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A balance that applies a compensating force to hold its load support near a reference position; the compensation signal measures the applied force.

A force probe measures an applied force through a calibrated response. A bending beam with strain gauges is one realization; orientation and measurement calibration matter.

Measurement calibration establishes a relation between an instrument's indications and reference values, with the relevant measurement uncertainty. It is distinct from changing the instrument until it reads correctly: an instrument can be calibrated by determining a response curve and applying a correction afterward.

A practical voltage experiment connects a meter to several reference values across its working range and repeats the sequence in both directions. Fitting $V_{\mathrm{indicated}}=a{V}_{\mathrm{reference}}+b$ estimates a scale factor and measurement zero offset; departures from that relation test the response model. Repetition at later times checks instrumental drift. Uncertainty in the reference, the fitted parameters, and unresolved departures belongs in the measurement uncertainty budget, even if the displayed readings appear very stable.^[ref]

A systematic error is a component of measurement error that remains constant or changes predictably when measurements are repeated under specified conditions. A ruler with an incorrect scale, a thermometer with a measurement zero offset, and an unaccounted instrumental drift can all produce such effects. Closely agreeing readings can therefore have poor measurement accuracy: repetition tests their scatter, not the correctness of the shared reference.

An estimated systematic error should be corrected when possible, with the correction's measurement uncertainty included in the measurement uncertainty budget. Suppose a balance reads $m+b$ and a reference gives an estimate $\hat{b}$ of its offset. The corrected result is $m_{\mathrm{read}}-\hat{b}$; uncertainty in $\hat{b}$ remains even if the same unknown offset affected every weighing. Changing the reference, using a blank, or testing systematic-error cancellation by reversal can reveal effects that a long sequence of identical readings misses. The Type A evaluation of uncertainty and Type B evaluation of uncertainty distinction concerns how uncertainty is evaluated, so neither label is synonymous with systematic error.^[ref]

Measurement uncertainty describes the spread of values reasonably attributable to a specified measurand, using the available information. It differs from measurement error, which is a difference from a reference value: the uncertainty can be quantified even when that difference is unknown. A useful result therefore states what was measured, the unit of measurement, the value, and how its uncertainty was evaluated.^[ref]

For a wire's electrical resistance obtained from $R=V/I$, uncertainty in the voltage and electric current propagates through that measurement model. Shared measurement calibration effects can make the inputs correlated, so adding percentages or treating every contribution as independent can be wrong. The measurement uncertainty budget records those assumptions. Report a standard uncertainty when a standard-deviation scale is intended, or an expanded uncertainty with its stated coverage factor; a bare plus-or-minus number does not specify which was used.

Pharmacology studies how drugs interact with living systems. It connects molecular recognition, changes in physiology and the time course of exposure. Distinguishing what a substance does from how it reaches its targets is central to interpreting experiments.

By Keith Conrad.

Online text: kconrad.math.uconn.edu/blurbs/linmultialg/extmod.pdf

Author-hosted expository notes, 47 pages. No date appears on the title page; no publication date is established.

Cited passages: §3, Definition 3.1 and Theorem 3.3; §4, Theorem 4.2; §10, exterior algebra.

Whole-book author: Oscar Levin. No separate chapter author is credited in the inspected section.

Online front matter: discrete.openmathbooks.org/dmoi3/frontmatter.html

The author's institutional publication list dates the third edition to 2018. The online front matter displays 12 August 2025 without identifying that date as the edition's original publication date.^{[ref] [ref]}

Online text: discrete.openmathbooks.org/dmoi3/sec_counting-binom.html

Cited passages: §1.2, Lattice Paths, step-position encoding; Binomial Coefficients, counting interpretations and recurrence; Pascal’s Triangle, indexing and edge values. The third edition’s 2018 publication year is distinct from the online front matter’s displayed date of 12 August 2025.

Senior contributing authors: Gilbert Strang and Edwin “Jed” Herman. OpenStax also credits additional contributing authors in its preface.

