PK8SQGcollection.anki2SQLite format 3@ :3 .WH  :OvH 'aindexix_notes_csumnotes CREATE INDEX ix_notes_csum on notes (csum)I 'aindexix_revlog_cidrevlog CREATE INDEX ix_revlog_cid on revlog (cid)U )yindexix_cards_schedcards CREATE INDEX ix_cards_sched on cards (did, queue, due)E %]indexix_cards_nidcards CREATE INDEX ix_cards_nid on cards (nid)I'aindexix_revlog_usnrevlog CREATE INDEX ix_revlog_usn on revlog (usn)E%]indexix_cards_usncardsCREATE INDEX ix_cards_usn on cards (usn)E%]indexix_notes_usnnotesCREATE INDEX ix_notes_usn on notes (usn)!tablegravesgravesCREATE TABLE graves ( usn integer not null, oid integer not null, type integer not null )ktablerevlogrevlogCREATE TABLE revlog ( id integer primary key, cid integer not null, usn integer not null, ease integer not null, ivl integer not null, lastIvl integer not null, factor integer not null, time integer not null, type integer not null )KutablecardscardsCREATE TABLE cards ( id integer primary key, /* 0 */ nid integer not null, /* 1 */ did integer not null, /* 2 */ ord integer not null, /* 3 */ mod integer not null, /* 4 */ usn integer not null, /* 5 */ type integer not null, /* 6 */ queue integer not null, /* 7 */ due integer not null, /* 8 */ ivl integer not null, /* 9 */ factor integer not null, /* 10 */ reps integer not null, /* 11 */ lapses integer not null, /* 12 */ left integer not null, /* 13 */ odue integer not null, /* 14 */ odid integer not null, /* 15 */ flags integer not null, /* 16 */ data text not null /* 17 */ )_tablenotesnotesCREATE TABLE notes ( id integer primary key, /* 0 */ guid text not null, /* 1 */ mid integer not null, /* 2 */ mod integer not null, /* 3 */ usn integer not null, /* 4 */ tags text not null, /* 5 */ flds text not null, /* 6 */ sfld integer not null, /* 7 */ csum integer not null, /* 8 */ flags integer not null, /* 9 */ data text not null /* 10 */ )wtablecolcolCREATE TABLE col ( id integer primary key, crt integer not null, mod integer not null, scm integer not null, ver integer not null, dty integer not null, usn integer not null, ls integer not null, conf text not null, models text not null, decks text not null, dconf text not null, tags text not null ) hlYKT KANKAJ { "activeDecks": [ 1 ], "addToCur": true, "collapseTime": 1200, "curDeck": 1, "curModel": "1425279151691", "dueCounts": true, "estTimes": true, "newBury": true, "newSpread": 0, "nextPos": 1, "sortBackwards": false, "sortType": "noteFld", "timeLim": 0 }{"5013213199628173081": {"css": "\n .card {\n font-famil87~tj`VLB8.$1c-A0#/,o'_)G(%&x#Z!B &x^@$t`L<&|j."j=  42 } 65 }! +*}}%$0Qkz쭡}  K~tj`VLB8.$zpf\RH>4*   v l b X N D : 0 &    | r h ^ T J @ 6 , "    x n d Z P F < 2 (   ~ t j ` V L B 8 . $    z p f \ R H > 4 *  vlbXND:0&|rh^TJ@6,"xndZPF<2( ~tj`VLB8.$zpf\RH>4*  vlbXND:0& }y }w }u }s }q }o }m }k }i }g }e }c }a }_ }] }[ }Y }W }U }S }Q }O }M }K }I }G }E }C }A }? }= }; }9 }7 }5 }3 }1 }/ }- }+ }) }' }% }# }! } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } }~ }| }z }x }v }t }r }p }n }l }j }h }f }d }b }` }^ }\ }Z }X }V }T }R }P }N }L }J }H }F }D }B }@ }> }< }: }8 }6 }4 }2 }0 }. }, }* }( }& }$ }" } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } } }~ }| }z }x }v }t }r }p }n }l }j }h }f }d }b }` }^ }\ }Z }X }V }T }R }P }N }L }J }H }F }D }B }@ }> }< }: }8 }6 }4 }2 }0 }. }, }* }( }& }$ }" } } } } } } } } } } } } } } } } } } } } } } } } } } } } } }ality. One nice way to edit cards is to use something like [HackMD](http://hackmd.io) where you can preview the output of all of the cards before actually "compiling" it into Anki deck: ![](figures/image_2020-05-15-23-23-30.png) - You can add Mathjax macros directly to the card template within Anki, but it's better to modify 'ankdown.py' so that your macros are included whenever a new deck is generated. ## Automating - Install the [ankdown Anki plugin](https://ankiweb.net/shared/info/109255569) (Tools->Addons->Get Add-ons, then enter the add-on code) ![](figures/image_2020-05-15-23-28-22.png) - Restart Anki, navigate back to the Ankdown add-on and click "Config" - Change the file path for "Ankdown Location" to wherever your installed version is, or the version in this repository. - Change the file path for "Markdown Deck Library Path" to the "Decks" path in this repository, or wherever you'd like to store yours. - For example, ``` { "Ankdown Location": "/home/zack/SparkleShare/github.com/Math-Flashcards/ankdown.py", "Markdown Deck Library Path": "/home/zack/SparkleShare/github.com/Math-Flashcards/Decks" } ``` - Now pressing F5 or selecting "Tools->Reload Markdown Decks" will automatically re-scan your directory and add any new cards it finds. Cool! - Note: it's hard to tell if/when this works. My modified ankdown script will attempt to send a push notification to your OS when it updates decks: ![](figures/image_2020-05-15-23-40-41.png) - To test, just modify any of the files under "Decks", add a new entry, press F5 in Anki, then browse the Deck to see if it appears: ![](figures/image_2020-05-15-23-42-51.png) ![](figures/image_2020-05-15-23-43-12.png) - It's also helpful to create an Ankiweb account and enable syncing -- then you can review your cards on the phone app: ![](figures/image_2020-05-15-23-51-35.png)

Instructions

Notes

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Euler Characteristics of Surfaces: \( S^n, \Sigma_g, {\mathbb{RP}}^2, K \).

 formulas

Euler Characteristics of Surfaces: \( S^n, \Sigma_g, {\mathbb{RP}}^2, K \).

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Classification of Closed Surfaces



There is a commutative monoid \( \Sigma = \left\langle{P, K, T, S {~\mathrel{\Big\vert}~}S=0, 3P = PT, PK = PT}\right\rangle \) where \( P = {\mathbb{RP}}^2, K \) is the Klein bottle, and \( T \) is a 2-torus of genus 1. Thus every surface is homeomorphic to either \( \Sigma_{g} \) for \( g\geq 0 \), i.e. \( S^2, \Sigma_1 = T^2, \Sigma_2 = T^2\# T^2, \cdots \) or if it is nonorientable, a surface \( \mkern 1.5mu\overline{\mkern-1.5mu\Sigma\mkern-1.5mu}\mkern 1.5mu_1 = {\mathbb{RP}}^2, \mkern 1.5mu\overline{\mkern-1.5mu\Sigma\mkern-1.5mu}\mkern 1.5mu_2 = {\mathbb{RP}}^2 \# {\mathbb{RP}}^2, \cdots \).

fact

Classification of Closed Surfaces

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Neighborhood Basis



A neighborhood basis about a point \( x \) is a collection of open sets \( \left\{{B_k}\right\}_{k\in J} \) such that for every neighborhood \( U_x \) of \( x \), there exists some \( j \) such that \( B_k \subseteq U_x \).

definition

Neighborhood Basis

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First Countable



A space \( X \) is first countable iff each \( x\in X \) admits a countable neighborhood basis.

definition

First Countable

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Locally homeomorphic



\( X \) is locally homeomorphic to \( Y \) iff every \( x\in X \) admits a neighborhood \( U_x \) that is homeomorphic to an open subset of \( Y \).

Note that \( X \) being locally homeomorphic to \( Y \) does not imply that there exists a local homeomorphism, which needs to be a single map.

Counterexample: \( S^2 \) locally homeomorphic to \( {\mathbb{R}}^2 \), but there is no single local homeomorphism \( f:S^2 \to {\mathbb{R}} \).

definition

Locally homeomorphic

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Local homeomorphism



A map \( f:X\to Y \) is a local homeomorphism iff for every \( x\in X \) there exists a neighborhood \( U_x \) such that \( f(U_x) \) is open in \( Y \) and \( f\mathrel{\Big|}_{U_x}: U_x \to f(U_x) \) is a homeomorphism.

Examples: etale spaces, covering spaces.

definition

Local homeomorphism

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Semilocally Simply Connected



Every point admits a neighborhood \( U \) such that \( \pi_1(U) = 0 \).

definition

Semilocally Simply Connected

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Hausdorff Space



A space \( X \) is Hausdorff iff for every \( x,y \in X \) there exist neighborhoods of \( x \) and \( y \) that are disjoint from each other.

(Implies uniqueness of limits.)

definition

Hausdorff Space

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Urysohn's Lemma



A space \( X \) is normal iff for every closed \( U, V \subset X \) there exists a continuous function \( f: X\to [0, 1] \) with \( f(U) = 0, f(V) = 1 \).

Equivalently \( \chi_U \leq f \leq \chi_V \) .

Equivalently, a topological space is separable and metrizable \( \iff \) it is regular, Hausdorff, and second-countable.

theorem

Urysohn's Lemma

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Egorov's Theorem



If \( E\subset {\mathbb{R}}^n \) is measurable, \( m(E) > 0 \), and \( \left\{{f_n}\right\} \) measurable with \( f_k \to f \) with \( f(x) < \infty \) existing and finite a.e., then \( f_n\to f \) almost uniformly, i.e. for all \( \varepsilon > 0 \) there exists a closed \( F\subset E \) such that \( m(E\setminus F)<\varepsilon \) and \( f\overset{u}\to f \) on \( F \).

theorem

Egorov's Theorem

Th 3%? 4048547499295785179E "ap

Fatou's Lemma



If \( \left\{{f_n}\right\} \subset L^+ \), then \[ \int \liminf f_n \leq \liminf \int f_n \] - Pulling the limit out makes it bigger - Pushing the limit in means integrating a smaller function, making the whole thing smaller.

theorem

Fatou's Lemma

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Instructions

Notes



Answer Field, with inline math \( cx+d \) and/or displaymath \[\sum_{i=1}^N c_n\]

tag1, tag2 ``` - See the files under 'Decks' for examples of how to typeset things. Generally basic inline math like `$ax+b$` and display style equations like `$$\int_0^1 f(x)$$` tend to work fine. - You can use markdown in your cards, including lists. I haven't tested images yet. See [here](https://github.com/adam-p/markdown-here/wiki/Markdown-Cheatsheet) for general markdown syntax. - Everything is rendered with Mathjax, which has only a small subset of Latex function  C| 1}G 157811323428998498E "ap

Limsup/Liminf of Sets



\[\begin{align*} \limsup_n A_n \coloneqq\cap_n \cup_{j\geq n} A_j&= \left\{{x {~\mathrel{\Big\vert}~}x\in A_n \text{ for inf. many $n$}}\right\} \\ \liminf_n A_n \coloneqq\cup_n \cap_{j\geq n} A_j &= \left\{{x {~\mathrel{\Big\vert}~}x\in A_n \text{ for all except fin. many $n$}}\right\} \\ \end{align*}\]

definition, important

Limsup/Liminf of Sets

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Proof of Borel-Cantelli Lemma



Proof of Borel Cantelli:

- If \( \sum_j m(E_j) < \infty \), then \( \sum_{j=N}^\infty m(E_j) \overset{N\to\infty}\to 0 \) as the tail of a convergent sequence.

