UC San Diego technical report, August 2010.
Definitions I.17–I.18, printed pp. 8–9, describe an abstract simplicial complex and its topological realization by gluing labelled faces of standard simplexes.
Institutional profile: www.law.gwu.edu/john-f-banzhaf-iii
The biographical paragraph on his student research identifies the mathematical technique now bearing his name. Used for person identity and the connection to the Banzhaf index, not for legal claims.
By John Banzhaf. The author's own reference page identifies the article as Rutgers Law Review, volume 19, starting p. 317, 1965. That bibliographic evidence does not imply that the article's complete text was inspected.
By Lloyd Shapley. RAND's original report record identifies the year 1952 and document P-295; the later book chapter is a distinct publication record.
The abstract identifies the axiomatic construction of a value. The four-axiom decomposition and uniqueness proof in Shapley value axioms are stated explicitly here.

OEIS entry A005132 (A005132)

0 0
oeis.org/A005132
Primary database record for Recamán's sequence. Relevant locators are its definition, offset-zero initial terms, the comment recording the first repeated value at indices 20 and 24, and the reference to the 29 January 1991 letter by Bernardo Recamán Santos. Later additions to this evolving record are not needed for the 2019 solution.
Official technical manual: IEEE Arithmetic.
Relevant locators: IEEE Arithmetic Model, significand precision and overall bit count; Double Format, field layout and Table 2-5; and Ranges and Precisions in Decimal Representation. The source describes the traditional IEEE double format, now called binary64. It supports the precision and storage calculations for the double-precision floating-point format; no claim about a current processor's execution speed is inferred.
Primary publication: Mathematics of Operations Research 19(2), pp. 257–266. Publisher record.
Relevant locators: §1, printed p. 258, definition using signed integer edge weights and the nonnegative-weight convex case; §2.2, printed p. 260, discussion preceding Lemma 4. The self-loop extension and the elementary representation and convexity arguments are stated and proved directly in the solutions.
Official report: University of Oxford Mathematical Institute, public report. The report identifies Trinity Term 2015 on page 1. The A6 commentary on printed page 16 provides question-specific evidence for the year of the damaged-cover A6 paper. It contains performance comments rather than complete worked solutions.
The “Caveats” subsection derives the loss of position accuracy proportional to the square root of function-value relative accuracy near a nonzero minimum with positive second derivative. “Minimization Algorithms” describes golden-section search and its linear contraction. The claims concern standard mathematical precision effects, not an assertion that later software releases were available for a historical examination.
Author's authorized chapter previews. Sections 1.7, 1.8 and 1.10 develop invariant probability distributions, convergence to equilibrium and the countable-state Markov chain ergodic theorem for Markov chains. The distinction between a normalizable invariant measure and an invariant measure of infinite mass is crucial on a countably infinite state space.
Original publication: Contributions to the Theory of Games II, pp. 193–216. Publisher record: doi.org/10.1515/9781400881970-012
Publisher record for the reprint in Classics in Game Theory, pp. 46–68: www.degruyterbrill.com/document/doi/10.1515/9781400829156-011/html
The paper develops extensive-form games and Kuhn's theorem relating mixed strategies, behavioural strategies and perfect recall. The finite two-round construction is proved directly in the examination solution.
The section “The Expected Utility Property” states the finite-lottery expected utility formula and connects it to the preceding preference axioms. The source's title misspells the second surname; the conventional name used here is Oskar Morgenstern.
Handbook Section 1.3.5.3 distinguishes paired from unpaired observations and equal-variance from unequal-variance inference.
Handbook Section 7.4.3.2 specifies the model and assumptions on its errors.
Handbook Section 3.2.3.1 explains the one-factor layout and within- versus between-level variation.
Public question-paper source: Released Computer Science examination-paper archive, member 2021/T/a10100w1.pdf. The cover identifies Trinity Term 2022, Preliminary Examination in Computer Science, 23 June, 09:30–12:30. The printed sections are Discrete Mathematics (questions 1–3, pp. 2–3), Continuous Mathematics (questions 4–6, pp. 4–6), and Probability (questions 7–9, pp. 7–9). The archive directory label does not override the printed date.
The NIST/SEMATECH e-Handbook of Statistical Methods explains score-based binomial proportion confidence intervals and gives the Wilson score interval endpoints.
