Online text: ocw.mit.edu/courses/6-441-information-theory-spring-2016/aaa8d18ddecde45f97134d3f3dcee4a3_MIT6_441S16_chapter_24.pdf

Cited passages: §24.1, Theorem 24.1, printed pp. 246–247; §24.2, Theorem 24.2, printed p. 247: memoryless source hypotheses, single-letterization and asymptotic achievability.

Online text: ocw.mit.edu/courses/6-441-information-theory-spring-2016/880c39878ba7dc35a5fbb35758d42a69_MIT6_441S16_chapter_23.pdf

Cited passages: §23.2, Definition 23.1 and Theorem 23.2, printed pp. 242–243: reconstruction alphabets, separable distortion and the information converse.

Online text: ocw.mit.edu/courses/6-441-information-theory-spring-2016/184197ca5d5418da2415d37e929860b9_MIT6_441S16_chapter_2.pdf

Cited passages: Section 2.4, Definition 2.4 and Theorem 2.5(1), printed pp. 26–27; Theorem 2.1, printed p. 20; Theorem 2.3(1,3), printed p. 23; Theorem 2.4(2) and Corollary 2.4, printed p. 24; Theorem 2.4(1) and its proof, printed p. 24: I(X;X) is infinite for a non-discrete distribution; Theorem 2.4(3), printed p. 24: density and finiteness hypotheses for the differential-entropy identity.

Cited passages: MIT 6.441, Chapter 2, Definition 2.4 and Theorem 2.5(1); MIT 6.441, Chapter 2, note following Theorem 2.5.

Online text: ocw.mit.edu/courses/6-441-information-theory-spring-2016/2243edffb30f57181ed97dcb77691580_MIT6_441S16_chapter_1.pdf

Cited passages: Chapter 1, Definition 1.3, printed p. 8; Theorem 1.1(4), printed p. 10.

Online text: pdos.csail.mit.edu/6.1810/2022/labs/cow.html

Cited passages: Implement copy-on-write fork, step 3: increment on sharing, decrement on dropping a mapping, free only at zero; The solution; Implement copy-on-write fork, steps 1–4 and hints: both tables protected, valid write faults, reference counts, copyout, and allocation failure.

By Massachusetts Institute of Technology, 6.006 course staff.

Online text: ocw.mit.edu/courses/6-006-introduction-to-algorithms-fall-2011/e59f8a55929028498953691891229a17_MIT6_006F11_lec14.pdf

Cited passages: Lecture 14: Graphs II: Depth-First Search, printed pp. 3–4: stack-duration marks and cycle detection; MIT 6.006, lecture 14: depth-first search, pp. 2–3.

Online text: ocw.mit.edu/courses/6-006-introduction-to-algorithms-fall-2011/160b3b5f9da2e03815ca1e6ee0dba62a_MIT6_006F11_lec09.pdf

Cited passages: Lecture 9: Hashing II, p. 2, rebuilding cost and geometric table doubling; Lecture 9: Hashing II, p. 3, expected bounds and shrinking at one-quarter occupancy.

Online text: ocw.mit.edu/courses/6-006-introduction-to-algorithms-fall-2011/d3e4d64266d481c74c9e7c15e09999fe_MIT6_006F11_lec08.pdf

Cited passages: Lecture 8: Hashing I, p. 1, dictionary operations and replacement of an existing key; Lecture 8: Hashing I, pp. 4–5, chaining worst case and simple uniform hashing assumption.

Cited passages: MIT 6.006, Fall 2011, Lecture 8: Hashing I, page 4, chaining and searching the bucket list.

Online text: ocw.mit.edu/ans7870/6/6.005/s16/classes/13-abstraction-functions-rep-invariants/index.html

Cited passages: Reading 13, How to Establish Invariants: creation, preservation and absence of representation exposure; Reading 13, Invariants, Immutability: mutable aliases through both returned values and constructor arguments; Reading 13, Rep Invariant and Abstraction Function: representation values, abstract values, AF and RI.

By Terence Tao.

Online text: terrytao.wordpress.com/2014/12/09/254a-notes-2-complex-analytic-multiplicative-number-theory/

Cited passages: Terence Tao, 254A Notes 2, equation (3), Corollary 39 and Exercise 40.

By Terence Tao.

Online text: terrytao.wordpress.com/2014/11/23/254a-notes-1-elementary-multiplicative-number-theory/

Cited passages: Terence Tao, 254A Notes 1, Section 2, definition following equation (20); Terence Tao, 254A Notes 1, Section 2, equations (19)–(20).

Online text: ocw.mit.edu/courses/18-303-linear-partial-differential-equations-fall-2006/22ead9d70b36836a68d13c7393e19649_waveeqni.pdf

Cited passages: Matthew J. Hancock, The 1-D Wave Equation, sections 5.3.1–5.4, printed page 14; The 1-D Wave Equation, sections 5.5–5.6, printed page 15, equations 28–33; the distributional extension follows by differentiation of translated profiles.

Online text: ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/e407e57ea631a29148ee94afecef7d33_MIT18_102s21_lec12.pdf

The course notes page attributes 2021 student notes to Andrew Lin and identifies Casey Rodriguez as instructor; the individual PDF has no author byline.

Cited passages: Lecture 12, Definition 120, Proposition 123(2), Remark 124 and Theorem 126; Lecture 12, printed page 59, Definitions 120–121 and Proposition 122.

Cited passages: MIT 18.102 lecture 12, Proposition 122: linearity for integrable functions.