\\[E = \limsup_j E_j = \cap_{k=1}^\infty \cup_{j=k}^\infty E_j \implies E \subseteq \cup_{j=k}^\infty E_k \quad \forall k \\]
proof

Proof of Borel-Cantelli Lemma

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Uniform Continuity



For \( f: (X, d_1) \to (Y, d_2) \), for every \( \varepsilon > 0 \) there exists \( \delta(\varepsilon) > 0 \) such that for every \( x,y\in X \), \[x\in B_\delta(y) \implies f(x) \in B_\varepsilon(f(y)).\]

Slogan: continuity, but \( \delta \) can be chosen independent of \( x \).

definition

Uniform Continuity

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Bernoulli's Inequality



\[ (1+x)^n \geq 1 + nx \] for all \( n\in {\mathbb{R}} \) and \( x\geq -1 \). If \( n\in 2{\mathbb{Z}} \), then this is valid for all \( x\in {\mathbb{R}} \). Prove by induction.

formula

Bernoulli's Inequality

(Lebesgue) Measurable Function



A function \( f:X\to\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{R}}\mkern-1.5mu}\mkern 1.5mu \) is measurable iff for all \( \alpha \in {\mathbb{R}} \), the following set is Lebesgue measurable: \[ S_\alpha \coloneqq\left\{{ x\in X {~\mathrel{\Big\vert}~}f(x) > \alpha}\right\} .\]

definition

(Lebesgue) Measurable Function

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Monotone Convergence Theorem



If \( f_n \in L^+ \) and \( f_n\nearrow f \) a.e. then \[ \lim \int f_n = \int \lim f_n \]

theorem

Monotone Convergence Theorem

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Dominated Convergence Theorem



If \( \left\{{f_n}\right\} \subset L^1 \) and \( f_n\to f \) a.e. with \( {\left\lvert {f_n} \right\rvert} \leq g \in L^1 \) for every \( n \), then \[ \lim \int f_n = \int \lim f_n \]

theorem

Dominated Convergence Theorem

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Borel set



The smallest \( \sigma{\hbox{-}} \)algebra generated by the topology, i.e. every element is obtained by countable unions/intersections/complements of open sets.

definition

Borel set

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Riesz Representation Theorem



For \( 1\leq p <\infty \), \( X \subset {\mathbb{R}}^n \) measurable, \( \Lambda \in L^p(X) {}^{ \vee } \), there exists a unique \( g\in L^q(X) \) such that \[\begin{align*} \forall f\in L^p(X), \quad \Lambda(f) &= \int_X fg \\ {\left\lVert {\Lambda} \right\rVert}_{L^p(X) {}^{ \vee }} &= {\left\lVert {g} \right\rVert}_{L^q(X)} \end{align*}\]

theorem

Riesz Representation Theorem

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Definition: Uniform convergence of a sequence of functions



\( \left\{{f_n}\right\} \overset{u}\to f \) on \( E \) iff for every \( \varepsilon>0 \) that exists an \( N(\varepsilon) \) such that for all \( n\geq N \) and for all \( x\in E \), \( {\left\lvert {f_n(x) - f(x)} \right\rvert} < \varepsilon. \)

Equivalently, \[{\left\lVert {f_n - f} \right\rVert}_\infty \coloneqq\sup_{x\in E}{\left\lvert {f_n(x) - f(x)} \right\rvert} < \varepsilon.\]

definition

Definition: Uniform convergence of a sequence of functions

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Showing uniform convergence of a series of functions



\( M{\hbox{-}} \)test: find \( M_n \) independent of \( x \) such that \( {\left\lVert {f_n} \right\rVert}_\infty < M_n \) where \( \sum M_n < \infty \).

technique

Showing uniform convergence of a series of functions

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Showing uniform convergence of a sequence of functions



Find \( M_n \) independent of \( x \) such that \[ {\left\lvert {f_n(x) - f(x)} \right\rvert} \leq M_n \to 0 \]

technique

Showing uniform convergence of a sequence of functions

2 3s- 8725320664461229852E "ap

Null Set



A set \( A \) is null iff for every \( \varepsilon>0 \) there exists a cover \( \left\{{U_j}\right\}\rightrightarrows A \) such that \( \sum \mu(U_j) < \varepsilon \), i.e. \( \mu(A) = 0 \).

definition

Null Set

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\( F_\sigma \) sets



\( X \) is \( F_\sigma \) iff \( X \) is a countable union of closed sets.

definition

\( F_\sigma \) sets

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\( G_\delta \) sets



\( X \) is \( G_\delta \) iff \( X \) is a countable intersection of open sets.

definition

\( G_\delta \) sets

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Riemann-Lebesgue Lemma



If \( f\in L^1 \), then the Fourier transform satisfies \[\widehat{f}(\xi) \overset{{\left\lvert {\xi} \right\rvert}\to \infty}\to 0.\]

theorem

Riemann-Lebesgue Lemma

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Parseval's Identity



Equality in Bessel's inequality, obtained when \( \left\{{e_k}\right\} \) is a basis:

\[ \sum {\left\lvert {{\left\langle {x},~{e_k} \right\rangle}} \right\rvert}^2 = {\left\lVert {x} \right\rVert}^2 \]

theorem

Parseval's Identity

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Bessel's Inequality



For \( x\in H \) a Hilbert spaces and \( \left\{{e_k}\right\} \) an orthonormal sequence, \[ \sum {\left\lvert {{\left\langle {x},~{e_k} \right\rangle}} \right\rvert}^2 \leq {\left\lVert {x} \right\rVert}^2 \]

Proof idea: \[ \left\|x-\sum_{n=1}^{N}\left\langle x, u_{n}\right\rangle u_{n}\right\|^{2}=\cdots .\]

formula

Bessel's Inequality

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Convolution



\[ (f\ast g)(\xi) = \int f(\xi - y)g(y) \, dy \]

formula

Convolution

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Holder's Inequality



\[{\left\lVert {fg} \right\rVert}_1 \leq {\left\lVert {f} \right\rVert}_p {\left\lVert {f} \right\rVert}_q\]

formula

Holder's Inequality

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Cauchy-Schwarz Inequality



\[ {\left\lVert {fg} \right\rVert}_1 \leq {\left\lVert {f} \right\rVert}_2 {\left\lVert {g} \right\rVert}_2 \\ \int{\left\lvert {fg} \right\rvert} \leq \sqrt{\int {\left\lvert {f} \right\rvert}^2} \sqrt{\int {\left\lvert {g} \right\rvert}^2} \] with equality iff \( f \in {\operatorname{span}}_{\mathbb{C}}(g) \).

formula

Cauchy-Schwarz Inequality

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Definition: Measurable Function



\( f:{\mathbb{R}}\to \mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{R}}\mkern-1.5mu}\mkern 1.5mu \) is Lebesgue/Borel measurable iff \[ \left.\{x \in E | f(x)>a\}=f^{-1}((a, \infty])\right) \in \mathcal{M}_L, \mathcal{M}_B ,\] the collection of Lebesgue/Borel measurable sets respectively.

Mnemonic: preimage of a ray should be a measurable set.

definition

Definition: Measurable Function

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Definition of Outer Measure



\[ m_*(E) = \inf \left\{{ \sum {\left\lvert {Q_i} \right\rvert} {~\mathrel{\Big\vert}~}\left\{{Q_i}\right\}\rightrightarrows E \text { closed cubes}}\right\} \]

definition

Definition of Outer Measure

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Example of a function that is Lebesgue integrable but not Riemann integrable



The Dirichlet function \( f(x) = \chi_{\mathbb{Q}} \), since \( D_f = {\mathbb{R}} \) is not null.

counterexample

Example of a function that is Lebesgue integrable but not Riemann integrable

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Definition: Measurability of a Set



A set \( E\subseteq {\mathbb{R}}^n \) is measurable iff for every \( \varepsilon>0 \) there exists an open \( G(\varepsilon) \supset E \) with \( m_*(G(\varepsilon)\setminus E)<\varepsilon \to 0 \) (outer regular).

definition

Definition: Measurability of a Set

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Equivalent characterizations of measurability of a set



\( E\subset {\mathbb{R}}^n \) is measurable iff any of these conditions hold

theorem

Equivalent characterizations of measurability of a set

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Continuity of measure from above/below

 theorem

Continuity of measure from above/below

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Relationship between continuity and differentiability



Differentiability \( \implies \) continuity: \[ f(x) - f(x_0) = (x-x_0) \qty{ f(x)-f(x_0) \over x-x_0 } = (x-x_0) f'(x_0) \to 0 .\] Not conversely: \( f(x) = {\left\lvert {x} \right\rvert} \).

proof

Relationship between continuity and differentiability

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Negating uniform convergence of a sequence of functions



Fix \( \varepsilon \), find \( x(\varepsilon, n) \) with \( {\left\lvert {f_n(x) - f(x)} \right\rvert} > \varepsilon \).

Example: \( {1 \over 1 + nx} \), take \( x={1\over n} \).

technique

Negating uniform convergence of a sequence of functions

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Example of a function that converges pointwise but not uniformly



\( f_n(x) \coloneqq x^n \).

counterexample

Example of a function that converges pointwise but not uniformly

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A sequence of functions that converges in \( L^1 \) but not uniformly, pointwise, or a.e.



counterexamples

A sequence of functions that converges in \( L^1 \) but not uniformly, pointwise, or a.e.

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A sequence of functions that converges a.e. but not in \( L^1 \), uniformly, or pointwise



\[ f_n \coloneqq n \chi_{(0, {1\over n} )} \]

counterexamples

A sequence of functions that converges a.e. but not in \( L^1 \), uniformly, or pointwise

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A sequence of functions that converges pointwise and a.e. but not uniformly or in \( L^1 \).



\[ f_n \coloneqq\chi_{(n, n+1)} \]

counterexamples

A sequence of functions that converges pointwise and a.e. but not uniformly or in \( L^1 \).

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Relative strengths of convergence

 fact

Relative strengths of convergence

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A sequence of functions that converges uniformly, pointwise, a.e., but not in \( L^1 \).



\[ f_n \coloneqq{1\over n} \chi_{[0, n]} \]

counterexamples

A sequence of functions that converges uniformly, pointwise, a.e., but not in \( L^1 \).

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Proposition: \( \sum {\left\lvert {f_n} \right\rvert} \in L^1 \implies \sum {\left\lvert {f_n(x)} \right\rvert} < \infty \) a.e.



Proof: by contradiction.

proof

Proposition: \( \sum {\left\lvert {f_n} \right\rvert} \in L^1 \implies \sum {\left\lvert {f_n(x)} \right\rvert} < \infty \) a.e.

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How to commute a sum and an integral



\( f_n \in L^1 \) with \( \sum \int {\left\lvert {f_n} \right\rvert} < \infty \).

Proof: Use Fubini-Tonelli

technique

How to commute a sum and an integral

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Proposition: \( \int f = 0 \implies f = 0 \) a.e.

 proof

Proposition: \( \int f = 0 \implies f = 0 \) a.e.

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Chebyshev's Inequality



\[ \mu\qty{\{x \in{\mathbb{R}}^n {~\mathrel{\Big\vert}~}{\left\lvert {f(x)} \right\rvert} \geq \alpha\}} \leq \qty{{\left\lVert {f} \right\rVert}_p \over \alpha }^p \quad \forall \alpha, p .\] Take \( p=1 \) to obtain \[ \mu\qty{\{x \in {\mathbb{R}}^n{~\mathrel{\Big\vert}~}{\left\lvert {f(x)} \right\rvert} \geq \alpha\}} \leq {1\over \alpha } \int {\left\lvert {f(x)} \right\rvert} \, dx \quad \forall \alpha .\]

definition

Chebyshev's Inequality

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Definition: The Lebesgue Integral



\[ \int f:=\sup \left\{\int \phi {~\mathrel{\Big\vert}~}0 \leq \phi \leq f, \,\, \phi \text { simple }\right\} .\]

definition

Definition: The Lebesgue Integral

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Lusin's Theorem



If \( f \) is measurable and finite-valued on a measurable \( E \) with \( m(E) < \infty \) then there exist closed sets \( F\subset E \) such that \( m(E\setminus F) < \varepsilon \to 0 \) such that \( f\mathrel{\Big|}_F \) is continuous.

theorem

Lusin's Theorem

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Give an example of a sequence of functions that converge uniformly but not in \( L^1 \).



The box of height \( 1\over n \) and width \( n \) uniformly converges to zero but integrates to 1.

example

Give an example of a sequence of functions that converge uniformly but not in \( L^1 \).

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Fubini-Tonelli



If \( f \)

then Tonelli on \( {\left\lvert {f} \right\rvert} \) yields \( f\in L^1 \) and \( \int f \) is equal to any iterated integral.

theorem

Fubini-Tonelli

H 3') 8900928383770193784E "ap

Fubini



Let \( f(x, y) \in L^1({\mathbb{R}}^n\times{\mathbb{R}}^k) \). Then for almost every \( y\in {\mathbb{R}}^k \),

  1. The slice function \( f^y(x)\coloneqq f(x ,y) \) is integrable, so \( f^y \in L^1({\mathbb{R}}^n) \).
  2. The function \( F(y) \coloneqq\int_{{\mathbb{R}}^n} f^y(x) \, dx \) is integrable, so \( F\in L^1({\mathbb{R}}^{k}) \).
  3. \[\int_{{\mathbb{R}}^{n+k}} f(\mathbf{u}) \, d\mathbf{u} = \int_{{\mathbb{R}}^n} \qty{ \int_{{\mathbb{R}}^k} f^y(x) \, dx} \, dy\] in any order.

Note: requires integrability, not just measurability, but doesn't require non-negativity.

theorem

Fubini

F 3M+ 8346868657301191464E "ap

Tonelli



Let \( f(x, y) \) be non-negative and measurable on \( {\mathbb{R}}^{n}\times{\mathbb{R}}^k \). Then for almost every \( y\in {\mathbb{R}}^k \),

  1. The slice function \( f^y(x) \coloneqq f(x, y) \) is measurable on \( {\mathbb{R}}^n \).
  2. The function \( F(y) \coloneqq\int_{{\mathbb{R}}^n} f^y(x) \, dx \) is measurable on \( {\mathbb{R}}^k \).
  3. \[\int_{{\mathbb{R}}^{n+k}} f(\mathbf{u}) \, d\mathbf{u} = \int_{{\mathbb{R}}^n} \qty{\int_{{\mathbb{R}}^k} f^y(x) \,dx}\, dy\] in any order (where the integral may be infinite.)