Publisher: OpenStax, Rice University.

Official text: openstax.org/books/calculus-volume-3/pages/1-introduction

The preface's citation information gives publication as 30 March 2016; the current website footer is a separate copyright notice.^[ref]

By Lynn Marecek, MaryAnne Anthony-Smith, Andrea Honeycutt Mathis.

By Kyle Siegrist.

Online text: www.randomservices.org/random/prob/Events.html

The linked section discusses events and random variables. Its terminology includes a measurable structure with the sample space; this wiki names the underlying set and its sigma-algebra separately.

Additional section: Expected Value, 2 Additional Properties.

Cited passages: Right Distribution Function; A General Definition.

Additional section: Conditional Probability.

Cited passages: Probability Spaces, §4 Conditional Probability, Definitions and Interpretations, The Basic Definition; Basic Theory, The Multiplication Rule; Conditioning and Bayes’ Theorem, countable partition and weighted-average interpretation. Whole-textbook authorship is credited to Kyle Siegrist; no separate section byline or publication or revision date is established. The University of Alabama in Huntsville is an affiliation and project supporter, distinct from authorship; Random Services hosts the text.

Cited section: Expected Value, §6 Generating Functions, Basic Theory, The Probability Generating Function: definition, power-series expansion and absolute convergence on [-1,1]; Inversion formula f(k)=P^(k)(0)/k! and its termwise differentiation proof; Factorial moments and variance under the source’s stronger radius > 1 hypothesis; Independent-sum identity and expected-product factorization; The Moment Generating Function: M(t)=G(exp(t)) and finite-domain requirement.

Cited section: Definition: sequence of independent identically distributed jumps independent of the Poisson process; Properties: conditional mean/variance proof, generating-function conditioning proof, and PGF version; Compound Poisson Distributions: random count independent of the whole sequence; fixed-count-parameter definition.

By Keith Conrad.

Online text: kconrad.math.uconn.edu/blurbs/grouptheory/dihedral.pdf

Authored expository notes by Keith Conrad. No document date is established.

Cited passages: §§1–2, Theorem 2.5, printed pp. 1–4: rigid motions, rotations, reflections and dihedral order conventions.

By Gaston Floquet.

Source information: www.numdam.org/item/ASENS_1883_2_12__47_0/

Online text: www.numdam.org/article/ASENS_1883_2_12__47_0.pdf

Research article credited in print to G. Floquet, Gaston Floquet, published in 1883 in Annales scientifiques de l’École Normale Supérieure, series 2, volume 12, pp. 47–88. The linked scan and archive metadata identify the article; the historical citation does not assert priority for every modern matrix formulation.

Cited passages: Printed p. 48, definition of a function periodic of the second kind and its multiplier; p. 49, transformation of a fundamental solution system after one period.

By John M. Lee.

Source information: sites.math.washington.edu/~lee/Books/ITM/

Source information: www.math.toronto.edu/mgualt/courses/MAT1300F-2012/

Collection of sources; individual chapter authors and dates are recorded below.

Collection of sources; individual chapter authors and dates are recorded below.

Collection of sources; individual chapter authors and dates are recorded below.

Collection of sources; individual chapter authors and dates are recorded below.

By Terence Tao.

Online text: terrytao.wordpress.com/2009/04/06/the-fourier-transform/

Authored website post credited to Terence Tao and published 6 April 2009. Its course category is 245C, Real Analysis; the byline applies to this post, not to a separately published book of every course resource.

Cited passages: §3, Definition 32 and Exercise 54: Schwartz hypotheses and Poisson summation on the integer lattice; §3, Exercise 41: transform of a scaled Gaussian; put r=1/sqrt(a) and adjust its prefactor.

Additional locator: section 2, the constant character and Corollary 22, Fourier expansion on compact abelian groups with probability Haar measure. The zero coefficient is the average against the constant character. terrytao.wordpress.com/2009/04/06/the-fourier-transform/