From df8c2c8f1e42cd2d9ab9368f1e1b182e4360ecb5 Mon Sep 17 00:00:00 2001 From: Claude Date: Mon, 2 Nov 2026 18:08:04 +0000 Subject: [PATCH 1/5] Promote body headings one level so sections number 1, 2, 3 instead of 0.1, 0.2 number-sections numbers by heading level and the page title is not a numbered level, so a page whose first body heading is level 2 numbers its sections 0.1, 0.2. Heading text is unchanged, so anchors and links are unchanged. Part of #568 Co-Authored-By: Claude Sonnet 5.5 Claude-Session: https://claude.ai/code/session_019xfc9mgFxbUyGpESuW5TVR --- _algebra.qmd | 4 ++-- _expressions-algorithms.qmd | 6 +++--- _functions.qmd | 2 +- _growth-rates.qmd | 6 +++--- _indicator-functions.qmd | 2 +- _induction.qmd | 4 ++-- _notation.qmd | 6 +++--- _number-systems.qmd | 4 ++-- _proof-writing.qmd | 18 +++++++++--------- _relations.qmd | 4 ++-- 10 files changed, 28 insertions(+), 28 deletions(-) diff --git a/_algebra.qmd b/_algebra.qmd index af9c2598f..8d7ce3d97 100644 --- a/_algebra.qmd +++ b/_algebra.qmd @@ -6,10 +6,10 @@ {{< include mds/_subfiles/_sec_algebra_exponentials.qmd >}} -## Cosine and sine {#sec-cosine-sine} +# Cosine and sine {#sec-cosine-sine} {{< include mds/_subfiles/_sec_algebra_trigonometry.qmd >}} -## Complex numbers {#sec-complex-numbers} +# Complex numbers {#sec-complex-numbers} {{< include mds/_subfiles/_sec_algebra_complex.qmd >}} diff --git a/_expressions-algorithms.qmd b/_expressions-algorithms.qmd index 9b306e57a..73fe5100f 100644 --- a/_expressions-algorithms.qmd +++ b/_expressions-algorithms.qmd @@ -9,7 +9,7 @@ the parse trees of expressions, and algorithms, which compute those functions. ::: -## Expressions +# Expressions {{< include mds/_subfiles/expressions-algorithms/_def-expression.qmd >}} @@ -53,7 +53,7 @@ the parse trees of expressions, and algorithms, which compute those functions. {{< include mds/_subfiles/expressions-algorithms/_rem-expression-parse-tree.qmd >}} -## Algorithms +# Algorithms {{< include mds/_subfiles/expressions-algorithms/_def-algorithm.qmd >}} @@ -81,7 +81,7 @@ the parse trees of expressions, and algorithms, which compute those functions. {{< include mds/_subfiles/expressions-algorithms/_exm-no-known-algorithm.qmd >}} -## Exercises +# Exercises {{< include mds/_subfiles/expressions-algorithms/_exr-expression-or-algorithm.qmd >}} diff --git a/_functions.qmd b/_functions.qmd index 05c91927b..ab305a251 100644 --- a/_functions.qmd +++ b/_functions.qmd @@ -201,7 +201,7 @@ The functions here go between the sets defined on the [Sets](sets.qmd) page. {{< include mds/_subfiles/functions/_fig-logistic-function.qmd >}} -## Countable sets +# Countable sets {{< include mds/_subfiles/functions/_def-finite-set.qmd >}} diff --git a/_growth-rates.qmd b/_growth-rates.qmd index 40e65a19b..8511ab3e7 100644 --- a/_growth-rates.qmd +++ b/_growth-rates.qmd @@ -8,7 +8,7 @@ This page defines the notation for that comparison. ::: -## Upper bounds +# Upper bounds {{< include mds/_subfiles/growth-rates/_def-big-o.qmd >}} @@ -16,7 +16,7 @@ This page defines the notation for that comparison. {{< include mds/_subfiles/growth-rates/_exm-big-o.qmd >}} -## Lower bounds +# Lower bounds {{< include mds/_subfiles/growth-rates/_def-big-omega.qmd >}} @@ -24,7 +24,7 @@ This page defines the notation for that comparison. {{< include mds/_subfiles/growth-rates/_exm-big-omega.qmd >}} -## Matching bounds +# Matching bounds {{< include mds/_subfiles/growth-rates/_def-big-theta.qmd >}} diff --git a/_indicator-functions.qmd b/_indicator-functions.qmd index 5b2f00762..7a575880f 100644 --- a/_indicator-functions.qmd +++ b/_indicator-functions.qmd @@ -1,5 +1,5 @@ {{< include latex-macros/macros.qmd >}} -## Indicator functions {#sec-indicator-functions} +# Indicator functions {#sec-indicator-functions} {{< include mds/_subfiles/indicator-functions/_sec-indicator-functions.qmd >}} diff --git a/_induction.qmd b/_induction.qmd index 6030cd84b..b2969915b 100644 --- a/_induction.qmd +++ b/_induction.qmd @@ -9,7 +9,7 @@ then adds the stronger form, [strong induction](induction.qmd#def-strong-inducti ::: -## Recursively defined sequences +# Recursively defined sequences {{< include mds/_subfiles/induction/_def-recursive-sequence.qmd >}} @@ -25,7 +25,7 @@ then adds the stronger form, [strong induction](induction.qmd#def-strong-inducti {{< include mds/_subfiles/induction/_exm-fibonacci-sequence.qmd >}} -## Strong induction +# Strong induction {{< include mds/_subfiles/induction/_def-strong-induction.qmd >}} diff --git a/_notation.qmd b/_notation.qmd index 16ab2a7b9..25796d4e6 100644 --- a/_notation.qmd +++ b/_notation.qmd @@ -43,7 +43,7 @@ which define those terms. of the predictor values $x$ with coefficients $\beta$: $g(\mu) = \eta$. -## Writing math in Quarto {#sec-writing-math-quarto} +# Writing math in Quarto {#sec-writing-math-quarto} The third column of @tbl-notation-collected gives the LaTeX command for each symbol. @@ -56,7 +56,7 @@ These notes also use shorthand macros, such as `\floor{x}` for $\floor{x}$, defined in the `latex-macros` submodule's `macros.qmd`. -## Subscripts and parentheses {#sec-subscripts-parentheses} +# Subscripts and parentheses {#sec-subscripts-parentheses} {{< include mds/_subfiles/notation/_rem-subscript-parentheses-convention.qmd >}} @@ -68,7 +68,7 @@ defined in the `latex-macros` submodule's `macros.qmd`. {{< include mds/_subfiles/notation/_exm-process-notation.qmd >}} -## Why is notation in probability and statistics so inconsistent and disorganized? +# Why is notation in probability and statistics so inconsistent and disorganized? In grad school, we are asked to learn diff --git a/_number-systems.qmd b/_number-systems.qmd index f2c4638e4..2ff8371be 100644 --- a/_number-systems.qmd +++ b/_number-systems.qmd @@ -10,7 +10,7 @@ even and odd integers, and rational and irrational real numbers. ::: -## Natural numbers {#sec-notation-naturals} +# Natural numbers {#sec-notation-naturals} {{< include mds/_subfiles/number-systems/_exr-naturals-below-three.qmd >}} @@ -50,7 +50,7 @@ even and odd integers, and rational and irrational real numbers. {{< include mds/_subfiles/number-systems/_note-natural-numbers-other-conventions.qmd >}} -## Rational, real, and irrational numbers {#sec-notation-reals} +# Rational, real, and irrational numbers {#sec-notation-reals} {{< include mds/_subfiles/number-systems/_def-rational-numbers.qmd >}} diff --git a/_proof-writing.qmd b/_proof-writing.qmd index cf97cdfef..ef7c4aff8 100644 --- a/_proof-writing.qmd +++ b/_proof-writing.qmd @@ -6,7 +6,7 @@ Each principle on this page serves that goal. The symbols for [logical entailment](#def-logical-entailment) are listed under [Definitions for proofs](#sec-proof-definitions) below. -## Definitions for proofs {#sec-proof-definitions} +# Definitions for proofs {#sec-proof-definitions} {{< include mds/_subfiles/proof-writing/_def-proposition.qmd >}} @@ -272,11 +272,11 @@ Let's save $\rightarrow$ (`\rightarrow`) for [convergence](algebra.qmd#def-conve {{< include mds/_subfiles/proof-writing/_exm-proof-by-contradiction-truth-table.qmd >}} -## Don't skip steps {#sec-dont-skip-steps} +# Don't skip steps {#sec-dont-skip-steps} {{< include mds/_subfiles/proof-writing/_sec-dont-skip-steps.qmd >}} -## Annotate each step {#sec-annotate-each-step} +# Annotate each step {#sec-annotate-each-step} {{< include mds/_subfiles/proof-writing/_sec-annotate-each-step.qmd >}} @@ -317,7 +317,7 @@ The probability notes derive the variance identity $\Var{X} = \E{X^2} - \sqf{\E{X}}$ [in the same annotated style](https://morrison-lab.github.io/pds/variance-covariance.html#thm-variance). -## Direct proof {#sec-direct-proof} +# Direct proof {#sec-direct-proof} {{< include mds/_subfiles/proof-writing/_def-direct-proof.qmd >}} @@ -325,7 +325,7 @@ $\Var{X} = \E{X^2} - \sqf{\E{X}}$ {{< include mds/_subfiles/proof-writing/_exm-direct-proof.qmd >}} -## Proof by contrapositive {#sec-proof-by-contrapositive} +# Proof by contrapositive {#sec-proof-by-contrapositive} {{< include