Note: requires non-negativity and measurability, but not integrability.

theorem

Tonelli

D3-' 5010249969121176565E "ap

Proposition: Translation/Dilation Invariance of the Lebesgue Integral

 proof

Proposition: Translation/Dilation Invariance of the Lebesgue Integral

xB 3Ge 1449352195152170180E "ap

Proposition: Continuity in \( L^1 \)



Proof: approximate with compactly supported functions, use uniform continuity to bound integrand.

proof

Proposition: Continuity in \( L^1 \)

-@ 3KK 8234728747725879220E "ap

Continuity in \( L^1 \)



\[ f \in L^{1} \Longrightarrow \lim _{h \rightarrow 0} \int|f(x+h)-f(x)|=0 .\]

Proof idea: \( \varepsilon/3 \) argument, approximate by compactly supported functions.

theorem

Continuity in \( L^1 \)

X> / c 38450551636234312E "ap

Small Tails and Absolute Continuity



Let \( f\in L^1 \) and \( \varepsilon> 0 \).

  1. Small Tails: there exists an \( N \) such that \( \int_{B_N^c} f < \varepsilon \).
  2. Absolute Continuity: there exists a \( \delta \) such that \( m(E) < \delta \implies \int_E {\left\lvert {f} \right\rvert} < \varepsilon \).
theorem, important

Small Tails and Absolute Continuity

O q 4  /COz`3?o 4841088744411290145E "ap

Example of a sequence of differentiable functions whose pointwise limit exists but is not differentiable.



\( f_n(x) = {x^2 \over \sqrt{x^2 + {1\over n}}} \to {\left\lvert {x} \right\rvert} \).

example

Example of a sequence of differentiable functions whose pointwise limit exists but is not differentiable.

j^35Y 5584012627851755214E "ap

Example of a sequence of differentiable functions whose derivatives do not converge pointwise.



\[ f_n(x) = {\sin(nx) \over n} \to 0 \quadtext{pointwise but}\quad f_n'(\pi) = (-1)^n \]

example

Example of a sequence of differentiable functions whose derivatives do not converge pointwise.

q\3c9 4621999422285244947E "ap

Example of a sequence of bounded functions whose pointwise limit is unbounded.



\[f_n(x) = {1 \over x + {1\over n}} \to {1\over x}\] on \( (0, 1) \), noting that \( {\left\lvert {f_n(x)} \right\rvert} \leq n \).

example

Example of a sequence of bounded functions whose pointwise limit is unbounded.

kZ3} 7755031548515760502E "ap

Replacing a sequence of sets by a sequence of disjoint sets



Given \( \left\{{E_j}\right\}_{j\in {\mathbb{N}}} \), set \( F_j\coloneqq E_j \setminus \qty{\cup_{k<j} E_k} \), then \( \cup E_j = {\textstyle\coprod}F_j \).

technique

Replacing a sequence of sets by a sequence of disjoint sets

KX 1w] 274594878782383914E "ap

Growth Rates of Common Functions



For \( c>1 \), \[ x! > e^x > x^{\varepsilon} > x\log(x) > x > \log(x) > \log(\log(nx) > \cdots > 1 .\]

facts

Growth Rates of Common Functions

xV3eE 4453845641379163923E "ap

Where is \( {1\over x^p} \) integrable in \( {\mathbb{R}} \)? (Depending on \( p \))

 facts

Where is \( {1\over x^p} \) integrable in \( {\mathbb{R}} \)? (Depending on \( p \))

*T 33] 5695036264227230812E "ap

Four Properties of Outer Measure


  1. Monotonicity
  2. Countable Subadditivity
  3. Approximation from above by opens
  4. Almost disjoint additivity
definition, theorem

Four Properties of Outer Measure

5R3o5 8871596254545177689E "ap

Give an example of a function that converges in \( L^1 \) but not pointwise.



The Cathode Ray:

example

Give an example of a function that converges in \( L^1 \) but not pointwise.

5P3E_ 1220191913355083227E "ap

Give an example of a function that converges almost everywhere but not pointwise or in \( L^1 \).



\[f_n = n\chi_{(0, {1\over n})}\]

example

Give an example of a function that converges almost everywhere but not pointwise or in \( L^1 \).

JN3gg 6015692305959842012E "ap

Give an example of a sequence of functions that converge pointwise but not uniformly or in \( L^1 \).



Skateboard to infinity: \( \chi_{[n, n+1]} \).

example

Give an example of a sequence of functions that converge pointwise but not uniformly or in \( L^1 \).

M c -/V[t 3=5 7650971493870834350E "ap

Normal Space



A space \( X \) is normal iff every two disjoint closed sets have disjoint open neighborhoods.

definition

Normal Space

8r 3SY 5905506512809024917E "ap

Are singletons open or closed?

 theorem

Are singletons open or closed?

np 3]; 6830042021781145791E "ap

Locally Compact



A space \( X \) is locally compact iff for every \( x\in X \) there exists an open \( U \) and compact \( K \) such that \( x\in U \subseteq K \).

Compact implies locally compact but not conversely: \( {\mathbb{R}}^n \).

Non locally-compact spaces:

definition, counterexample

Locally Compact

[n 3G+ 3736250298386639216E "ap

Retract



Retract: A subspace \( A \subset X \) is a retract of \( X \) iff there exists a continuous map \( f: X\to A \) such that \( f\mathrel{\Big|}_{A} = \operatorname{id}_A \).

Equivalently it is a left inverse to the inclusion.

definition

Retract

gl 3+_ 6926911180140753758E "ap

Weierstrass Approximation Theorem



If \( f: I\to {\mathbb{R}} \) is continuous, then for every \( \varepsilon \) there exists a polynomial \( p_\varepsilon(x) \) such that \( {\left\lVert {f - p_\varepsilon} \right\rVert}_\infty < \varepsilon \).

Slogan: polynomials are dense in \( C([0, 1], {\left\lVert {{-}} \right\rVert}_\infty) \).

theorem

Weierstrass Approximation Theorem

j 3ee 2563902786780464424E "ap

\( p{\hbox{-}} \)test for integrals.



\[ \int_0^1 {1\over x^p} < \infty \iff p < 1 \\ \int_1^\infty {1\over x^p} < \infty \iff p > 1 .\]

fact

\( p{\hbox{-}} \)test for integrals.

th 3UO 7397921027813186534E "ap

Definition: Infinity Norm



The least upper bound that holds almost everywhere: \[ {\left\lVert {f} \right\rVert}_\infty \coloneqq\inf_{\alpha \geq 0} \left\{{\alpha {~\mathrel{\Big\vert}~}m\left\{{{\left\lvert {f} \right\rvert} \geq \alpha}\right\} = 0}\right\} .\]

definition

Definition: Infinity Norm

2f 3?a 6332897986786222027E "ap

Inclusions among \( L^p \) spaces.



\[\begin{align*} m(X) = \infty \implies &\text{No inclusions (use $p{\hbox{-}}$test)} \\ m(X) < \infty \implies &L^\infty(X) \subset L^2(X) \subset L^1(X) \\ &\ell^1({\mathbb{Z}}) \subset \ell^2({\mathbb{Z}}) \subset \ell^\infty({\mathbb{Z}}) \end{align*}\]

theorem, important

Inclusions among \( L^p \) spaces.

bd 3#] 5847322203474055164E "ap

Conditions for Tonelli vs Fubini

 theorem

Conditions for Tonelli vs Fubini

+b3'i 2942430679189620267E "ap

Example of a sequence of differentiable functions \( f_n \to f \) uniformly with \( f_n' \to g \) pointwise for some \( g \), but \( g' \neq \lim f_n' \).

 example

Example of a sequence of differentiable functions \( f_n \to f \) uniformly with \( f_n' \to g \) pointwise for some \( g \), but \( g' \neq \lim f_n' \).

   C a a^ 3w 4601283431782459685E "ap

Proposition: Every free module is projective.



proposition

Proposition: Every free module is projective.

r\3  7199202289004500835E "ap

Classification of Finitely Generated Modules over a PID



theorem

Classification of Finitely Generated Modules over a PID

Z 3-7 8375586960035856716E "ap

Cyclic Module



For \( M \) an \( R{\hbox{-}} \)module, \( M \) is cyclic iff \( M = \left\langle{m}\right\rangle \) iff \( M \cong R/I \) for some ideal \( I{~\trianglelefteq~}R \).

definition

Cyclic Module

%X 33S 6819838853539152696E "ap

Definition: Diagonalizable.



For a matrix \( A \), similar to a diagonal matrix. I.e. there exists a diagonal \( D \) and some \( P\in \mathrm{GL}(n, R) \) such that \( A = PDP^{-1} \)

definition

Definition: Diagonalizable.

DV 3]g 4658820885861084478E "ap

Definition: Characteristic polynomial



\[ p_A(x) = \mathrm{det}(xI - A) .\]

definition

Definition: Characteristic polynomial

T 3mc 6970366073872014590E "ap

Definition: Rational Canonical Form



For \( \phi:V\to V \), corresponds to invariant factor decomposition of \( V \) as a \( k[x]{\hbox{-}} \)module, \[ V \cong \bigoplus k[x]/ (r_i) \qquad r_1 \divides r_2 \divides \cdots .\]

The \( r_i \) are the minimal polynomials of \( \phi \) restricted to \( V_i \). The matrix of \( \phi \) consists of blocks of companion matrices for the \( \phi_i \).

definition

Definition: Rational Canonical Form

:R 1kG 155901638386672605E "ap

Indecomposable Module



Can not be written as a direct sum of two nonzero submodules.

definition

Indecomposable Module

-P 3UA 3309081577580531244E "ap

Irreducible Module



Simple module, i.e. no nontrivial proper submodules.

definition

Irreducible Module

N 31K 6970641775854004150E "ap

Annihilator of a module



For \( M \) an \( R{\hbox{-}} \)module, \[ \mathrm{ann}_R(M) = \left\{{r\in R{~\mathrel{\Big\vert}~}\forall m\in M,\, rm=0}\right\} {~\trianglelefteq~}R .\]

definition

Annihilator of a module

2L 3UK 2358118077866120710E "ap

Definition: torsionfree



\( \operatorname{Tor}(M) = \left\{{0}\right\} \)

definition

Definition: torsionfree

JJ 3 G 1098485189366003561E "ap

Rank of a Free module



Maximal number of \( R{\hbox{-}} \)linearly independent elements of \( M \).

definition

Rank of a Free module

wH 3QY 7653595049708109083E "ap

Definition: Torsion submodule.



\[ \operatorname{Tor}(M) = \{m \in M {~\mathrel{\Big\vert}~}\exists r \in R, ~r \neq 0, ~rm = 0\} .\]

definition

Definition: Torsion submodule.

=F 3cS 6936900409649460348E "ap

Definition: Torsion element



\( m\in M \) is torsion iff \( \operatorname{Tor}(m)\neq 0 \).



Definition: Torsion element

LD 3? 2884708560964295752E "ap

Noetherian Module



Any strictly increasing chain of submodules \( M_1 \subsetneq M_2 \cdots \) is finite.

definition

Noetherian Module

KB3M 1609683440308005854E "ap

\( \left\langle{p}\right\rangle{\hbox{-}} \)primary



For every \( m\in M \) and \( t\in R \) with \( t\not\in \left\langle{p}\right\rangle \), there exists a \( a\in R \) such that \( atm = m \).

definition

\( \left\langle{p}\right\rangle{\hbox{-}} \)primary

@Q e P @V$ 3K 1952982043296903325E "ap

\( H_* \mathbb{RP}^4 \)



\[[\mathbb{Z}, \mathbb{Z}_2, 0, \mathbb{Z}_2, 0, 0\rightarrow ]\]

math

\( H_* \mathbb{RP}^4 \)

T" 3K 6492366409414849284E "ap

\( H_* \mathbb{RP}^3 \)



\[[\mathbb{Z}, \mathbb{Z}_2, 0, \mathbb{Z}, 0, 0\rightarrow ]\]

math

\( H_* \mathbb{RP}^3 \)

K 3K 2448239540010337335E "ap

\( H_* \mathbb{RP}^2 \)



\[[\mathbb{Z}, \mathbb{Z}_2, 0, 0, 0, 0\rightarrow ]\]

math

\( H_* \mathbb{RP}^2 \)

5 37o 4617627639012000461E "ap

Mayer Vietoris LES for \( X = A \cup B \)



\[\ldots H_n(A \cap B) \xrightarrow{(i^*,~ j^*)} H_n(A) \oplus H_n(B) \xrightarrow{l^* - r^*} H_n(X) \xrightarrow{\delta} H_{n-1}(A\cap B)\ldots\]

math

Mayer Vietoris LES for \( X = A \cup B \)

> 3eS 2035271409075692789E "ap

Definition: Totally Bounded



A metric space \( (M, d) \) is totally bounded iff \( \forall \varepsilon \) there exists a finite collection of open balls of radius \( \varepsilon \) whose union contains \( M \).



Definition: Totally Bounded

B 3]c 5924940400204816014E "ap

Definition: Limit Point Compactness



Every infinite subset has a limit point.

definition

Definition: Limit Point Compactness

@ 3_] 3355975767650188554E "ap

Definition: Sequentially Compact



Every sequence has a convergent subsequence.

definition

Definition: Sequentially Compact

v3  7042721393227514216E "ap

Give an example of a function \( f: {\mathbb{R}}^n \to {\mathbb{R}} \) that is continuous in each variable but not continuous.