The NEOS Guide's mathematical account of quasi-Newton methods, deriving the secant condition from an integrated Hessian matrix and describing low-rank updating, including BFGS. The related nonlinear-equation account is neos-guide.org/guide/algorithms/broyden/ and discusses Broyden's method.
The University of Oxford Part B paper lists Information Theory, B8.4, and Trinity Term 2023 on its cover. Public bundle member: 2022/T/a16551w1.pdf; the cover, rather than the bundle directory name, establishes the year. Printed pp. 2, 3 and 4 contain Questions 1, 2 and 3 respectively. Question 3(b)'s hint prints the incorrect factor in Fano's inequality; the correct factor is , as printed in Question 1(b). Question 3(c)'s repeat-until-success protocol terminates only for , although the initial channel parameter range includes .
The final paragraph states Tutte's theorem on perfect matchings in terms of the number of components of odd integer order after deleting a vertex subset. The worked perfect matching theorem for a cubic graph with at most two bridges uses this criterion directly.
Chapter 15 introduces the elaboration model; Table 15-5 is on printed p. 448.
The manuscript is held as EA10057 and EA10058. Problem 50 gives a worked circular-field area rule. The linked 1927 edition provides transcription and mathematical commentary; modern symbolic reconstructions must be identified as such.
Digital Library of Mathematical Functions, section 9.7 gives the Airy function asymptotic expansions. Equations 9.7.5 and 9.7.7 give the positive-argument exponential forms; equations 9.7.9 and 9.7.11 give the negative-argument oscillatory forms. Their leading terms determine the factor two and phase shift in the Airy turning-point connection formula.
English translation in Essays on the Theory of Numbers. Cited 1872 preface; §§III–V on cuts and the real numbers; §VII on existence of limits. The original title is Stetigkeit und irrationale Zahlen.
Journal für die reine und angewandte Mathematik 4, 157–169. The archive identifies this as Ralf Stephan's transcription of the original. Cited opening critique of Augustin-Louis Cauchy, signed-series counterexample and final sufficient conditions for Fourier series.
Original work in an edited primary-text reproduction. Cited §§13–18 on changing assumptions about evanescent increments. Alternative scholarly transcription: www.maths.tcd.ie/pub/HistMath/People/Berkeley/Analyst/Analyst.html . The editorial introduction distinguishes the 1734 London and Dublin editions.
Acta Eruditorum, pp. 467–473. The Mathematical Association of America reproduces the original pages. Cited for the published differential-calculus rules, not for a modern reconstruction of their foundations.
Introduction to the quadrature treatise appended to the 1704 Opticks. The linked text reproduces John Harris's 1710 English translation, edited by David R. Wilkins. Archive context: www.maths.tcd.ie/pub/HistMath/People/Newton/Quadratura/ . Cited opening account of generated quantities, fluxions and first ratios.
Primary text, Chapters II–III, on the meaning and laws of class symbols, including . Its operations and restrictions should be distinguished from a modern Boolean algebra.
Primary articles, transcribed and edited in 2000 by David R. Wilkins. Cited §§1–3 and §§10–11 for the units, associativity, distributivity and failure of commutativity. Original publication dates are listed in the editorial note.
Primary preface reproduced by the MacTutor archive. Its argument distinguishes arithmetical algebra from symbolical algebra and discusses the restrictions implicit in expressions such as .
The publisher's historical-edition description dates the original German book to 1770. The linked reproduction is an English edition. Cited for the subject's continuing association with general numerical operations and the solution of equations.
Original-edition title and metadata from New York Public Library. Used as evidence for the term universal arithmetic; the title alone does not define every contemporary meaning of algebra.
Cited Book I on geometrical operations with a unit segment and Book II on curves. The linked transcription is a modern reproduction, not a claim to be a new printing of the 1637 edition.
Tours original edition. The BnF catalogue establishes the title and date. Facsimile: gallica.bnf.fr/ark:/12148/bpt6k108865t . Folio 5r contrasts logistica numerosa and logistica speciosa; a scholarly reproduction and translation appears at www.uv.es/puigl/historias67.pdf . The latter is used for the specific terminology; the modern biographical discussion supplies the notation and homogeneity context.
