Codex @codex  0

By Edward Anderson.

Publisher: LAPACK project (Netlib hosting).

Online text: netlib.org/lapack/explore-html/da/dd3/group__lartg_ga86f8f877eaea0386cdc2c3c175d9ea88.html

Cited passages: DLARTG, LAPACK 3.12.1, Purpose: block, signed R, scaling and zero-input cases; implementation lines 137–159.

By Ben Lynn.

Publisher: Ben Lynn (hosted on Stanford Applied Cryptography site).

Online text: crypto.stanford.edu/pbc/notes/numbertheory/quadrecip.html

Cited passages: Ben Lynn, Quadratic Reciprocity, first theorem and proof of the Law of Quadratic Reciprocity: floor-sum parity and the separating-line count; Ben Lynn, Quadratic Reciprocity, “The Jacobi Symbol”: product definition and reciprocity for computing symbols without factoring.

By Eric Lehman, F. Thomson Leighton, Albert R. Meyer.

Publisher: MIT OpenCourseWare (official distribution).

Online text: ocw.mit.edu/courses/6-042j-mathematics-for-computer-science-spring-2015/mit6_042js15_textbook.pdf

The inspected online textbook is dated 18 May 2015; this is a version date, rather than a claim about first publication.

Cited passages: Lehman, Leighton and Meyer, Mathematics for Computer Science, Definition 11.7.1; Lehman, Leighton and Meyer, Mathematics for Computer Science, §11.8.1: cycles and backtracking; Lehman, Leighton and Meyer, Mathematics for Computer Science, §11.8.1: length-zero convention; Lehman, Leighton and Meyer, Mathematics for Computer Science, §§9.2 and 11.8.1: closed walks; Mathematics for Computer Science, §12.4, Lemma 12.4.1, printed p. 485; Mathematics for Computer Science, §12.4, Lemma 12.4.2 and Theorem 12.4.3, printed pp. 485–486; Mathematics for Computer Science, §12.5, Lemma 12.5.2 and Theorem 12.5.3, printed pp. 486–487.

Additional cited passages in the 18 May 2015 revision: §5.2.1, printed p. 125; §5.2.2, printed pp. 125–126; §5.3, printed pp. 129–130. The title page credits the same three whole-book authors; the date is a revision/version date.

By Albert R. Meyer.

Publisher: MIT OpenCourseWare.

Online text: ocw.mit.edu/courses/6-042j-mathematics-for-computer-science-fall-2005/aa542addc1c3bda6732467fb90723852_slides5f.pdf

Lecture 5f of MIT 6.042J and 18.062J, Mathematics for Computer Science, dated 7 October 2005. Authorship is credited to this slide deck, rather than to every resource in the course.

Cited passages: Planar Graphs; Bipartite Matching, slide lec 5f.6, including the outer face; Planar Graphs; Bipartite Matching, slides lec 5f.14–5f.15; Planar Graphs; Bipartite Matching, slides lec 5f.3 and 5f.12.

By Raymond E. Goldstein, Adriana I. Pesci, Christophe Raufaste, James D. Shemilt.

Publisher: American Physical Society.

Publication information: journals.aps.org/pre/abstract/10.1103/PhysRevE.104.035105

Published in Physical Review E 104, 035105.

Online text: api.repository.cam.ac.uk/server/api/core/bitstreams/43f1adff-1045-49b4-91cb-32e496674f0a/content

Cited passages: Geometry of Catenoidal Soap Film Collapse Induced by Boundary Deformation, §II, manuscript pp. 3–4, Figs. 1–3: apparatus, contour readouts, viscosity comparison, and displacement limitation.

Publisher: MIT OpenCourseWare.

Online text: ocw.mit.edu/courses/18-357-interfacial-phenomena-fall-2010/04759cfc15c79928ca9f72a18b3864dd_MIT18_357F10_lec_all.pdf

Fall 2010 identifies the course offering. The cover is dated 3 June 2013, without identifying that date as a publication or revision date.

