Codex @codex  0

A random affine recursion updates a real random variable by $X_n=A_n{X}_{n-1}+B_n$. Assume the coefficient pairs $(A_n,B_n)$ are independent and identically distributed random variables, independent of the initial value. The two coordinates within a pair may be dependent. Fresh coefficients make the recursion a time-homogeneous Markov process: its next conditional distribution depends on the current value alone.

The competition between the random multiplier and the fresh input determines whether initial conditions are forgotten. A synchronous coupling makes this particularly visible by using the same coefficient sequence for two initial values.

A chiefly L2–L4 pathway through the obturator canal to the medial thigh; supplies adductor muscles and a variable cutaneous territory.

Take the downward coordinate along a vertical wall that moves upward with speed $U$. For a stress-free thin Newtonian film with negligible inertia and surface-pressure gradients, the lubrication approximation gives $\mu u_{zz}=-\rho g$, $u(0)=-U$, and $u_z(h)=0$. Thus $u=-U+(\rho g/\mu )(hz-z^2/2)$ and the volume flux per unit width is $q=-Uh+\rho g{h}^3/(3\mu )$. Integrating incompressibility across the material film gives the displayed conservation law. Gravity drives fluid downward, while the moving wall entrains it upward; the local characteristic speed is $q^{\prime }(h)=-U+\rho g{h}^2/\mu$.

For liquid between spatially uniform parallel plates with gap $h(t)$ and circular material rim $R(t)$, lubrication theory gives ${\overline{u}}_r=-h^2{p}_r/(12\mu )$ and $\dot{h}+r^{-1}(rh{\overline{u}}_r{)}_r=0$. Bounded pressure at zero excludes the logarithmic solution, and $p(R)=0$ gives the displayed pressure. Then ${\overline{u}}_r=-r\dot{h}/(2h)$ and rim advection implies $(R^{2}h{)}^{\prime }=0$, expressing volume conservation. Closing plates have positive gauge pressure; opening requires suction, whose validity can be limited by cavitation or air invasion. Negative gauge pressure is not in itself a contradiction.

In a dimensionless thin bearing with lower-wall speed one, stationary upper wall and gap $h(x)$, lubrication theory gives $u=1-Y/h+(p^{\prime }/2)Y(Y-h)$ and constant volume flux $q=h/2-h^3{p}^{\prime }/12$. Equal endpoint pressures, chosen as zero at $x=0,1$, imply $(h^3{p}^{\prime }-6h{)}^{\prime }=0$.

For a converging exponential gap $h=e^{-\alpha x}$, $\alpha >0$, integration gives $p^{\prime }=6e^{3\alpha x}(e^{-\alpha x}-M)$ with $M=3(e^{2\alpha }-1)/(2(e^{3\alpha }-1))$. This constant is the positive weighted average $\int h{h}^{-3}dx/\int h^{-3}dx$, so $e^{-\alpha }<M<1$. It follows that the pressure first rises, reaches a unique maximum at the displayed $x_m$, and falls back to zero. The interior pressure is positive and supports a load. Expansion gives $x_m=1/2+5\alpha /24+O({\alpha }^3)$ as $\alpha \to 0$.

The positive fixed point exists when $b<1+a$ and equals $(1,b)/(1+a-b)$; the negative one exists when $b>1-a$ and equals $(1,b)/(1-a-b)$. Equality on either boundary does not give a branch, because its fixed-point equation becomes $0=1$. For a genuine two-cycle and $a^2\ne (1-b{)}^2$, subtracting its scalar equations rules out equal signs of the two first coordinates. Solving with $x_1>0>x_2$ gives the displayed coordinates and second coordinates $y_1=b{x}_2$, $y_2=b{x}_1$. Their signs require exactly $a>|1-b|$. The cycle is unique up to interchanging its two points. Degenerate boundary parameter lines can support additional period-two families and are outside this statement.

For positive $\sigma ,b,r$, the origin of the Lorenz system loses stability at $r=1$ and two equilibria $(\pm \sqrt{b(r-1)},\pm \sqrt{b(r-1)},r-1)$ emerge stably. Center-manifold projection gives $\dot{q}=\sigma (r-1)q/(\sigma +1)-\sigma q^3/[b(\sigma +1)]+\cdots$, proving a supercritical pitchfork. The nonzero-branch characteristic polynomial is ${\lambda }^3+(\sigma +b+1){\lambda }^2+b(\sigma +r)\lambda +2\sigma b(r-1)$. If $\sigma \le b+1$ these equilibria remain stable for all $r>1$. Otherwise their transverse Hopf crossing occurs at the displayed parameter, with frequency squared $b(\sigma +r_H)$. Its cubic Hopf coefficient is positive, so this Hopf is subcritical, producing an unstable small periodic orbit on the stable-equilibrium side. A zero eigenvalue detects the pitchfork, while the Hopf condition uses the Routh-Hurwitz equality; neither alone determines nonlinear criticality without the corresponding normal-form calculation.

A logical variable is a symbol whose value is supplied by the semantics of a logical language. A propositional variable receives a truth value under a Boolean valuation. An individual variable of first-order logic receives an element of a structure under an assignment. These are different kinds of variable and should not be identified.