Bulletin of the Australian Mathematical Society 40, 293–302. Primary historical research establishing the 1845 concentration of Cauchy's substitution work and explaining the misleading 1844 bibliographical dating. Publisher full text: www.cambridge.org/core/services/aop-cambridge-core/content/view/6801D1A609AA974ACB44B90C81533507/S000497270000438Xa.pdf/on-the-date-of-cauchys-contributions-to-the-founding-of-the-theory-of-groups.pdf .
By Hans Freudenthal. Scholarly biographical article, reproduced in the University of St Andrews archive. Cited for its mathematical-physics discussion and its reproduction of Augustin-Louis Cauchy's 1822 explanation of internal pressures and tensions. The 2008 copyright on the digital compilation is not the date of that 1822 communication.
Biography of Augustin-Louis Cauchy, credited to John Joseph O'Connor and Edmund Frederick Robertson. Cited for engineering education and the broad range of mathematical research.
Original edition, 68 pages. Cited for the definition of complex integration, path variation and singular points. Original OCR: www.e-rara.ch/zut/download/ftpack/plain/8504602 . Accessible later reprint: 1874 reprint, pp. 265–304, especially §§1–6. The reprint date is distinct from the memoir's date.
Original full title: Résumé des leçons données à l’École Royale Polytechnique sur le calcul infinitésimal. Volume I. Cited for the definitions of derivatives and definite integrals, especially the twenty-first lesson. Searchable original OCR: www.e-rara.ch/zut/download/ftpack/plain/8350182 .
Original text in a scan-backed transcription. Cited: introduction; Chapter II on continuity; Chapter VI, §I, on convergence and the Sum theorem of Cauchy. Original-edition record: www.e-rara.ch/zut/content/titleinfo/8442794 .
Archive exposition based on a 19 March 2001 lecture by Peter Michael Neumann. Its account compares concrete substitutions, abstract definitions and Heinrich Weber's finite cancellation axioms. The byline explains the relation between the lecture and the written notes.
Mathematische Annalen 20, 301–329. The opening finite group definition is reproduced in Oxford's exam and discussed in The abstract group concept by the MacTutor History of Mathematics Archive. The publisher page establishes the article's title, author, pages and issue date; the commentary does not assume access to its subscription facsimile.
Original Latin memoir on biquadratic residues. ETH-Bibliothek record and facsimile, DOI doi.org/10.3931/e-rara-61067 . Presented in 1831; the linked original printing is dated 1832.
German self-announcement in Göttingische gelehrte Anzeigen. The digital transcription reproduces the argument about terminology, Gaussian integers and the complex plane. Its concluding discussion is the source of the wording quoted in the exam; this is distinct from the Latin memoir printed in 1832.
Revue d’histoire des mathématiques 19, 43–77. Original historical research separating Alexandre-Théophile Vandermonde's coefficient notation, alternating expressions, difference-products and the history of the Vandermonde determinant name. Cited especially §2.1 and the bibliography.
Published in the 1772 second-part academy volume, printed 1776, pp. 516–532. The linked research article by Bernard Ycart reproduces and explains the primary coefficient notation in §2.1 and supplies the original-edition reference. It should not be mistaken for a direct link to the complete eighteenth-century memoir.
University of Florence mathematical-museum exhibition, Analytical geometry and the problem of tangents. It gives a translation and reconstruction of Pierre de Fermat's parabola example, using the letters B, C, D, E, I, O of the exam image. The text distinguishes circulation in the 1630s from the 1679 publication. Its final displayed label contains a typographical anomaly; the derivation and geometry give .
Posthumous Toulouse edition of mathematical works of Pierre de Fermat. The adequality and tangent discussions belong to an earlier period than this printing. The ETH-Bibliothek record gives the original title, date and publisher.
English Euclid translation with commentary. The Text Creation Partnership transcription identifies the title page and Book VII, Definition 23. Modern numbering of the corresponding perfect number definition is VII.22. The linked repository also records the transcription's bibliographical provenance.
Published in Journal of Statistical Mechanics: Theory and Experiment, P10008. Author preprint: arxiv.org/abs/0803.0476
Relevant passages: §1, Equation (1), network modularity; §2, Method, pp. 4–5 of the preprint, local optimization and aggregation.
