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An incompressible Mooney-Rivlin solid has strain energy density $W=C_1(I_1-3)+C_2(I_2-3)$, with right Cauchy-Green deformation tensor $C=F^{T}F$, $I_1=\mathrm{tr}C$ and $I_2=\mathrm{tr}{C}^{-1}$ when $\det C=1$. It extends the neo-Hookean solid by a second invariant. A convention $W={\mu }_1(I_1-3)/2-{\mu }_2(I_2-3)/2$ means $C_1={\mu }_1/2$, $C_2=-{\mu }_2/2$; the minus sign must be retained when comparing material constants.

For a regular material constraint $F(A)=0$, admissible rates of the deformation gradient lie in the kernel of $F_{,A}$. The difference between the nominal stress and the derivative of the strain energy density annihilates that tangent space by virtual work, so it is a Lagrange multiplier times $F_{,A}$. For incompressibility, the derivative of the determinant is its cofactor matrix, and the material-first reaction is $q{A}^{-1}$ when $\det A=1$.

For unconstrained hyperelastic material, the Hessian of the strain energy density gives $\delta N_{\alpha i}=c_{\alpha i\beta j}\delta A_{j\beta }$. Commuting the second derivatives gives major symmetry $c_{\alpha i\beta j}=c_{\beta j\alpha i}$. These material-coordinate moduli linearize about a finite base deformation; they need not have the minor symmetries of a small-strain elastic stiffness tensor. Material constraints require their additional reaction increments and admissibility equations.

At leading thin-shell order, a spherical balloon with reference radius $a_0$, thickness $h_0$ and current radius $a=\lambda a_0$ has reference elastic energy $4\pi a_0^2{h}_{0}W(\lambda ,\lambda ,{\lambda }^{-2})$. Equating its differential to pressure work $p4\pi a^{2}da$ gives the displayed pressure difference. Its thickness becomes $h_0{\lambda }^{-2}$ by incompressibility. The same result follows from the spherical membrane force balance $p=2h{\sigma }_t/a$, with current tangential Cauchy stress tensor component ${\sigma }_t=\lambda N$ and the equibiaxial nominal tension of an incompressible sheet. Thin-shell validity requires $h/a\ll 1$.

At a stress-free natural reference state, an objective isotropic hyperelastic material with energy $W$ per unit mass has ${\rho }_0[W(I+H)-W(I)]=\lambda (\mathrm{tr}e{)}^2/2+\mu e:e+O(|H{|}^3)$, where $e=\mathrm{sym}H$. Twice differentiating gives the displayed Lamé parameters and the usual elastic stiffness tensor. A prestressed reference requires initial-stress corrections.

This objective isotropic constitutive model uses the Green-Lagrange strain tensor $E=(A^{\mathsf{T}}A-I)/2$. It reproduces linear elasticity at small strain. Under simple shear $A=[(1,q,0{)}^{\mathsf{T}},e_2,e_3]$, its longitudinal nominal stress is $(\lambda +2\mu )q^2/2$. Nonuniform finite shear therefore generates longitudinal acceleration, even though the corresponding infinitesimal transverse mode is uncoupled.

For a plane material-coordinate elastic wave with $u=(0,w(X_1,t),0)$, set $q=w_{,X_1}$. The longitudinal equation is $0={\partial }_{X_1}{P}_{11}(1,q,0)$. Arbitrary nonuniform pure transverse profiles thus require the displayed constitutive compatibility, which does not follow from isotropy alone. If it holds and ${\tau }^{\prime }(q)>0$, the shear equation closes as ${\rho }_0{w}_{tt}={\partial }_{X_1}\tau (q)$ with speed $c(q{)}^2={\tau }^{\prime }(q)/{\rho }_0$.

The composition of hyperbolic reflections in two ultraparallel hyperbolic lines has infinite order. Map their common perpendicular to the imaginary axis in the Poincaré half-plane model. The two reflecting hyperbolic lines become semicircles centred at zero, with distinct radii $r_1,r_2$. Reflection in the radius-$r$ semicircle is $z\mapsto r^2/\overline{z}$, so their composition is $z\mapsto (r_2/r_1{)}^{2}z$. Every positive power remains nonidentity, because $(r_2/r_1{)}^{2k}\ne 1$ for $k\ge 1$.

A reflection in the perpendicular bisector of a point and its image corrects that point. A second bisector reflection through the corrected point can correct another point without moving the first. An isometry fixing two distinct points is either the identity or reflection in their joining geodesic: points with prescribed distances from both centres have at most two possibilities, exchanged by that reflection. Fixing a third noncollinear point then fixes every point. This constructs a factorization with at most three hyperbolic reflections.

The curvature-minus-one hyperbolic metric on the punctured disk assigns distance $2\log n$ along the radial geodesic from $e^{-1}$ to $e^{-n^2}$. Its exponential distances therefore sum to a finite value; the triangle inequality gives the same conclusion for any fixed basepoint. Nevertheless no nonzero bounded holomorphic function has these zeros: the Riemann removable singularity theorem extends it at zero, and the identity theorem then makes it identically zero. Simple connectivity is necessary in the hyperbolic characterization of bounded analytic zero sets.

For points $ib$ and $a+ib$ with $a,b>0$, the hyperbolic distance is $2\mathrm{arsinh}(a/(2b))$. It equals their Euclidean distance $a$ exactly when the displayed relation holds. The function of $a$ decreases continuously from one to zero: its derivative has the sign of $\sinh (a/2)-(a/2)\cosh (a/2)<0$. Hence every height $b\in (0,1)$ has exactly one positive matching horizontal separation.