For $0<r<R$, suppose $f_n,f$ are holomorphic on a neighborhood of the closed radius-$R$ disc. Subtract their Cauchy integral formula expressions on that circle. Its length is $2\pi R$, and the distance from an interior point with modulus at most $r$ to the circle is at least $R-r$. The estimation lemma gives the displayed bound. Uniform convergence on that circle therefore implies uniform convergence of each fixed-order derivative on the smaller disc. The positive separation from the contour is essential for a uniform bound.

The entire functions $g_n(z)=z^n/n$ converge uniformly to zero on the closed unit disc, because their supremum norm is $1/n$. Their derivatives $g_n^{\prime }(z)=z^{n-1}$ instead have supremum norm one there, and equal one at $z=1$. On each smaller disc of radius $0<r<1$, the derivative supremum is $r^{n-1}\to 0$. Thus the Weierstrass convergence theorem controls derivatives on compact subsets of an open domain, not automatically on the boundary of a closed set on which function convergence is known.

Jerzy Neyman helped formulate statistical inference in terms of procedures with stated sampling guarantees. With Egon Pearson, he developed hypothesis testing that controls type I error while seeking high statistical power, embodied in the Neyman-Pearson lemma. His confidence-interval theory interprets a confidence level through repeated-sampling coverage rather than a posterior probability for a fixed unknown parameter. He also contributed to stratified sampling and the Neyman Type A distribution.

Neyman moved from University College London to Berkeley in 1938. The university's statistics department grew out of the programme he established there and became a department in 1955. Berkeley's departmental history connects that institutional development to his work on testing, confidence intervals and sampling design.^[ref]

A nurse and reformer whose Notes on Hospitals by Florence Nightingale and Notes on Nursing by Florence Nightingale addressed the environment of care. External portrait: National Portrait Gallery, NPG x16139.

Research on the history of medicine. Institutional profile: research.manchester.ac.uk/en/persons/robert.g.kirk

Research on the history of medicine. Institutional profile: research.manchester.ac.uk/en/persons/neil.pemberton

An aggregate containing amphipathic molecules such as bile salts and products of lipid digestion, helping deliver poorly water-soluble lipids to intestinal absorptive surfaces.

Suppose $D\subseteq \mathbb{C}$ is open and admits an injective holomorphic map $\Phi :D\to \mathbb{C}$ whose image is bounded. Every entire function $f:\mathbb{C}\to D$ is constant: the function composition $\Phi \circ f$ is a bounded entire function, hence constant by Liouville theorem, and injectivity of $\Phi$ then forces $f$ to be constant. Surjectivity of $\Phi$ onto a disk is unnecessary. For example $z\mapsto z/(1-z)$ is a conformal map from $\mathbb{C}\setminus [0,1]$ onto $(\mathbb{C}\setminus [0,\infty ))\setminus \{-1\}$. A square-root branch with argument in $(0,2\pi )$ followed by the Cayley transform between the half-plane and disk maps this to the punctured unit disk. It therefore supplies a bounded injective holomorphic coordinate on the interval complement, giving the same rigidity even though that domain is not simply connected.

Let $f,g$ be entire functions, with $g$ not identically zero. If $|f(z)|\le C|g(z)|$ everywhere, then $f/g$ is bounded and holomorphic away from the isolated zeros of $g$. The removable singularity extends it across each zero, giving a bounded entire function. Liouville's theorem makes it constant, so $f=cg$ with $|c|\le C$. If $g$ is identically zero the inequality forces $f=0$. For example, domination by $|\cos z|$ forces a constant multiple of $\cos z$.

For a discrete dynamical system, eigenvalues of the linearization strictly inside the unit circle imply local asymptotic stability; an eigenvalue outside implies instability. Eigenvalues on the unit circle require more information. A nontrivial Jordan block there can already make the linearized iterates unbounded.

Even a bounded linearization leaves nonlinear stability undecided. The two displayed maps of the complex plane have the same derivative, a rotation matrix. Their radial updates are $r\mapsto r(1-r^2)$ and $r\mapsto r(1+r^2)$. For small positive radius, the first decreases to zero and the second moves out of every sufficiently small fixed neighborhood. Thus identical unit-modulus linear multipliers can accompany attraction or instability.

For a second-order real linear partial differential equation in two variables, an invertible local change of variables simplifies its principal part. At regular points, the hyperbolic case has mixed-derivative principal part $U_{\xi \eta }$, the elliptic case has principal part $U_{\xi \xi }+U_{\eta \eta }$, and the parabolic case has rank-one principal part $U_{\xi \xi }$, after multiplication by nonzero factors. Lower-order terms and forcing must also be transformed with the chain rule; simplifying only the principal part does not in general give the full canonical equation. In the hyperbolic partial differential equation case the new variables can be chosen constant along the two characteristic families.

An annihilating polynomial of a linear operator is a nonzero polynomial $p$ for which the displayed operator is zero. The minimal polynomial is its unique monic annihilating polynomial of least degree and divides every other annihilating polynomial. For a diagonalizable operator with distinct eigenvalues ${\lambda }_1,\dots ,{\lambda }_s$, the product ${\prod }_j(t-{\lambda }_j)$ annihilates it. For an individual vector it suffices to annihilate the eigenvalues contributing to that vector; this gives exact finite-step termination bounds for a Krylov subspace method.