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Let $L>0$ and consider holomorphic functions $f$ from the unit disc into $e^{-L}<|w|<e^L$, normalized by $f(0)=1$. Their largest possible derivative modulus is $4L/\pi$. Lift $f$ to its holomorphic logarithm $h$ with $h(0)=0$, so $|\mathrm{Re}h|<L$. The function $\tan (\pi h/(4L))$ maps the disc into itself and fixes zero. The Schwarz lemma gives $|h^{\prime }(0)|\le 4L/\pi$. Equality holds precisely forHere $\arctan$ is its holomorphic branch vanishing at zero; $(4L/\pi )\arctan z$ is a conformal map from the disc onto the vertical strip. Thus the maximal derivative modulus is attained, with a circle of distinct extremizers rather than a unique map.

A holomorphic function between hyperbolic Riemann surfaces lifts to a disk map. If the lift is not an automorphism, strict Schwarz-Pick lemma derivative inequality makes every nonconstant geodesic shorter. Equality of distance for one distinct pair therefore forces the map to be a covering map and a local isometry. Covering maps attain equality for sufficiently close pairs in a single lift of a small convex metric ball; they need not preserve all global distances. A punctured disk covers a thrice-punctured sphere by including a cyclic parabolic subgroup into $\Gamma (2)$, so equality is possible in that example.

// Destination: analysis.bigb

For holomorphic functions $f:\mathbb{D}\to \mathbb{D}$ with $f(z_0)=w_0$, put $r=|(z-z_0)/(1-{\overline{z}}_{0}z)|$. Their possible values at $z$ form the closed disc with centre $(1-r^2)w_0/(1-r^2|w_0{|}^2)$ and radius $r(1-|w_0{|}^2)/(1-r^2|w_0{|}^2)$. Indeed ${\psi }_{w_0}\circ f={\psi }_{z_0}h$ with holomorphic $|h|\le 1$; all constant choices of $h$ in the closed unit disc are admissible. Completing the square in the displayed pseudohyperbolic inequality gives the centre and radius.

If a holomorphic function $f$ on the unit disc satisfies $|f|\le M$ and $f(a)=0$, the Schwarz-Pick lemma applied to $f/M$ gives $|f(0)|/M\le |a|$. Equality for a nonzero central value forces a scaled automorphism of the unit disk. If $f$ extends past the unit circle and $|f(0)|$ is smaller than its minimum boundary modulus, the maximum modulus principle applied to $1/f$ proves that a zero occurs inside the circle.

For $0<k<1$, principal upper-half-plane branches give a conformal map onto the upper half-plane cut along $[0,\tau ]$, where $\tau =2k^k(1-k{)}^{1-k}{e}^{i\pi k}$. Its derivative is $(z-1+2k)(z-1{)}^{k-1}(z+1{)}^{-k}$. The two prevertices $-1,1$ map to the slit base, and $1-2k$ maps to its tip. Boundary traversal goes out along the slit and back, while $g(z)=z+1-2k+O(1/z)$ at infinity. The argument principle gives one preimage of every point off the slit.

A boundary point of the source half-plane that maps to a polygon vertex. If that corner has angle $\pi \alpha$, the Schwarz-Christoffel mapping has local behavior $F(z)-F(a)\sim C(z-a{)}^{\alpha }$ at its prevertex, so the derivative has exponent $\alpha -1$. Infinity may also be a prevertex, expressed in the local coordinate $1/z$.

Sum the weight monomials of all Semistandard Young tableaux on the skew Young diagram $\lambda /\mu$, with weak rows and strict columns. A Bender-Knuth involution interchanges the counts of adjacent letters without changing the shape, so the sum is a symmetric function. Taking $\mu$ empty gives the Schur function.

Let $a_{\gamma }=\det [x_j^{{\gamma }_i}]$ and $\delta =(N-1,\dots ,0)$. Multiply the Jacobi–Trudi identity matrix by $B_{jk}=(-1{)}^{N-j}{e}_{N-j}(X\setminus x_k)$. The generating-function identity $H(t){\prod }_{i\ne k}(1-x_{i}t)=(1-x_{k}t{)}^{-1}$ makes the product matrix $[x_k^{{\lambda }_i+N-i}]$. Setting $\lambda =0$ shows $\det B=a_{\delta }$, proving the formula as a polynomial identity.

A pure-state source emits independent states from a fixed quantum state ensemble. Its average block density operator is the tensor power of the one-use average. Unassisted reliable quantum source compression preserves the emitted quantum states, rather than only their classical labels. Schumacher compression gives the optimum asymptotic qubit rate $S(\rho )$ for such a source, with every larger rate achievable. Nonorthogonal equally likely pure states need not have entropy one: their classical labels have entropy one but their average density operator can have lower Von Neumann entropy.

