A wall displacement $\eta$ with spring law $p^{\prime }={\rho }_{0}k\eta$, combined with linear Bernoulli $p^{\prime }=-{\rho }_0{\varphi }_t$ and kinematic condition ${\varphi }_n={\eta }_t$, gives the displayed dynamic boundary condition. For a rectangular cavity of width $a$, height $b$, rigid remaining walls and sound speed $c_0$, real vertical modes obey ${\omega }^2=c_0^2[(n\pi /a{)}^2+(\beta /b{)}^2]$ and $\beta \tan \beta =-{\omega }^{2}b/k$. On each interval $((j+1/2)\pi ,(j+1)\pi )$ the tangent equation has a real root. As $c_0^2/(kb)\to \infty$ at fixed mode numbers, these branches tend to $\beta =(j+1/2)\pi$, the pressure-release upper-wall acoustic spectrum. Imaginary vertical exponents can describe additional branches and are not counted by this real-root argument.

A velocity potential obeying the wave equation with constant speed of sound $c_0$ has spherical normal modes $\varphi =e^{-i\omega t}[A\cos kr+B\sin kr]/r$, $k=\omega /c_0$. Zero radial velocity at radii $a,b$ gives Neumann conditions on the potential, not on its numerator. The inner condition is built into $F(r)=C[\cos k(r-a)+(ka{)}^{-1}\sin k(r-a)]/r$. The outer condition gives $k(b-a)\cos k(b-a)-(1+k^{2}ab)\sin k(b-a)=0$, equivalent to the displayed equation for nonzero oscillatory modes. The formal zero root is a constant potential gauge, rather than a nonzero-frequency acoustic wave. Applying a one-dimensional sine condition to the numerator would miss the spherical geometric factors.

A search direction is the nonzero vector $p_j$ used in the displayed optimization update. In a line search with positive step size, its orientation matters: a descent direction has negative directional derivative of the objective function. Minimization over a full line instead permits either sign of ${\alpha }_j$.

Suppose $f$ is bounded below on its initial level set, continuously differentiable with a Lipschitz gradient nearby, and each descent direction $p_k$ uses a step satisfying fixed Wolfe conditions. If $\cos {\theta }_k=-\nabla f(x_k{)}^T{p}_k/(\Vert \nabla f(x_k)\Vert \Vert p_k\Vert )$, the displayed series converges. In particular a uniform positive lower bound on these cosines gives $\nabla f(x_k)\to 0$.

For a gradient Lipschitz constant $L$, the curvature condition gives ${\alpha }_k\ge (1-\eta )(-\nabla f^T{p}_k)/(L\Vert p_k{\Vert }^2)$. Sufficient decrease then gives $f(x_k)-f(x_{k+1})\ge \sigma (1-\eta ){\cos }^2{\theta }_k\Vert \nabla f(x_k){\Vert }^2/L$. Summing and using the lower bound on $f$ proves the assertion.

A rectangular upper bound $u,v\le M$, $M>1$, is inward pointing for the Leslie-Gower predator-prey model. To avoid the singular edge $u=0$, use $P=u{e}^{-v}$ and the inequality $\dot{P}/P\ge 1-u[1+(a/d+b{)}^2/(4b)]$, obtained by completing a square. Choose $\epsilon e^M$ sufficiently small and $0<\eta <\epsilon$. Then $u\le M$, $\eta \le v\le M$, $P\ge \epsilon$ is a compact forward-invariant set in the positive quadrant. Removing an outward-flow neighborhood of a repelling equilibrium leaves a trapping region without equilibria, so the Poincaré-Bendixson theorem yields a periodic omega-limit orbit.

Time-dependent elastance models the pressure generated by a cardiac ventricle at its current stressed volume. The reference $V_0$ is its zero-pressure volume. The inverse of elastance is compliance at fixed time. Rising elastance increases pressure without volume change during isovolumetric contraction; an approximately constant systolic value $E_S$ gives compliance $C_s=1/E_S$ during a simplified ejection stage.

Return the successive accumulator values of a left fold, including the initial accumulator: scanl f e [] = [e] and scanl f e (x:xs) = e : scanl f (f e x) xs. The $k$th result is the fold of the first $k$ elements, proved by the fold equation. Thus the scan equals map (foldl f e) . inits, but computes $n$ updates instead of $n(n+1)/2$ on a finite input of length $n$. Accumulator strictness and the cost of an update remain separate from this operation count.

