when $h(w,w)>0$; when $h(w,w)=0$, varying $z$ forces $h(v,w)=0$. Thus its zero-norm vectors are exactly the radical of a Hermitian form. Quotienting this radical of a Hermitian form gives a positive inner product; taking its Hilbert space completion then gives a Hilbert space. This is the final positivity step in the Gupta-Bleuler null-state quotient.

The radical of a Hermitian form is the vector subspace orthogonal to the entire space. The form descends to the quotient vector space by its radical: adding a radical vector to either argument does not change the value. For a positive semidefinite Hermitian form, the Cauchy-Schwarz inequality identifies this radical with its zero-norm vectors, so the quotient form is positive definite. This conclusion does not hold for a general indefinite Hermitian form: a zero-norm vector need not belong to its radical.

Pairing a far-field pattern with the conjugate density of a Herglotz wave function converts the angular integral into boundary data:This follows by substituting the surface representation of an obstacle far-field pattern and interchanging the integrals.

Suppose a Non-Archimedean absolute value on $K$ has a unique extension of an absolute value to every finite extension. Factor a monic $g\in {\mathcal{O}}_K[X]$ into monic irreducible factors over $K$. All factors have integral coefficients because their roots have absolute value at most one. By pure-power reduction of a monic irreducible polynomial, each factor reduces to a power of a single irreducible polynomial. If $\overline{g}={\overline{g}}_1{\overline{g}}_2$ is a coprime monic factorization, assign each whole irreducible factor of $g$, with its multiplicity, to the unique side containing its residue factor. The resulting monic $g_1,g_2\in {\mathcal{O}}_K[X]$ satisfy $g=g_1{g}_2$ and reduce to the prescribed factors. Completeness of $K$ is not required for this proof.

With the convention $d_{\mathrm{Hell}}^2=\frac{1}{2}\int (\sqrt{p}-\sqrt{q}{)}^2$, the Hellinger distance between two normal distributions with ${\sigma }_1,{\sigma }_2>0$ isTo prove it, integrate the geometric mean of their probability density functions. Completing the square makes that geometric mean the density of a normal distribution times the displayed prefactor and exponential. For example, writing $S={\sigma }_1^2+{\sigma }_2^2$, that normal distribution has mean $({\theta }_1{\sigma }_2^2+{\theta }_2{\sigma }_1^2)/S$ and variance $2{\sigma }_1^2{\sigma }_2^2/S$. The identity $d_{\mathrm{Hell}}^2=1-\int \sqrt{pq}$ then proves the formula.

For equal unit variances, this becomes $1-e^{-({\theta }_1-{\theta }_2{)}^2/8}$. More generally, put $B=\sqrt{2{\sigma }_1{\sigma }_2/S}\le 1$ and $x=({\theta }_1-{\theta }_2{)}^2/(4S)$. Then $1-B{e}^{-x}=(1-B)+B(1-e^{-x})$, $1-B=({\sigma }_1-{\sigma }_2{)}^2/[S(1+B)]$, and $1-e^{-x}\le x$, proving

If $f$ is a measurable function into a separable Banach space and has finite second moments under probability measures $\mu ,{\mu }^{\prime }$, then their Bochner integral expected values satisfyHere the Hellinger distance includes the factor $1/2$ in its square. For a proof, use $\nu =(\mu +{\mu }^{\prime })/2$, $p=d\mu /d\nu$, and $q=d{\mu }^{\prime }/d\nu$. Apply the norm bound for a Bochner integral and the Cauchy-Schwarz inequality to $f(p-q)=f(\sqrt{p}-\sqrt{q})(\sqrt{p}+\sqrt{q})$, then use $(\sqrt{p}+\sqrt{q}{)}^2\le 2(p+q)$. This explains why stability of a Bayesian posterior in Hellinger distance controls posterior expected values when the relevant second moments remain bounded.

The Hellinger distance convention $h^2(P,Q)=\int (\sqrt{p}-\sqrt{q}{)}^{2}d\nu$, with no factor $1/2$, for densities under a common dominating measure. Its range is $[0,\sqrt{2}]$. This convention makes $\mathrm{TV}(P,Q)\le h(P,Q)\le \sqrt{\mathrm{KL}(P,Q)}$ hold with the displayed constants.

For the unnormalized Hellinger distance, $\mathrm{TV}(P,Q)\le h(P,Q)\le \sqrt{\mathrm{KL}(P,Q)}$. Factor $|p-q|=|\sqrt{p}-\sqrt{q}|(\sqrt{p}+\sqrt{q})$ and use the Cauchy-Schwarz inequality to bound the total variation distance. For the other bound, $\log t\le t-1$ gives $p\log (p/q)\ge 2(p-\sqrt{pq})$, whose integral is the squared unnormalized Hellinger distance. Infinite Kullback-Leibler divergence is allowed.

Left $S{L}_2(\mathbb{Z})$-orbits of integral matrices of positive determinant $n$ have unique representatives $\left(\begin{array}{cc}a& b\\ 0& d\end{array}\right)$ with $a,d>0$, $ad=n$ and $0\le b<d$. Bezout's identity reduces the first column to $(a,0)$; a row shear reduces $b$ modulo $d$.

For the determinant-normalized slash operator for modular forms, $T_{n}f=n^{k/2-1}{\sum }_{\gamma \in {\Pi }_n}f{|}_k\gamma$. If $f={\sum }_{r\ge 0}{a}_r{q}^r$, then $a_r(T_{n}f)={\sum }_{a\mid \gcd (n,r)}{a}^{k-1}{a}_{nr/a^2}(f)$, with $\gcd (n,0)=n$. In particular $a_1(T_{n}f)=a_n(f)$, integer coefficients are preserved, and cusp forms remain cusp forms.

This is the subring generated by the Hecke operators $T_n$ and the integer scalars acting on level-one cusp forms. It preserves the lattice of cusp forms with integral Fourier coefficients of a composite-index Hecke operator. It is distinct from the generic Hecke algebra attached to a Coxeter system.

A simultaneous eigenfunction of all normalized level-one Hecke operators in positive even weight, with nonzero constant Fourier coefficient, is a multiple of the normalized Eisenstein series. The constant coefficient forces its $T_n$ eigenvalue to be ${\sigma }_{k-1}(n)$, and $a_1(T_{n}f)=a_n(f)$ forces $a_n(f)=a_1(f){\sigma }_{k-1}(n)$. Subtracting the matching constant multiple of the Eisenstein series leaves a cusp form. Unless every coefficient vanishes, its prime-index coefficients grow like $p^{k-1}$, contradicting the Fourier coefficient bound for a cusp form $O(p^{k/2})$ when $k\ge 4$. Weight two has no nonzero modular forms, by vanishing of weight-two level-one modular forms.

A body loses internal energy by emitting thermal radiation faster than it absorbs radiation or receives other heating. In a gravitating gas this can remove pressure support and allow further contraction.

For a real smooth solution $w_t=\Delta w$ on a bounded regular domain with $w_t=0$ on its boundary, differentiation and Green's first identity yieldThe boundary term is zero because the boundary values are fixed in time, not because the normal derivative vanishes. Equality at a particular time is equivalent to $\Delta w=0$ throughout the domain at that time.