Take limit along \( y=x \) and compare to \( y=0 \): \[ f(x, y) = {\begin{cases} xy \over x^2 +y^2} & (x, y) \neq \mathbf{0} \\ 0 & \text{else} \end{cases} .\]

example

Give an example of a function \( f: {\mathbb{R}}^n \to {\mathbb{R}} \) that is continuous in each variable but not continuous.

~ 3Uc 1141762295905425477E "ap

Definition: the indiscrete topology



For \( X \) a space, the indiscrete topology is given by \( \tau = \left\{{\emptyset, X}\right\} \).

definition

Definition: the indiscrete topology

 3g_ 8198320370406399955E "ap

Definition: the discrete topology



For \( X \) a space, the discrete topology is given by \( \tau = \mathcal{P}(X) \), i.e. every subset is open.

definition

Definition: the discrete topology

 33G 4603769381531068198E "ap

Definition: separable



Contains a countable dense subset

definition

Definition: separable

C 3? 2116725649434339231E "ap

Definition: Dense



A subset \( A\subset X \) is dense in \( X \) iff \( \mathrm{cl}_X(A) = X \).

definition

Definition: Dense

 1K 950384383196012692E "ap

Definition: Limit Point



A point \( x\in X \) is a limit point of \( A\subseteq X \) iff every open \( U \ni x \) contains a point \( y\in A\setminus\left\{{x}\right\} \).

definition

Definition: Limit Point

I 3sY 7056577476928484549E "ap

Show that a continuous bijection from a compact space to a Hausdorff space is a homeomorphism.

 theorem, proof

Show that a continuous bijection from a compact space to a Hausdorff space is a homeomorphism.

' 37S 5737433069014965928E "ap

Definition: Closed Surfaces



Compact and without boundary.

definition

Definition: Closed Surfaces

VyLk> ] 0  | O " n A  ` 3   R %qDc6 U(tGf9 X+wJi<& }0Qkz쭡ap& }0Qkz쭡ap&  }0Qkz쭡ap&  }0Qkz쭡ap&  }0Qkz쭡ap& }0Qkz쭡ap& }0Qkz쭡ap& }0Qkz쭡ap& }0Qkz쭡ap& }~0Qkz쭡ap&} }|0Qkz쭡ap&{ }z0Qkz쭡ap&y }x0Qkz쭡ap&w }v0Qkz쭡ap&u }t0Qkz쭡ap&s }r0Qkz쭡ap&q }p0Qkz쭡ap&o }n0Qkz쭡ap&m }lOW핳ap&k }jOW핳ap&i }hOW핳ap&g }fOW핳ap&e }dOW핳ap&c }bOW핳ap&a }`OW핳ap&_ }^OW핳ap&] }\OW핳ap&[ }ZOW핳ap&Y }XOW핳ap&W }VOW핳ap&U }TOW핳ap&S }ROW핳ap&Q }POW핳ap&O }NOW핳ap&M }LOW핳ap&K }JOW핳ap&I }HOW핳ap&G }FOW핳ap&E }DOW핳ap&C }BOW핳ap&A }@OW핳ap&? }>OW핳ap&= } ] 0  | O " n A  ` 3   R %qDc6 U(tGf9 X+wJi<&= }<0oap&; }:0oap&9 }80oap&7 }60oap&5 }40oap&3 }20oap&1 }00oap&/ }.0oap&- },0oap&+ }*0oap&) }(0oap&' }&0oap&% }$0oap&# }"0oap&! } 0oap& }0oap& }0oap& }0oap& }0oap& }0oap& }0oap& }0oap& }0oap& }0oap&  } 0oap&  } 0oap&  }0oap& }0oap& }0oap& }0oap& }0oap& }0oap&} }0oap&{ }0oap&y }0oap&w }0oap&u }0oap&s }0oap&q }0oap&o }0oap&m }0oap&k }0oap&i }0oap&g }0oap&e }0oap&c }0oap&a }0oap&_ }0oap&] }0oap&[ }0oap&Y }0oap&W }0oap&U }0oap&S }0oap&Q }0oap&O }0oap&M }0oap&K }0oap&I }0oap&G }0oap&E }0oap&C }0oap&A }0oap&? }0oap&= }0oap&; }0oap&9 }0oap&7 }0Qkz쭡ap&5 }0Qkz쭡ap&3 }0Qkz쭡ap&1 }0Qkz쭡ap&/ }0Qkz쭡ap&- }0Qkz쭡ap&+ }0Qkz쭡ap&) }0Qkz쭡ap&' }0Qkz쭡ap&% }0Qkz쭡ap&# }0Qkz쭡ap&! }0Qkz쭡ap& }0Qkz쭡ap& }0Qkz쭡ap& }0Qkz쭡ap& }0Qkz쭡ap& }0Qkz쭡ap& }0Qkz쭡ap& }0Qkz쭡ap 53Y V )A59@ 3u9 6483971336166793434E "ap

Submodule test



If \( r\in R,\, m,n \in M \implies rm + n \in M \) then \( M \) is a submodule.

fact

Submodule test

C> 3kW 3751249011246158558E "ap

Definition: Projective module



Equivalently, for \( R{\hbox{-}} \)modules, \( M \) is projective \( \iff \) \( M \) is a direct summand of a free module.

definition

Definition: Projective module

4< 3YK 6235520896834672797E "ap

Definition: Free module



definition

Definition: Free module

F:3' 3486317333890462616E "ap

Characterizations of Diagonalizability of a Square Matrix \( M \)

 theorem

Characterizations of Diagonalizability of a Square Matrix \( M \)

z8 3Mc 7191087232111922468E "ap

Does \( A^n=B^n \) imply \( A=B \)?



No, counterexample: \[ M^2 = \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix}^2 = I .\]

counterexample

Does \( A^n=B^n \) imply \( A=B \)?

T6 3!C 2701792849761746232E "ap

Deformation Retract



Deformation Retract: A subspace \( A \subset X \) is a deformation retract of \( X \) iff there exists a deformation retraction: a continuous map \( F:X\times I \) to \( X \) such that

\[ F(x, 0) &= x \iff F_0 = \operatorname{id}_X \\ F(x, 1) &\in A \iff F_1(X) \subseteq A \\ F(a, 1) &= a \iff F_1\mathrel{\Big|}_A = \operatorname{id}_A .\]

Equivalently it is a homotopy between a retraction \( X\to A \) and \( \operatorname{id}_X \).

definition

Deformation Retract

=4 3?w 6907994241612656927E "ap

Euler Characteristic: Formula Involving Genus



\( \chi(X) = 2-2g \)

math

Euler Characteristic: Formula Involving Genus

-2 3KK 6828981369847374294E "ap

Euler Characteristic -2



\( \chi X = -2 \implies X \cong \mathbb{RP}^2 \)

math

Euler Characteristic -2

+0 3II 8849550787379606642E "ap

Euler Characteristic 0



\( \chi X = 0 \implies X \cong T^2 \) or \( K \)

math

Euler Characteristic 0

. 33I 3641773345194433221E "ap

Euler Characteristic 2



\( \chi X = 2 \implies X \cong S^2 \)

math

Euler Characteristic 2

0, 3SI 2047897590909668917E "ap

Nonorientable Surfaces



\( \mathbb{RP}^\text{even}, \mathbb{M}, \mathbb{K} \)

math

Nonorientable Surfaces

* 3/C 1059898922582070795E "ap

Orientable Surfaces



\( S^n, T^n, \mathbb{RP}^\text{odd} \)

math

Orientable Surfaces

R( 3K 3952923392717134320E "ap

\( H_* \mathbb{CP}^2 \)



\[[\mathbb{Z}, 0, \mathbb{Z}, 0, \mathbb{Z}, 0\rightarrow ]\]

math

\( H_* \mathbb{CP}^2 \)

E& 33 4233879057858581496E "ap

\( H_* K \)



\[[\mathbb{Z}, \mathbb{Z} \oplus \mathbb{Z}_2, 0, 0, 0, 0\rightarrow ]\]

math

\( H_* K \)

 L  MG% 3q 7869295301108696171E "ap

Double angle formulas involving \( \tan \)



The double angle formulas: \[ \sin(2t) = {2\tan(t) \over 1+\tan^2(t)} && \cos(2t) = {1-\tan^2(2t) \over 1+\tan^2(t)} .\]

formula

Double angle formulas involving \( \tan \)

q 3YE 6805882619462079059E "ap

Maximum Length Lemma



\[ {\left\lvert {\int _\gamma f} \right\rvert} \leq \sup_{z\in \gamma} {\left\lvert {f(z)} \right\rvert} \cdot \ell(\gamma) .\]



Maximum Length Lemma

M 3au 3108478433118215402E "ap

Cauchy Integral Formula (Higher Derivatives)



For \( f \) holomorphic in \( U\supseteq \mkern 1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu \) and \( C \) is a circle such that \( C^\circ \subset U \) then for any \( z\in C^\circ \), \[ f^{(n)}(z)=\frac{n !}{2 \pi i} \int_{C} \frac{f(\zeta)}{(\zeta-z)^{n+1}} d \zeta .\]

formula, theorem

Cauchy Integral Formula (Higher Derivatives)

 3Uq 2986497425147306674E "ap

Cauchy Integral Formula (First Derivative)



For \( f \) holomorphic in \( U\supseteq \mkern 1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu \), then for any \( z\in D \), \[ f(z) = {1 \over 2\pi i} \int _{{{\partial}}D} {f(\xi) \over \xi - z} \,d\xi .\]

formula, theorem

Cauchy Integral Formula (First Derivative)

~ 1wC 243893621006107381E "ap

Cauchy Inequalities



\[ \left|f^{(n)}\left(z_{0}\right) \over n! \right| \leq \frac{ \sup_{z\in\gamma} {\left\lvert {f(z)} \right\rvert} {R^{n}} .\]

formula, theorem

Cauchy Inequalities

 3[  3361609432812762828E "ap 

If \( f, g \) are holomorphic on \( \mkern 1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu(z_0) \) and \[ {\left\lvert {f - g} \right\rvert} < {\left\lvert {f} \right\rvert} + {\left\lvert {g} \right\rvert} \quad\text{on}\quad {{\partial}}D ,\] then \( f,g\neq 0 \) on \( {{\partial}}D \) and have the same number of zeros.

complex, theorem t 3-w 8818678336160127929E "ap

Cauchy's Integral Formula for Derivatives



\[ \left|f^{(n)}(0)\right| \leq \frac{n !}{r^{n}} \sup _{|z|=r}|f(z)| .\]

formula

Cauchy's Integral Formula for Derivatives

1q  552533663126759066E "ap

Series expansion for \( \csch(z) = {1\over \sinh(z)} \)



\[ \operatorname{csch} x=x^{-1}-\frac{x}{6}+\frac{7 x^{3}}{360}-\frac{31 x^{5}}{15120}+\cdots .\]

formula

Series expansion for \( \csch(z) = {1\over \sinh(z)} \)

53# 2860863595650105815E "ap

Series expansion for \( { \mathrm{sech} }(z) = {1\over \cosh(z)} \)



\[ \operatorname{sech} x=1-\frac{x^{2}}{2}+\frac{5 x^{4}}{24}-\frac{61 x^{6}}{720}+\cdots .\]

formula

Series expansion for \( { \mathrm{sech} }(z) = {1\over \cosh(z)} \)

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Angle addition formulas: \[ \cosh(x+iy) &= \cdot \sinh(x+iy) &= \cdots .\]



\[ \cosh (x+i y) &=\cosh (x) \cos (y)+i \sinh (x) \sin (y) \\ \sinh (x+i y) &=\sinh (x) \cos (y)+i \cosh (x) \sin (y) .\]

formulas

Angle addition formulas: \[ \cosh(x+iy) &= \cdot \sinh(x+iy) &= \cdots .\]

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Series expansion for \( \sinh(z) \)



\[ \sinh x=x+\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}+\frac{x^{7}}{7 !}+\cdots=\sum_{n=0}^{\infty} \frac{x^{2 n+1}}{(2 n+1) !} .\]

formula

Series expansion for \( \sinh(z) \)

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Series expansion for \( \cosh(z) \)



\[ \cosh x=1+\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}+\frac{x^{6}}{6 !}+\cdots=\sum_{n=0}^{\infty} \frac{x^{2 n}}{(2 n) !} .\]

formula

Series expansion for \( \cosh(z) \)

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Four Characterizations of Galois extensions \( K/F \).


  1. \( K \) is a splitting field of a separable polynomial in \( F[x] \).
  2. \( K \) such that the fixed field of \( \mathop{\mathrm{Aut}}(K/F) \) is exactly \( K \).
  3. \( K \) such that \( [K:F] = {\left\lvert {\mathop{\mathrm{Aut}}(K/F)} \right\rvert} \).
  4. \( K \) such that \( K/F \) is finite, normal, and separable.
theorem

Four Characterizations of Galois extensions \( K/F \).

