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For a slender tube of length $L$ supplied from a large reservoir with excess pressure $P$, negligible gravity and entrance inertia, the Unsteady Bernoulli equation gives $L\dot{U}+U^2/2=P/\rho$. A stationary initial liquid column has the displayed speed and volume flux $\pi a^{2}U$ for tube radius $a$. The inertial term controls the early linear acceleration; the kinetic-energy term sets the terminal inviscid speed. This is a leading slender-tube model while the reservoir remains large and the tube full.

A spherical balloon feeding an inviscid uniform tube plug, with entrance excess pressure $2\gamma /R$, satisfies $L\dot{u}=2\gamma /(\rho R)$ and $4\pi R^2\dot{R}=-\pi a^{2}u$. Eliminating $u$ yields the displayed equation. Its first integral is $R^4{\dot{R}}^2+K{R}^2=C$. From rest at radius $R_0$, $C=K{R}_0^2$; integration gives $t=R_0^2{K}^{-1/2}(\pi /4-\theta /2+\sin (2\theta )/4)$, where $\theta =\arcsin (R/R_0)$. The ideal emptying time is $\pi R_0^2/(4\sqrt{K})$. Independently prescribed initial flow changes $C$ and therefore this time.

A locally uniform convergence limit of univalent functions on a domain is either constant or univalent. For a nonconstant limit $f$, suppose distinct $x,y$ satisfy $f(x)=f(y)$. Choose a small closed disk around $x$ excluding $y$ with $f-f(y)$ nonzero on its boundary. The Rouche theorem then forces $f_j-f_j(y)$ to have a zero in that disk for large $j$, contradicting injectivity of $f_j$. This supplies the injectivity step in an extremal proof of the Riemann mapping theorem without assuming it as a special lemma.

For an injective holomorphic function $f$ on an open subset of $\mathbb{C}$, $f^{\prime }(z)$ is nowhere zero. If $f(z)-f(z_0)=(z-z_0{)}^{m}h(z)$ with $h(z_0)\ne 0$, choose a small circle where both $h$ and $mh+(z-z_0)h^{\prime }$ have no zeros. The argument principle counts $m$ zeros of $f(z)-f(z_0)-w$ inside that circle for sufficiently small $w\ne 0$: deforming $w$ to zero cannot change the count. All these zeros are simple, since $f^{\prime }$ only vanishes at $z_0$ inside the circle and $z_0$ is not a zero when $w\ne 0$. Injectivity forces $m=1$, and hence $f^{\prime }(z_0)\ne 0$.

A normalized univalent function is a holomorphic function on the unit disc which is injective and has the displayed normalization. Its Taylor series begins $z+a_2{z}^2+\cdots$. Normalization makes coefficient bounds and image-size estimates independent of translation and dilation; the Koebe quarter theorem is the sharp universal image bound.

For a univalent exterior map $G(\zeta )=\zeta +b_0+{\sum }_{n\ge 1}{b}_n{\zeta }^{-n}$, the image of $|\zeta |=R>1$ is a positively oriented Jordan curve. Green's theorem gives its enclosed area as $\pi (R^2-{\sum }_{n\ge 1}n|b_n{|}^2{R}^{-2n})$. Nonnegativity and $R\downarrow 1$ give the displayed bound. Laurent-series differentiation and integration on a circle of radius greater than one are justified by uniform convergence there.

For a compact group, the matrix coefficients of its finite-dimensional irreducible unitary representations span a dense subspace of $C(G)$ and form a complete orthogonal decomposition of $L^2(G)$ with Haar measure. If $K$ is a closed subgroup and $W$ is a finite-dimensional unitary $K$-representation, the space of L2 sections of the associated homogeneous bundle has multiplicity $\dim {\mathrm{Hom}}_K(E{|}_K,W)$ for each irreducible $G$-representation $E$. This is the compact-group form of Frobenius reciprocity and determines homogeneous-bundle multiplicities from isotropy weights.

A strongly continuous unitary representation of the circle group splits as an orthogonal sum of character subspaces. The projections are $P_n={\int }_{\mathbb{T}}{z}^{-n}{U}_{z}dz$. Character orthogonality makes them mutually orthogonal, and Fejér sum shows their ranges span the represented Hilbert space. The self-adjoint generator has integer eigenvalues, with domain given by ${\sum }_n{n}^2\Vert P_n\xi {\Vert }^2<\infty$.

A strongly continuous unitary representation of the circle has integer spectral weights: $H$ is the orthogonal sum of spaces on which $U_z=z^m$. Positive energy means that these weights are bounded below, equivalently its self-adjoint generator has spectrum bounded below. Multiplication by an integer character shifts the lower bound to zero. A unitary operator with a nonzero rotation weight cannot exist in its operator algebra: repeated application of it or its inverse would move a nonzero vector to arbitrarily low energy.

For a strongly continuous unitary representation, integration against $f\in L^1(G)$ defines an operator satisfying $\Vert \pi (f)\Vert \le \Vert f{\Vert }_1$. Fubini's theorem gives $\pi (f\ast g)=\pi (f)\pi (g)$, and the group-algebra involution gives $\pi (f^{\ast })=\pi (f{)}^{\ast }$. On an Abelian group, the operator-norm closure is a commutative C-star algebra. Its algebra characters produce continuous unitary group characters, allowing the Commutative Gelfand--Naimark theorem and Riesz-Markov-Kakutani representation theorem to turn a positive vector functional into a finite spectral measure.

