For a perfect-gas normal shock wave reflected from a stationary rigid wall, apply the Rankine-Hugoniot conditions for a perfect gas using signed velocity in each shock frame. The once-shocked state is the common left state for both applications. Its two relative speeds are roots of $v^2-(\gamma +1)u_{s}v/2-\gamma p_s/{\rho }_s=0$. Their product and the pressure jump eliminate the velocity, giving the displayed relation. A strong incident shock wave gives $p_1/p_s\to (3\gamma -1)/(\gamma -1)$.

For $a,v,m_j>0$ and independent normal observations conditional on $\Theta$, with mean $\Theta$ and variances $v/m_j$, and normal prior mean $\mu$ and variance $a$, write $W={\sum }_j{m}_j$. Multiplying likelihood and prior adds precisions $W/v+1/a$. completing the square gives the displayed posterior distribution. Its posterior mean is $Z{\overline{X}}_w+(1-Z)\mu$, where $Z=aW/(v+aW)$ and ${\overline{X}}_w={\sum }_j{m}_j{X}_j/W$. Thus the Bayes estimator under squared error loss is exactly the Bühlmann–Straub credibility estimate, not merely an approximation to it. Its posterior variance is $av/(v+aW)$.

For repeated independent $N(\theta ,{\sigma }^2)$ measurements with prior $\theta \sim N(\mu ,{\tau }^2)$ and known scales, normal-normal conjugacy gives $v_k=({\tau }^{-2}+k{\sigma }^{-2}{)}^{-1}$ and $m_k=v_k(\mu {\tau }^{-2}+{\sigma }^{-2}{\sum }_{j=1}^k{y}_j)$. The next measurement has posterior predictive distribution $N(m_k,v_k+{\sigma }^2)$. Its one-sided conditional posterior-predictive false-alarm level $\alpha$ yields the displayed limit; this is a per-measurement probability, not a simultaneous false-alarm guarantee across repeated monitoring. Predictive width tends to the irreducible measurement width, while the center adapts to the individual's data. To assess a held-out measurement, fit with earlier measurements only.

A unilateral deviation changes only one player's strategy, holding the other players' strategies fixed. It is profitable if it strictly increases that player's payoff, or expected payoff for mixed strategies. The comparison does not allow the opponents to adjust in response to the deviation. In a sequential game, however, holding an opponent's complete pure strategy fixed still allows their prescribed actions to differ after different observed histories.

A strategy profile specifies one strategy for every player. A pure strategy strategy profile lies in the Cartesian product $S_1\times \cdots \times S_n$ of the players' strategy sets. A mixed strategy strategy profile specifies one probability distribution ${\sigma }_i$ over each $S_i$. Write $s_{-i}$ or ${\sigma }_{-i}$ for the choices of all players except player $i$.

A near-identity transformation is a local change of variables whose derivative at the reference point is the identity. In a normal form of a dynamical system, substituting $x=y+h(y)$ and solving $(I+Dh(y))\dot{y}=f(y+h(y))$ determines which nonlinear coefficients can be removed. The transformation is locally invertible by the inverse function theorem. Terms whose removal equations are singular are resonant and must be retained; a computed zero resonant coefficient cannot be assigned an arbitrary nonzero value by normalization.

A reflection acting as $(A,B)\mapsto (A,-B)$ forces an equivariant vector field to have $\dot{A}$ even and $\dot{B}$ odd in $B$. With the zero solution preserved, its quadratic normal form is $\dot{A}={\lambda }_{1}A+a_1{A}^2+a_2{B}^2$, $\dot{B}={\lambda }_{2}B+a_{3}AB$. For positive coefficients, every mixed equilibrium with $B\ne 0$ has Jacobian determinant $-2a_2{a}_3{B}^2<0$, hence is a saddle equilibrium. The two invariant-axis equilibria meet in a transcritical bifurcation, and the mixed branches meet them in pitchfork bifurcations.

A family of meromorphic functions on a domain that omits three fixed points of the Riemann sphere is normal. For holomorphic functions with values in the complex plane, omitting two fixed values suffices.

If a bounded holomorphic function $g$ on the complex upper half-plane has $g(iy)\to \ell$ as $y\downarrow 0$, the rescaled functions $g(z/n)$ form a normal family. Every subsequential limit equals $\ell$ on the imaginary axis, hence everywhere by the identity theorem for holomorphic functions. The entire sequence converges locally uniformly to $\ell$. For a nontangential sequence $z_j=x_j+i{y}_j\to 0$, put $n_j=\lfloor 1/y_j\rfloor$; then $n_j{z}_j$ remains in a fixed compact subset of the half-plane. This gives $g(z_j)\to \ell$. The bounded example $e^{-i/z}$ shows that unrestricted boundary convergence can fail.

// Destination: analysis.bigb

Suppose a family of meromorphic functions is a normal family on a neighbourhood of every point of a planar domain. Choose a countable cover by open discs with that property. From any sequence extract successive subsequences converging locally uniformly in the chordal metric on the first disc, second disc, and so on; take the diagonal subsequence argument.