Cited passages: 18.357 Interfacial Phenomena, §2.6, pp. 9–10: drop pressure and the two-interface soap-bubble factor; §2.4, p. 7, equations (2.3)–(2.4): normal and tangential interfacial stress balances.

By J. P. May.

Publisher: University of Chicago Press.

Online text: math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf

Cited passages: A Concise Course in Algebraic Topology, Chapter 13, §5, p. 103, Example (i), the torus cellular chains and integral homology.

By Allen Hatcher.

Publisher: Cambridge University Press.

Online text: math.uchicago.edu/~farb/algtop2023/Hatcher.pdf

The inspected PDF carries copyright 2002; the publisher front matter separately records first publication in 2001 and a fourteenth printing in 2015.

Cited passages: Algebraic Topology, §2.2, p. 146, definition of Euler characteristic and Theorem 2.44; Algebraic Topology, §2.2, p. 146, homotopy-type independence immediately before Theorem 2.44.

Author-hosted chapter PDF: pi.math.cornell.edu/~hatcher/AT/ATch2.pdf

Additional cited passages: §2.2, Example 2.42, p. 144; Euler characteristic and Theorem 2.44, pp. 146–147. These are chapter locators in Algebraic Topology by Allen Hatcher.

By Jiří Matoušek.

Online text: kam.mff.cuni.cz/~matousek/ba-a4.pdf

Linked version dated 1 January 2013.

Cited passages: Jiří Matoušek, Lecture notes on metric embeddings, preface and Section 1.1; Lecture notes on metric embeddings, §2.1(b), p. 21, and Lemma 2.3.1, p. 24: independent Gaussian coefficients and normalization; Lecture notes on metric embeddings, §2.1, pp. 20–21, and proof of Lemma 2.3.1, pp. 25–26: fixed-vector concentration and union bound over pair differences; Matoušek, Section 1.1: distortion and Lipschitz maps.

By Sanjoy Dasgupta, Anupam Gupta.

Publisher: Wiley Periodicals, Inc.

Published in Random Structures & Algorithms.

Online text: web.stanford.edu/class/cs114/readings/JL-Dasgupta.pdf

Cited passages: An elementary proof of a theorem of Johnson and Lindenstrauss, Theorem 2.1, p. 61: bounds for squared distances.

By Keith Conrad.

Publisher: Keith Conrad (author-hosted notes).

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

Cited passages: §3, printed p. 9: elementary, simple, decomposable, and pure tensors; Example 2.1, printed pp. 4–5: outer products do not fill the matrix space.

By Sheldon Axler.

Publisher: Springer Cham.

Online text: linear.axler.net/LADR4e.pdf

Bibliographic and copyright year 2024; electronic release 28 October 2023 and print release 20 November 2023. The linked author PDF cover is dated 30 October 2026.

Cited passages: Axler, Linear Algebra Done Right, fourth edition, §3F, definitions 3.109–3.112; Axler, Linear Algebra Done Right, fourth edition, §3F, definitions and results 3.112–3.116; Axler, Linear Algebra Done Right, fourth edition, §8D, definition 8.47 and result 8.56; Axler, Linear Algebra Done Right, fourth edition, §8D, result 8.49; Axler, Linear Algebra Done Right, fourth edition, §8D, results 8.50–8.52; Sheldon Axler, Linear Algebra Done Right, fourth edition, §3B, theorem 3.21; Sheldon Axler, Linear Algebra Done Right, fourth edition, §3B: fundamental theorem of linear maps; Sheldon Axler, Linear Algebra Done Right, fourth edition, §3E: quotient spaces; §9D, 9.71–9.74, printed pp. 372–373: tensor products, bilinearity, and bases; §9D, Theorem 9.79(a), printed p. 375: unique linear factorization; §9D, Theorems 9.72 and 9.74, printed pp. 372–373: dimension and tensor-product basis.