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Conjugates of a root



For \( \alpha\in L \), a conjugate of \( \alpha \) is any other root of the minimal polynomial of \( \alpha \).

definition

Conjugates of a root

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\( K(\alpha) \) for \( \alpha \in F\supset K \)



The minimal subfield of \( F \) generated by \( K \) and \( \alpha \)

definition

\( K(\alpha) \) for \( \alpha \in F\supset K \)

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Definition: \( K[\alpha] \) for \( \alpha \in F \supset K \)



The minimal subring of \( F \) generated by \( K \) and \( \alpha \).

definition

Definition: \( K[\alpha] \) for \( \alpha \in F \supset K \)

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Cyclotomic Polynomial: \( \Phi_{2p}(x) \).



\[\Phi_{2p}(x) = 1 - x + x^2 - \cdots + x^{p-1}.\]

formula

Cyclotomic Polynomial: \( \Phi_{2p}(x) \).

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Cyclotomic Polynomial: \( \Phi_p(x) \).



\[\Phi_p(x) = 1 + x + x^2 + \cdots + x^{p-1}.\]

formula

Cyclotomic Polynomial: \( \Phi_p(x) \).

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Irreducible



An element \( p \) in a ring \( R \) is irreducible \( \iff \) \( p=ab \implies a \in R^{\times} \) or \( b\in R^{\times} \).

definition

Irreducible

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\( K[x] \)



The algebra of polynomials in the variable \( x \), i.e.  \[K[x] = \left\{{\sum_{i=0}^n a_ix^i {~\mathrel{\Big\vert}~}n\in {\mathbb{Z}}^{\geq 0}}\right\}.\]

notation, definition

\( K[x] \)

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\( K(x) \)



The field of rational functions, i.e. ratios of polynomials, in the variable \( x \); \[K(x) = \left\{{{P(x) \over Q(x)} {~\mathrel{\Big\vert}~}P,Q \in K[x]}\right\}.\]

Example: \( {\mathbb{C}}(x) \) is the field of meromorphic functions on the Riemann sphere.

notation, definition

\( K(x) \)

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Separable Extension



A field extension \( L/K \) is separable \( \iff \) for every \( \alpha \in K \), the \( \min_\alpha(x)/K \) is separable (equivalently, has nonzero derivative).

definition

Separable Extension

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Normal Extension



A field extension \( L/K \) is normal \( \iff \) every irreducible \( p(x) \in K[x] \) either has no roots in \( L \) or splits into linear factors in \( L[x] \).

definition

Normal Extension

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Algebraic Extension



A field extension \( L/K \) is algebraic \( \iff \) every \( \alpha \in L \) is the root of some polynomial \( f(x)\in K[x] \).

definition

Algebraic Extension

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Eisenstein's Criterion



If \( f(x) = \displaystyle\sum_{i=0}^n \alpha_i x^i \in {\mathbb{Q}}[x] \) and \( \exists p \) such that

then \( f \) is irreducible.

theorem

Eisenstein's Criterion

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Zorn's Lemma



If every chain in a poset \( P \) has an upper bound in \( P \), then \( P \) contains a maximal element.

Note: a chain is a totally ordered subset.

theorem

Zorn's Lemma

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Characteristic of a ring



The smallest number \( n \) such that \( \sum_{j=1}^n 1_R = 0_R \).

Equivalently, the kernel \( n{\mathbb{Z}} \) of the unique map \( {\mathbb{Z}}\to R \).

definition

Characteristic of a ring

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Definition: perfect field.



Every irreducible polynomial is separable.

Equivalently, the characteristic is zero, or is \( p \) where Frobenius is an automorphism.

All characteristic zero or finite fields are perfect.

definition

Definition: perfect field.

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Definition: algebraically closed.



Every polynomial has a root.

definition

Definition: algebraically closed.

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Definition: splitting field of a polynomial \( f\in k[x] \)



The smallest extension \( F/k \) such that \( f \) splits into linear factors in \( F[x] \).

definition

Definition: splitting field of a polynomial \( f\in k[x] \)

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Definition: irreducible polynomial



Nonconstant and can not be factored into two polynomials of smaller degrees.

definition

Definition: irreducible polynomial

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Definition: Algebraic Element



If \( L/K \) is a field extension then \( \alpha \in L \) is algebraic iff there exists a \( g\in K[x] \) such that \( g(\alpha) = 0 \).

definition

Definition: Algebraic Element

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Fundamental Theorem of Galois Theory



Let \( K/F \) and \( G = \mathrm{Gal}(K/F) \), then there is a bijection \[ E {~\mathrel{\Big\vert}~}K/E/F \qquad&\iff\qquad H{~\mathrel{\Big\vert}~}1\leq H \leq G \\ E &\mapsto \tau\in G,\, \tau\mathrel{\Big|}_E = \mathrm{id} \\ \text{Fixed field of } H &\mapsto H ,\] such that

  1. Inclusion reversing: \( E_1 \subseteq E_2 \iff H_2 \leq H_ 1 \).
  2. \( {\left\lvert {H} \right\rvert} = [K: E] \) and \( [E:F] = [G: H] \)
  3. \( K/E \) is always Galois with \( G(K/E) = H \).
  4. \( E/F \) is Galois \( \iff \) \( H{~\trianglelefteq~}G \), in which case \( G(E/F) \cong G/H \).

  5. Compositum correspondences:

theorem

Fundamental Theorem of Galois Theory

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Definition: Primitive Root of Unity.



\( \zeta \) is a primitive \( n \)th root of unity iff \( \zeta^n = 1 \) and \( n \) is the smallest such integer for which this holds.

definition

Definition: Primitive Root of Unity.

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Example of a field extension that is not Galois.



\( K = {\mathbb{Q}}(2^{1\over 3})/{\mathbb{Q}} \), since \( {\left\lvert {\mathop{\mathrm{Aut}}(K/{\mathbb{Q}})} \right\rvert} = 1 \) but this is a degree 3 extension.

example

Example of a field extension that is not Galois.

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Definition of Galois group for a finite extension \( K/F \).



\( K/F \) is Galois \( \iff \) \( {\left\lvert {\mathop{\mathrm{Aut}}(K/F)} \right\rvert} = [K: F] \)

definition

Definition of Galois group for a finite extension \( K/F \).

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Sylow Theorems



Write \( {\left\lvert {G} \right\rvert} = p^n m \) with \( (p, m) = 1 \).

Sylow 1

Writing \( {\left\lvert {G} \right\rvert} = \prod p_i^{n_i} \), there exist Sylow \( p_i{\hbox{-}} \)subgroups \( P_i \) of order \( p_i^{n_i} \) for every \( i \).

Sylow 2

All Sylows are conjugate.

Sylow 3

  1. \( n_p \) divides \( m \)
  2. \( n_p \equiv 1 \operatorname{mod}p \)
  3. \( n_p = [G: N_G(P)] \)
theorem

Sylow Theorems

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Simple group



No nontrivial normal subgroups

definition

Simple group

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Definition: Unique Factorization Domain



Every nonzero element \( x\in R \) can be written as \( x = u\prod p_i \) where the \( p_i \) are irreducible and \( u \in R^\times \).

This decomposition is unique up to multiplication by a unit, and a permutation of the factors \( \sigma \) that makes each \( p_i \) associate to \( p_{\sigma(i)} \)

definition

Definition: Unique Factorization Domain

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Definition: Euclidean domain.



Admits a "remainder measuring" function \( f: R\setminus\left\{{0}\right\} \to {\mathbb{N}} \) such that \( a\in R, b\in R^\bullet \) implies there exist \( q, r\in R \) such that \( a = bq + r \) with either \( r=0 \) or \( f(r) < f(b) \).

definition

Definition: Euclidean domain.

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Definition: Maximal ideal.



An ideal \( I{~\trianglelefteq~}R \) is maximal iff \( I\subseteq J {~\trianglelefteq~}R \) implies \( J=R \).

definition

Definition: Maximal ideal.

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Definition: prime ideal.



An ideal \( \mathfrak{p}{~\trianglelefteq~}R \) is prime iff it is proper and \( ab\in \mathfrak{p} \implies a\in \mathfrak{p} \) or \( b\in \mathfrak{p} \).

definition

Definition: prime ideal.

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Gauss' Lemma



Let \( R \) be a UFD with fraction field \( F \) and \( p \in R[x] \). If \( p \) is reducible in \( F[x] \), then \( p \) is reducible in \( R[x] \).

Implies that \( R \) a UFD implies \( R[\left\{{x_i}\right\}] \) is again a UFD.

theorem

Gauss' Lemma

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Definition: integral domain.



A nonzero commutative ring with no nonzero zero divisors.

definition

Definition: integral domain.

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Definition: prime.



A nonzero nonunit ring element \( r \) is prime iff \( r\divides ab \) implies \( r\divides a \) or \( r\divides b \).

definition

Definition: prime.

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Definition: irreducible.



A nonzero nonunit ring element \( r\in R \) is irreducible iff \( r=ab \) implies \( a \) or \( b \) is a unit.

definition

Definition: irreducible.

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Definition: simple ring.



A ring \( R \) is simple iff its only two-sided ideals are \( (0) \) and \( R \).

definition

Definition: simple ring.

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Zero Divisor



An element \( r\in R \) is a zero divisor if there exists an \( x\in X \) such that \( rx=0 \).

Equivalently, the map \( x\mapsto rx \) is not injective.

definition

Zero Divisor

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Lagrange's Theorem



Any \( H\leq G \) has size dividing \( {\left\lvert {G} \right\rvert} \).

theorem

Lagrange's Theorem

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\( p{\hbox{-}} \)group



A group of order \( p^n \) for some \( n\geq 1 \).

definition

\( p{\hbox{-}} \)group

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Class Equation



\[ \#{G} = \#{Z(G)} + \sum_{i} [G: Z(x)] \] where \( Z(x) \coloneqq\left\{{ g\in G {~\mathrel{\Big\vert}~}[gx] = e }\right\} \) is the centralizer of \( x \), sometimes denoted \( C_G(x) \).

definition, notation

Class Equation

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Notation and definition of fixed points of a group action.



\( X^g = \left\{{x\in X {~\mathrel{\Big\vert}~}g.x = x}\right\} \subseteq X \)

definition, notation

Notation and definition of fixed points of a group action.

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\( G_x \)



The Stabilizer subgroup: \( G_x = \left\{{g\in G {~\mathrel{\Big\vert}~}g.x = x}\right\} \leq G \).

definition, notation

\( G_x \)

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Stabilizer



A subgroup: \( G_x = \left\{{g\in G {~\mathrel{\Big\vert}~}g.x = x}\right\} \).

definition, notation

Stabilizer

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Burnside's Formula



The number of orbits is equal to the average number of fixed points: \[{\left\lvert {G} \right\rvert} {\left\lvert {X/G} \right\rvert} = \sum {\left\lvert {X^g} \right\rvert}\] where

formula

Burnside's Formula

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What is the sign of the cycle \( \sigma=(123456)(789)(10~11)(12~13~14~15)(16~17~18) \)?



\( \varepsilon(\sigma) = -1 \), since it has 3 (an odd number) cycles of odd cycles (even length).

problem

What is the sign of the cycle \( \sigma=(123456)(789)(10~11)(12~13~14~15)(16~17~18) \)?

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How to determine sign of a permutation



A cycle is odd iff it has an odd number of odd cycles (i.e. even number of elements in cycle).i

definition

How to determine sign of a permutation

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Even vs Odd Permutations

 definition

Even vs Odd Permutations

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Alternating Group



The kernel of the sign homomorphism, i.e. the set of even permutations (sign = 1).

definition

Alternating Group

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Composition Series



A composition series of \( G \) is a sequence of subgroups \[ 1=N_{0} \leq N_{1} \leq N_{2} \leq \cdots \leq N_{k-1} \leq N_{k}=G \] such that \( N_i {~\trianglelefteq~}N_{i+1} \) and the composition factors \( N_{i+1}/N_i \) are simple.

definition

Composition Series

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Solvable Group



A group is solvable iff it has a composition series with abelian composition factors.

definition

Solvable Group

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Recognizing Direct Products



When there exist two subgroups \( H, K \leq G \) such that

  1. \( H\cap K = \emptyset \)
  2. \( G = HK \)
  3. \( H \) and \( K \) are both normal.
theorem

Recognizing Direct Products

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Groups of Order 16



5 Abelian, 9 Nonabelian

  1. \( {\mathbb{Z}}/16{\mathbb{Z}} \)
  2. \( ({\mathbb{Z}}/4{\mathbb{Z}})^2 \)
  3. \( {\mathbb{Z}}/8{\mathbb{Z}}\times {\mathbb{Z}}/2{\mathbb{Z}} \)
  4. \( {\mathbb{Z}}/4{\mathbb{Z}}\times ({\mathbb{Z}}/2{\mathbb{Z}})^2 \)
  5. \( ({\mathbb{Z}}/2{\mathbb{Z}})^4 \)
  6. \( ({\mathbb{Z}}/4{\mathbb{Z}}\times{\mathbb{Z}}/2{\mathbb{Z}}) \rtimes{\mathbb{Z}}/2{\mathbb{Z}} \)
  7. \( {\mathbb{Z}}/4{\mathbb{Z}}\rtimes{\mathbb{Z}}/4{\mathbb{Z}} \)
  8. \( {\mathbb{Z}}/8{\mathbb{Z}}\rtimes{\mathbb{Z}}/2{\mathbb{Z}} \)
  9. \( D_8 \)
  10. \( D_4\times{\mathbb{Z}}/2{\mathbb{Z}} \)
  11. \( Q_8 \times{\mathbb{Z}}/2{\mathbb{Z}} \)
  12. \( Q_{16} = \left\langle{a,b,c {~\mathrel{\Big\vert}~}a^4=b^2=c^2=abc}\right\rangle \)
  13. \( QD_{16} = \left\langle{r,s{~\mathrel{\Big\vert}~}r^8, s^2, srs^{-1}s^{-3}}\right\rangle \)
  14. \( ({\mathbb{Z}}/4{\mathbb{Z}}\times{\mathbb{Z}}/2{\mathbb{Z}})\rtimes{\mathbb{Z}}/2{\mathbb{Z}} \) (Pauli matrices)
fact