The diagonal element $\mathrm{diag}(m,m^{-1})$ conjugates $U(t)$ to $U(m^{2}t)=U(t{)}^{m^2}$. In a finite-dimensional unitary representation, conjugacy therefore makes its eigenvalue multiset invariant under every square power. Square-power rigidity of a finite nonzero spectrum makes the image of every upper unipotent the identity. Its kernel is a normal subgroup; elementary unipotent generators of SL2R then force the entire group representation to be trivial. This algebraic argument does not require continuity.

A local unit has valuation zero in the local field, or equivalently is an invertible element of its valuation ring. A principal unit additionally reduces to one in the residue field. The distinction between local units and principal units retains the prime-to-p residue roots of unity in a cyclotomic p-adic field.

Vary the unobserved outcome probabilities for item and unit nonresponders, and any unknown auxiliary-variable mixture, over scientifically plausible ranges. Recompute the population estimate and identify values at which substantive conclusions change. This probes untestable transport or missing at random assumptions; it is distinct from sampling uncertainty represented by a confidence interval. Letting each missing-outcome probability vary from zero to one gives worst-case identification bounds.

For a uniform random permutation $\pi$ of $n\ge 1$ objects, let $M_n$ count its fixed points. Write $M_n={\sum }_{i=1}^n{1}_{\{\pi (i)=i\}}$. Each indicator random variable has expected value $1/n$, so linearity of expectation gives $\mathbb{E}[M_n]=1$ for every $n\ge 1$. This average does not require independence: for $n\ge 2$, fixing two specified objects has probability $1/[n(n-1)]$, greater than the product $1/n^2$ of their individual probabilities.

For four objects, exact enumeration gives counts $9,8,6,0,1$ for $0,1,2,3,4$ fixed points among the $24$ equally likely permutations. Their weighted arithmetic mean is $(0\cdot 9+1\cdot 8+2\cdot 6+4\cdot 1)/24=1$. Exactly three fixed points are impossible: the last object must then also be fixed.

Choose the $m$ fixed objects and derange the remaining $n-m$. Their count is $\left(\genfrac{}{}{0ex}{}{n}{m}\right)D_{n-m}$ among $n!$ equally likely permutations. The inclusion-exclusion principle for $D_{n-m}$ proves the displayed law, valid for $0\le m\le n$ and zero otherwise. With $D_0=1$, it includes the all-fixed case and the impossibility of exactly $n-1$ fixed objects. For fixed $m$, the law tends to $e^{-1}/m!$, the Poisson distribution of mean one.

For a uniform random permutation $a_1,\dots ,a_n$, define $R_i$ as the rank of $a_i$ among $a_1,\dots ,a_i$, with rank one assigned to the smallest value. Each sequence $r_i\in \{1,\dots ,i\}$ corresponds to exactly one permutation: insert index $i$ in position $r_i$ of the current increasing-value list, then assign the final list the values $1,\dots ,n$. The $n!$ equally likely permutations therefore correspond to the ${\prod }_{i=1}^{n}i=n!$ equally likely rank sequences. Consequently the $R_i$ are independent random variables, each with the displayed uniform distribution.

When $L$ has full column rank, its Gram matrix $L^{T}L$ is positive-definite and $BL{L}^{T}L(L^{T}L{)}^{-1}=BL=0$. The stacked normal vector $(BY,L^{T}Y)$ therefore has zero cross covariance, so its blocks are independent. Since $Y^{T}AY=\Vert L^{T}Y{\Vert }^2$, the linear and quadratic statistics are independent. This is the projection argument behind independence of regression coefficient and variance estimators in a normal linear model.

For $r>0$, the forward average is $r^{-1}{\int }_x^{x+r}f(t)dt$ and the backward average is $r^{-1}{\int }_{x-r}^{x}f(t)dt$, both with positive interval length in the denominator. They are linear operators. Their absolute values are bounded by the uncentered maximal function: enlarge the interval to put $x$ in its interior and pass to the limit. The same boundary-enlargement argument works regardless of the value of the chosen representative at $x$.

For normalized linearly independent pure states, their Gram matrix $G_{ij}=⟨{\psi }_i|{\psi }_j⟩$ is positive definite. The displayed reciprocal vectors satisfy $⟨{\chi }_i|{\psi }_j⟩={\delta }_{ij}$. The operator $R={\sum }_i|{\chi }_i⟩⟨{\chi }_i|$ has nonzero eigenvalues equal to those of $G^{-1}$: use the polar factorization of the matrix whose columns are ${\psi }_i$. Choosing $0<ϵ\le {\lambda }_{\min }(G)$ gives the POVM effects $E_i$ and $E_{?}=I-ϵR$. Each conclusive outcome has probability $ϵ$ on its own state and zero on every other input. The inconclusive effect preserves positivity, so the construction is a physical measurement in quantum mechanics.

An ultraproduct over a nonprincipal ultrafilter on $\omega$ realizes each countable finitely satisfiable type. For the first $n$ formulas, their coordinate simultaneous-witness set $E_n$ belongs to the ultrafilter by the Łoś theorem. Replace it by $E_n\cap [n,\infty )$ and choose at coordinate $i$ a witness to as many initial formulas as that coordinate permits. This finite number tends to infinity modulo the ultrafilter, so the resulting product class realizes every formula. More generally countably incomplete ultrafilters admit the same descending-set diagonalization. This statement does not assert saturation at every larger cardinal.

A phosphotyrosine residue is a tyrosine residue modified by phosphorylation. Its location and surrounding protein primary structure help determine which SH2 domains can recognize it.