If $\alpha$ is a real algebraic irrational number and $\epsilon >0$, thenhas only finitely many reduced rational solutions $p/q$.

Relative to a Schauder basis, a block sequence consists of nonzero finitely supported vectors with successive disjoint supports: $\max \mathrm{supp}{x}_i<\min \mathrm{supp}{x}_{i+1}$. It is a basic sequence, with basis constant at most that of the original basis.

Relative to a Schauder basis $(e_j)$, a block basic sequence consists of nonzero finite combinations supported on successive disjoint coordinate intervals. In the l-p sequence space, disjointly supported vectors $u_n$ of norm one satisfy $\Vert \sum a_n{u}_n{\Vert }_p^p=\sum |a_n{|}^p$. Their closed span is consequently isometric to the l-p sequence space.

Write the S-matrix as $S=1+iT$. Unitarity gives $-i(T-T^{†})=T^{†}T$, so the diagonal forward matrix element obeys $2\mathrm{Im}{T}_{ii}={\sum }_f\int d{\Phi }_f|T_{fi}{|}^2$. Thus the imaginary part of a forward scattering amplitude measures the total inclusive transition rate. For an electromagnetic source, the hadronic electromagnetic-current spectral density supplies the inclusive spectral sum.

With this half-variance convention, the unforced scalar equation gives $d⟨C^2/2⟩/dt=-{ϵ}_c$ in a homogeneous incompressible flow. Authors defining dissipation of the full variance instead use $2\alpha ⟨|\nabla C{|}^2⟩$. Stating the convention is necessary when relating the dissipation rate to a scalar structure function.

Newtonian gauge removes scalar shear and writes scalar metric perturbations in terms of gravitational potentials. The fluid Euler equation then contains an explicit gradient of the Newtonian potential.

For classical nondegenerate electrons, the free-particle Boltzmann occupancy has the form $z_e\exp [-p^2/(2mkT)]$, with chemical potential measured relative to its rest energy. Counting the two spin states in phase space gives $n_e=2z_e(2\pi mkT/h^2{)}^{3/2}$. A bound level of energy $-I_n$ has relative occupancy $(g_n/g_{+})z_e{e}^{I_n/kT}$ against its parent ion. Eliminating $z_e$ gives the level-specific Saha ionization equation.

Suppose a function near a critical point has Taylor expansion $f(p+h)=f(p)+P_m(h)+o(\Vert h{\Vert }^m)$, where $P_m$ is its first nonzero homogeneous polynomial term. If $P_m$ takes both positive and negative values on directions, then $p$ is a saddle point of a scalar function. Indeed, on rays $h=rv$ the leading term is $r^m{P}_m(v)$; for each chosen direction its sign dominates the remainder for small positive $r$. This works even when the Hessian matrix is zero. For example, $P_4(\xi ,\eta )=-{\xi }^3(\xi +\eta )$ is negative on $(1,0)$ and positive on $(1,-2)$, hence detects a saddle despite having degree four.

When a saddle-node lies on a returning invariant circle, removing the equilibria creates passage around a periodic orbit through a slow bottleneck. For $\dot{x}=x^2+\eta$ with $\eta >0$, crossing a fixed interval $[-h,h]$ takes $2\arctan (h/\sqrt{\eta })/\sqrt{\eta }\sim \pi /\sqrt{\eta }$. If the remaining orbit takes bounded time, this is also its leading period. Strong transverse contraction makes the newborn periodic orbit attracting. This global saddle-node mechanism differs from a local fold occurring away from an already existing cycle.

Let $S$ be a finite set of prime numbers and let $f_S(x)$ count S-smooth numbers $1\le n\le x$, where $x\ge 1$. Each such $n$ has a unique expression $n=a{b}^2$, where $a$ is a positive squarefree integer and $b$ is a positive integer. Split every prime exponent into its parity and an even remainder to obtain this decomposition.

Every prime factor of $a$ belongs to $S$, so $a$ is the product of a subset of $S$, giving at most $2^{|S|}$ possibilities. Also $b\le \sqrt{x/a}\le \sqrt{x}$. Ignoring further restrictions on $b$ only increases the count, provingIn particular, each fixed finite-prime family has natural density zero. This differs from allowing the smoothness bound, and hence the prime set, to grow with $x$.

For radial quantum scattering, the incoming coefficient of the regular solution defines the Jost function, with convention $S(k)=F(-k)/F(k)$. A zero at $k=i\kappa$, $\kappa >0$, removes the exponentially growing component and produces a bound state, subject to regularity and normalizability. Poles of the outgoing numerator alone need not produce bound states.