Groups of Order 16

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Groups of Order 15



\( {\mathbb{Z}}/15{\mathbb{Z}} \)

fact

Groups of Order 15

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Groups of Order 14



1 Abelian, 1 Nonabelian

  1. \( {\mathbb{Z}}/14{\mathbb{Z}} \)
  2. \( D_7 \)
fact

Groups of Order 14

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Groups of Order 12



2 Abelian, 3 Nonabelian

  1. \( {\mathbb{Z}}/4{\mathbb{Z}}\times {\mathbb{Z}}/3{\mathbb{Z}} \)
  2. \( ({\mathbb{Z}}/2{\mathbb{Z}})^2 \times {\mathbb{Z}}/3{\mathbb{Z}} \)
  3. \( A_4 \)
  4. \( D_6 \)
  5. \( \left\langle{a,b,c{~\mathrel{\Big\vert}~}a^2, b^2, c^2, abc}\right\rangle \)
fact

Groups of Order 12

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Groups of Order 10



1 Abelian, 1 Nonabelian

  1. \( {\mathbb{Z}}/10{\mathbb{Z}} \)
  2. \( D_{5} \)
fact

Groups of Order 10

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Groups of Order 9


  1. \( {\mathbb{Z}}/9{\mathbb{Z}} \)
  2. \( ({\mathbb{Z}}/3{\mathbb{Z}})^2 \)
fact

Groups of Order 9

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Groups of Order 8



3 Abelian, 2 Nonabelian

  1. \( {\mathbb{Z}}/8{\mathbb{Z}} \)
  2. \( {\mathbb{Z}}/4{\mathbb{Z}}\times {\mathbb{Z}}/2{\mathbb{Z}} \)
  3. \( ({\mathbb{Z}}/2{\mathbb{Z}})^3 \)
  4. \( D_4 \)
  5. \( Q_8 \)
fact

Groups of Order 8

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Groups of Order 6



1 Abelian, 1 Nonabelian

  1. \( {\mathbb{Z}}/6{\mathbb{Z}} \)
  2. \( D_3 \)
fact

Groups of Order 6

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Definition: Transitive Group Action



A group action \( G\curvearrowright X \) is transitive iff for every pair \( x, y\in X \) there exists a \( g\in G \) such that \( g.x = y \).

definition

Definition: Transitive Group Action

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\( \phi(p^k) = ? \)



\( \phi(p^k) = p^{k-1}(p-1) = p^k \qty{1 - {1\over p}} \)

formula

\( \phi(p^k) = ? \)

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Euler's Theorem



\( a^{\phi(n)} \cong 1 \operatorname{mod}n \).

theorem

Euler's Theorem

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Fermat's Little Theorem



\( a^p \cong a \operatorname{mod}p \), and if \( p \) does not divide \( a \), \( a^{p-1} \cong 1 \operatorname{mod}p \).

theorem

Fermat's Little Theorem

 dlXD0t`L8$p|hT@,xdP<(t`L8$p\H4  lXD0|hT@, p \ H 4 \H4  l X D 0   | h T @ ,   x t ` L 8 $  P < (  0o}W0o}U0o}S0o}Q0o}O0o}M0o}K0o}I0o}G0o}E0o}C0o}A0o}?0o}=0o};0o}90o}70o}50o}30o}10o}/0o}-0o}+0o})0o}'0o}%0o}#0o}!0o}0o}0o}0o}0o}0o}0o}0o}0o}0o} 0o} 0o} 0o}0o}0o}0o}0o}0o}0o}0o}0o}0o}0o}0o}0o}0o}0o}0o}0o}0o}0o}0o}0o}0o}0o}0o}0o}0o}0o}0o}0o}0o}0o}0o}0o}0o}0o}0o}0o}0o}0o}0o}}}}}}}}}}{}y}w}u}s}q}o0o}m0o}k0o}i0o}g0o}e0o}c0o}a0o}_0o}]0o}[0o}Y0Qkz쭡}0Qkz쭡}0Qkz쭡}0Qkz쭡}0Qkz쭡}}0Qkz쭡}{0Qkz쭡}y0Qkz쭡}w0Qkz쭡}u0Qkz쭡}s0Qkz쭡}q0Qkz쭡}o8Za[}z+Uu-_}p+Uu-_}n+Uu-_}l+Uu-_}j+Uu-_}h+Uu-_}f+Uu-_}d+Uu-_}b+Uu-_}`+Uu-_}^+Uu-_}\+Uu-_}Z+Uu-_}X+Uu-_}V+Uu-_}T+Uu-_}Rb}:b}8b}6b}4b}2b}0b}.b},b}*b}(b}&b}$b}"b} b}b}b}b}b}b}b}b}}}}}}}}}}}}}}}}}}}}}  t`L8$p\H4 |hT@,xdP<(t`L8H4  $p\ l X D 0   | h T @ ,   x d P < (  t ` L 8 $  p \ H 4 lXD07Hwa}x7Hwa}v7Hwa}t7Hwa}rFa܂}PFa܂}NFa܂}LFa܂}JFa܂}HFa܂}FFa܂}DFa܂}BFa܂}@Fa܂}>Fa܂}<1}1} 1} 1}1}1}1}1}1}1}1}1}1}1}1}1}1}1}1}1}1}1}1}1}1}1}1}1}1}1}WD}WD}WD}WD}WD}WD}WD}WD}WD}WD}WD}WD}WD}WD}WD}WD}OW핳}mOW핳}kOW핳}iOW핳}gOW핳}eOW핳}cOW핳}aOW핳}_OW핳}]OW핳}[OW핳}YOW핳}WOW핳}UOW핳}SOW핳}QOW핳}OOW핳}MOW핳}KOW핳}IOW핳}GOW핳}EOW핳}COW핳}AOW핳}?OW핳}=OW핳};OW핳}9OW핳}7OW핳}5OW핳}3OW핳}1OW핳}/OW핳}-OW핳}+OW핳})OW핳}'OW핳}%OW핳}#OW핳}!OW핳}OW핳}OW핳}OW핳}OW핳}OW핳}OW핳}OW핳}OW핳}OW핳} OW핳} OW핳} OW핳}OW핳}OW핳}OW핳}OW핳}OW핳}OW핳}OW핳}OW핳}OW핳}OW핳}OW핳}OW핳}OW핳}OW핳}OW핳}M}0Qkz쭡}0Qkz쭡}0Qkz쭡}0Qkz쭡}0Qkz쭡}0Qkz쭡}0Qkz쭡}0Qkz쭡}0Qkz쭡}0Qkz쭡}0Qkz쭡}0Qkz쭡}0Qkz쭡}0Qkz쭡}0Qkz쭡}0Qkz쭡}0Qkz쭡}0Qkz쭡}0Qkz쭡}0Qkz쭡}0Qkz쭡}0Qkz쭡}0Qkz쭡}0Qkz쭡} :  % Y jU:ex3g 8566583700087976957E "ap

Relating hyperbolic functions to usual ones: \[ \cos(z) &= \cosh(?)\\ \sin(z) &= \sinh(?) .\]



\[ \cos(z) &= \cosh(iz)\\ \sin(z) &= {1\over i} \sinh(iz) .\]

definition

Relating hyperbolic functions to usual ones: \[ \cos(z) &= \cosh(?)\\ \sin(z) &= \sinh(?) .\]

!v3s  4198531501002717422E "ap

Exponential definitions of \( \cosh \) and \( \sinh \)



\[ \cosh(z) &= {e^{iz} + e^{-iz}\over 2} \\ \sinh(z) &= {e^{iz} - e^{-iz}\over 2} \\ .\]

definition

Exponential definitions of \( \cosh \) and \( \sinh \)

5t 1q7 412362867093732988E "ap

Conformal Map



A holomorphic map with nowhere vanishing derivative (locally injective).

definition

Conformal Map

r 1We 521697941264870068E "ap

Definition: An essential singularity



An isolated singularity that is not a pole (or removable). \[ \lim_{z\to z_0 } f(z) \text{ DNE } .\]

definition

Definition: An essential singularity

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Definition: A removable singularity



A pole of order zero, so \[ \lim_{z\to z_0}f(z) < \infty \] and \( f \) is bounded on some neighborhood of \( z_0 \).

definition

Definition: A removable singularity

0n 3)s 5042135286423393409E "ap

Definition: A pole \( a \) of order \( m \)



The smallest \( m \) such that \[ \lim_{z\to a}(z-a)^{m+1}f(z) < \infty \text{ but } \lim_{z\to a}(z-a)^{k\leq m} f(z) = \infty .\]

definition

Definition: A pole \( a \) of order \( m \)

hl 3]1 7938164582073206330E "ap

Normalizer



?

definition

Normalizer

jj 3_3 4971140825990801328E "ap

Centralizer



?

definition

Centralizer

h 35? 6878738190406415993E "ap

Invariant Factors



\( r_1 \divides r_2 \divides \cdots \)

definition

Invariant Factors

gf 3GC 7824113018893846090E "ap

Elementary Divisors



\( G \cong \bigoplus {\mathbb{Z}}/p_i^{\alpha_i} {\mathbb{Z}} \) with the \( p_i \) not necessarily distinct.

definition

Elementary Divisors

Dd 3{I 8459417995598758111E "ap

One step subgroup test



\( a, b\in H \implies ab^{-1} \in H \) implies that \( H \) is a subgroup.

fact

One step subgroup test

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Groups of Order 4



\( {\mathbb{Z}}/4{\mathbb{Z}}, ({\mathbb{Z}}/2{\mathbb{Z}})^2 \)

fact

Groups of Order 4

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Order of the smallest nonabelian group



Order six: \( D_3 \)

fact

Order of the smallest nonabelian group

I^ 3 A 2120258823952846885E "ap

Groups of Order 20



2 Abelian, 3 Nonabelian

  1. \( {\mathbb{Z}}/20{\mathbb{Z}} \)
  2. \( ({\mathbb{Z}}/2{\mathbb{Z}})^2\times {\mathbb{Z}}/5{\mathbb{Z}} \)
  3. \( D_{10} \)
  4. \( {\mathbb{Z}}/5{\mathbb{Z}}\rtimes{\mathbb{Z}}/4{\mathbb{Z}} \)
  5. \( \left\langle{a,b,c {~\mathrel{\Big\vert}~}a^5, b^2,c^2}\right\rangle \)
fact

Groups of Order 20

Z\ 3/A 2754195392877229262E "ap

Groups of Order 18



2 Abelian, 2 Nonabelian

  1. \( {\mathbb{Z}}/18{\mathbb{Z}} \)
  2. \( ({\mathbb{Z}}/3{\mathbb{Z}})^2\times {\mathbb{Z}}/2{\mathbb{Z}} \)
  3. \( D_9 \)
  4. \( S_3 \times{\mathbb{Z}}/3{\mathbb{Z}} \)
fact

Groups of Order 18

H u`55'_ 3CG 8261322400433154839E "ap

Arzela-Ascoli Theorem



A sequence of functions \( \left\{{f_n}\right\} \) has a uniformly convergent subsequence \( \iff \left\{{f_n}\right\} \) is uniformly bounded and uniformly equicontinuous.

theorem

Arzela-Ascoli Theorem

] 3ku 5741499895266256071E "ap

Equivalent Characterizations of Completeness

 definition

Equivalent Characterizations of Completeness

y[ 3kE 8287154831588557886E "ap

Closed Graph Theorem



?

theorem

Closed Graph Theorem

yY 3kE 7473295271696287592E "ap

Open Mapping Theorem



?

theorem

Open Mapping Theorem

#W 3E= 3016293988497825250E "ap

Compact Operator

 definition

Compact Operator

U 3!U 4714487008357169592E "ap

Riesz Representation Theorem



For \( H \) a Hilbert space and \( \varphi \in H {}^{ \vee } \), there exists an \( f\in H \) such that \( x\in H \implies \varphi(x) = {\left\langle {f},~{x} \right\rangle} \) with \( {\left\lVert {f} \right\rVert}_H = {\left\lVert {\varphi} \right\rVert}_{H {}^{ \vee }} \).

theorem

Riesz Representation Theorem

hS 1KC 123968236531261456E "ap

Hahn-Banach Theorem



If \( p: V\to {\mathbb{R}} \) is a sublinear function and \( \phi: U\to {\mathbb{R}} \) a linear functional on \( U\leq V \) with \( \phi \leq p \), then there exists a \( p{\hbox{-}} \)sublinear extension \( \tilde \phi: V\to {\mathbb{R}} \)

theorem

Hahn-Banach Theorem

OQ3S 6746818848717453320E "ap

Functional characterization of surjective functions.



\[ \forall y, \exists X \text{ such that } f(x) = y \iff f\text{ has a right inverse, } \exists g \text{ such that } f(g(y)) = y\, \forall y .\]

definition

Functional characterization of surjective functions.

