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If a linear system $Ax=b$ has one solution $x_0$, every other solution differs from it by an element of the null space: $Ax=b$ is equivalent to $A(x-x_0)=0$. Its solution set is therefore the displayed affine subspace, of dimension equal to the nullity. If the system is inconsistent, the solution set is empty, rather than a translate of the kernel.

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.

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.

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.

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.