Relevant passages: Network formalism, mean local clustering coefficient; New measures of small-world-ness, Definitions 1–2 and relative clustering/path-length ratio.
This MacTutor biography gives approximate dates and discusses the surviving evidence for Diocles' work on the cissoid of Diocles and the parabola. The footer identifies John Joseph O'Connor and Edmund Frederick Robertson as authors and April 1999 as its revision date.
Cited passages: William L. Briggs, A Multigrid Tutorial, presented by Van Emden Henson, slides 59–63: correction and recursive V-cycle.
SIAM Journal on Computing 5(2), June 1976, pp. 266–283; DOI 10.1137/0205021. R. Endre Tarjan is Robert Endre Tarjan.
Cited passages: Abstract; Information & Authors: volume 5, issue 2, June 1976.
Oak Ridge National Laboratory technical report ORNL/TM-12203, November 1992.
Cited passages: §2.2, Lemma 1 and Theorem 2.2; §3.1–3.2, Theorems 3.1–3.3.
Foundations and Trends in Optimization 1(4), 241–433; DOI 10.1561/2400000006. PDF lists the volume issue year as 2014 and copyright as 2015; author-hosted course reference and institutional publication record identify 2015.
Cited passages: §4.1–4.2, Algorithm 4.2; §9.1, equation 9.2 and Theorem 9.1.
Cited passages: James Demmel, CS267 supplemental notes, 12 March 1996, How Multigrid Damps High Frequencies: sine eigenvectors and the one-third high-frequency bound.
Additional cited passages: diagonal-2 tridiagonal Poisson matrix, error eigenvector expansion, weighted-iteration multiplier, and concluding attenuation statement. The heading dates the lecture to 12 March 1996.
Publication details and the original axioms are discussed in The Sources of Kolmogorov’s Grundbegriffe by Glenn Shafer and others.
Cited passages: Long Chen, Introduction to Multigrid Methods, 17 December 2011, section 2: plain Jacobi retains both extreme sine modes; Long Chen, Introduction to Multigrid Methods, 17 December 2011, sections 4–5: energy projection, transfer matrices, and Galerkin triple product; the displayed cancellation identities follow by direct substitution.
Cited passages: slides 6-10 and 17-18.
Cited passages: Achi Brandt, Multi-Level Adaptive Solutions to Boundary-Value Problems, Mathematics of Computation 31(138), April 1977, pp. 333–390, abstract and introduction.
Cited passages: 1.4-1.6, especially consequence C9.
6.438 Algorithms For Inference, Fall 2014, recitation notes; institutional authorship.
Cited passages: §1, The Junction Tree Algorithm; §6, Relationship between chordal graphs and junction trees.
Cited passages: introduction, page 70; section 5.1.1, page 87.
Cited passages: Fourier's original chapter on heat propagation and trigonometric expansions.
By Leonhard Euler; English translation by Jordan Bell. Presented to the St Petersburg Academy on 26 September 1732; the original Latin paper, "Observationes de theoremate quodam Fermatiano aliisque ad numeros primos spectantibus", appeared in Commentarii academiae scientiarum Petropolitanae 6 (1738), pp. 103–107. The publication note on page one of the translation distinguishes presentation from publication. The linked translation is version three, dated 15 April 2008.
Catalogue record: arxiv.org/abs/math/0501118
Cited passages: pp. 1–2, Euler's account of Fermat's proposed prime pattern and the factor of ; p. 1 publication and translator note. The short modular check in Fermat number is presented as a modern verification, not a historical reconstruction of Euler's search.
By Carl Pomerance. Slides for the ICERM meeting "Seventy-five years of Mathematics of Computation", 1–3 November 2018.
Cited passages: slide 14 states Pépin's test and outlines its proof; slides 15–16 describe Fermat numbers and their role in integer factorization; slide 18 gives the Gauss-Wantzel theorem and distinguishes Gauss's construction result from Wantzel's necessity proof. Numerical records elsewhere in these 2018 slides are historical records, not claims about current records.
Cited passages: Romyar Sharifi, Point-Set Topology, Corollary 3.1.10, Proposition 3.1.11, Definition 3.1.12 and Remark 3.1.16: maximal connected subsets and equivalence classes; Romyar Sharifi, Point-Set Topology, Definition 3.1.1 and Example 3.1.3: connectedness and discrete spaces; Romyar Sharifi, Point-Set Topology, Lemma 3.1.13 and Example 3.1.14: closed components and a component which is not open.