O35 8003936765144333350E "ap

Functional characterization of injective functions.



\[ f(x) = f(y) \implies x=y \iff f \text{ has a left inverse.} .\]

definition

Functional characterization of injective functions.

5M 3]I 6945497940497084924E "ap

Factor \( x^n + y^n \)



\[ (x+y)(x^{n-1} + x^{n-2}(-y) + \cdots + (-y)^{n+1}) \]

formula

Factor \( x^n + y^n \)

0K 3SI 7817419314828801262E "ap

Factor \( x^n - y^n \)



\[ (x-y)(x^{n-1} + x^{n-2} y + \cdots + y^{n-1}) \]

formula

Factor \( x^n - y^n \)

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Descartes' Rule of Signs



The number \( r^+ \) of positive real roots (counted with multiplicity) of a polynomial \( p(x) \) are at most the number \( n \) of sign changes in the coefficients. Moreover, \( r^+ \) is exactly \( n-2k \) for some \( k\in {\mathbb{Z}}^{\geq 0} \). Similarly, the number \( r^- \) of negative real roots is the same idea applied to \( p(-x) \). The minimal number of nonreal roots is thus \( n - (r^+ + r^-) \).

Examples:

fact

Descartes' Rule of Signs

s m G s% 3oU 5958321358438939185E "ap

Casorati-Weierstrass Theorem

 Theorem

Casorati-Weierstrass Theorem

8# 39s 5243737973842640325E "ap

Riemann's Removable Singularity Theorem



Let \( U\subset {\mathbb{C}} \) be open, \( a\in U \), and \( f \) holomorphic on \( U\setminus\left\{{a}\right\} \). Then TFAE

- There exists a neighborhood of \( a \) on which \( f \) is bounded.

\\[\lim_{z\to a} (z-a)f(z) = 0.\\]
theorem

Riemann's Removable Singularity Theorem

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Morera's Theorem



If \( f \) is continuous in an open disc \( {\mathbb{D}} \) and \[ \text{For all triangles } T\subset {\mathbb{D}}, \qquad \int_T f = 0 ,\] then \( f \) is holomorphic.

theorem

Morera's Theorem

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Dirichlet's Test



If \( \left\{{a_n}\right\}, \left\{{b_n}\right\} \) satisfy

Then \[ \sum_{n=1}^\infty a_n b_n < \infty .\]

theorem

Dirichlet's Test

J3]q 7708999002946019036E "ap

An analytic function with convergence radius 1 that converges at every point on \( S^1 \) except \( z=1 \)



\[ \sum_{n=1}^\infty {z^n\over n} \]

example

An analytic function with convergence radius 1 that converges at every point on \( S^1 \) except \( z=1 \)

,3AQ 1231572468213235845E "ap

An analytic function with convergence radius 1 which converges at every point on \( S^1 \)



\[ \sum_{n=1}^\infty {z^n\over n^2} \]

example

An analytic function with convergence radius 1 which converges at every point on \( S^1 \)

.39] 6879995054531133523E "ap

An analytic function with convergence radius 1 which fails to converge at any point on \( S^1 \)



\( \sum_{n=1}^\infty nz^n \)

example

An analytic function with convergence radius 1 which fails to converge at any point on \( S^1 \)

 3y 5780199920711641771E "ap

Example of a conformal map that is not injective.



\[z\mapsto e^z\]

Not injective because it is periodic, not surjective because it's never zero.

examples

Example of a conformal map that is not injective.

 3 [ 3339778494547789293E "ap

The generalized residue formula



\[ \mathop{\mathrm{Res}}_{z=z_0} f = \lim_{z\to z_0} {1 \over (n-1)!} \qty{{\frac{\partial }{\partial z}\,}}^{n-1} (z-z_0)^n f(z) .\]

formula

The generalized residue formula

D 1 ; 709020337658694305E "ap

Cross-ratio map



\[ R(z) \coloneqq(z, z_2, z_3, z_4) &\coloneqq{z - z_3\over z-z_4}{z_2 - z_4 \over z_2 - z_3} \\ .\]

Sends

formula

Cross-ratio map

#3m 3573524308737296949E "ap

Standard parameterization of a circle in \( {\mathbb{C}} \)



\[ F(t) = {1-x^2\over 1+x^2} + i{2x\over 1+x^2} \quad x = \tan(t), t\in (-\pi/2, \pi/2) .\]

formula

Standard parameterization of a circle in \( {\mathbb{C}} \)

2 < i XFG<2G3) 3500802847695111632E "ap

\( e^{2\pi i \over 6} = \cdots \) in rectangular coordinates



\[ e^{2\pi i \over 6} = {1\over 2}\qty{1 + i\sqrt 3} .\]

formula

\( e^{2\pi i \over 6} = \cdots \) in rectangular coordinates

E3+ 1006424574577889144E "ap

\( e^{2\pi i \over 3} = \cdots \) in rectangular coordinates



\[ e^{2\pi i \over 3} = {1\over 2}\qty{-1 + i\sqrt 3} .\]

formula

\( e^{2\pi i \over 3} = \cdots \) in rectangular coordinates

!C 3-Q 1055818088854968422E "ap

\( \tan(\pi/3) = \cdots \)



\[ \tan(\pi/3) = \sqrt 3 .\]

formula

\( \tan(\pi/3) = \cdots \)

A 3!Q 8048186507949420487E "ap

\( \tan(\pi/4) = \cdots \)



\[ \tan(\pi/4) = 1 .\]

formula

\( \tan(\pi/4) = \cdots \)

+? 3AQ 4455369571005901365E "ap

\( \tan(\pi/6) = \cdots \)



\[ \tan(\pi/6) = {\sqrt 3 \over 3} .\]

formula

\( \tan(\pi/6) = \cdots \)

$= 15Q 265612215786136626E "ap

\( \cos(\pi/3) = \cdots \)



\[ \cos(\pi/3) = {1 \over 2} .\]

formula

\( \cos(\pi/3) = \cdots \)

+; 3AQ 8908966494958771861E "ap

\( \cos(\pi/4) = \cdots \)



\[ \cos(\pi/4) = {\sqrt 2 \over 2} .\]

formula

\( \cos(\pi/4) = \cdots \)

+9 3AQ 7633752610512111332E "ap

\( \cos(\pi/6) = \cdots \)



\[ \cos(\pi/6) = {\sqrt 3 \over 2} .\]

formula

\( \cos(\pi/6) = \cdots \)

*7 1AQ 497301278609316008E "ap

\( \sin(\pi/3) = \cdots \)



\[ \sin(\pi/3) = {\sqrt 3 \over 2} .\]

formula

\( \sin(\pi/3) = \cdots \)

+5 3AQ 4187739397685068908E "ap

\( \sin(\pi/4) = \cdots \)



\[ \sin(\pi/4) = {\sqrt 2 \over 2} .\]

formula

\( \sin(\pi/4) = \cdots \)

$3 33Q 1752222610145943595E "ap

\( \sin(\pi/6) = \cdots \)



\[ \sin(\pi/6) = {1\over 2} .\]

formula

\( \sin(\pi/6) = \cdots \)

F13%! 1826519987638214014E "ap

General form of maps in \( \mathop{\mathrm{Aut}}({\mathbb{D}}) \).



\[f_\alpha(z) \coloneqq{\alpha - z \over 1 - \mkern 1.5mu\overline{\mkern-1.5mu\alpha \mkern-1.5mu}\mkern 1.5muz}\]

fact

General form of maps in \( \mathop{\mathrm{Aut}}({\mathbb{D}}) \).

}/ 3[[ 7910542508294816066E "ap

Types of isolated singularities

  complex, fact

Types of isolated singularities

J- 3I 3176706323105474127E "ap

Cauchy-Goursat Theorem



If \( f \) is holomorphic on a simply connected region \( \Omega \) containing a contour \( \gamma \), then \[\int_\gamma f = 0.\] Moreover, this holds for any contour \( \gamma \subset \Omega \).

complex, theorem

Cauchy-Goursat Theorem

:+ 3Ek 1080895952571885474E "ap

Exponential expansions of \( \sin(z) \)



\[\begin{align*} \sin(\theta) &= \frac{e^{i\theta} - e^{-i\theta}}{2i} = \frac{z - z^{-1}}{2i} \\ d\theta &= \frac{dz}{iz} \end{align*}\]

complex, identity

Exponential expansions of \( \sin(z) \)

s) 3]E 6831193321903192589E "ap

Open Mapping Theorem



If \( f: \Omega \to {\mathbb{C}} \) is holomorphic and not constant on \( \Omega \), then \( f \) is an open map.

complex, theorem

Open Mapping Theorem

-' 1IO 278262218668222941E "ap

Maximum Modulus Principle



If \( f: \Omega \to {\mathbb{C}} \) is holomorphic and not constant on \( \Omega \), then \( {\left\lvert {f} \right\rvert} \) is unbounded in \( \Omega^\circ \).

complex, theorem

Maximum Modulus Principle

rp`P@0 p`P@0  p ` P @ 0  p ` P @ 0  p ` P @ 0  p ` P @ 0  p ` P @ 0  p`P@0 p`P@0 p`P@0 p`P@0 p`P@0 p`P@0 p`P@0 }}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}~}}|}}}z}{}x}y}v}w}t}u}r}s}p}q}n}o}l}m}j}k}h}i}f}g}d}e}b}c}`}a}^}_}\}]}Z}[}X}Y}V}W}T}U}R}S}P}Q}N}O}L}M}J}K}H}I}F}G}D}E}B}C}@}A}>}?}<}=}:};}8}9}6}7}4}5}2}3}0}1}.}/},}-}*}+}(})}&}'}$}%}"}#} }!}}}}}}}}}}}}}}}}}}} } } } }} }}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} p`P@0 p`P@0  p ` P @ 0  p ` P @ 0  p ` P @ 0  p ` P @ 0  p ` P @ 0  p`P@0 p`P@0 p`P@0 p`P@0 p`P@0 p`P@0 }y}z}w}x}u}v}s}t}q}r}o}p}m}n}k}l}i}j}g}h}e}f}c}d}a}b}_}`}]}^}[}\}Y}Z}W}X}U}V}S}T}Q}R}O}P}M}N}K}L}I}J}G}H}E}F}C}D}A}B}?}@}=}>};}<}9}:}7}8}5}6}3}4}1}2}/}0}-}.}+},})}*}'}(}%}&}#}$}!}"}} }}}}}}}}}}}}}}}}} }} } } } }}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}~}}|}}}z}{}x}y}v}w}t}u}r}s}p}q}n}o}l}m}j}k}h}i}f}g}d}e}b}c}`}a}^}_}\}]}Z}[}X}Y}V}W}T}U}R}S}P}Q}N}O}L}M}J}K}H}I}F}G}D}E}B}C}@}A}>}?}<}=}:};}8}9}6}7}4}5}2}3}0}1}.}/},}-}*}+}(})}&}'}$}%}"}#} }!}}}}}}}}}}}}}}}}}}} } } } }} }}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}  l > k3o 3q1 4675468216838026893E "ap

Meagre Set



A set is meagre iff it is a countable union of nowhere dense sets.

definition

Meagre Set

m 3S 1820626244349559840E "ap

Reverse Triangle Inequality



\( {\left\lvert {{\left\lVert {x} \right\rVert} - {\left\lVert {y} \right\rVert}} \right\rvert} \leq {\left\lVert {x-y} \right\rVert} \)

formula

Reverse Triangle Inequality

sk 3i9 5249276912697577699E "ap

Equicontinuous



For \( X, Y \) metric spaces and \( \mathcal{F} \) a family of functions, \( F \) is equicontinuous at \( x_0 \) iff for every \( \varepsilon > 0 \) there exists a \( \delta(\varepsilon, x_0)>0 \) such that \[x\in B_\delta(x_0) \implies f_i(x) \in B_\varepsilon(f_i(x_0))\] for all \( f_i \in \mathcal{F} \). The family \( F \) is uniformly equicontinuous iff \( \delta(\varepsilon) \) only depends on \( \varepsilon \) and holds for any pair \( x_1, x_2 \) with \( x_1 \in B_\delta(x_2) \).

definition

Equicontinuous

i 3!7 8502802991206848134E "ap

Nowhere Dense



A set is \( A \) nowhere dense if its closure has empty interior \( \qty{\mkern 1.5mu\overline{\mkern-1.5muA\mkern-1.5mu}\mkern 1.5mu}^\circ \), equivalently it is not dense in any nonempty open set. For \( {\mathbb{R}} \), every interval \( I \) contains a subinterval \( S\subset I \) with \( S\cap A = \emptyset \), i.e. its closure contains no intervals.

Intuition: elements are not tightly clustered, set is full of holes.

