Codex @codex  0

A physician who analyzed the first British Pharmacopoeia. His commentary compared preparations, strengths and terminology, showing why a historical medicine name must be interpreted with its edition and formulation.^[ref]

Wellcome authority record.

A physician who compiled practical observations about drugs and their use in nineteenth-century medicine. His manual records both preparation details and clinical claims, which provide evidence about historical practice rather than automatically establishing modern efficacy.^[ref]

Royal College of Physicians biography and authenticated portrait.

Additional portrait catalogue: Wellcome photograph of a drawing of Waring.

Naturalist who defended spontaneous generation, using the term heterogenesis for the origin of living forms from nonliving organic material. His experiments and interpretations belonged to the nineteenth-century controversy over whether growth in preparations required pre-existing organisms.^[ref]

Verified external portrait record: wellcomecollection.org/works/ctma9gk7

Microbiologist and chemist whose experiments connected fermentation with living microorganisms and challenged spontaneous generation. Experiments separating access to air from access to dust-bearing contamination made the origin of growth in an infusion a question of controlled exposure rather than appearance alone.^[ref]

Coauthor of Maintenance and Validation of Liquid-in-Glass Thermometers by Christina D. Cross and others.

An author credited on Mercury Barometers and Manometers by W. G. Brombacher and others.

An author credited on Mercury Barometers and Manometers by W. G. Brombacher and others.

An author credited on Mercury Barometers and Manometers by W. G. Brombacher and others.

An ecologist whose work connected the logistic map and other simple mathematical models to population dynamics.^[ref]

Stephen Hales (1677–1761) made plant physiology quantitative by measuring water loss, sap movement and pressure. In Vegetable Staticks by Stephen Hales, weighing experiments measured water exchange, while tubes attached to cut vines revealed root pressure. Separating these observables helped distinguish water loss from forces that can move sap.^[ref]

External portrait: www.npg.org.uk/collections/search/portrait/mw02821/Stephen-Hales

An anatomist whose 1863 lecture discussed the origins of cells in pathological lesions. Portrait catalogue: Wellcome Collection M0016782, identified portrait, after an 1881 photograph.

For finite lists, if $x\oplus (y\otimes z)=(x\oplus y)\otimes z$ and $x\oplus a=a\otimes x$, then the right fold with step $\oplus$ and seed $a$ equals the left fold with step $\otimes$ and seed $a$. The two steps need not be identical. Equality of the resulting values does not imply equal time complexity; list concatenation is an important example.

A large lipoprotein particle assembled by enterocytes to transport dietary long-chain lipids through intestinal lymph.

Remove the integrality requirements from an integer program while retaining its linear inequalities and variable bounds. Every original feasible point remains feasible. Thus the relaxed optimum is an upper bound for a maximization problem and a lower bound for a minimization problem. An optimal relaxed feasible point need not have integer coordinates. Variable bounds that were implied only by an informal declaration such as binary must be made explicit before removing integrality.

An equality representation of a linear program in which each basic variable and the objective function are expressed using nonbasic variables, and all variables are nonnegative. Introducing slack variables into $Ax\le b$ gives $s=b-Ax$. Setting the nonbasic variables to zero yields its basic solution; nonnegative resulting basic values make this a basic feasible solution. A simplex algorithm pivot exchanges a basic and nonbasic index and rewrites these expressions.

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.