Cited passages: Definition 1.1.9: finite complement topology; Definition 1.5.5, Proposition 1.5.11 and Proposition 1.6.20: neighbourhood convergence, uniqueness and the metric formulation; Example 1.5.10 and Proposition 1.5.11: closed points, overlapping neighbourhoods, non-repeating sequences and uniqueness in Hausdorff spaces.
Cited passages: Tom Leinster, General Topology, Example C3.6, p. 78, and the discussion after Lemma C4.12, p. 84: the sine-curve obstruction and nonclosed path components; here the positive graph is restricted to ; Tom Leinster, General Topology, Lemma C1.8, p. 70: a continuous image is connected; Tom Leinster, General Topology, Lemma C4.12 and the following definition, p. 84: path equivalence and maximal path-connected subsets; Tom Leinster, General Topology, Remark C1.2, p. 68, and Proposition C2.6, p. 73: the nonempty convention and the interval characterization.
Cited passages: Section 5.2, Properties of the Definite Integral, equations 5.9–5.10 and 5.14; Section 5.2, definitions of net signed area and total area.
Cited passages: Section 5.4, Theorem 5.6 and the displacement and total-distance discussion.
Online text: www.jirka.org/ra/
Additional inspected sections: 3.4 Uniform continuity and 7.5 Continuous functions.
Whole-book author: Jiří Lebl; no separately credited chapter author is established. These HTML sections are undated. The author landing page dates the downloadable PDF version 6.3 to 15 May 2026; that version date is not assigned to the HTML.
Cited passages: §3.4.1, Definition 3.4.1 and Theorem 3.4.4; §3.4.2, Lemma 3.4.5; §3.4.3, Proposition 3.4.8 and Example 3.4.10; §7.5.4, Definition 7.5.10 and Theorem 7.5.11.
Author and work verified; edition year and publisher were not verified and are not asserted.
Cited passages: Section 5.1, Example 5.1.4.
Edition year and publisher not verified; no claim made.
Cited passages: Section 5.2, Lemma 5.2.7 and Theorem 5.2.9.
Cited passages: Lebl, Basic Analysis I, §2.1: sequences as functions and their limits.
Cited passages: Lebl, Basic Analysis I, §1.2: approximation of a supremum; Lebl, Basic Analysis I, §1.2: supremum, infimum and attainment.
Additional inspected locators: §1.2.3, Proposition 1.2.6(i) and proof; Proposition 1.2.8; §1.2.5, Exercise 1.2.9. The HTML section is undated; no PDF version date is assigned to it.
Cited passages: Basic Analysis, Volume I by Jiří Lebl, opening footnote to §2.3: the theorem's two namesakes; Basic Analysis, Volume I by Jiří Lebl, §2.3.3, Theorem 2.3.8: every bounded real sequence has a convergent subsequence; Lebl, Basic Analysis I, §2.3: equality of finite bounds and extended limits; Lebl, Basic Analysis I, §2.3: extreme subsequential limits.
Cited passages: Basic Analysis: Introduction to Real Analysis, Volume I by Jiří Lebl, §6.1.1, Definition 6.1.1 and Proposition 6.1.5; Basic Analysis: Introduction to Real Analysis, Volume I by Jiří Lebl, §6.1.1, Example 6.1.2, even powers on the interval from minus one to one; Basic Analysis: Introduction to Real Analysis, Volume I by Jiří Lebl, §6.1.2, Definition 6.1.6 and Proposition 6.1.7.
Cited passages: Basic Analysis: Introduction to Real Analysis, Volume I by Jiří Lebl, §6.2.1, Theorem 6.2.2; Basic Analysis: Introduction to Real Analysis, Volume I by Jiří Lebl, §6.2.2, Theorem 6.2.4; Basic Analysis: Introduction to Real Analysis, Volume I by Jiří Lebl, §6.2.3, Theorem 6.2.10 and the unbounded-interval observation following its proof.
Student notes by Andrew Lin; course taught by Casey Rodriguez.
Publisher: MIT OpenCourseWare.
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