mds/_subfiles/proof-writing/_def-proof-by-contrapositive.qmd >}} @@ -333,7 +333,7 @@ $\Var{X} = \E{X^2} - \sqf{\E{X}}$ {{< include mds/_subfiles/proof-writing/_exm-proof-by-contrapositive.qmd >}} -## Proof by cases {#sec-proof-by-cases} +# Proof by cases {#sec-proof-by-cases} {{< include mds/_subfiles/proof-writing/_def-proof-by-cases.qmd >}} @@ -341,11 +341,11 @@ $\Var{X} = \E{X^2} - \sqf{\E{X}}$ {{< include mds/_subfiles/proof-writing/_exm-proof-by-cases.qmd >}} -## Follow the golden rule {#sec-golden-rule} +# Follow the golden rule {#sec-golden-rule} {{< include mds/_subfiles/proof-writing/_sec-golden-rule.qmd >}} -## Proof by induction {#sec-proof-by-induction} +# Proof by induction {#sec-proof-by-induction} Many proofs in these notes show that a statement holds for every [natural number](number-systems.qmd#def-natural-numbers) $n$, @@ -365,7 +365,7 @@ The following example uses [summation notation](algebra.qmd#def-summation). {{< include mds/_subfiles/proof-writing/_exm-induction-sum.qmd >}} -## Formal logic {#sec-formal-logic} +# Formal logic {#sec-formal-logic} {{< include mds/_subfiles/proof-writing/_def-propositional-logic.qmd >}} diff --git a/_relations.qmd b/_relations.qmd index 4cf17c06e..6060b7a9d 100644 --- a/_relations.qmd +++ b/_relations.qmd @@ -8,7 +8,7 @@ and the groups of elements that such a relation puts together. ::: -## Properties of relations +# Properties of relations {{< include mds/_subfiles/relations/_def-reflexive.qmd >}} @@ -24,7 +24,7 @@ and the groups of elements that such a relation puts together. {{< include mds/_subfiles/relations/_exm-symmetric-relation.qmd >}} -## Equivalence relations +# Equivalence relations {{< include mds/_subfiles/relations/_def-equivalence-relation.qmd >}} From 8d2fa89aa3f5d48603000d42b10854339e7aa1dd Mon Sep 17 00:00:00 2001 From: Claude Date: Mon, 2 Nov 2026 18:08:06 +0000 Subject: [PATCH 2/5] Promote body headings one level so sections number 1, 2, 3 instead of 0.1, 0.2 number-sections numbers by heading level and the page title is not a numbered level, so a page whose first body heading is level 2 numbers its sections 0.1, 0.2. Heading text is unchanged, so anchors and links are unchanged. Part of #568 Co-Authored-By: Claude Sonnet 5.5 Claude-Session: https://claude.ai/code/session_019xfc9mgFxbUyGpESuW5TVR --- _sets-functions.qmd | 4 ++-- _sets.qmd | 4 ++-- algebra.qmd | 4 ++-- calculus-double-integrals.qmd | 2 +- calculus.qmd | 4 ++-- functions.qmd | 2 +- growth-rates.qmd | 2 +- index.qmd | 8 ++++---- induction.qmd | 4 ++-- linear-algebra.qmd | 4 ++-- 10 files changed, 19 insertions(+), 19 deletions(-) diff --git a/_sets-functions.qmd b/_sets-functions.qmd index c332ea8c4..12d0906ea 100644 --- a/_sets-functions.qmd +++ b/_sets-functions.qmd @@ -8,11 +8,11 @@ The sets and functions fragments no longer open with a section heading, so a host page that includes them writes its own Sets and Functions headings; the shim writes them here, with ids that differ from the fragments' own anchors. --> -## Sets {#sec-shim-sets} +# Sets {#sec-shim-sets} {{< include mds/_sets.qmd >}} -## Functions {#sec-shim-functions} +# Functions {#sec-shim-functions} {{< include mds/_functions.qmd >}} diff --git a/_sets.qmd b/_sets.qmd index 96472f6c5..34e0720f1 100644 --- a/_sets.qmd +++ b/_sets.qmd @@ -123,7 +123,7 @@ the expressions that write functions down and the algorithms that compute them. {{< include mds/_subfiles/sets/_exm-indexed-family.qmd >}} -## Combining sets +# Combining sets {{< include mds/_subfiles/sets/_def-union.qmd >}} @@ -191,7 +191,7 @@ the expressions that write functions down and the algorithms that compute them. {{< include mds/_subfiles/sets/_exm-cartesian-product.qmd >}} -## Extended non-negative real numbers +# Extended non-negative real numbers {{< include mds/_subfiles/sets/_def-extended-nonneg-reals.qmd >}} diff --git a/algebra.qmd b/algebra.qmd index 79db14bbc..528af44d6 100644 --- a/algebra.qmd +++ b/algebra.qmd @@ -39,7 +39,7 @@ goes on to the page that now holds it. ::: -## Further reading {#sec-further-reading} +# Further reading {#sec-further-reading} - @abramson2021algebra is a free online textbook on algebra and trigonometry. @@ -49,7 +49,7 @@ goes on to the page that now holds it. develops the real numbers, including the least upper bound property behind the infimum and supremum. -## References {.unnumbered} +# References {.unnumbered} ::: {#refs} ::: diff --git a/calculus-double-integrals.qmd b/calculus-double-integrals.qmd index 7933782f8..5f9e4afbd 100644 --- a/calculus-double-integrals.qmd +++ b/calculus-double-integrals.qmd @@ -14,6 +14,6 @@ format: {{< include mds/_subfiles/_sec_calculus_double_integrals.qmd >}} -## Further reading {.unnumbered} +# Further reading {.unnumbered} - @wp:fubini --- Fubini's theorem, in the Riemann and Lebesgue settings diff --git a/calculus.qmd b/calculus.qmd index 3a1022f76..51169824f 100644 --- a/calculus.qmd +++ b/calculus.qmd @@ -35,7 +35,7 @@ goes on to the page that now holds it. ::: -## Further reading {#sec-additional-resources} +# Further reading {#sec-additional-resources} - @mosaiccalc - @khuri2003advanced @@ -49,7 +49,7 @@ goes on to the page that now holds it. - [Essence of Calculus](https://www.youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr) is a 12-video YouTube playlist by Grant Sanderson (3Blue1Brown) that develops geometric and visual intuition for limits, derivatives, product and chain rules, implicit differentiation, integration, and Taylor series. -## References {.unnumbered} +# References {.unnumbered} ::: {#refs} ::: diff --git a/functions.qmd b/functions.qmd index 495d96c70..1af638aa0 100644 --- a/functions.qmd +++ b/functions.qmd @@ -13,7 +13,7 @@ format: {{< include mds/_functions.qmd >}} -## References {.unnumbered} +# References {.unnumbered} ::: {#refs} ::: diff --git a/growth-rates.qmd b/growth-rates.qmd index 344147106..37f129e69 100644 --- a/growth-rates.qmd +++ b/growth-rates.qmd @@ -12,7 +12,7 @@ format: {{< include mds/_growth-rates.qmd >}} -## References {.unnumbered} +# References {.unnumbered} ::: {#refs} ::: diff --git a/index.qmd b/index.qmd index c3b4d352c..1b2b84c71 100644 --- a/index.qmd +++ b/index.qmd @@ -10,7 +10,7 @@ format: output-file: index.docx --- -## Welcome {.unnumbered} +# Welcome {.unnumbered} > Math is not just a way of calculating numerical answers; it is a way of thinking, @@ -68,7 +68,7 @@ organized by topic: - [Proof Writing](proof-writing.qmd): propositions, quantifiers, proofs and derivations, and showing and annotating every step of a [derivation](proof-writing.qmd#def-derivation) -## Further reading {.unnumbered} +# Further reading {.unnumbered} These resources cover related material. @@ -111,7 +111,7 @@ These resources cover related material. - [Neural Networks](https://www.youtube.com/playlist?list=PLZHQObOWTQDNU6R1_67000Dx_ZCJB-3pi) covers gradient descent and backpropagation calculus, visualizing multivariable optimization in connection with our [vector calculus](vector-calculus.qmd) and [optimization](optimization.qmd) pages. -## Using these notes in another site {.unnumbered} +# Using these notes in another site {.unnumbered} Course sites link to these pages by URL; they do not include this repository as a git submodule. @@ -148,7 +148,7 @@ Quarto resolves `@id` cross-references only within one rendered page, so a host site that links to a result here uses an explicit link, `[text](notation.qmd#id)`. -## References {.unnumbered} +# References {.unnumbered} ::: {#refs} ::: diff --git a/induction.qmd b/induction.qmd index d8c12d803..793187d0a 100644 --- a/induction.qmd +++ b/induction.qmd @@ -12,12 +12,12 @@ format: {{< include mds/_induction.qmd >}} -## Further reading {#sec-further-reading} +# Further reading {#sec-further-reading} - @velleman2019prove is a structured introduction to proof techniques, including [proof by induction](proof-writing.qmd#def-proof-by-induction). -## References {.unnumbered} +# References {.unnumbered} ::: {#refs} ::: diff --git a/linear-algebra.qmd b/linear-algebra.qmd index 5e4d7bf2e..7e33ffcfb 100644 --- a/linear-algebra.qmd +++ b/linear-algebra.qmd @@ -57,7 +57,7 @@ goes on to the page that now holds it. {{< include mds/_subfiles/linear-algebra/_rem-matrix-analysis.qmd >}} -## Further reading {#sec-additional-resources} +# Further reading {#sec-additional-resources} - @strang2023linear is a widely used first course in linear algebra. @@ -86,7 +86,7 @@ goes on to the page that now holds it. - [Essence of Linear Algebra](https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab) is a 16-video YouTube playlist by Grant Sanderson (3Blue1Brown) that develops visual and geometric intuition for vectors, linear combinations, spans, linear transformations, matrix multiplication, determinants, inverses, dot and cross products, change of basis, and eigenvectors and eigenvalues. -## References {.unnumbered} +# References {.unnumbered} ::: {#refs} ::: From 7d8f83750e7f90d72cab2a51e635eb567a55cbed Mon Sep 17 00:00:00 2001 From: Claude Date: Mon, 2 Nov 2026 18:08:08 +0000 Subject: [PATCH 3/5] Promote body headings one level so sections number 1, 2, 3 instead of 0.1, 0.2 number-sections numbers by heading level and the page title is not a numbered level, so a page whose first body heading is level 2 numbers its sections 0.1, 0.2. Heading text is unchanged, so anchors and links are unchanged. Part of #568 Co-Authored-By: Claude Sonnet 5.5 Claude-Session: https://claude.ai/code/session_019xfc9mgFxbUyGpESuW5TVR --- measures.qmd | 4 ++-- notation.qmd | 4 ++-- optimization.qmd | 4 ++-- proof-writing.qmd | 4 ++-- relations.qmd | 2 +- sets-functions.qmd | 2 +- sets.qmd | 4 ++-- vector-calculus.qmd | 4 ++-- 8 files changed, 14 insertions(+), 14 deletions(-) diff --git a/measures.qmd b/measures.qmd index 26c9b8836..20adf60d0 100644 --- a/measures.qmd +++ b/measures.qmd @@ -43,7 +43,7 @@ goes on to the page that now holds it. ::: -## Further reading {#sec-further-reading} +# Further reading {#sec-further-reading} - @billingsley1995probability develops measure theory and builds probability on it. @@ -55,7 +55,7 @@ goes on to the page that now holds it. is a graduate course in probability that opens with a chapter on introductory measure theory. -## References {.unnumbered} +# References {.unnumbered} ::: {#refs} ::: diff --git a/notation.qmd b/notation.qmd index 64997c216..39735db99 100644 --- a/notation.qmd +++ b/notation.qmd @@ -36,7 +36,7 @@ format: {{< include mds/_notation.qmd >}} -## Further reading {#sec-further-reading} +# Further reading {#sec-further-reading} For the logical symbols and proof conventions used in these notes (whose definitions are on the [Proof Writing](proof-writing.qmd#sec-proof-definitions) page): @@ -52,7 +52,7 @@ For the logical symbols and proof conventions used in these notes For set and function notation, see the [Sets](sets.qmd), [Functions](functions.qmd), and [Expressions and Algorithms](expressions-algorithms.qmd) pages. -## References {.unnumbered} +# References {.unnumbered} ::: {#refs} ::: diff --git a/optimization.qmd b/optimization.qmd index 8e22ada80..efb92dbbc 100644 --- a/optimization.qmd +++ b/optimization.qmd @@ -51,7 +51,7 @@ goes on to the page that now holds it. ::: -## Further reading {#sec-additional-resources} +# Further reading {#sec-additional-resources} - @boyd2004convex is the standard reference on convex optimization, covering convex sets, convex functions, convex optimization problems, duality, unconstrained minimization, and interior-point methods. @@ -78,7 +78,7 @@ goes on to the page that now holds it. its chapters on gradient descent and backpropagation calculus visually illustrate how gradients of multivariate cost functions are computed and used for optimization. -## References {.unnumbered} +# References {.unnumbered} ::: {#refs} ::: diff --git a/proof-writing.qmd b/proof-writing.qmd index 785053dc6..cf2758ef8 100644 --- a/proof-writing.qmd +++ b/proof-writing.qmd @@ -14,7 +14,7 @@ format: {{< include mds/_proof-writing.qmd >}} -## Further reading {#sec-further-reading} +# Further reading {#sec-further-reading} - @velleman2019prove is a structured introduction to proof techniques @@ -35,7 +35,7 @@ Free course materials on formal logic: - Eric Pacuit's [Phil 151: First Order Logic](https://cs.stanford.edu/~epacuit/classes/phil151winter09.html) (Stanford, Winter 2009) posts a syllabus, a modal logic study guide and problem sets. -## References {.unnumbered} +# References {.unnumbered} ::: {#refs} ::: diff --git a/relations.qmd b/relations.qmd index 196f3a78c..217dfe7f3 100644 --- a/relations.qmd +++ b/relations.qmd @@ -12,7 +12,7 @@ format: {{< include mds/_relations.qmd >}} -## References {.unnumbered} +# References {.unnumbered} ::: {#refs} ::: diff --git a/sets-functions.qmd b/sets-functions.qmd index 8fcc952e0..33f54ec6b 100644 --- a/sets-functions.qmd +++ b/sets-functions.qmd @@ -33,7 +33,7 @@ format: ``` -## The pages {#sec-split-pages} +# The pages {#sec-split-pages} The definitions that were on this page are now on three pages: diff --git a/sets.qmd b/sets.qmd index 0996b5e3f..e62413a65 100644 --- a/sets.qmd +++ b/sets.qmd @@ -12,7 +12,7 @@ format: {{< include mds/_sets.qmd >}} -## Further reading {#sec-further-reading} +# Further reading {#sec-further-reading} These books and articles treat sets and the logic behind them in more depth. @@ -31,7 +31,7 @@ These books and articles treat sets and the logic behind them in more depth. teaches propositional and first-order logic, the language behind statements such as "for every set $A$ there exists...". -## References {.unnumbered} +# References {.unnumbered} ::: {#refs} ::: diff --git a/vector-calculus.qmd b/vector-calculus.qmd index 79394f5ea..c90b14ab4 100644 --- a/vector-calculus.qmd +++ b/vector-calculus.qmd @@ -14,7 +14,7 @@ format: {{< include mds/_subfiles/_sec_vector_calc.qmd >}} -## Further reading {#sec-additional-resources} +# Further reading {#sec-additional-resources} - @marsden2013vector is a standard textbook on multivariable and vector calculus. @@ -38,7 +38,7 @@ further reading](linear-algebra.qmd#sec-additional-resources). its chapters on gradient descent and backpropagation calculus visually illustrate how gradients of multivariate cost functions are computed and used for optimization. -## References {.unnumbered} +# References {.unnumbered} ::: {#refs} ::: From 8e213a306185d187e9016268ca5c41a41ede792b Mon Sep 17 00:00:00 2001 From: Claude Date: Mon, 2 Nov 2026 21:46:41 +0000 Subject: [PATCH 4/5] Promote all section headings to level 1 across mds Pages and the fragments they include started their sections at level 2, so number-sections rendered 0.1, 0.2 and so on. This moves every page body and included fragment up one level together, so sections number 1, 2, 3, splits the indicator section into one div per subfile (a pure move) so the one-div-per-subfile check can pass, updates split-page-headings.lua for the new level, and sets the revealjs heading shift to 1 so each section is one slide. Replaces the partial PRs, whose reviews saw a half-converted tree. Co-Authored-By: Claude Sonnet 5.5 Claude-Session: https://claude.ai/code/session_019xfc9r2idFsJ4pHCsRGyGx --- _calc-derivatives.qmd | 12 +- _calc-integrals.qmd | 6 +- _quarto-website.yml | 2 +- _subfiles/_sec_algebra_basics.qmd | 12 +- _subfiles/_sec_algebra_exponentials.qmd | 10 +- _subfiles/_sec_algebra_sums.qmd | 18 +- _subfiles/_sec_calculus_derivatives.qmd | 2 +- _subfiles/_sec_calculus_double_integrals.qmd | 2 +- _subfiles/_sec_calculus_integration.qmd | 2 +- _subfiles/_sec_critical_points.qmd | 2 +- _subfiles/_sec_linear_algebra.qmd | 2 +- .../_sec_linear_algebra_diagonalization.qmd | 4 +- _subfiles/_sec_linear_algebra_direct_sums.qmd | 6 +- .../_sec_linear_algebra_inner_products.qmd | 10 +- _subfiles/_sec_linear_algebra_matrices.qmd | 18 +- _subfiles/_sec_linear_algebra_projections.qmd | 8 +- .../_sec_linear_algebra_rank_nullity.qmd | 4 +- .../_sec_linear_algebra_special_matrices.qmd | 12 +- _subfiles/_sec_linear_algebra_subspaces.qmd | 4 +- _subfiles/_sec_linear_algebra_vectors.qmd | 6 +- _subfiles/_sec_measures.qmd | 10 +- _subfiles/_sec_measures_measures.qmd | 4 +- _subfiles/_sec_optimization.qmd | 10 +- _subfiles/_sec_optimization_optimality.qmd | 2 +- _subfiles/_sec_vector_calc.qmd | 2 +- .../_def-indicator-function.qmd | 18 ++ .../_def-iverson-bracket.qmd | 15 ++ .../_def-kronecker-delta.qmd | 15 ++ .../_def-predicate-indicator.qmd | 17 ++ .../_exm-indicator-numerical.qmd | 22 +++ .../_exm-iverson-bracket.qmd | 8 + .../_exm-kronecker-delta.qmd | 32 ++++ .../_rem-iverson-indicator.qmd | 27 +++ .../_sec-indicator-functions.qmd | 174 ++---------------- _subfiles/vector-calc/_sec-gradient-shape.qmd | 2 +- split-page-headings.lua | 14 +- 36 files changed, 261 insertions(+), 253 deletions(-) create mode 100644 _subfiles/indicator-functions/_def-indicator-function.qmd create mode 100644 _subfiles/indicator-functions/_def-iverson-bracket.qmd create mode 100644 _subfiles/indicator-functions/_def-kronecker-delta.qmd create mode 100644 _subfiles/indicator-functions/_def-predicate-indicator.qmd create mode 100644 _subfiles/indicator-functions/_exm-indicator-numerical.qmd create mode 100644 _subfiles/indicator-functions/_exm-iverson-bracket.qmd create mode 100644 _subfiles/indicator-functions/_exm-kronecker-delta.qmd create mode 100644 _subfiles/indicator-functions/_rem-iverson-indicator.qmd diff --git a/_calc-derivatives.qmd b/_calc-derivatives.qmd index cdce28843..ac46b91a7 100644 --- a/_calc-derivatives.qmd +++ b/_calc-derivatives.qmd @@ -1,4 +1,4 @@ -### Limits and derivatives {#sec-limits-derivatives} +## Limits and derivatives {#sec-limits-derivatives} {{< include mds/_subfiles/derivatives/_def-limit.qmd >}} @@ -62,7 +62,7 @@ {{< include mds/_subfiles/derivatives/_exm-second-derivative.qmd >}} -### Derivative rules {#sec-derivative-rules} +## Derivative rules {#sec-derivative-rules} :::{#thm-deriv-const} #### Constant rule @@ -313,7 +313,7 @@ $$ {{< include mds/_subfiles/calculus-review/_exr-miller-deriv-second-log-constant.qmd >}} -### Studying a new function {#sec-function-profile} +## Studying a new function {#sec-function-profile} {{< include mds/_subfiles/derivatives/_def-function-profile.qmd >}} @@ -323,7 +323,7 @@ $$ {{< slidebreak >}} -### Tangent lines {#sec-tangent-lines} +## Tangent lines {#sec-tangent-lines} {{< include mds/_subfiles/derivatives/_def-tangent-line.qmd >}} @@ -333,7 +333,7 @@ $$ {{< slidebreak >}} -### Linear approximation {#sec-linear-approximation} +## Linear approximation {#sec-linear-approximation} For a differentiable function $f$ and a small step $\smallconst$, @@ -411,7 +411,7 @@ as an affine function for functions of several variables as well. {{< slidebreak >}} -### Taylor series {#sec-taylor-series} +## Taylor series {#sec-taylor-series} {{< include mds/_subfiles/derivatives/_def-taylor-polynomial.qmd >}} diff --git a/_calc-integrals.qmd b/_calc-integrals.qmd index 0a265a05f..7e1a001fb 100644 --- a/_calc-integrals.qmd +++ b/_calc-integrals.qmd @@ -10,14 +10,14 @@ Calculus and a worked example from probability. {{< slidebreak >}} -### Antiderivatives +## Antiderivatives {{< include mds/_subfiles/_calc_antiderivatives.qmd >}} -### Regularity Conditions +## Regularity Conditions {{< include mds/_subfiles/_calc_integrals_regularity.qmd >}} -### Fundamental Theorem of Calculus +## Fundamental Theorem of Calculus {{< include mds/_subfiles/_calc_integrals_ftc.qmd >}} diff --git a/_quarto-website.yml b/_quarto-website.yml index 1b0434eab..914447684 100644 --- a/_quarto-website.yml +++ b/_quarto-website.yml @@ -349,7 +349,7 @@ format: incremental: false df-print: paged number-depth: 2 - shift-heading-level-by: 0 + shift-heading-level-by: 1 slide-level: 2 scrollable: true echo: true diff --git a/_subfiles/_sec_algebra_basics.qmd b/_subfiles/_sec_algebra_basics.qmd index 84a692eb6..a3fb4df15 100644 --- a/_subfiles/_sec_algebra_basics.qmd +++ b/_subfiles/_sec_algebra_basics.qmd @@ -1,4 +1,4 @@ -## Equalities +# Equalities {{< include mds/_subfiles/algebra/_note-elementary-algebra-prereq.qmd >}} @@ -24,7 +24,7 @@ {{< include mds/_subfiles/algebra/_exm-identity.qmd >}} -## Inequalities +# Inequalities {{< include mds/_subfiles/algebra/_def-strict-inequality.qmd >}} @@ -48,7 +48,7 @@ {{< include mds/_subfiles/algebra/_thm-negative-one.qmd >}} -## Powers, square roots, and absolute values {#sec-powers-roots-abs} +# Powers, square roots, and absolute values {#sec-powers-roots-abs} {{< include mds/_subfiles/algebra/_def-power.qmd >}} @@ -84,7 +84,7 @@ {{< include mds/_subfiles/algebra/_rem-absolute-value-distance.qmd >}} -## Percent {#sec-percent} +# Percent {#sec-percent} The word "percent" comes from the Latin "per centum"; "centum" is Latin for 100, @@ -102,7 +102,7 @@ so "percent" means "per hundred" You are welcome to switch between decimal and percent notation freely; just make sure you execute it correctly. -## Minimum, maximum, argmin and argmax {#sec-min-max} +# Minimum, maximum, argmin and argmax {#sec-min-max} {{< include mds/_subfiles/algebra/_exr-min-and-argmin.qmd >}} @@ -142,7 +142,7 @@ just make sure you execute it correctly. {{< include mds/_subfiles/algebra/_exm-negative-part.qmd >}} -## Global and local minimizers {#sec-minimizers} +# Global and local minimizers {#sec-minimizers} {{< include mds/_subfiles/algebra/_exr-local-vs-global-min.qmd >}} diff --git a/_subfiles/_sec_algebra_exponentials.qmd b/_subfiles/_sec_algebra_exponentials.qmd index 045bceb26..444d6cc27 100644 --- a/_subfiles/_sec_algebra_exponentials.qmd +++ b/_subfiles/_sec_algebra_exponentials.qmd @@ -1,4 +1,4 @@ -## Polynomials {#sec-polynomials} +# Polynomials {#sec-polynomials} {{< include mds/_subfiles/algebra/_exr-polynomial-degree.qmd >}} @@ -58,7 +58,7 @@ {{< include mds/_subfiles/algebra/_exm-monomial-degree.qmd >}} -## Affine and linear functions of one variable {#sec-affine-linear-functions} +# Affine and linear functions of one variable {#sec-affine-linear-functions} {{< include mds/_subfiles/algebra/_def-affine-function.qmd >}} @@ -90,7 +90,7 @@ {{< include mds/_subfiles/algebra/_thm-affine-function-props.qmd >}} -## Limits of sequences {#sec-sequence-limits} +# Limits of sequences {#sec-sequence-limits} {{< include mds/_subfiles/algebra/_def-sequence-limit.qmd >}} @@ -98,7 +98,7 @@ {{< include mds/_subfiles/algebra/_def-diverges-to-infinity.qmd >}} -## Series {#sec-series} +# Series {#sec-series} {{< include mds/_subfiles/algebra/_def-non-decreasing.qmd >}} @@ -126,7 +126,7 @@ {{< include mds/_subfiles/algebra/_lem-nonneg-series.qmd >}} -## Exponentials and Logarithms +# Exponentials and Logarithms {{< include mds/_subfiles/algebra/_def-exponential-function.qmd >}} diff --git a/_subfiles/_sec_algebra_sums.qmd b/_subfiles/_sec_algebra_sums.qmd index 70822e062..b1e820461 100644 --- a/_subfiles/_sec_algebra_sums.qmd +++ b/_subfiles/_sec_algebra_sums.qmd @@ -1,4 +1,4 @@ -## Convex functions {#sec-convexity} +# Convex functions {#sec-convexity} {{< include mds/_subfiles/algebra/_exr-convex-chord.qmd >}} @@ -46,7 +46,7 @@ {{< include mds/_subfiles/algebra/_exm-jensen.qmd >}} -## Infimum and supremum +# Infimum and supremum {{< include mds/_subfiles/algebra/_def-bounded.qmd >}} @@ -78,7 +78,7 @@ {{< include mds/_subfiles/algebra/_exm-supremum.qmd >}} -## Sums +# Sums {{< include mds/_subfiles/algebra/_def-term.qmd >}} @@ -122,7 +122,7 @@ {{< include mds/_subfiles/algebra/_thm-sum-assoc.qmd >}} -## Products +# Products @@ -136,11 +136,11 @@ {{< include mds/_subfiles/algebra/_thm-prod-assoc.qmd >}} -## Division +# Division {{< include mds/_subfiles/algebra/_thm-prod-div.qmd >}} -## Sums and products together +# Sums and products together @@ -190,7 +190,7 @@ {{< include mds/_subfiles/algebra/_thm-square-of-a-sum.qmd >}} -## Summation notation {#sec-summation} +# Summation notation {#sec-summation} {{< include mds/_subfiles/algebra/_exr-expand-a-sum.qmd >}} @@ -266,7 +266,7 @@ {{< include mds/_subfiles/algebra/_exm-sum-over-empty-set.qmd >}} -## Quotients +# Quotients {{< include mds/_subfiles/algebra/_def-quotient.qmd >}} @@ -286,6 +286,6 @@ {{< include mds/_subfiles/algebra/_prose-proportion-reference.qmd >}} -## Limits of sums and notational shorthands {#sec-notational-shorthands} +# Limits of sums and notational shorthands {#sec-notational-shorthands} {{< include mds/_subfiles/algebra/_sec-notational-shorthands.qmd >}} diff --git a/_subfiles/_sec_calculus_derivatives.qmd b/_subfiles/_sec_calculus_derivatives.qmd index 4c767d983..1981ef4d4 100644 --- a/_subfiles/_sec_calculus_derivatives.qmd +++ b/_subfiles/_sec_calculus_derivatives.qmd @@ -1,3 +1,3 @@ -## Derivatives {#sec-derivatives} +# Derivatives {#sec-derivatives} {{< include mds/_calc-derivatives.qmd >}} diff --git a/_subfiles/_sec_calculus_double_integrals.qmd b/_subfiles/_sec_calculus_double_integrals.qmd index 348832f51..2ef58b7a7 100644 --- a/_subfiles/_sec_calculus_double_integrals.qmd +++ b/_subfiles/_sec_calculus_double_integrals.qmd @@ -1,4 +1,4 @@ -## Double Integrals {#sec-double-integrals} +# Double Integrals {#sec-double-integrals} {{< include mds/_subfiles/_fubini_intro_thm.qmd >}} diff --git a/_subfiles/_sec_calculus_integration.qmd b/_subfiles/_sec_calculus_integration.qmd index 1bae991d2..122e1a615 100644 --- a/_subfiles/_sec_calculus_integration.qmd +++ b/_subfiles/_sec_calculus_integration.qmd @@ -1,3 +1,3 @@ -## Integration {#sec-integration} +# Integration {#sec-integration} {{< include mds/_calc-integrals.qmd >}} diff --git a/_subfiles/_sec_critical_points.qmd b/_subfiles/_sec_critical_points.qmd index 13aec1321..02e18e008 100644 --- a/_subfiles/_sec_critical_points.qmd +++ b/_subfiles/_sec_critical_points.qmd @@ -1,4 +1,4 @@ -## Critical points and optimization {#sec-critical-points} +# Critical points and optimization {#sec-critical-points} {{< include mds/_subfiles/derivatives/_def-flat-point.qmd >}} diff --git a/_subfiles/_sec_linear_algebra.qmd b/_subfiles/_sec_linear_algebra.qmd index c396cfb70..1a1b0d1a0 100644 --- a/_subfiles/_sec_linear_algebra.qmd +++ b/_subfiles/_sec_linear_algebra.qmd @@ -18,6 +18,6 @@ {{< include mds/_subfiles/_sec_linear_algebra_diagonalization.qmd >}} -## Complex vectors and matrices {#sec-complex-vectors-matrices} +# Complex vectors and matrices {#sec-complex-vectors-matrices} {{< include mds/_subfiles/_sec_linear_algebra_complex.qmd >}} diff --git a/_subfiles/_sec_linear_algebra_diagonalization.qmd b/_subfiles/_sec_linear_algebra_diagonalization.qmd index f4bf6fc51..0c848b72e 100644 --- a/_subfiles/_sec_linear_algebra_diagonalization.qmd +++ b/_subfiles/_sec_linear_algebra_diagonalization.qmd @@ -1,4 +1,4 @@ -## Similarity and diagonalization {#sec-diagonalization} +# Similarity and diagonalization {#sec-diagonalization} {{< include mds/_subfiles/linear-algebra/_prose-diagonalization-attribution.qmd >}} @@ -46,7 +46,7 @@ {{< include mds/_subfiles/linear-algebra/_thm-diagonalizable.qmd >}} -## Gram matrices, Schur complements and Cholesky {#sec-cholesky} +# Gram matrices, Schur complements and Cholesky {#sec-cholesky} {{< include mds/_subfiles/linear-algebra/_prose-pd-attribution.qmd >}} diff --git a/_subfiles/_sec_linear_algebra_direct_sums.qmd b/_subfiles/_sec_linear_algebra_direct_sums.qmd index 103e04b6b..4960a0986 100644 --- a/_subfiles/_sec_linear_algebra_direct_sums.qmd +++ b/_subfiles/_sec_linear_algebra_direct_sums.qmd @@ -1,4 +1,4 @@ -## Sums and direct sums {#sec-direct-sums} +# Sums and direct sums {#sec-direct-sums} {{< include mds/_subfiles/linear-algebra/_prose-direct-sums-attribution.qmd >}} @@ -30,7 +30,7 @@ {{< include mds/_subfiles/linear-algebra/_def-projection.qmd >}} -## Orthogonal complements {#sec-orthogonal-complements} +# Orthogonal complements {#sec-orthogonal-complements} {{< include mds/_subfiles/linear-algebra/_thm-dot-linear.qmd >}} @@ -58,6 +58,6 @@ {{< include mds/_subfiles/linear-algebra/_thm-double-complement.qmd >}} -## The fundamental theorem of linear algebra {#sec-fundamental-theorem} +# The fundamental theorem of linear algebra {#sec-fundamental-theorem} {{< include mds/_subfiles/linear-algebra/_thm-fundamental-subspaces.qmd >}} diff --git a/_subfiles/_sec_linear_algebra_inner_products.qmd b/_subfiles/_sec_linear_algebra_inner_products.qmd index fd2260cff..8ee8c0af4 100644 --- a/_subfiles/_sec_linear_algebra_inner_products.qmd +++ b/_subfiles/_sec_linear_algebra_inner_products.qmd @@ -1,4 +1,4 @@ -## Inner products {#sec-inner-products} +# Inner products {#sec-inner-products} {{< include mds/_subfiles/linear-algebra/_def-inner-product.qmd >}} @@ -26,7 +26,7 @@ {{< include mds/_subfiles/linear-algebra/_rem-orthogonal-inner-product-dot-product.qmd >}} -## Cauchy-Schwarz and the triangle inequality {#sec-cauchy-schwarz} +# Cauchy-Schwarz and the triangle inequality {#sec-cauchy-schwarz} {{< include mds/_subfiles/linear-algebra/_prose-cauchy-schwarz-attribution.qmd >}} @@ -106,7 +106,7 @@ {{< include mds/_subfiles/linear-algebra/_exm-pearson-angle.qmd >}} -## Orthonormal bases and Gram-Schmidt {#sec-gram-schmidt} +# Orthonormal bases and Gram-Schmidt {#sec-gram-schmidt} {{< include mds/_subfiles/linear-algebra/_prose-gram-schmidt-attribution.qmd >}} @@ -138,7 +138,7 @@ {{< include mds/_subfiles/linear-algebra/_cor-orthonormal-basis-exists.qmd >}} -## Outer product +# Outer product {{< include mds/_subfiles/linear-algebra/_def-outer-product.qmd >}} @@ -146,7 +146,7 @@ {{< include mds/_subfiles/linear-algebra/_thm-outer-product-rank.qmd >}} -## Hilbert spaces {#sec-hilbert-spaces} +# Hilbert spaces {#sec-hilbert-spaces} {{< include mds/_subfiles/linear-algebra/_def-cauchy-sequence.qmd >}} diff --git a/_subfiles/_sec_linear_algebra_matrices.qmd b/_subfiles/_sec_linear_algebra_matrices.qmd index 90727e85a..846f09cf5 100644 --- a/_subfiles/_sec_linear_algebra_matrices.qmd +++ b/_subfiles/_sec_linear_algebra_matrices.qmd @@ -1,4 +1,4 @@ -## Matrices +# Matrices {{< include mds/_subfiles/linear-algebra/_def-matrix.qmd >}} @@ -8,11 +8,11 @@ {{< include mds/_subfiles/linear-algebra/_def-tensor.qmd >}} -### Matrix transpose {#sec-matrix-transpose} +## Matrix transpose {#sec-matrix-transpose} {{< include mds/_subfiles/linear-algebra/_def-matrix-transpose.qmd >}} -### Matrix addition +## Matrix addition {{< include mds/_subfiles/linear-algebra/_def-zero-matrix.qmd >}} @@ -36,11 +36,11 @@ {{< include mds/_subfiles/linear-algebra/_thm-matadd-inverse.qmd >}} -### Scalar multiplication +## Scalar multiplication {{< include mds/_subfiles/linear-algebra/_def-scalar-mult.qmd >}} -### Matrix multiplication +## Matrix multiplication {{< include mds/_subfiles/linear-algebra/_def-matrix-mult.qmd >}} @@ -72,7 +72,7 @@ {{< include mds/_subfiles/linear-algebra/_sol-shape-rule.qmd >}} -### Square and identity matrices +## Square and identity matrices {{< include mds/_subfiles/linear-algebra/_def-square-matrix.qmd >}} @@ -90,7 +90,7 @@ {{< include mds/_subfiles/linear-algebra/_thm-identity.qmd >}} -### Matrix-vector multiplication +## Matrix-vector multiplication {{< include mds/_subfiles/linear-algebra/_def-matvec-mult.qmd >}} @@ -100,7 +100,7 @@ {{< include mds/_subfiles/linear-algebra/_def-linear-system.qmd >}} -### Linear and affine maps +## Linear and affine maps {{< include mds/_subfiles/linear-algebra/_def-linear-map.qmd >}} @@ -202,7 +202,7 @@ {{< include mds/_subfiles/linear-algebra/_sol-which-are-linear.qmd >}} -### Transposes of sums and products +## Transposes of sums and products diff --git a/_subfiles/_sec_linear_algebra_projections.qmd b/_subfiles/_sec_linear_algebra_projections.qmd index d3fb0a09c..5e4d15691 100644 --- a/_subfiles/_sec_linear_algebra_projections.qmd +++ b/_subfiles/_sec_linear_algebra_projections.qmd @@ -1,4 +1,4 @@ -## Design Matrix +# Design Matrix {{< include mds/_subfiles/linear-algebra/_def-design-matrix.qmd >}} @@ -16,7 +16,7 @@ {{< include mds/_subfiles/linear-algebra/_note-thm-hat-matrix.qmd >}} -### Orthogonal projection onto a subspace {#sec-orthogonal-projection} +## Orthogonal projection onto a subspace {#sec-orthogonal-projection} {{< include mds/_subfiles/linear-algebra/_prose-orthogonal-projection-attribution.qmd >}} @@ -40,7 +40,7 @@ {{< include mds/_subfiles/linear-algebra/_thm-hat-matrix-projects.qmd >}} -### Generalized inverses and linear systems {#sec-generalized-inverse} +## Generalized inverses and linear systems {#sec-generalized-inverse} {{< include mds/_subfiles/linear-algebra/_prose-generalized-inverse-attribution.qmd >}} @@ -124,7 +124,7 @@ {{< include mds/_subfiles/linear-algebra/_thm-projector-generalized-inverse.qmd >}} -### Solving linear systems and least squares {#sec-solving-linear-systems} +## Solving linear systems and least squares {#sec-solving-linear-systems} {{< include mds/_subfiles/linear-algebra/_prose-linear-systems-attribution.qmd >}} diff --git a/_subfiles/_sec_linear_algebra_rank_nullity.qmd b/_subfiles/_sec_linear_algebra_rank_nullity.qmd index 7a1582779..377249429 100644 --- a/_subfiles/_sec_linear_algebra_rank_nullity.qmd +++ b/_subfiles/_sec_linear_algebra_rank_nullity.qmd @@ -1,4 +1,4 @@ -## Column space and null space {#sec-column-null-space} +# Column space and null space {#sec-column-null-space} {{< include mds/_subfiles/linear-algebra/_def-column-space.qmd >}} @@ -32,7 +32,7 @@ {{< include mds/_subfiles/linear-algebra/_thm-null-gram.qmd >}} -## Rank-nullity and rank factorization {#sec-rank-nullity} +# Rank-nullity and rank factorization {#sec-rank-nullity} {{< include mds/_subfiles/linear-algebra/_prose-rank-nullity-attribution.qmd >}} diff --git a/_subfiles/_sec_linear_algebra_special_matrices.qmd b/_subfiles/_sec_linear_algebra_special_matrices.qmd index b0f9d88de..7f1f84f0b 100644 --- a/_subfiles/_sec_linear_algebra_special_matrices.qmd +++ b/_subfiles/_sec_linear_algebra_special_matrices.qmd @@ -1,4 +1,4 @@ -## Special Matrices +# Special Matrices ```{r} #| include: false @@ -89,7 +89,7 @@ reticulate::py_require("numpy") {{< include mds/_subfiles/linear-algebra/_thm-orthogonal-norm.qmd >}} -## Quadratic Forms +# Quadratic Forms {{< include mds/_subfiles/linear-algebra/_def-quadratic-form.qmd >}} @@ -107,7 +107,7 @@ reticulate::py_require("numpy") {{< include mds/_subfiles/linear-algebra/_thm-quadratic-form-symmetric.qmd >}} -## Trace and Matrix Inner Product +# Trace and Matrix Inner Product {{< include mds/_subfiles/linear-algebra/_def-trace.qmd >}} @@ -131,7 +131,7 @@ reticulate::py_require("numpy") {{< include mds/_subfiles/linear-algebra/_thm-frobenius-norm-columns.qmd >}} -## Matrix Decompositions +# Matrix Decompositions {{< include mds/_subfiles/linear-algebra/_def-eigenvalue.qmd >}} @@ -163,7 +163,7 @@ reticulate::py_require("numpy") {{< include mds/_subfiles/linear-algebra/_thm-svd-evd.qmd >}} -## Definite Matrices +# Definite Matrices {{< include mds/_subfiles/linear-algebra/_def-positive-semidefinite.qmd >}} @@ -249,7 +249,7 @@ reticulate::py_require("numpy") {{< include mds/_subfiles/linear-algebra/_rem-inner-products-machine-learning.qmd >}} -## Determinants +# Determinants {{< include mds/_subfiles/linear-algebra/_def-determinant.qmd >}} diff --git a/_subfiles/_sec_linear_algebra_subspaces.qmd b/_subfiles/_sec_linear_algebra_subspaces.qmd index ee496c4f8..0a395cbb5 100644 --- a/_subfiles/_sec_linear_algebra_subspaces.qmd +++ b/_subfiles/_sec_linear_algebra_subspaces.qmd @@ -1,4 +1,4 @@ -## Rank +# Rank {{< include mds/_subfiles/linear-algebra/_def-linearly-independent.qmd >}} @@ -6,7 +6,7 @@ {{< include mds/_subfiles/linear-algebra/_def-rank.qmd >}} -## Subspaces {#sec-subspaces} +# Subspaces {#sec-subspaces} {{< include mds/_subfiles/linear-algebra/_prose-subspaces-attribution.qmd >}} diff --git a/_subfiles/_sec_linear_algebra_vectors.qmd b/_subfiles/_sec_linear_algebra_vectors.qmd index f1867ee39..073347469 100644 --- a/_subfiles/_sec_linear_algebra_vectors.qmd +++ b/_subfiles/_sec_linear_algebra_vectors.qmd @@ -1,4 +1,4 @@ -## Vectors +# Vectors ```{r} #| include: false @@ -45,7 +45,7 @@ reticulate::py_require("numpy") {{< include mds/_subfiles/_def_lincom.qmd >}} -### Special vectors +## Special vectors {{< include mds/_subfiles/linear-algebra/_def-zero-vector.qmd >}} @@ -87,7 +87,7 @@ reticulate::py_require("numpy") {{< include mds/_subfiles/linear-algebra/_thm-standard-basis-expansion.qmd >}} -### Orthogonality +## Orthogonality {{< include mds/_subfiles/linear-algebra/_def-orthogonal-vectors.qmd >}} diff --git a/_subfiles/_sec_measures.qmd b/_subfiles/_sec_measures.qmd index d7b53b5ba..7db1c8e6e 100644 --- a/_subfiles/_sec_measures.qmd +++ b/_subfiles/_sec_measures.qmd @@ -1,23 +1,23 @@ {{< include latex-macros/macros.qmd >}} -## $\sigma$-algebras +# $\sigma$-algebras {{< include mds/_subfiles/_sec_measures_sigma_algebras.qmd >}} -## Additivity +# Additivity {{< include mds/_subfiles/_sec_measures_additivity.qmd >}} {{< include mds/_subfiles/_sec_measures_measures.qmd >}} -## Measurable functions and integrals +# Measurable functions and integrals {{< include mds/_subfiles/_sec_measures_integrals.qmd >}} -## $L^p$ spaces +# $L^p$ spaces {{< include mds/_subfiles/_sec_measures_lp_spaces.qmd >}} -## Product measures +# Product measures {{< include mds/_subfiles/_sec_measures_product.qmd >}} diff --git a/_subfiles/_sec_measures_measures.qmd b/_subfiles/_sec_measures_measures.qmd index 8c7881638..850b2d975 100644 --- a/_subfiles/_sec_measures_measures.qmd +++ b/_subfiles/_sec_measures_measures.qmd @@ -1,4 +1,4 @@ -## Measures +# Measures ```{r} #| include: false @@ -82,7 +82,7 @@ reticulate::py_require(py_packages) {{< include mds/_subfiles/measures/_exm-sigma-finite.qmd >}} -## Probability measures +# Probability measures {{< include mds/_subfiles/measures/_def-probability-measure.qmd >}} diff --git a/_subfiles/_sec_optimization.qmd b/_subfiles/_sec_optimization.qmd index e974036d1..5647861f6 100644 --- a/_subfiles/_sec_optimization.qmd +++ b/_subfiles/_sec_optimization.qmd @@ -4,22 +4,22 @@ {{< include mds/_subfiles/_sec_optimization_optimality.qmd >}} -## Convexity in several variables {#sec-convexity-vector} +# Convexity in several variables {#sec-convexity-vector} {{< include mds/_subfiles/_sec_optimization_convexity.qmd >}} -## Minimizing a quadratic under linear constraints {#sec-lagrange} +# Minimizing a quadratic under linear constraints {#sec-lagrange} {{< include mds/_subfiles/_sec_optimization_constrained.qmd >}} -## Newton's method {#sec-newton-method} +# Newton's method {#sec-newton-method} {{< include mds/_subfiles/_sec_optimization_newton.qmd >}} -## Gradient descent {#sec-gradient-descent} +# Gradient descent {#sec-gradient-descent} {{< include mds/_subfiles/_sec_optimization_gradient_descent.qmd >}} -## Quasi-Newton methods {#sec-quasi-newton} +# Quasi-Newton methods {#sec-quasi-newton} {{< include mds/_subfiles/_sec_optimization_quasi_newton.qmd >}} diff --git a/_subfiles/_sec_optimization_optimality.qmd b/_subfiles/_sec_optimization_optimality.qmd index b3ca2621b..e535da34f 100644 --- a/_subfiles/_sec_optimization_optimality.qmd +++ b/_subfiles/_sec_optimization_optimality.qmd @@ -1,4 +1,4 @@ -## Optimality conditions in several variables {#sec-optimality-vector} +# Optimality conditions in several variables {#sec-optimality-vector} {{< include mds/_subfiles/vector-calc/_prose-optim-attribution.qmd >}} diff --git a/_subfiles/_sec_vector_calc.qmd b/_subfiles/_sec_vector_calc.qmd index 3b9cc275c..2bf07be51 100644 --- a/_subfiles/_sec_vector_calc.qmd +++ b/_subfiles/_sec_vector_calc.qmd @@ -239,7 +239,7 @@ {{< include mds/_subfiles/vector-calc/_thm-gradient-of-linear.qmd >}} -## Second partial derivatives and the Hessian {#sec-second-derivatives} +# Second partial derivatives and the Hessian {#sec-second-derivatives} {{< include mds/_subfiles/vector-calc/_def-ball.qmd >}} diff --git a/_subfiles/indicator-functions/_def-indicator-function.qmd b/_subfiles/indicator-functions/_def-indicator-function.qmd new file mode 