For the cylinder-plane image charges geometry, the attractive force is the displayed quantity per unit length. One derivation uses the effective energy including the voltage source: at fixed cylinder voltage $V$, the force toward increasing centre height is $F_d=V^2(d{C}^{\prime }/dd)/2$. Since $C^{\prime }=2\pi {\epsilon }_0/\mathrm{arcosh}(d/a)$ decreases with $d$, this force points toward the plane. For $d\gg a$, $\lambda \simeq 2\pi {\epsilon }_{0}V/\ln (2d/a)$ and $f\simeq \pi {\epsilon }_0{V}^2/[d{\ln }^2(2d/a)]$. A total force additionally requires the cable length.

Unsigned lecture notes in CS 4820: Algorithms, Spring 2011. The masthead identifies Cornell University and dates the document 2 May 2011. No individual author or course instructor is established. Cornell University is the institutional issuer and web host.

Online text: www.cs.cornell.edu/courses/cs4820/2011sp/handouts/approx_algs.pdf

Cited passages: Printed p. 1, heading and §1.1: institutional issuer, course, explicit date, independent random vertex placement and half-total-weight expectation.; §1.1.2, printed pp. 2–3: partial-assignment expectation, Algorithms 1–2, nondecreasing expectation, half-total-edge-weight guarantee and linear-time adjacency-list implementation..

Unsigned lecture in CS 312 at Cornell University, hosted by Cornell University’s Department of Computer Science. No individual author or publication date is credited on the inspected page.

Lecture notes: www.cs.cornell.edu/courses/cs312/2008sp/lectures/lec20.html

Cited passages: Resizable hash tables and amortized analysis; Amortized analysis.

Unsigned institutional lecture notes in CS 2112 Fall 2019. No individual author, editor or transcriber is credited, and no publication or revision date is established.

Online text: www.cs.cornell.edu/courses/cs2112/2019fa/lectures/loopinv/

Cited passages: opening definition; Using loop invariants to show code is correct, Establishment, Preservation, Postcondition and Termination; Binary search loop invariant; Debugging and choosing loop invariants.

The characteristic outward shape of a crystal; distinct from its internal crystal structure.

An atomic basis of a crystal is the motif of chemical species $s_j$ and relative positions $r_j$ attached identically to every lattice point $R$ of a Bravais lattice $\Lambda$. The ideal periodic structure is the labeled setThe primitive lattice translation vectors generate $R$ by integer combinations; the atomic basis lists what is repeated at $R$. These are different meanings of the word basis. In particular, the offsets $r_j$ need not be linearly independent vectors, and one of them may be zero. A primitive unit cell represents $N$ motif sites, even though it represents only one lattice point.

Changing the origin, choosing another primitive unit cell, or moving a motif site by a Bravais lattice translation changes the coordinates used to describe the same infinite crystal structure. The grouping into a motif does not, by itself, identify bonded molecules. In solid sodium chloride, grouping one sodium cation and one chlorine anion into an atomic basis does not create a collection of discrete NaCl molecules; the ionic crystal has an extended network of equivalent neighboring ions.

A lattice plane is a plane containing a two-dimensional periodic array of lattice points. Parallel lattice planes form equally spaced geometrical sheets; an atomic basis can place chemical atoms between them.

With primitive lattice translation vectors $a_i$, write $r=x{a}_1+y{a}_2+z{a}_3$. For integers $h,k,l$ with greatest common divisor one, the plane equationsgive a complete family labeled by Miller indices $(hkl)$. In a primitive cubic lattice, $(110)$ means $x+y=m$, parallel to $a_3$.^[ref]

Crystallography studies crystal structures and their symmetries. A structure determination connects measurements such as crystal scattering patterns to a model of the unit cell and its atomic basis of a crystal.

X-ray crystallographic structure inference makes that connection explicit: compare proposed crystal structures through the positions and wave intensities of their predicted reflections. The main difficulty is that a diffraction pattern encodes wave interference, so it must be interpreted through a model rather than read as a direct picture of the chemical atoms.

A surface reconstruction rearranges chemical atoms at a crystal surface relative to a simple termination of its bulk crystal structure.

By Nicholas Harvey, posted under the byline nickhar on 1 February 2012. The institutional course page links this post as lecture 9 of CPSC 536N: Randomized Algorithms; its scheduled lecture date, 2 February 2012, is distinct from the blog publication date. WordPress.com is the host.

Online text: nickhar.wordpress.com/2012/02/01/lecture-9-polynomial-identity-testing-by-the-schwartz-zippel-lemma/

Cited passages: §1, formal coefficient-based zero and evaluation access; §1.2, Lemma 1 and proof; §1.3, evaluation rule and independent repetition; §1.4, formal polynomials versus induced functions, extension fields and representation limits.

Physical cosmology connects the universe's history to the measurable behaviour of its contents. The Friedmann equations describe cosmic expansion; atomic physics, radiation pressure and gravity determine what that expanding universe looks like. A particularly revealing example is the route from cosmological recombination through the photon-baryon fluid to the Cosmic microwave background acoustic peaks. The universe supplies the initial conditions and the laboratory; the prediction must still pass through a model of the telescope before it can be compared with observations.

The practical reward is that one sky contains several different tests. The Cosmic microwave background power spectrum carries information about the sound horizon, the cosmological baryon density and cosmic geometry. These are related quantities with different observational signatures. Angular sound-horizon degeneracy explains why measuring a characteristic angle alone cannot count the universe's baryons.