Here the residual map $R$ may be nonlinear. Its Jacobian matrix $J=DR$ gives $\nabla f=J^{T}R$ and ${\nabla }^{2}f=J^{T}J+{\sum }_j{R}_j{\nabla }^2{R}_j$. The Gauss-Newton algorithm drops the second term to obtain an approximation using only first derivatives of the residuals. Full column rank of $J$ makes $J^{T}J$ positive definite; at a zero-residual solution this approximation equals the exact Hessian.

The +2 oxidation state of lead. The actual aqueous species can depend on pH, hydrolysis and ligands; a bare ionic formula is an analytical shorthand.

A yellow, sparingly soluble lead–iodide material used in historical lead recognition chemistry.

For a finite nonempty collection of exact real multipliers greater than one, the multiplicative closure of one can be enumerated by a shared lazy list: emit one, then merge its elementwise multiples by all generators. Every value after one has a smaller predecessor obtained by removing one factor, proving productivity by increasing value. There are only finitely many exponent tuples below any fixed bound, because every generator is greater than one; thus locally finite sorted-stream merge eventually emits every generated value. Equal products are emitted once.

A multiplier below one contradicts increasing order starting at one. A multiplier equal to one adds no new values and should be removed from the recursive streams; an empty remaining generator collection has the finite closure consisting of one. These qualifications concern exact arithmetic: finite precision or overflow can violate the order and growth assumptions.

A polymeric osmotic laxative; 3350 denotes its approximate average molecular mass. Retention of water in the intestinal lumen accounts for its local action.^[ref]

A medicine producing bowel evacuation, historically often implying a more vigorous effect than a mild laxative.

The cricothyroid muscle changes the relation of the cricoid cartilage and thyroid cartilage, helping lengthen and tense the vocal folds. This contributes to pitch control during phonation. It is supplied by the external branch of the superior laryngeal nerve, a distinctive relation among intrinsic laryngeal muscles.

The caecum is a blind-ended initial region of the large intestine near the small-intestinal entry. Its size and importance in fermentation vary substantially among mammals. The modest human structure should not be assumed representative of every herbivorous digestive system.

For a maximization problem, every feasible point's finite objective is bounded above by every valid Lagrangian relaxation value. If a sequence of nonnegative multipliers makes those upper bounds tend to minus infinity, no feasible point can exist. Thus a minimization Lagrangian dual problem unbounded below certifies primal infeasibility. Finite dual value alone does not certify primal feasibility.

For conjugate exponents $p,q>1$, the constraint set is compact, so the continuous product has a maximum. Boundary endpoints have product zero, while $(1,1)$ is feasible with product one. An interior maximum therefore satisfies the Lagrange multiplier equations $y=\lambda x^{p-1}$ and $x=\lambda y^{q-1}$. Multiplying by $x$ and $y$ respectively gives $x^p=y^q$, since the multiplier is nonzero. Substitution in the constraint makes both powers equal one. Thus the maximum is one, attained uniquely at $(1,1)$. The boundary check and compactness are what turn the necessary multiplier equations into a global conclusion.

An integral constraint need not admit a normal Lagrange multiplier unless its first variation is nonzero. For the quadratic normalization in the title, the variation in direction $y$ has derivative $2\int w{y}^2=2$, so the constraint is always regular at an admissible function. With free endpoint values this justifies the Euler-Lagrange equation for $F-\lambda w{y}^2$ and its natural endpoint conditions. For $F=p{y}^{\prime 2}-q{y}^2$, multiplication of the differential equation by $y$ gives $\lambda =J[y]$.

For phases ${\theta }_1,\dots ,{\theta }_N$, the Kuramoto order parameter is their complex centroid on the unit circle:The triangle inequality gives $0\le r\le 1$. Equality $r=1$ requires all phase vectors to point in the same direction. When $r>0$, $\psi$ is the complex argument of $Z$; it is undefined at $r=0$.

Zero does not specify a unique phase distribution. For example, the four phases $0,0,\pi ,\pi$ give $Z=(1+1-1-1)/4=0$, while $0,\pi /2,\pi ,3\pi /2$ give $Z=(1+i-1-i)/4=0$. The first has two clusters and the second is evenly spread. A first complex moment cannot recover every detail of a distribution.

Multiplying $Z$ by $e^{-i{\theta }_i}$ and taking the imaginary part givesThis identity rewrites all pairwise terms of the Kuramoto model using two collective real variables. It remains valid at $r=0$ when expressed as $\mathrm{Im}(Z{e}^{-i{\theta }_i})$, without assigning a nonexistent complex argument to zero.

A microscopic account of gas properties using the motion and collisions of its constituents. For a dilute monatomic ideal gas, translational kinetic energy gives $p=Nm⟨v^2⟩/(3V)$ and $U=3N{k}_{\mathrm{B}}T/2$. Interactions and additional molecular modes require extensions.