The local Riemann sphere valued limits agree on overlaps, since the same subsequence converges there. They therefore define one continuous global limit. Locally it is meromorphic or identically infinity, so on a connected domain it is globally meromorphic or identically infinity. A finite cover of each compact subset transfers the local uniform estimates to that subset. Thus the original family is a normal family on the entire domain. The argument handles poles using the chordal metric; it does not require every local limit to be a finite holomorphic function.

For a holomorphic normal family, limits are holomorphic or identically infinity. A uniform bound at one interior point excludes infinity on its connected domain. Every sequence therefore has a finite locally uniform subsequential limit. The Cauchy integral formula makes the derivatives converge locally uniformly too, so the derivative family is normal. Without the point bound this can fail: $n(1+z^2/2)$ tends uniformly to infinity on the unit disk, whereas its derivatives $nz$ are not normal at zero.

A fixed-aperture approach region at a boundary point of the unit disc. Its points approach the boundary point without becoming tangent to the circle. Convergence inside every such fixed region is a nontangential limit; the corresponding supremum defines a non-tangential maximal function.

Put a flex at $[0:1:0]$ with tangent line $Z=0$. The tangent section has a triple zero there, so $F(X,Y,0)=a{X}^3$ with $a\ne 0$. Smoothness gives $b\ne 0$. In characteristic different from two, replacing $Y$ by $Y+(cX+eZ)/(2b)$ completes the square and produces a Weierstrass equation of an elliptic curve using only projective linear coordinate changes.

For two independent samples of sizes $m,n$, rank their pooled observations and let $W_X$ be the sum of the ranks of the first sample. With continuous observations, the null hypothesis of a common distribution makes every allocation of $m$ ranks to that sample equally probable. A one-sided alternative that the first sample is stochastically smaller rejects for small $W_X$, equivalently small $U_X$. In a common-shape location model this tests a negative location difference. Without the common-shape assumption it is not simply a test of medians. The shifted statistic $U_X$ counts cross-sample pairs for which the first observation exceeds the second. Ties use average ranks and a conditional permutation test or a tie-adjusted normal approximation.

For independent paired observations, form differences $D_i=U_i-V_i$. Assume the differences are identically distributed, continuous and symmetric about a location $\delta$. Under the zero-location null hypothesis, the signs are independent fair signs conditional on their absolute values. Rank the absolute differences and add ranks with positive signs. A negative-location alternative rejects for small $W_{+}$. Its exact conditional distribution is that of a sum of an independently selected subset of ranks, each of the $2^n$ subsets equally likely. Thus its mean is $n(n+1)/4$ and its variance is $n(n+1)(2n+1)/24$. Symmetry is essential: a zero median alone does not justify this signed-rank null law. Zero differences are omitted in the usual version, and tied absolute differences receive average ranks.

A rank test is a hypothesis test whose statistic is a function of the ranks of observations, possibly together with fixed sample labels. Under independent observations from a common continuous probability distribution, exchangeable rank allocations give its exact null law without specifying that distribution. The Wilcoxon rank sum test is an example. Invariance under common increasing transformations follows because ranks do not change.

A sum of squares polynomial is a real polynomial expressible as $p={\sum }_i{q}_i^2$ for real polynomials $q_i$. In the homogeneous degree-$2m$ case the summands may be taken homogeneous of degree $m$. Every such polynomial is nonnegative on real inputs; the converse fails for multivariate polynomials in general.

A real polynomial $p$ is a sum of squares polynomial if $p={\sum }_{\ell }{q}_{\ell }^2$ for real polynomials $q_{\ell }$. Such a representation certifies global nonnegativity. If $m(z)$ is the vector of relevant monomials, the identity $p(z)=m(z{)}^{T}Gm(z)$ with a positive semidefinite matrix $G$ encodes a sum of squares through a Gram matrix. Equating coefficients is linear in $G$, so finding a certificate is a semidefinite program.

The nonlinear Schrödinger equation adds an amplitude-dependent interaction to Schrödinger-type wave evolution. Its Laplacian term disperses the complex wave amplitude, while the sign and form of the nonlinear term determine focusing or defocusing behavior. A one-dimensional cubic specialization is the focusing nonlinear Schrodinger equation; members of this broader family need not be integrable systems.

For unknown-location Pauli operators of weight at most $t$, there are $N(t)={\sum }_{j=0}^t{3}^j\left(\genfrac{}{}{0ex}{}{n}{j}\right)$ physically distinct errors modulo phase, including the identity. A nondegenerate quantum code assigns them mutually orthogonal error-image subspaces, each of dimension $2^k$, within the $2^n$-dimensional physical Hilbert space. Hence $N(t)2^k\le 2^n$. The hypothesis of nondegeneracy is essential to this dimension count.

For a complex scalar field with global $U(1)$ invariance and Lagrangian $\mathcal{L}=\frac{1}{2}|{\partial }_t\varphi {|}^2-\frac{1}{2}|\nabla \varphi {|}^2-U(|\varphi |)$, a Q-ball is a localized phase-rotating configuration with conserved Noether charge $Q=\int \mathrm{Im}(\bar{\varphi }{\varphi }_t)$. At fixed charge its real profile can minimizeSuch solutions require a suitable potential and charge range; the complex nature of a field alone does not ensure existence. A common condition is ${\inf }_{f>0}2U(f)/f^2<U^{\prime \prime }(0)$.