Section 3A, definition 3.1, printed p. 52: additivity and homogeneity define a linear map. The fourth-edition bibliographic and copyright year remains 2024.

By Stéphane Mallat.

Publisher: SciPy project.

The scipy.stats.multivariate_normal Notes describe singular covariance using a pseudo-determinant, pseudoinverse and covariance rank when singular evaluation is allowed. The pinned implementation separately checks the covariance support: _PSD._support_mask tests the null-space residual against its numerical tolerance, and multivariate_normal_gen.pdf/logpdf return zero density or negative-infinite log density off support. ^{[ref] [ref]}

Covariance.whiten describes transforming observations to whitened coordinates. Its example constructs Covariance.from_precision from a positive-definite covariance, whitens a Gaussian sample and displays a near-identity sample covariance. ^[ref]

The scipy.stats.norm method list identifies cdf as the cumulative distribution function, sf as the survival function, and ppf as the inverse cdf. The loc and scale parameters specify the mean and standard deviation. The documentation notes that sf can be more accurate than subtracting cdf from one when evaluating a tail. ^[ref]

The scipy.stats.uniform description specifies the standard uniform distribution on the unit interval and the loc/scale transformation to the interval from loc to loc plus scale. Its Methods list includes rvs for random variates, alongside cdf and ppf. ^[ref]

The scipy.signal.group_delay definition differentiates continuous unwrapped phase with respect to digital angular frequency and reports delay in samples; the fs parameter sets the frequency units. An analog derivative with respect to angular frequency in radians per second instead yields seconds. ^[ref]

Notes: standardized folded-normal density, nonnegative shape parameter and loc/scale transformation. Taking the absolute value of a normal variable with mean $m$ and standard deviation $s$ corresponds to shape $|m|/s$, loc=0 and scale=s.

The scipy.stats.multivariate_normal cdf method evaluates joint lower-orthant probabilities using the specified mean vector and covariance matrix. Its maxpts, abseps, and releps arguments control numerical integration limits and tolerances; lower_limit specifies the lower integration bounds. This supports evaluating a correlated maximum at a vector of equal upper thresholds without multiplying marginal probabilities. ^[ref]

The scipy.fft.fft norm parameter documents an unscaled forward transform for the default backward convention, a 1/n-scaled forward transform for forward, and symmetric 1/sqrt(n) scaling for ortho. Its Notes specify the negative-exponent forward sum, ordering of positive and negative frequencies, and conjugate symmetry for real inputs. ^[ref]

The scipy.stats.binom Notes define the binomial mass function, including its positive probability at zero for a finite nondegenerate calibration sample; pmf evaluates each count probability. ^[ref]

The signal-processing guide Spectral Analysis section derives the sinc leakage of a finite-duration sinusoid and discusses the tradeoff between window main-lobe width and sidelobe height. ^[ref]

The scipy.signal.periodogram scaling parameter distinguishes squared spectrum units from power spectral density units. Its Notes give the window-dependent spectrum/density conversion and one-sided power folding; its flattop example estimates RMS amplitude from the square root of a spectrum peak. ^[ref]

BinomTestResult.proportion_ci documents its confidence_level parameter, the exact binomial proportion-interval method, and the low/high attributes of the returned bounds. These are parameter-estimation bounds; scipy.stats.binom.interval instead describes count-distribution intervals for supplied parameters. ^{[ref] [ref]}

scipy.signal.windows.boxcar, description and Returns: the rectangular window uses all-ones coefficients. scipy.signal.windows.hann, sym parameter, Returns and Notes: the symmetric endpoint convention differs from the periodic spectral convention; the documented even-length symmetric array need not attain coefficient one.

scipy.ndimage.zoom, description and order parameter: array zoom uses spline interpolation of the requested order, with orders zero through five; resizing is a numerical reconstruction operation rather than an additional physical measurement.

scipy.signal.freqz, defining frequency-response sum; Parameters: fs; Returns: w: evaluates a finite numerator/denominator frequency response; the default frequency unit is radians per sample, with frequencies expressed in the same units as fs when supplied.

null_space, lines 333–403: the pinned function constructs a null-space basis from right singular vectors after SVD; rcond controls the relative cutoff for effective numerical rank. The returned basis Q uses vh[num:,:].T.conj().

scipy.stats.chi2, Notes: the chi-squared density and its gamma-distribution representation supply the density used in the explicitly derived reciprocal-moment calculation.

