For weak drive in this homogeneous model, half maximum occurs at $|\Delta |=1/T_2$. The absorption full width at half maximum is thereforeThe example gives $2.653\mathrm{Hz}$ or $0.06230\mu \mathrm{T}$. At finite drive the model width is multiplied by $\sqrt{1+s}$. Field inhomogeneity and unresolved components can also broaden an actual line, so its width alone does not determine pure dephasing.

The same width appears by an original Fourier transform calculation. For $a=1/T_2$ and a free homogeneous signal demodulated exactly at resonance,This connects exponential memory to a Lorentzian absorptive line. It is a mathematical connection, not evidence that the 1946 teams performed pulsed Fourier transform spectroscopy. Instrumental signal bandwidth is a separate choice: narrowing a receiver band can reject thermal electrical noise, but too narrow a band distorts a changing signal. A sample linewidth and a receiver bandwidth are not interchangeable.

Drive reduces the longitudinal population imbalance when excitation competes with recovery. The dimensionless $s$ in the steady Bloch equations measures that competition.

By Jeremy Orloff, credited on the first page. Authored course notes in Complex Variables with Applications, MIT 18.04, Spring 2018. Spring 2018 identifies the course offering; no separate note publication or revision date is verified.

Online text: ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/330e301bd727c7bdaa679cf44cb75fe3_MIT18_04S18_topic2.pdf

Cited passages: §2.5, printed pp. 5–6, one point at infinity and exterior neighbourhoods; §2.5.2, printed p. 7, equatorial unit-sphere projection and north-pole cap. The first page gives the title, author and heading 2 Analytic functions.

Allocate $n$ distinct objects to $k$ labelled bins of specified nonnegative sizes $n_1,\dots ,n_k$, with ${\sum }_i{n}_i=n$. Order within a bin is ignored. The number of allocations is the multinomial coefficientTo prove this, order all the objects in a permutation and assign the first $n_1$ to the first bin, the next $n_2$ to the second, and so on. Every allocation comes from exactly $n_1!\cdots n_k!$ internal orderings. These disjoint classes exhaust the $n!$ permutations, proving the formula by the multiplication principle. Empty bins are allowed because $0!=1$.

For two bins of sizes $m$ and $n-m$, the count is $\left(\genfrac{}{}{0ex}{}{n}{m}\right)$ and every allocation has multiplicity $m!(n-m)!$ in the permutation sample space. Exchanging two equal-sized labelled bins generally gives a different allocation. In contrast, a set partition uses unlabelled nonempty blocks; forgetting bin labels changes the counting problem and must not be done silently.

A screened Coulomb pair potential models the effective electrostatic interaction of particles in an electrolyte. For identical weakly charged spheres of radius $a$, charge $Q$, and Debye–Hückel screening length ${\kappa }^{-1}$, a common pairwise approximation is $U(R)=Q^2{e}^{-\kappa (R-2a)}/[4\pi \epsilon (1+\kappa a{)}^{2}R]$ for $R\ge 2a$. The near-contact interaction can require more detailed boundary conditions or nonlinear screening. Charge, screening length and temperature all matter to a resulting phase diagram.

DLVO theory combines screened electrostatic repulsion from overlapping electric double layers with attractive Van der Waals forces to model colloid stability. A repulsive barrier can inhibit aggregation even when a deep short-distance attractive minimum exists. Increasing electrolyte concentration reduces the screening length and can lower that barrier. Molecular surface forces and polymer-mediated interactions require additions to this description.

A depletion force is an effective interaction caused by exclusion of smaller particles or macromolecules from regions around larger objects. Overlap of the excluded regions increases the accessible volume for the smaller species. At fixed reservoir osmotic pressure ${\Pi }_p$, the associated pair free energy is $-{\Pi }_p$ times the overlap volume in the ideal dilute-depletant approximation. It is an entropic attraction for two large hard particles. Surface adsorption can introduce different interactions, so adding an arbitrary polymer does not by itself guarantee depletion attraction.

Steric stabilization is resistance to particle aggregation supplied by attached or adsorbed molecular layers. Bringing two coated particles close compresses or overlaps the layers, increasing their free energy through conformational and osmotic effects. It differs from the depletion force of nonadsorbing dissolved chains.

Polymer bridging occurs when one polymer chain attaches to more than one particle, producing an effective attraction or network connection. Surface affinity and incomplete coverage are essential to this mechanism. It should be distinguished from both steric stabilization by densely coated particles and the depletion force produced by nonadsorbing chains.

A colloidal gel is a mechanically persistent, space-spanning particle network arrested on the observation timescale. Attractions can produce such a network during aggregation or arrested phase separation. A gel is a dynamical, often nonequilibrium state; its appearance does not establish an additional equilibrium thermodynamic phase.

A colloidal glass is a disordered dense particle assembly with structural relaxation too slow on the observation timescale. Crowding, repulsion or short-range attraction can impede rearrangement. Kinetic arrest must be distinguished from an equilibrium phase transition to a crystal.

Particle aggregation is association of initially separate particles into larger clusters. For colloids, both the rate of Brownian motion encounters and the probability that an encounter produces attachment matter. Strong attraction with negligible repulsive barriers can give diffusion-limited cluster aggregation; a barrier that makes attachment unlikely can give reaction-limited cluster aggregation. Reversible association and bond rearrangement require additional kinetic assumptions.

By Valentine Kabanets, explicitly credited on PDF p. 1. Authored lecture notes within CMPT 710 and CMPT 407 - Computational Complexity - Fall 2007, dated 8 November 2007. Simon Fraser University hosts the notes; the instructor credit is separate from the lecture byline.

Online text: www2.cs.sfu.ca/~kabanets/710_2007/new_lectures/l18.pdf

Cited passages: §3.1, printed pp. 3–4, Lemma 4, hypotheses and inductive proof; §3.2, printed p. 4, formula equality, one-sided error and degree/bit-cost discussion.