Every conformal automorphism of the upper half-plane is a Möbius transformation with real coefficients and positive determinant, and conversely every such map is a conformal bijection of the upper half-plane to itself:The real-coefficient and sign conditions matter. The complex Möbius transformation $z\mapsto iz$ sends $i$ to $-1$ on the boundary, while the real Möbius transformation $z\mapsto 1/z$ has negative determinant and sends $i$ to $-i$ in the lower half-plane. ^[ref]

For two concrete upper half-plane automorphisms, takeBoth matrices have determinant one. Here $T(i)=1+i$, $S(i)=i$, and $S(1+i)=(-1+i)/2$. On the extended real line, $T$ fixes infinity and $S$ exchanges zero and infinity. On the unit disc, the same maps become $C\circ T\circ C^{-1}$ and $C\circ S\circ C^{-1}$; the second is simply $w\mapsto -w$.

Multiplying all four coefficients by a nonzero real scalar leaves $f$ unchanged. Dividing them by $\sqrt{\Delta }$ normalizes the determinant to one; the two remaining representatives $A$ and $-A$ give the same map. Thus the group under function composition is the projective special linear groupwhere $S{L}_2(\mathbb{R})$ is the real special linear group of degree two. Composition of the Möbius transformations corresponds to multiplication of their matrices. ^[ref]

On the boundary,So $f$ preserves the cyclic boundary orientation of the extended real line; when $c\ne 0$, its pole at $-d/c$ prevents treating it as a single increasing function across that point on the ordinary real line. Three distinct boundary images determine a Möbius transformation uniquely, but an upper half-plane automorphism can realize only triples with matching cyclic boundary orientation.

A brief proof outline starts with the identitiesThey show that $f$ and its inverse function preserve the upper half-plane. For completeness of the classification, conjugate an arbitrary conformal bijection by the Cayley transform. Normalize its image of zero with an automorphism of the unit disk, then apply the Schwarz lemma to the resulting map and its inverse function; both inequalities force a rotation. Conjugating back gives the real Möbius transformation above.

Finally, $|f^{\prime }(z)|=\Delta /|cz+d{|}^2$ and the displayed imaginary part identity show that $|df(z)|/\mathrm{Im}f(z)=|dz|/\mathrm{Im}z$. Every upper half-plane automorphism is therefore also an isometry of the Poincaré half-plane model. The same formulas describe both holomorphic coordinate symmetries and motions of the hyperbolic plane.

The unit signature map records the signs of a unit under all real field embeddings. Its kernel consists of totally positive units. Its image determines the kernel of the homomorphism from the narrow ideal class group to the ordinary ideal class group: this kernel is $\{\pm 1{\}}^{r_1}/\mathrm{sgn}({\mathcal{O}}_K^{\times })$.

The bosons are the metric, two-form, dilaton, RR one-form and RR three-form, with physical dimensions $35+28+1+8+56=128$. Two opposite-chirality gravitinos and two dilatinos supply $2(56+8)=128$ fermionic degrees of freedom. They form the ten-dimensional type IIA supergravity multiplet. Counts refer to on-shell polarizations, not raw field components.

Level matching combines two chiral massive sectors, each with 128 NS and 128 R states. The NS-NS and R-R combinations give $2(128{)}^2=32768$ bosons, and the mixed sectors give the same fermion count. Their total $2^{16}$ matches a massive uncharged long multiplet with 32 real supercharges and sixteen fermionic creation operators.

After the GSO projection, the first positive shifted level is one: $N_{\mathrm{NS}}=3/2$ and $N_R=1$. In a closed-string mass convention it corresponds to $M^2=4/{\alpha }^{\prime }$. The transverse oscillator basis has 128 NS states and 128 R states. They form the bosonic and fermionic pieces of a massive supersymmetry multiplet.

Two point masses released from rest at separation $a$ have negative mechanical energy $-G{m}_1{m}_2/a$ but collide in time $\pi \sqrt{a^3/[8G(m_1+m_2)]}$. Their separation approaches zero along a regular trajectory before collision. The energy sign alone therefore does not supply a positive lower separation bound.

Apply the Selberg upper-bound sieve to the numbers $p+2$ with $p\le x$ prime. For odd squarefree integers $d$, the local count is $\pi (x;d,-2)$ and the density is $1/\varphi (d)$. With sieve level $D=x^{1/3}$, the normalizing sum is $\gg \log x$ by the truncated Euler-product lower bound. The Selberg least-common-multiple weights have absolute values at most $3^{\omega (d)}$. The weighted arithmetic-progression error bound and Bombieri–Vinogradov theorem make their total error $O(x/{\log }^{3}x)$. Consequently the number of twin prime pairs with smaller member at most $x$ is $O(x/{\log }^{2}x)$.

A trigonometric polynomial is real on the unit circle precisely when its finite coefficients have the displayed symmetry. To prove necessity, conjugate its values on the circle, use $\overline{z}=z^{-1}$, and compare coefficients. Multiplying a coefficient difference by $z^d$ gives an ordinary polynomial vanishing at infinitely many points, so it is zero. Symmetry also gives $p(z^{-1})=\overline{p(\overline{z})}$ for every nonzero complex $z$.