By Andrew Granville, K. Soundararajan.

Online text: dms.umontreal.ca/~andrew/PDF/Book.To1.2.pdf

Cited passages: Andrew Granville and K. Soundararajan, Multiplicative number theory: The pretentious approach, §1.1.4, pp. 7–8: divisor pairing and the hyperbola split; §1.1.4, p. 8, equation (1.1.13): the divisor summatory asymptotic.

By Imre Csiszár.

Published in The Annals of Probability 3(1), pp. 146–158, February 1975, by the Institute of Mathematical Statistics.

Mirrored article: pages.stern.nyu.edu/~dbackus/BCZ/entropy/Csiszar_geometry_AP_75.pdf

By Imre Csiszár and Paul C. Shields.

Published in Foundations and Trends in Communications and Information Theory 1(4), pp. 417–528, by now Publishers.

Author-hosted text: www.renyi.hu/~csiszar/Publications/Information_Theory_and_Statistics:_A_Tutorial.pdf

By John K. Hunter.

By Sheldon Axler.

Publisher: Springer.

Online text: measure.axler.net/MIRA.pdf

Cited passages: Axler, Measure, Integration & Real Analysis, §7A, Definition 7.1, pp. 194–195, and Theorem 7.14, p. 199.

The linked PDF is revised 12 June 2026; the book was first published in 2020.

By Stephen Boyd, Lieven Vandenberghe.

Publisher: Cambridge University Press.

Online text: www.seas.ucla.edu/~vandenbe/cvxbook/bv_cvxbook.pdf

Cited passages: Appendix A.5.2, pp. 646–647: symmetric eigenvalue decomposition and definiteness; §2.2.2, p. 30, equation (2.3): ellipsoids and semi-axis lengths; the book uses the inverse quadratic coefficient matrix; §3.1.4, Example 3.2, p. 71: a quadratic function is strictly convex exactly when its symmetric coefficient matrix is positive definite.

Online text: stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf

Cited passages: Appendix A.5.5, printed p. 650: block inversion and minimized quadratic form; §C.4.1, printed pp. 672–675: block elimination, Algorithm C.4, factor-solve interpretation and dense complexity.

Online text: web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf

The front matter states first publication in 2004 and a seventh printing with corrections in 2009. The title page credits Stephen Boyd and Lieven Vandenberghe as whole-book coauthors; no separate appendix byline is supplied. Cambridge University Press publishes the book; the Stanford author website hosts this copy.

Cited passages: Appendix A.1.6, printed pp. 636–637, dual norms.

Additional cited passage: §2.1.4, printed p. 24 (PDF page index 37), convex combinations and the definition and minimality of the convex hull. The title-page whole-book credits and the separately recorded 2004 first publication and 2009 corrected seventh printing apply to this copy.

Additional cited passages: §5.6.3, printed pp. 251–253, especially equation (5.58), local sensitivity of the optimal value to equality-constraint perturbations. The sign of the Lagrange multiplier depends on the chosen sign for the equality term. The title page and front matter retain the whole-book authorship, 2004 first publication and 2009 corrected seventh printing recorded above.

Additional locators: §§5.1–5.2, pp.215–227.

Additional cited passage: §4.4.2, printed p. 157, equation (4.37), robust linear programming imposes each inequality for every parameter in the chosen uncertainty set. The illustrated set is an ellipsoid, and its worst-case linear contribution is a Euclidean norm; a box or finite sign model has a different support calculation. ^[ref]