For a proper arc $I$ of the boundary of the unit disc, $Q(I)$ is the side of the hyperbolic geodesic between its endpoints adjacent to $I$. For short arcs this is a curved boundary cap. For arcs longer than a semicircle it contains the center. With endpoints $e^{\pm i\alpha }$ and $0<\alpha <\pi /2$, the geodesic is the circle with center $\sec \alpha$ and radius $\tan \alpha$, orthogonal to the unit circle. It is a diameter at $\alpha =\pi /2$.

A fixed stellar homology profile with $P=\rho \mathcal{R}T/{\mu }_i+K_{\mathrm{nr}}{\rho }^{5/3}$ giveswhere $A,B>0$. As a function of $1/R$ this is a concave quadratic; its maximum is $A^2{G}^2{\mu }_i{M}^{4/3}/(4B{K}_{\mathrm{nr}}\mathcal{R})$, at $R=2B{K}_{\mathrm{nr}}{M}^{-1/3}/(AG)$. At the maximum, the central electron degeneracy pressure and thermal pressure are equal. If this temperature is insufficient for sustained hydrogen burning, the object becomes a brown dwarf. Only the positive-temperature segment represents a physical sequence.

Consider two widely separated rigid spheres of equal radius $a$, with separation $R\mathbf{n}$, in Stokes flow. Apply a force $6\pi \mu a{\mathbf{U}}_0$ to the first sphere and let the second be force-free and torque-free. The second sphere samples the rate-of-strain tensor of the first sphere's Stokeslet, $\mathbf{E}=3a({\mathbf{U}}_0\cdot \mathbf{n})(\mathbf{I}-3\mathbf{nn})/(4R^2)$. Its leading disturbance is the stresslet of a sphere in a uniform straining Stokes flow. Evaluating that disturbance back at the first sphere givesThe hydrodynamic mobility matrix decreases at fixed force, consistently with the increase of resistance caused by an additional rigid inclusion.

For the nonzero spectral decomposition $\rho ={\sum }_{a=1}^r{\lambda }_a|e_a⟩⟨e_a|$, every pure-state quantum state ensemble of $\rho$ is specified by the displayed column linear isometry. To prove this, write $w_j=\sqrt{p_j}{\varphi }_j$. Positivity forces each $w_j$ into the support of $\rho$, and the coefficients $U_{ja}=⟨e_a|w_j⟩/\sqrt{{\lambda }_a}$ have orthonormal columns because ${\sum }_j|w_j⟩⟨w_j|=\rho$. Conversely this column identity reconstructs $\rho$; the squared vector norms are the probabilities and sum to one. Complete the columns to unitary bases after adding zero ensemble vectors to obtain the usual unitary-mixing formulation.

Replace sibling leaves in a code tree labelled $a,b$ by their common parent in a rooted tree, now a leaf labelled $m$ of weight $w_m=w_a+w_b$. All other codewords stay fixed; each of $a,b$ loses one bit. For any such code tree $T$ and the contracted code tree $T^{\prime }$, their weighted costs satisfyIf $a,b$ have two least weights, the Huffman sibling property gives an optimal prefix code containing this pair. Its contracted prefix code must be optimal for the reduced weights $w^{\prime }$: any cheaper reduced prefix code could be expanded at $m$ by appending $0,1$, giving a cheaper original prefix code. Conversely, expanding any optimal reduced prefix code adds precisely $w_a+w_b$ and attains the original minimum. This is optimal substructure, the justification for repeating the merge in Huffman coding. ^[ref]

In the exchange diagram, $C,D$ contract to weight $0.3$ at leaf depth in a code tree $2$. The reduced expected codeword length is $0.4\cdot 1+0.3\cdot 2+0.3\cdot 2=1.6$ bits; expanding restores $1.6+0.3=1.9$ bits. For $n=2$, the reduction leaves one symbol: allow its temporary codeword to be empty, of cost zero, so expansion gives the two one-bit codewords. This convention is only the terminal reduced problem; the original source has at least two symbols.

Two leaves of a code tree with the same parent in a rooted tree. Their codewords have a common initial segment $u$ and differ only in the final bit: $u0,u1$. Their leaf depth in a code tree is equal. The Huffman sibling property places two chosen least weights at such leaves in some optimal prefix code.

For $m\ge 2$ equiprobable symbols, the maximum codeword length of a binary Huffman code is $s=\lceil {\log }_{2}m\rceil$. Exactly $2^s-m$ codewords have length $s-1$ and $2m-2^s$ have length $s$. The balancing exchange for a uniform optimal prefix code restricts the depths to these two levels; counting leaves and using equality in the Kraft inequality determine the numbers.

// Target: probability-and-statistics.bigb

An orthogonal similarity preserves skew symmetry, so upper-Hessenberg reduction of a real skew-symmetric matrix is automatically tridiagonal with zero diagonal. For a Householder reflection, the scalar $v^{T}Av$ vanishes, eliminating the fourth term of the general similarity update. Storing only one triangle and using the displayed skew rank-two update costs $2n^3/3+O(n^2)$ scalar products, the same leading product count as optimized symmetric tridiagonalization; the simplification saves lower-order work.

The displayed pointwise control limits follow from the normal approximation to an observed binomial proportion. Clip plotted limits to the probability range. Their nominal coverage is approximate and can be poor for small expected counts; exact binomial tails give a discrete alternative. Pointwise coverage does not provide simultaneous coverage of all hospitals; use Bonferroni correction or another familywise error rate procedure when that is the required guarantee.