If $P=z^{d}p$ and $p$ is nonnegative on the unit circle, a root of a polynomial $\zeta =e^{i{\theta }_0}$ there has even multiplicity of a root. Indeed $p(e^{i\theta })$ is a real analytic nonnegative function, whose first nonzero term in its Taylor series at a zero must have even order. The map $\theta \mapsto e^{i\theta }-\zeta$ has a simple zero, and multiplication by $e^{-id\theta }$ is nonvanishing, so the order equals the polynomial multiplicity.

A trigonometric polynomial nonnegative on the unit circle has a polynomial modulus-square factor of polynomial degree at most its trigonometric order. For a nonzero trigonometric polynomial of actual order $d$, conjugate symmetry of trigonometric polynomial coefficients yields reciprocal-conjugate root pairing for $P=z^{d}p$. Even multiplicity of unit-circle roots of a nonnegative trigonometric polynomial permits pairing all $2d$ roots of a polynomial. Select representatives ${\zeta }_1,\dots ,{\zeta }_d$ and useIt follows that $p(z)=c{\prod }_i|z-{\zeta }_i{|}^2$ with $c=(-1{)}^d{p}_d/{\prod }_i\overline{{\zeta }_i}$. Evaluating away from the roots of a polynomial gives $c>0$. Thus $q(z)=\sqrt{c}{\prod }_i(z-{\zeta }_i)$ works. Constant and zero trigonometric polynomials have constant or zero factors. Different selections of roots of a polynomial and constant phases can give different factors.

Under the displayed diagonal-sum convention, set $w(z)=(1,z^{-1},\dots ,z^{-d}{)}^T$ on the unit circle. A Hermitian matrix $M$ then givesConsequently $M⪰0$ proves nonnegativity. Conversely the Fejér–Riesz theorem produces the rank-one spectral-factor Gram matrix. This gives a semidefinite programming representation of nonnegative trigonometric polynomials. The reversed convention $j-i=k$ instead uses the unconjugated monomial vector.

An upper triangular matrix system with nonzero diagonal is solved from the last row upward. Once $x_{i+1},\dots ,x_n$ are known, use $x_i=(b_i-{\sum }_{j>i}{a}_{ij}{x}_j)/a_{ii}$. The nonzero diagonal guarantees a unique solution; this is the final solve after a QR factorization.

An apparent horizon is the outer boundary of the future-trapped region on a chosen spatial slice, under the usual regularity conditions. Its location depends on the slice; it is not the globally defined event horizon. At a regular boundary the outward null expansion vanishes while the inward null expansion is negative. Its area cannot be substituted into every version of the Penrose inequality without the appropriate area and initial-data hypotheses.

Particles keep their velocity and move along straight trajectories. The method of characteristics gives $f(t,x,v)=f_0(x-tv,v)$. The shear preserves Lebesgue measure and every Lp norm. Compact initial support remains compact at each finite time, but need not remain in one compact set for all times.

For a density operator $\rho$ and pure state $P=|\psi ⟩⟨\psi |$, the diagonal absolute-sum bound for the trace norm in a basis containing $|\psi ⟩$ gives $D(\rho ,P)\ge 1-⟨\psi |\rho |\psi ⟩=1-F(\rho ,P{)}^2$. The quantum fidelity is unsquared. Block-diagonal states with respect to the target and its complement attain equality.

The nonnegative integer $\Omega (n)-\omega (n)$ counts repeated occurrences beyond the first of each prime factor. Here $\Omega$ is the total number of prime factors, counted with multiplicity, and $\omega$ is the prime omega function, which counts distinct prime factors. For example, $72=2^3{3}^2$ has excess $5-2=3$.

If $n$ is sampled with the uniform distribution on $\{1,\dots ,N\}$, the expectation is at most one, independently of $N$. Indeed,and the probability that $p^k$ divides $n$ is $\lfloor N/p^k\rfloor /N\le p^{-k}$. ConsequentlyThe final equality is a telescoping series, since $1/(m(m-1))=1/(m-1)-1/m$.

A group is a torsion group when every element is a torsion element, meaning that each element has finite order. The orders need not have a common finite bound. A torsion group has no nontrivial torsion-free group as a subgroup; in particular it cannot contain a nonabelian free group. Infinite finitely generated groups of this kind can be constructed using a torsion group construction by p-power relators.

A subset $B$ of a topological vector space is bounded when every neighborhood $U$ of zero absorbs it: $B\subseteq tU$ for all sufficiently large positive $t$. In a locally convex space defined by seminorms $p_{\lambda }$, this is equivalent to ${\sup }_{x\in B}{p}_{\lambda }(x)<\infty$ for every $\lambda$. In particular, a bounded subset of the Schwartz space has a uniform bound for every Schwartz seminorm; it need not have uniformly bounded supports.

A linear map from a metrizable space carrying a vector-space topology is continuous if it takes every null sequence to a null sequence. If continuity fails, choose a point in each shrinking basic neighborhood whose image stays outside a fixed neighborhood of zero. These points converge to zero but their images do not. For a Fréchet space described by increasing seminorms $p_j$, a discontinuous functional admits $p_j(f_j)\le 1/j$ and $|u(f_j)|\ge 1$.

A point of a topological space is closed when its singleton is a closed set. In a T1 space every point is closed. The closed points of an affine scheme $\mathrm{Spec}A$ are its maximal ideals; the closure of the point corresponding to $\mathfrak{p}$ is $V(\mathfrak{p})$.