Glossary Glossary

Composition.

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Vectors whose components correspond with relative proportions of a whole.

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Distribution.

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A mathematical description of how a quantity (e.g., a probability) is allocated across a set of outcomes or values.

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First-Principles Reasoning.

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A method of reasoning in which claims, hypotheses, models, or systems are explicitly decomposed into their most fundamental, independently verifiable assumptions, and conclusions are derived via transparent, stepwise inference based on those assumptions. This helps ensure the fidelity of the reasoning process and that conclusions are traceable and reproducible by the reader.

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Function.

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A mapping \(f:A\rightarrow B\) from a domain \(A\) to a codomain \(B\text{.}\) Such mappings assign exactly one element \(b\in B\) (the output) to each element of the domain \(a\in A\) (the input). Importantly, these mappings may be surjective, injective, both (i.e., bijective), or neither.

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Mechanics.

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The branch of physics concerned with describing the states, evolution, and interactions of physical systems.

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Classical and quantum mechanics are two major subfields of mechanics that provide distinct mathematical frameworks for describing the behavior of physical systems. They share fundamental concepts such as states, dynamics, observables, and conservation laws.

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Numerical Integration.

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Methods for approximating definite integrals.

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Poisson Bracket.

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Provides a fundamental operation in Hamiltonian mechanics and symplectic geometry. Specifically, it tells you how an observable \(A\) changes due to some other observable \(B\) acting as a generator.

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Power Spectral Density.

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A function \(S_{xx}(f)\) describing how the power of a signal is distributed over frequency \(f\text{.}\)

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Regression Methods.

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Methods for fitting models to data.

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Root-Finding Algorithms.

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Methods for solving equations of the form \(f(x) = 0\text{.}\)

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Significant Figure.

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A digit of a numerical value that contributes to the precision of a measurement and is therefore crucial for accurate interpretation.

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Spherical Harmonics.

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A set of orthogonal functions defined on the surface of a sphere that form a complete basis for representing angular dependence.

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Widely used in physics to represent angular structure in physical systems with spherical or approximately spherical symmetry, including gravitational and electromagnetic fields, atomic orbitals, and quantum-mechanical wavefunctions.

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In quantum mechanics, spherical harmonics are indexed using two quantum numbers, commonly denoted \(\ell\) and \(m\text{,}\) and arise as the angular solutions of the Laplace equation and other related differential equations in spherical coordinates.

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Spurious Correlations.

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Non-causal correlations between variables that manifest due to third party factors.

Support.

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The subset of the domain \(X\) of a function \(f: X\rightarrow \mathbb{R}\) on which the function is nonzero.

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\begin{equation*} \operatorname{supp}(f) = \{x \in X : f(x)\neq 0\}. \end{equation*}

Variational Mechanics.

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A formulation of mechanics in which the behavior of a physical system is characterized through the variation of a functional over possible states, trajectories, or histories.

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Variational mechanics provides a provides a variational framework applicable to both classical mechanics and quantum mechanics. In classical mechanics, this is commonly expressed through the principle of stationary action, in which the functional is the action and is required to be stationary under admissible variations. The resulting variational conditions are then used to derive systems of equations that describe the behavior of the physical system.

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