A concurrent object behaves as if its operations took effect at individual instants between invocation and response, respecting nonoverlapping operations' real-time order. A read overlapping a replacement can return the old version while remaining linearizable.

A construct combining mutual exclusion for operations on shared state with condition-based waiting. A waiting caller releases the monitor while asleep and must recheck its predicate after regaining access.

A permit-based synchronization object. A down operation waits until it can consume a permit; an up operation supplies one. Unlike ownership-based mutex use, signalling can pass between different software threads.

Many waiting software threads wake for a state change that can benefit only one or a few of them. Repeated monitor reacquisition, failed predicate checks and context switches add overhead; selective notification can reduce it.

The singular real symmetric matrices of size two form the double cone $z^2=x^2+y^2$ in these coordinates. Its complement has three connected components: $z>\sqrt{x^2+y^2}$ is positive definite, $z<-\sqrt{x^2+y^2}$ is negative definite, and $|z|<\sqrt{x^2+y^2}$ is indefinite. This follows also from the eigenvalues $z\pm \sqrt{x^2+y^2}$. The definite cones are convex. In the indefinite region, first move $z$ to zero, then move the nonzero planar vector $(x,y)$ within the punctured plane, proving path-connectedness. Continuous determinant and trace signs prevent joining different regions without crossing singular matrices.

Sweat is fluid secreted onto the skin by sweat glands. Evaporation can remove heat, but fluid remaining on the surface does not provide the same cooling as fluid that evaporates. Composition and secretion rate also affect water and electrolyte losses.

The tractoid is the surface of revolution of a unit tractrix. For $X(\theta ,v)=(h(\theta ),\sin \theta \cos v,\sin \theta \sin v)$ with $0<\theta <\pi /2$, its coordinate tangent vectors are orthogonal and have squared lengths ${\cot }^2\theta ,{\sin }^2\theta$. The coordinates $x=v$, $y=\csc \theta$ transform this first fundamental form into $(d{x}^2+d{y}^2)/y^2$, giving a local isometry to the hyperbolic plane. Thus its Gaussian curvature is minus one. The angular coordinate is periodic globally, while its unwrapped coordinate is a real variable; this distinguishes the local metric identification from a global embedding of the whole hyperbolic plane.

The trochlea of superior oblique is a pulley-like structure in the anterosuperomedial orbit. It redirects the tendon of the superior oblique muscle. The redirection explains why the muscle’s effective pull differs from a straight line between its belly and the eyeball.

A smoothed form of the summatory totient function satisfiesUse the Dirichlet series $\zeta (s-1)/\zeta (s)$ and the kernel $x^s/[s(s+1)]$ to write the left side as a vertical contour integral. Move the line from real part greater than two to $3/2$. The only crossed pole is at two, with residue $x^3/(6\zeta (2))=x^3/{\pi }^2$; the remaining vertical integral is $O(x^{5/2})$ by the standard zeta bounds in the half-strip. If $A(x)={\sum }_{n\le x}\phi (n)$ and $S(x)={\int }_0^{x}A(t)dt$, positivity of the summands makes $A$ nondecreasing. The backward and forward difference quotients of $S$ bound $A(x)$. Insert $S(x)\sim x^3/{\pi }^2$, then let the relative difference-quotient step tend to zero, proving the displayed asymptotic. This monotone integral Tauberian lemma justifies the passage from a smoothed estimate to an unsmoothed one.

Suicidal ideation refers to thoughts about or consideration of suicide. Thoughts, intent and behavior are related but distinct measurements; the presence of one does not uniquely specify the others.^[ref]

In research, the wording of questions and the period being assessed affect what is counted. A lifetime measure and a report about the past week answer different questions and should not be treated as interchangeable prevalence estimates.

Effective pressure is ice-overburden pressure minus water pressure beneath the ice. It controls creep closure of channels and cavities; smaller positive $N$ weakens that closure and favors separation of ice from bed obstacles. It is therefore central to both basal sliding and subglacial drainage.

Frictional dissipation of water flow melts a channel's ice walls, while ice creep under effective pressure in subglacial drainage closes them. The resulting equilibrium has increasing effective pressure with discharge in the basic channel theory. A route gaining water can lower its water pressure and attract further discharge, widening at the expense of neighbors. This positive redistribution feedback produces arterial drainage.

Sliding opens water-filled cavities behind bed obstacles and ice creep closes them; narrow connections allow drainage between cavities. In a decreasing effective-pressure/discharge relation, a route carrying extra flux needs higher water pressure. That pressure diverts water to neighboring routes, favoring a distributed network instead of concentration into one channel.

At fixed total discharge $Q_R+Q_K$, remove a small amount $\epsilon$ from a channel and send it through connected cavities. The equilibrium channel effective pressure drops by $N_R^{\prime }\epsilon$, while the cavity pressure difference drops by $|N_K^{\prime }|\epsilon$. Under the displayed inequality the cavity system leaves a common effective pressure higher than the reduced-flux channel can sustain. Excess creep closure then decreases channel discharge further, amplifying the perturbation.

A glacial lake outburst flood is rapid release of water stored in or behind glacial ice or its associated dam. Melt-driven enlargement of a subglacial channel can create positive feedback between discharge, frictional heating and further channel opening. Lake depletion and pressure-dependent material creep closure then alter the event. Channel models explain one mechanism and should not be taken as a universal description of every glacial flood.

The water pressure at a glacier-bed interface affects the effective normal pressure $N$, the difference between ice overburden pressure $p_i$ and water pressure $p_w$. Higher water pressure at fixed overburden reduces effective compression of the bed contact and can alter cavitation and basal resistance. It is not identical to basal shear stress, and a sliding law depending only on shear suppresses this independent hydraulic influence.

The left subclavian vein returns blood from the left upper limb and joins the left internal jugular vein. The neighboring jugulosubclavian region also receives major lymphatic return. Shared location connects distinct venous and lymphatic transport pathways.

A predicate on possibly partial lists is admissible when it holds at the limit of every increasing chain of finite approximations on which it holds. Increasing here means replacing a bottom value by more defined information while retaining already defined constructors and values. This closure condition permits a property established for finite approximations to pass to an infinite list. The predicate “the list is finite” lacks this closure: every finite prefix ending in an undefined tail is finite in extent, but their limit may have infinitely many constructors.

For an admissible predicate on lazy lists $P$, prove $P(\perp )$ and $P([])$, and prove $P(xs)\Rightarrow P(x:xs)$ for every possibly partial head $x$. Constructor induction then establishes $P$ on every finite partial approximation, and admissibility extends it to limits, including infinite lists. Without the bottom value case and the limit condition, ordinary finite structural induction does not justify a claim about infinite data.^[ref]

With fixed endpoint values on a finite interval, the displayed derivative coercivity makes a finite-energy strong local minimizer of a variational integral also a local minimizer in $L^1$. Otherwise choose better competitors converging in $L^1$. The upper energy bound controls their derivatives in $L^p$, and a common endpoint controls their function norms. Compact Sobolev embedding gives a uniformly convergent subsequence; its limit is the $L^1$ limit. This contradicts strong local minimality. Convexity of the integrand is unnecessary for this neighborhood-transfer argument.