100644 index 000000000..a6776dc43 --- /dev/null +++ b/_subfiles/indicator-functions/_def-indicator-function.qmd @@ -0,0 +1,18 @@ +:::{#def-indicator-function} +#### Indicator function, set indicator + +For any subset $A \subseteq \universe$ of a universal set $\universe$, +the **indicator function** of $A$, +[@wikipedia_indicator_function] +or **set indicator**, +is the function +$\indic{A} : \universe \to \{0, 1\}$ defined by: + +$$ +\indic{A}(x) \eqdef \begin{cases} +1, & x \in A \\ +0, & x \notin A +\end{cases} +$$ + +::: diff --git a/_subfiles/indicator-functions/_def-iverson-bracket.qmd b/_subfiles/indicator-functions/_def-iverson-bracket.qmd new file mode 100644 index 000000000..925a3cff4 --- /dev/null +++ b/_subfiles/indicator-functions/_def-iverson-bracket.qmd @@ -0,0 +1,15 @@ +:::{#def-iverson-bracket} +#### Iverson bracket + +For any logical proposition $P$, +the **Iverson bracket** of $P$ is +[@wikipedia_iverson_bracket] + +$$ +[P] \eqdef \begin{cases} +1, & \text{if } P \text{ is true} \\ +0, & \text{if } P \text{ is false} +\end{cases} +$$ + +::: diff --git a/_subfiles/indicator-functions/_def-kronecker-delta.qmd b/_subfiles/indicator-functions/_def-kronecker-delta.qmd new file mode 100644 index 000000000..af411fb48 --- /dev/null +++ b/_subfiles/indicator-functions/_def-kronecker-delta.qmd @@ -0,0 +1,15 @@ +:::{#def-kronecker-delta} +#### Kronecker delta + +For integers $i$ and $j$, +the **Kronecker delta** is +[@wikipedia_kronecker_delta] + +$$\kron_{ij} \eqdef [i = j]$$ + +that is, +$\kron_{ij} = 1$ when $i = j$ +and $\kron_{ij} = 0$ when $i \neq j$ +(@def-iverson-bracket). + +::: diff --git a/_subfiles/indicator-functions/_def-predicate-indicator.qmd b/_subfiles/indicator-functions/_def-predicate-indicator.qmd new file mode 100644 index 000000000..d1d1b829d --- /dev/null +++ b/_subfiles/indicator-functions/_def-predicate-indicator.qmd @@ -0,0 +1,17 @@ +:::{#def-predicate-indicator} +#### Predicate indicator + +For any logical proposition or predicate $P$, +the **indicator** of $P$, +or **predicate indicator**, +takes the value $1$ when $P$ is true +and $0$ when $P$ is false: + +$$ +\indicp{P} \eqdef \begin{cases} +1, & \text{if } P \text{ is true} \\ +0, & \text{if } P \text{ is false} +\end{cases} +$$ + +::: diff --git a/_subfiles/indicator-functions/_exm-indicator-numerical.qmd b/_subfiles/indicator-functions/_exm-indicator-numerical.qmd new file mode 100644 index 000000000..3e97dd664 --- /dev/null +++ b/_subfiles/indicator-functions/_exm-indicator-numerical.qmd @@ -0,0 +1,22 @@ +:::{#exm-indicator-numerical} +#### Evaluating set and predicate indicators + +Consider the real line $\universe = \reals$, +the set of nonnegative numbers $A = [0, \infty)$, +and a continuous +[random variable](https://morrison-lab.github.io/pds/random-variables.html#def-random-variable) +$Y$ (defined in the Morrison Lab's probability notes). + +1. Set indicator $\indic{A}(x)$: + - For $x = 3.5$: since $3.5 \in [0, \infty)$, $\indic{A}(3.5) = 1$. + - For $x = -2.1$: since $-2.1 \notin [0, \infty)$, $\indic{A}(-2.1) = 0$. + +2. Predicate indicator $\indicp{Y > 5}$: + - If an observation yields $Y = 7.2$, + the predicate $7.2 > 5$ is true, + so $\indicp{7.2 > 5} = 1$. + - If an observation yields $Y = 4.1$, + the predicate $4.1 > 5$ is false, + so $\indicp{4.1 > 5} = 0$. + +::: diff --git a/_subfiles/indicator-functions/_exm-iverson-bracket.qmd b/_subfiles/indicator-functions/_exm-iverson-bracket.qmd new file mode 100644 index 000000000..ebb20a5f9 --- /dev/null +++ b/_subfiles/indicator-functions/_exm-iverson-bracket.qmd @@ -0,0 +1,8 @@ +:::{#exm-iverson-bracket} +#### Evaluating Iverson brackets + +- $[3 > 2] = 1$, because $3 > 2$ is true. +- $[2 > 3] = 0$, because $2 > 3$ is false. +- $[4 \in \cb{1, 2}] = 0$, because $4$ is not an element of $\cb{1, 2}$. + +::: diff --git a/_subfiles/indicator-functions/_exm-kronecker-delta.qmd b/_subfiles/indicator-functions/_exm-kronecker-delta.qmd new file mode 100644 index 000000000..5a818cbbd --- /dev/null +++ b/_subfiles/indicator-functions/_exm-kronecker-delta.qmd @@ -0,0 +1,32 @@ +:::{#exm-kronecker-delta} +#### Evaluating the Kronecker delta + +- $$ + \begin{aligned} + \kron_{11} &= [1 = 1] \\ + &= 1. + \end{aligned} + $$ + +- $$ + \begin{aligned} + \kron_{12} &= [1 = 2] \\ + &= 0. + \end{aligned} + $$ + +- $$ + \begin{aligned} + \kron_{22} &= [2 = 2] \\ + &= 1. + \end{aligned} + $$ + +- $$ + \begin{aligned} + \kron_{23} &= [2 = 3] \\ + &= 0. + \end{aligned} + $$ + +::: diff --git a/_subfiles/indicator-functions/_rem-iverson-indicator.qmd b/_subfiles/indicator-functions/_rem-iverson-indicator.qmd new file mode 100644 index 000000000..b8b604b3f --- /dev/null +++ b/_subfiles/indicator-functions/_rem-iverson-indicator.qmd @@ -0,0 +1,27 @@ +:::{#rem-iverson-indicator} +#### The Iverson bracket is the predicate indicator + +The Iverson bracket $[P]$ is the predicate indicator $\indicp{P}$ +(@def-indicator-function) in different notation. +Under this notation, +set membership is written $[x \in A]$, +so $[x \in A] = \indic{A}(x)$. +For example, the values in @exm-indicator-numerical become + +$$ +\begin{aligned} +[7.2 > 5] &= \indicp{7.2 > 5} \\ +&= 1 +\end{aligned} +$$ + +and, with $A = [0, \infty)$, + +$$ +\begin{aligned} +[-2.1 \in A] &= \indic{A}(-2.1) \\ +&= 0. +\end{aligned} +$$ + +::: diff --git a/_subfiles/indicator-functions/_sec-indicator-functions.qmd b/_subfiles/indicator-functions/_sec-indicator-functions.qmd index 56915c528..302f637c1 100644 --- a/_subfiles/indicator-functions/_sec-indicator-functions.qmd +++ b/_subfiles/indicator-functions/_sec-indicator-functions.qmd @@ -23,69 +23,15 @@ This section summarizes the principal notational conventions. {{< slidebreak >}} -:::{#def-indicator-function} -#### Indicator function, set indicator - -For any subset $A \subseteq \universe$ of a universal set $\universe$, -the **indicator function** of $A$, -[@wikipedia_indicator_function] -or **set indicator**, -is the function -$\indic{A} : \universe \to \{0, 1\}$ defined by: - -$$ -\indic{A}(x) \eqdef \begin{cases} -1, & x \in A \\ -0, & x \notin A -\end{cases} -$$ - -::: +{{< include mds/_subfiles/indicator-functions/_def-indicator-function.qmd >}} {{< slidebreak >}} -:::{#def-predicate-indicator} -#### Predicate indicator - -For any logical proposition or predicate $P$, -the **indicator** of $P$, -or **predicate indicator**, -takes the value $1$ when $P$ is true -and $0$ when $P$ is false: - -$$ -\indicp{P} \eqdef \begin{cases} -1, & \text{if } P \text{ is true} \\ -0, & \text{if } P \text{ is false} -\end{cases} -$$ - -::: +{{< include mds/_subfiles/indicator-functions/_def-predicate-indicator.qmd >}} {{< slidebreak >}} -:::{#exm-indicator-numerical} -#### Evaluating set and predicate indicators - -Consider the real line $\universe = \reals$, -the set of nonnegative numbers $A = [0, \infty)$, -and a continuous -[random variable](https://morrison-lab.github.io/pds/random-variables.html#def-random-variable) -$Y$ (defined in the Morrison Lab's probability notes). - -1. Set indicator $\indic{A}(x)$: - - For $x = 3.5$: since $3.5 \in [0, \infty)$, $\indic{A}(3.5) = 1$. - - For $x = -2.1$: since $-2.1 \notin [0, \infty)$, $\indic{A}(-2.1) = 0$. - -2. Predicate indicator $\indicp{Y > 5}$: - - If an observation yields $Y = 7.2$, - the predicate $7.2 > 5$ is true, - so $\indicp{7.2 > 5} = 1$. - - If an observation yields $Y = 4.1$, - the predicate $4.1 > 5$ is false, - so $\indicp{4.1 > 5} = 0$. - -::: +{{< include mds/_subfiles/indicator-functions/_exm-indicator-numerical.qmd >}} {{< slidebreak >}} @@ -95,7 +41,7 @@ $Y$ (defined in the Morrison Lab's probability notes). {{< include mds/_subfiles/indicator-functions/_exm-binary-variable.qmd >}} -### Two primary notational paradigms: set vs. predicate notation +## Two primary notational paradigms: set vs. predicate notation The vast majority of indicator notations belong to one of two families: set notation or predicate notation. @@ -163,7 +109,7 @@ or a geometric region). Predicate notation is more natural when the condition involves compound inequalities, such as $\indicp{0 \le t \le u}$. -### Iverson bracket notation +## Iverson bracket notation In 1962, Kenneth Iverson introduced a compact notation in the programming language APL, @@ -173,117 +119,25 @@ the Iverson bracket {{< slidebreak >}} -:::{#def-iverson-bracket} -#### Iverson bracket - -For any logical proposition $P$, -the **Iverson bracket** of $P$ is -[@wikipedia_iverson_bracket] - -$$ -[P] \eqdef \begin{cases} -1, & \text{if } P \text{ is true} \\ -0, & \text{if } P \text{ is false} -\end{cases} -$$ - -::: +{{< include mds/_subfiles/indicator-functions/_def-iverson-bracket.qmd >}} {{< slidebreak >}} -:::{#exm-iverson-bracket} -#### Evaluating Iverson brackets - -- $[3 > 2] = 1$, because $3 > 2$ is true. -- $[2 > 3] = 0$, because $2 > 3$ is false. -- $[4 \in \cb{1, 2}] = 0$, because $4$ is not an element of $\cb{1, 2}$. - -::: +{{< include mds/_subfiles/indicator-functions/_exm-iverson-bracket.qmd >}} {{< slidebreak >}} -:::{#rem-iverson-indicator} -#### The Iverson bracket is the predicate indicator - -The Iverson bracket $[P]$ is the predicate indicator $\indicp{P}$ -(@def-indicator-function) in different notation. -Under this notation, -set membership is written $[x \in A]$, -so $[x \in A] = \indic{A}(x)$. -For example, the values in @exm-indicator-numerical become - -$$ -\begin{aligned} -[7.2 > 5] &= \indicp{7.2 > 5} \\ -&= 1 -\end{aligned} -$$ - -and, with $A = [0, \infty)$, - -$$ -\begin{aligned} -[-2.1 \in A] &= \indic{A}(-2.1) \\ -&= 0. -\end{aligned} -$$ - -::: +{{< include mds/_subfiles/indicator-functions/_rem-iverson-indicator.qmd >}} {{< slidebreak >}} -:::{#def-kronecker-delta} -#### Kronecker delta - -For integers $i$ and $j$, -the **Kronecker delta** is -[@wikipedia_kronecker_delta] - -$$\kron_{ij} \eqdef [i = j]$$ - -that is, -$\kron_{ij} = 1$ when $i = j$ -and $\kron_{ij} = 0$ when $i \neq j$ -(@def-iverson-bracket). - -::: +{{< include mds/_subfiles/indicator-functions/_def-kronecker-delta.qmd >}} {{< slidebreak >}} -:::{#exm-kronecker-delta} -#### Evaluating the Kronecker delta - -- $$ - \begin{aligned} - \kron_{11} &= [1 = 1] \\ - &= 1. - \end{aligned} - $$ - -- $$ - \begin{aligned} - \kron_{12} &= [1 = 2] \\ - &= 0. - \end{aligned} - $$ - -- $$ - \begin{aligned} - \kron_{22} &= [2 = 2] \\ - &= 1. - \end{aligned} - $$ - -- $$ - \begin{aligned} - \kron_{23} &= [2 = 3] \\ - &= 0. - \end{aligned} - $$ - -::: +{{< include mds/_subfiles/indicator-functions/_exm-kronecker-delta.qmd >}} -### Strengths and limitations of the Iverson bracket +## Strengths and limitations of the Iverson bracket The primary advantage of the Iverson bracket is algebraic conciseness: it turns a sum or integral over a subset @@ -309,7 +163,7 @@ To prevent visual confusion with expectation brackets or intervals, statistical literature predominantly uses $\mathbb{1}$ or $I$ rather than the bare Iverson bracket. -### Summary of indicator notations +## Summary of indicator notations @tbl-indicator-notations compares the major notations encountered across the literature. @@ -324,7 +178,7 @@ rather than the bare Iverson bracket. : Notations for indicator functions across mathematical and statistical literature {#tbl-indicator-notations} -### Conventions in this book +## Conventions in this book In these notes, we standardize on blackboard bold $\mathbb{1}$ @@ -340,7 +194,7 @@ it is visually distinct from the scalar $1$, the identity matrix $I$, and the observed and expected Fisher information matrices ($\oinf$, $\einf$). -### Key algebraic properties +## Key algebraic properties Indicator functions translate the logical connectives ([definition](proof-writing.qmd#def-logical-connective)) "and", "or", and "not", applied to set membership, diff --git a/_subfiles/vector-calc/_sec-gradient-shape.qmd b/_subfiles/vector-calc/_sec-gradient-shape.qmd index 20fe63c6c..287298aee 100644 --- a/_subfiles/vector-calc/_sec-gradient-shape.qmd +++ b/_subfiles/vector-calc/_sec-gradient-shape.qmd @@ -1,4 +1,4 @@ -## Checking a gradient by its shape {#sec-gradient-shape} +# Checking a gradient by its shape {#sec-gradient-shape} The gradient always has the same shape as the input it is taken with respect to, because it holds exactly one partial derivative per component of that diff --git a/split-page-headings.lua b/split-page-headings.lua index 459243deb..b2e57830d 100644 --- a/split-page-headings.lua +++ b/split-page-headings.lua @@ -1,16 +1,16 @@ -- For a page that renders one section fragment on its own. --- The fragment opens with a level 2 heading that repeats the page title, +-- The fragment opens with a level 1 heading that repeats the page title, -- because the combined Linear Algebra page includes the same fragment as a section. --- Drop that heading and move every later heading deeper than level 2 up one level, --- so the page body starts at level 2 under the title. --- The fragments that use this filter have no other level 2 heading. +-- Drop that heading and move every later heading deeper than level 1 up one level, +-- so the page body starts at level 1 under the title. +-- The fragments that use this filter have no other level 1 heading. function Pandoc(doc) local title = pandoc.utils.stringify(doc.meta.title or "") local dropped = false local blocks = pandoc.List() for _, block in ipairs(doc.blocks) do if block.t == "Header" then - if not dropped and block.level == 2 + if not dropped and block.level == 1 and pandoc.utils.stringify(block.content):lower() == title:lower() then dropped = true -- Keep the heading's id as an empty anchor so links to it still work. @@ -18,7 +18,7 @@ function Pandoc(doc) blocks:insert(pandoc.Para({pandoc.Span({}, pandoc.Attr(block.identifier))})) end else - if dropped and block.level > 2 then + if dropped and block.level > 1 then block.level = block.level - 1 end blocks:insert(block) @@ -28,7 +28,7 @@ function Pandoc(doc) end end if not dropped then - io.stderr:write("split-page-headings.lua: no level 2 heading matches the title '" .. title .. "'\n") + io.stderr:write("split-page-headings.lua: no level 1 heading matches the title '" .. title .. "'\n") end doc.blocks = blocks return doc From 849e3e94ec4630fde630fd9ca53947899fb0b4a4 Mon Sep 17 00:00:00 2001 From: Claude Date: Mon, 2 Nov 2026 22:30:30 +0000 Subject: [PATCH 5/5] Promote the three indicator-function subsection headings so they number 1.1 to 1.3 The subsections sat one level too deep after the page promotion, so they rendered as 1.0.1 to 1.0.3. Co-Authored-By: Claude Sonnet 5.5 Claude-Session: https://claude.ai/code/session_019xfc9r2idFsJ4pHCsRGyGx --- _subfiles/indicator-functions/_sec-indicator-functions.qmd | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/_subfiles/indicator-functions/_sec-indicator-functions.qmd b/_subfiles/indicator-functions/_sec-indicator-functions.qmd index 302f637c1..1a9125dc5 100644 --- a/_subfiles/indicator-functions/_sec-indicator-functions.qmd +++ b/_subfiles/indicator-functions/_sec-indicator-functions.qmd @@ -54,7 +54,7 @@ set notation or predicate notation. {{< include mds/_subfiles/indicator-functions/_exm-set-predicate-notation.qmd >}} -#### Set notation +### Set notation In set notation, the indicator is tied to a set $A$, @@ -72,7 +72,7 @@ the set of functions $f$ for which $\int \abs{f(x)}^p \, dx$ is finite), authors often omit the argument $x$, writing simply $\mathbf{1}_A$, $\indic{A}$, or $I_A$. -#### Predicate notation +### Predicate notation In predicate notation, the indicator takes a logical condition, @@ -90,7 +90,7 @@ where indicators frequently depend on inequalities ([definition](proof-writing.q such as $\indicp{T_i \le t}$ (an event occurring before time $t$) or $\indicp{Y_i = 1}$ (a binary outcome). -#### Equivalence between paradigms +### Equivalence between paradigms The two paradigms are connected by evaluating the predicate indicator at the membership statement $x \in A$: