A greedy-choice property says that some optimal feasible solution contains the locally selected choice. Together with optimal substructure in the remaining instance, it allows repeated commitments. An exchange argument often establishes this property.^[ref]

For interval scheduling, a safe first choice need not belong to every optimal feasible solution. When several optima exist, a commitment may exclude one while retaining another. The useful loop invariant is that the accepted prefix can still be extended to an optimum. Merely keeping the accepted intervals compatible proves feasibility, which is a weaker claim.

Online text: www.math.uni-hamburg.de/home/diestel/books/graph.theory/preview/Ch6.pdf

Chapter 6 of Graph Theory by Reinhard Diestel, by Reinhard Diestel. Springer publishes the book; the author’s University of Hamburg site hosts the preview. No edition, publication year or revision date is established from this preview.

Cited passages: Chapter 6 opening, printed p. 155 (PDF p. 1): signed amounts and nonterminal conservation.; §6.2, printed p. 158 (PDF p. 4): independently specified orientation capacities, conditions F1/F2′/F3 and source/sink cut definition.; §6.2, printed p. 159 (PDF p. 5): Proposition 6.2.1, cancellation proof, total-value definition, footnote 4 on negative values, capacity upper bound and Theorem 6.2.2..

The upper part of the terrestrial atmosphere includes regions where ionization makes the response to electromagnetic waves depend strongly on mobile electrons. The ionosphere describes that electrical structure; its boundaries need not coincide with the boundaries defined by temperature in the mesosphere and thermosphere.

Geometry studies shapes, positions and spatial relationships through the structures and transformations under consideration. Euclidean geometry measures lengths and angles; projective geometry emphasizes incidence; differential geometry studies smooth spaces using tangent spaces and curvature. A circle can be stretched into an ellipse by an affine map, but generally not by a Euclidean isometry. Their curves are also homeomorphic, illustrating how topology preserves a different level of structure.

A geometric envelope is a curve tangent to the members of a smoothly varying family of curves. For a regular implicit family $F(x,y,s)=0$, candidate contact points also satisfy the partial derivative condition ${\partial }_{s}F=0$; regularity is needed to turn these candidates into an actual envelope. The Mach cone is a three-dimensional analogue for expanding spheres.

A coordinate system assigns numerical labels to positions so that the labels determine the point in the region described. A Cartesian coordinate system uses components along chosen axes; polar coordinates use a radius and an angle. For example, $(1,1)$ has polar coordinates $(\sqrt{2},\pi /4)$ on the usual angular branch. On a manifold, a coordinate chart gives a homeomorphism from an open region to an open subset of Euclidean space. Coordinates describe points but do not alone specify a metric.

Chapter 6 of Genomes, 2nd edition by Terence A. Brown. Terence A. Brown, printed as T. A. Brown, is the whole-book author; no independent chapter-author byline is established. The second edition is dated 2002. NCBI Bookshelf hosts the online text.

Online text: www.ncbi.nlm.nih.gov/books/NBK21117/

Cited passages: Official indexed chapter text, section 6.1.1, Chain termination sequencing in outline; Figure 6.1 and Figure 6.2 captions; single-nucleotide resolution and bottom-to-top autoradiographic interpretation; Technical Note 6.1, polyacrylamide versus agarose separation.

A selectable marker is a gene or other inherited feature that permits recovery of carriers under specified growth conditions. Selection for a marker does not identify every other nucleotide sequence in its carrier.

A gravitational singularity theorem gives sufficient conditions for geodesic incompleteness: some freely falling worldline or light-ray geodesic cannot continue through an unlimited proper time or affine parameter within the stated spacetime. The conclusion concerns the failure of complete geodesic evolution; it does not by itself establish a curvature singularity.

A typical argument combines a focusing assumption such as the null energy condition, the focusing of a geodesic congruence, and assumptions on the global causal structure. The Penrose singularity theorem uses a trapped surface to show that collapse need not preserve exact spherical symmetry to force null incompleteness under its other hypotheses. This is useful because highly symmetric exact solutions alone cannot establish what happens during generic collapse.^[ref]

In synchronous coordinates, fixed spatial-coordinate worldlines are timelike geodesics, and $t$ is their proper time. The metric has unit lapse function and zero shift vector in time units consistent with the displayed factor $c^2$. Indeed $g_{tt}=c^2$ and $g_{ti}=0$ give ${\Gamma }^{\mu }{}_{tt}=0$. Locally, an irrotational freely falling congruence permits this construction before crossings of the coordinate worldlines. Comoving observers in a Friedmann-Lemaître-Robertson-Walker metric provide the standard cosmological example.

The later angular test used an evacuated chamber with a radon source and scattering foil fixed in place. A diaphragm served as a collimator; a zinc sulfide screen and attached optical microscope rotated about the foil on a graduated platform. Particle scintillation counts sampled successive angles. Foil-out counts supplied background subtraction, with corrections for source radioactive decay and finite beam width. Shielding reduced interfering radiation, but screen glow limited large angles. This tested the Rutherford differential cross-section.^[ref]