Glossary

Terms used in this book.

A reference list of the mathematical and physical vocabulary used in Math Foundations. Inline occurrences in the chapters are auto-tooltipped.

43 terms from this book.

A

acoustic impedance: The ratio of acoustic pressure to particle velocity in a propagating wave (Z = p/v). For a plane wave in a medium of density ρ and wave speed c, Z = ρc.
amplitude: The magnitude of a wave’s departure from equilibrium. For sound, the size of the pressure fluctuation.

basilar membrane: The membrane separating scala media from scala tympani. Its position-dependent stiffness gives different places different natural frequencies.
Bayes: Bayes’ theorem: P(M|S) = P(S|M)·P(M)/P(S). The brain’s posterior over hypotheses M given sensory data S combines the likelihood with the prior.

Cauchy–Schwarz inequality: For any vectors u, v in an inner-product space: |⟨u,v⟩| ≤ ‖u‖ · ‖v‖, with equality iff u and v are parallel. The single most-used inequality in functional analysis.
CFL condition: The stability bound for explicit finite-difference schemes of the wave equation: the Courant number C = c·Δt/Δx must satisfy C ≤ 1. Numerical signal speed (Δx/Δt) must not exceed physical signal speed c.
characteristic frequency: The frequency at which a given place on the basilar membrane (or auditory-nerve fibre, or cortical neuron) responds most strongly.
cochlea: The spiral, fluid-filled organ of the inner ear that performs frequency analysis on incoming sound and transduces it into neural signals.
convolution: A mathematical operation that combines two signals: the output at time t is the weighted sum of one signal across all times, weighted by the other shifted to t.
Courant number: The dimensionless ratio C = c·Δt/Δx in a finite-difference discretisation of the wave equation. The CFL condition requires C ≤ 1 for stability.

eigenstate: An eigenfunction of a quantum-mechanical operator, interpreted as a state of the system. Energy eigenstates are stationary states of a quantum system and form a complete basis for any wavefunction.

formant: A resonant peak in the spectrum of a vowel. The configuration of mouth and tongue produces F1, F2, F3… which together identify the vowel.
Fourier series: Decomposition of a periodic signal into a sum of sinusoids at multiples of its fundamental frequency.
Fourier transform: A mathematical operation that decomposes a signal into its sinusoidal components. Time-domain ↔ frequency-domain pair.
frequency: The number of oscillation cycles per second, measured in hertz (Hz). For sound, this is what the brain perceives as pitch.

harmonic function: A function satisfying Laplace’s equation ∇²u = 0. The value of a harmonic function at any interior point equals the average of its values on a surrounding sphere — the mean-value property.
Hermitian: A complex matrix or operator that equals its own conjugate transpose (A† = A). The complex generalisation of a symmetric matrix. Hermitian operators have real eigenvalues and orthogonal eigenvectors; their eigenstates form a complete basis.
HRTF: Head-Related Transfer Function. The full frequency-dependent filter the body applies between a sound source in space and the listener’s eardrum.

impedance: In acoustics, the ratio of pressure to particle velocity. A measure of how strongly a medium resists being moved by a wave.
impulse response: A system’s output when given a single, brief impulse as input. Fully characterises any linear time-invariant system.

likelihood: In Bayesian inference, P(S|M): how probable the observed sensory input is given that hypothesis M is true.
linear time-invariant: A system whose output obeys superposition (linear) and whose behaviour does not change over time (time-invariant). Sinusoids are its eigenfunctions.

McGurk effect: A multisensory illusion in which the visual articulation of one syllable and the audio track of another produce a third percept.

phonemic restoration: A perceptual effect in which a phoneme replaced by noise is heard as intact, restored by context-driven priors.
plane wave: A wave whose phase fronts are infinite parallel planes; idealisation of a wave from a distant source, valid locally near the listener.
posterior: In Bayesian inference, P(M|S): the probability of hypothesis M given the sensory input S. This is what the brain perceives.
predictive coding: A theory of cortical processing in which higher layers predict the activity of lower layers, and only the prediction errors propagate upward.
prior: In Bayesian inference, P(M): the probability of a hypothesis before any sensory input. Context, expectation, learned experience shape this.

quality factor: Q. A dimensionless measure of resonance sharpness. Equal to the peak amplitude (relative to DC) of a damped driven oscillator.

Rayleigh quotient: For a matrix A and vector v, the scalar R(v) = vᵀAv / vᵀv. For a self-adjoint A, R(v) is bounded above and below by the eigenvalues of A; iterating R(v) on power-method iterates converges to the dominant eigenvalue.
reflection: When a wave hits a boundary between two media, part of its energy turns back into the first medium. The reflection coefficient R = (Z2 − Z1)/(Z1 + Z2).
resonance: The condition where a driving frequency matches a system’s natural frequency, producing maximum response amplitude.

self-adjoint: A linear operator that equals its own adjoint. For real matrices: A = A^T (symmetric). For complex matrices: A = A† (Hermitian). For differential operators on a function space: ⟨f, Lg⟩ = ⟨Lf, g⟩. Guarantees real eigenvalues and orthogonal eigenfunctions.
Shepard tone: An auditory illusion of a continuously rising (or falling) pitch, created by stacking octaves with a fading amplitude envelope.
Sommerfeld radiation condition: The boundary condition at infinity for an unbounded Helmholtz problem: φ at large r must look like an outgoing spherical wave (~ e^{ikr}/r), not incoming. Ensures uniqueness of radiation and scattering solutions.
spectral theorem: A self-adjoint linear operator has a complete orthonormal basis of eigenvectors (or eigenfunctions), and all its eigenvalues are real. The reason mode expansions for PDEs work; the algebraic core of separation of variables.
speed of sound: The propagation speed of small-amplitude pressure disturbances. ≈343 m/s in air at room temperature, ≈1480 m/s in water.
spherical wave: A wave radiating outward from a point source, with amplitude falling as 1/r.
Sturm–Liouville theorem: For a self-adjoint differential operator on a bounded interval with appropriate boundary conditions, there is a countably infinite sequence of real eigenvalues and a corresponding complete orthonormal basis of eigenfunctions. Underwrites mode decomposition for PDEs.

Taylor expansion: An expansion of a smooth function f near a base point x0 as an infinite sum f(x0) + f′(x0)(x−x0) + ½f″(x0)(x−x0)² + … . Truncating gives a polynomial approximation of any required order.
Taylor series: An expansion of a smooth function f near a base point x0 as an infinite sum f(x0) + f′(x0)(x−x0) + ½f″(x0)(x−x0)² + … . Truncating gives a polynomial approximation of any required order.

vector strength: A measure of phase-locking strength: |⟨exp(iφ)⟩| averaged over many spikes. 1 = perfect, 0 = random.

wave equation: A second-order partial differential equation describing how a disturbance propagates. For pressure in air: ∂²p/∂t² = c²∇²p.