Online text: ocw.mit.edu/courses/18-03sc-differential-equations-fall-2011/0eccec5385a788c4359bde0ded7b4227_MIT18_03SCF11_s2_1text.pdf

Cited passages: Direction Fields, Isoclines, Integral Curves, pp. 1–2: constant-slope loci and the example y prime equals x minus y.

By J. Peraire, S. Widnall.

Online text: ocw.mit.edu/courses/16-07-dynamics-fall-2009/dd277ec654440f4c2b5b07d6c286c3fd_MIT16_07F09_Lec26.pdf

Cited passages: J. Peraire and S. Widnall, Lecture L26, printed pages 2–3, moments and products of inertia; Lecture L26, printed page 1, velocities and angular momentum relative to the centre of mass.

The wave phase specifies position within an oscillation cycle. For a sinusoidal travelling wave,where $k$ is the wavenumber, $\omega$ the angular frequency and ${\varphi }_0$ an initial phase shift. Adding $2\pi$ to $\Phi$ leaves the field unchanged; one complete cycle advances the wave phase by $2\pi$ radians.

A surface of equal wave phase is a wavefront. Tracking one such surface gives the phase velocity, whereas comparing two fields at the same point gives their phase difference. Two equal-frequency disturbances add constructively when their wave phases agree and can cancel when their phase difference is $\pi$. The sign assigned to a phase lag depends on the chosen time convention. In a complex amplitude, the complex argument carries this information while the modulus sets the wave amplitude.

The angular spectrum is the transverse Fourier transform of a monochromatic field on a reference plane. In a uniform medium, each transverse wavevector component propagates with a longitudinal wavenumber satisfying the Helmholtz equation. Components beyond the propagating wavenumber range are evanescent waves. The outgoing angular spectrum chooses the longitudinal branch that propagates or decays away from the plane.

A wave period is the time for a monochromatic wave to complete one oscillation. For angular frequency $\omega$, it is $2\pi /|\omega |$. In cascade order-of-magnitude estimates the factor $2\pi$ is normally suppressed, giving a linear interaction time of order $|\omega {|}^{-1}$.

An oscillation is repeated variation around a central state, such as the displacement of a mass on a spring, the pressure near a vibrating source or the electric field in a light signal. In simple harmonic motion,the amplitude $A$, angular frequency $\omega$ and wave phase $\varphi$ specify the motion. The period of an oscillation is $T=2\pi /\omega$ for $\omega >0$.

A mechanical oscillator repeatedly exchanges kinetic energy and potential energy. Damping removes mechanical energy, while a periodic applied force can replenish it and produce resonance. Not all oscillations are sinusoidal or keep constant amplitude; anharmonicity can make the period of an oscillation depend on amplitude. Measuring successive peaks in a recorded trace distinguishes the repetition time from the rate at which the envelope decays.

Polarization of waves describes the direction and time-dependent geometry of a transverse wave's vector during its oscillation at a fixed position. In three spatial dimensions, a transverse wave can have two independent directions of oscillation, both perpendicular to its direction of propagation. For polarization of an electromagnetic wave, the relevant vector is the transverse electric field; for a mechanical transverse wave, it can be the transverse displacement. A purely longitudinal wave has no choice between these transverse directions, and the orientation of a static electric field is not polarization of waves.

Choose transverse axes $x,y$ in a Cartesian coordinate system. A monochromatic wave with a fixed phase difference has the real electric fieldHere $A_x,A_y$ are wave amplitudes, $\omega$ is the angular frequency, and $\delta$ is the phase difference of the $y$ oscillation relative to the $x$ oscillation. The vector's tip traces its path during the wave period $T=2\pi /\omega$. Both plotted coordinates are real electric field components; neither axis represents the imaginary part of a complex amplitude.

For nonzero wave amplitudes, $\delta =0$ or $\pi$ modulo $2\pi$ gives linear polarization: the vector moves back and forth along a fixed line. Equal nonzero wave amplitudes and $\delta =\pm \pi /2$ modulo $2\pi$ give circular polarization: the tip traces a circle of constant radius. Unequal nonzero wave amplitudes with $\delta =\pm \pi /2$ give a noncircular ellipse whose axes coincide with $x,y$. More generally, nonzero wave amplitudes and $\sin \delta \ne 0$ give elliptical polarization, with circular polarization as a special case. If exactly one wave amplitude vanishes, the nonzero wave has linear polarization for every $\delta$; if both vanish, there is no nonzero wave to assign a polarization of waves. ^[ref].

For example, take $\omega =2\pi {\mathrm{s}}^{-1}$, so the wave period is $T=1\mathrm{s}$. The choices $(A_x,A_y,\delta )=(2,1,0)$, $(1,1,\pi /2)$ and $(2,1,\pi /2)$, with electric field wave amplitudes in $V/m$, produce respectively the line segment $E_y=E_x/2$, the circle $E_x^2+E_y^2=1$ and the ellipse $E_x^2/4+E_y^2=1$, when coordinates are measured in $V/m$. At $t=T/4$, their tips are respectively $(0,0)$, $(0,-1)$ and $(0,-1)V/m$. The last two share a point without having the same path.

Resolving the electric field into two transverse components makes polarization of waves useful: the same frequency and wave amplitudes can describe different paths when their phase difference changes. An ideal linear polarizer selects the component along its pass axis, letting its orientation probe the direction of linear polarization through Malus's law. A fixed pair of wave amplitudes and a fixed phase difference describe one stable path; unpolarized light requires a statistical description rather than one fixed ellipse.