Counterexample: \( \left\{{1 \over n}\right\}, {\mathbb{Z}} \) are nowhere dense, \( {\mathbb{Q}}, {\mathbb{Z}}\cup\qty{(a, b)\cap{\mathbb{Q}}} \) is not nowhere dense

definition, example, counterexample

Nowhere Dense

#g 15O 510429646724020712E "ap

First and Second Category

 definition

First and Second Category

&e 3U3 5974108772438640231E "ap

Baire Space



\( X \) is a Baire space iff whenever \( \left\{{U_n}\right\} \) is a countable collection of open dense subsets of \( X \), then their intersection \( \cap U_n \) is again dense.

definition

Baire Space

kc 3II 1008227625972604451E "ap

Young's Inequality



For \( 1\leq p, q\leq r \leq \infty \) with \( {1\over p} + {1\over q} - {1\over r} = 1 \), then \( {\left\lVert {f\ast g} \right\rVert}_r \leq {\left\lVert {f} \right\rVert}_p {\left\lVert {g} \right\rVert}_q \)

Useful cases: \[\begin{align*} {\left\lVert {f\ast g} \right\rVert}_1 & \leq {\left\lVert {f} \right\rVert}_1 {\left\lVert {g} \right\rVert}_1 \\ {\left\lVert {f\ast g} \right\rVert}_p & \leq {\left\lVert {f} \right\rVert}_1 {\left\lVert {g} \right\rVert}_p \\ {\left\lVert {f\ast g} \right\rVert}_\infty & \leq {\left\lVert {f} \right\rVert}_p {\left\lVert {g} \right\rVert}_q \\ {\left\lVert {f\ast g} \right\rVert}_\infty & \leq {\left\lVert {f} \right\rVert}_2 {\left\lVert {g} \right\rVert}_2 \end{align*}\]

formula

Young's Inequality

a 3Q 5014391438324521146E "ap

Minkowski's Inequality



For \( 1 \leq p < \infty \), \[ {\left\lVert {f + g} \right\rVert}_p \leq {\left\lVert {f} \right\rVert}_p + {\left\lVert {g} \right\rVert}_p .\]

theorem

Minkowski's Inequality

  V `f;)hA 3w 2846370719102928534E "ap

True or false: \( X =_? (f\circ f^{-1})(X) \)



Only when \( f \) is surjective, otherwise just \( \subseteq \).

misc

True or false: \( X =_? (f\circ f^{-1})(X) \)

i? 3y 3361289577063262278E "ap

True or false: \( X =_? (f^{-1} \circ f)(X) \)



Only when \( f \) is injective, otherwise just \( \subseteq \).

misc

True or false: \( X =_? (f^{-1} \circ f)(X) \)

T=3C 3747426254626751301E "ap

\( f^{-1} \left( \bigcap U_i \right) =_? \bigcap_i f^{-1} U_i \)?



Yes

misc

\( f^{-1} \left( \bigcap U_i \right) =_? \bigcap_i f^{-1} U_i \)?

];1W 687053063179551081E "ap

\( f^{-1} \left( \coprod U_i \right) =_? \coprod_i f^{-1} U_i \)?



Yes: \[ \begin{align} f^{-1}\left[\bigcup_{i \in I} Y_{i}\right] &=\left\{x \in X \mathrel{\Big|}f(x) \in \bigcup_{i \in I} Y_{i}\right\} \\ &=\left\{x \in \lambda \mathrel{\Big|}\quad \exists i \in I \text { such that } f(x) \in Y_{i}\right\} \\ &=\bigcup_{i \in I}\left\{x \in X \mathrel{\Big|}f(x) \in Y_{i}\right\} \\ &=\bigcup_{i \in I} f^{-1}\left[Y_{i}\right] \end{align} \]

misc

\( f^{-1} \left( \coprod U_i \right) =_? \coprod_i f^{-1} U_i \)?

9 3mS 8506628919450530649E "ap

Definition: Algebraic Group

definition

Definition: Algebraic Group

7 3mS 8197830993306648976E "ap

Definition: Reductive Group

definition

Definition: Reductive Group

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Definition: Height of an Ideal



?

definition

Definition: Height of an Ideal

3 3S 8641389267450876236E "ap

Definition: Specializations



?

definition

Definition: Specializations

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Definition: Global Field



?

definition

Definition: Global Field

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Definition: Ideal Class Group of an algebraic number field \( K \)



The group of fractional ideals of the ring of integers of \( K \) mod the subgroup of principal ideals. Trivial iff all ideals are principal.

definition

Definition: Ideal Class Group of an algebraic number field \( K \)

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Definition: Picard Group



The group of invertible sheaves (or line bundles) on \( X \) up to isomorphism equipped with \( \otimes \)

definition

Definition: Picard Group

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\[ \alpha\beta \]



Something new. \[ a^b = c \]

 topics

\[ \alpha\beta \]

) 33; 3794731890294249713E "ap

\[ \Lambda_r \]



The (integral) root lattice.

\[ \Lambda_r = \mathrm{span}_{\mathbb{Z}}\Phi \subset E \]

Has a basis given by \( \Phi \) or \( \Delta \).

 topics

\[ \Lambda_r \]

' 3{O 2491491216921134187E "ap

\[ \mathfrak{g}_\alpha \]



The root space.

\[ \mathfrak{g}_\alpha = \{g\in \mathfrak{g} \mathrel{\Big|}[h, g] = \alpha(h) g\} \]

 lie algebra

\[ \mathfrak{g}_\alpha \]

Z% 3?1 7185930555184464958E "ap

\[ \Phi \]



The root system.

\[ \alpha \in \Phi \subset \mathfrak{h}^* \iff [h, x] = \alpha(h) x \text{ for all } h\in \mathfrak{h} \]

 lie algebra

\[ \Phi \]

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Bolzano Weierstrass Property



Every sequence has a convergent subsequence

definition

Bolzano Weierstrass Property

4 3w- 2681814038142706545E "ap

Diameter



\[ \mathrm{diam}(A) = \sup_{x, y\in A} {\left\lvert {d} \right\rvert}(x, y) .\]

definition

Diameter

7 3' 7686965247173062033E "ap

Dense



A subset \( A\subseteq X \) is dense in \( X \) iff \( \mathrm{cl}_X(A) = X \).

definitions

Dense

3# 5777378167930936953E "ap

Is a composition of Lebesgue measurable functions measurable?



No:

example

Is a composition of Lebesgue measurable functions measurable?

c 3m 8408275043503016289E "ap

Limit definition of exponential function



\( e^x = \lim_{n \to \infty} \qty{1 + {x\over n}}^n \)

definition, formula

Limit definition of exponential function

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Definition: separable



Has a countable dense subset

definition

Definition: separable

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Proposition: Convergence in measure is equivalent to a.e. convergence



Proof: ? Use Egorov's Theorem

proof

Proposition: Convergence in measure is equivalent to a.e. convergence

fy 3?I 3101645151459865493E "ap

Convergence in Measure



\[ \lim _{k \rightarrow \infty} m\left(\left\{x \in E|| f_{k}(x)-f(x) |>\alpha\right\}\right)=0 .\]

definition

Convergence in Measure

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Definition: Almost Disjoint



\( A^\circ \cap B^\circ = \emptyset \)

definition

Definition: Almost Disjoint

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Caratheodory Characterization



\( E\subseteq {\mathbb{R}}^n \) is measurable \( \iff \) for all \( A\subset {\mathbb{R}}^n \), \( m_*(A) = m_*(E\cap A) + m_*(E\cap A^c) \)

theorem

Caratheodory Characterization

Fs 3I 2385127791805552580E "ap

Baire Category Theorem



If \( X \) is a complete metric space or a locally compact Hausdorff space, then \( X \) is a Baire space. A (non-empty) complete metric space is not the countable union of nowhere dense sets.

theorem

Baire Category Theorem

cq3U+ 7104647696949172601E "ap

Characterizations of \( D_f \), the set of discontinuities of functions

 fact

Characterizations of \( D_f \), the set of discontinuities of functions

RK * rHfR8# 3Eg 5507145239217268237E "ap

Which adjoints preserve which limits?



Left adjoints preserve colimits.

fact

Which adjoints preserve which limits?

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Match up:

to

 definition

Match up:

to

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Definition: Primary Decomposition



?

definition

Definition: Primary Decomposition

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Idempotent



An element \( x \) in a set with a closed binary operation \( \cdot \) is idempotent iff \( x\cdot x = x \).

algebra

Idempotent

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Dedekind Rings



Every ideal is projective.

commutative algebra

Dedekind Rings

3 3m5 7849171726784791236E "ap

Regular Ring



\( R \) is regular \( \iff R \) is Noetherian and every finitely generated \( R{\hbox{-}} \)module has a finite resolution by finitely-generated projective \( R{\hbox{-}} \)modules.

algebra, commutative algebra

Regular Ring

Y 3+C 6998453987048260889E "ap

Zariski's Lemma



If a field extension is finitely generated as a \( k{\hbox{-}} \)algebra then it is finite degree.

theorem

Zariski's Lemma

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Noether Normalization



?

theorem

Noether Normalization

i1MA 869557089468810997E "ap

Definition: A natural transformation between \( F, G: \mathcal{C}\to\mathcal{D} \)



A natural transformation \( \eta \) is a class of morphism \( \eta = \left\{{\eta_x: F(x) \to G(x)~\forall x\in \mathcal{C}}\right\} \) such that the following square commutes:

definition

Definition: A natural transformation between \( F, G: \mathcal{C}\to\mathcal{D} \)

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Nakayama's Lemma



If \( M \in {\mathsf{R}{\hbox{-}}\mathsf{Mod}} \) is finitely generated and \( I{~\trianglelefteq~}R \) with \( I \subseteq {\mathcal{J}}(R) \), then \[IM = M \implies M = 0.\]

theorem

Nakayama's Lemma

r 3YG 6431608861694277039E "ap

Reduced ring



A ring \( R \) is reduced \( \iff R \) has no nonzero nilpotent elements, i.e. \( x^n = 0 \implies x=0 \).

definition

Reduced ring

 / ; 92162172743638032E "ap

Interpretation of \( \sup, \inf \) and \( \limsup, \liminf \) as set operations

 analysis

Interpretation of \( \sup, \inf \) and \( \limsup, \liminf \) as set operations

 3W 6650003278245213962E "ap

Uniform Boundedness Principle



If \( \mathcal{F} \) is a family of bounded operators \( T_n:X\to Y \) between Banach spaces with \[ \forall x\in X, \qquad \sup_{T_n \in \mathcal{F}} {\left\lVert {T_n(x)} \right\rVert}_Y < \infty ,\] then \( \sup_{T_n\in \mathcal{F}} {\left\lVert {T_n} \right\rVert}_X < \infty \).

Slogan: pointwise bounded sequences of operators are uniformly bounded.

theorem

Uniform Boundedness Principle

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Complete Measure



A measure whose domain includes all subsets of null sets.

definition

Complete Measure

y * k / 8`h@8<@y?c 3wC 5879758072438244854E "ap

Kunneth (nice case)



\[H_{k}(X\times Y;F) \cong \bigoplus_{i+j=k}H_{i}(X;F)\otimes H_{j}(Y;F)\]

Math

Kunneth (nice case)

ta 31s 6452520366406393143E "ap

Homology in terms of cohomology (nice case)



\[H_i(X; \mathbb{Z}) = F(H^i(X; \mathbb{Z})) \times T(H^{i+1}(X; \mathbb{Z}))\]

math

Homology in terms of cohomology (nice case)

t_ 31s 2562196850759030133E "ap

Cohomology in terms of homology (nice case)



\[H^i(X; \mathbb{Z}) = F(H_i(X; \mathbb{Z})) \times T(H_{i-1}(X; \mathbb{Z}))\]

math

Cohomology in terms of homology (nice case)

] 3 3 4399846563711962863E "ap

Kunneth SES



\[0\to \bigoplus_{i+j=k}H_{i}(X;R)\otimes _{R}H_{j}(Y;R)\to H_{k}(X\times Y;R)\to \bigoplus_{i+j=k-1}{\mathrm {Tor}}_{R}^{1}(H_{i}(X;R),H_{j}(Y;R))\to 0\]

math

Kunneth SES

[ 3] 3625360538477120472E "ap

UCT: Cohomology and Homology SES



\[{ 0\to \mathrm{Ext}_{\mathbb{Z}}^{1}(H_{i-1}(X; \mathbb{Z}),A)\to H^{i}(X; A)\to \mathrm{Ext}_{\mathbb{Z}}^{0}(H_{i}(X; \mathbb{Z}),A) \to 0}\]

math

UCT: Cohomology and Homology SES

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UCT: Change of Coefficients SES



\[{\displaystyle 0\to \mathrm{Tor}_\mathbb{Z}^0 (H_{i}(X;\mathbb{Z}), A)\,{\to }\,H_{i}(X;A)\to \mathrm{Tor}_\mathbb{Z}^1 (H_{i-1}(X;\mathbb{Z} ),A)\to 0}\]

math

UCT: Change of Coefficients SES

cW 3q 8321920181651958914E "ap

Long Exact Sequence of a Pair \( (A, B) \)



\[\ldots H_n(B) \to H_n(A) \to H_n(A,B) \to H_{n-1}(B) \ldots\]

 math

Long Exact Sequence of a Pair \( (A, B) \)

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Submersion



?

definition

Submersion

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Immersion



?

definition

Immersion

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Embedding



Homeomorphism onto its image

definition

Embedding

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What inverses exist for injective (resp. surjective) functions?

 unsorted

What inverses exist for injective (resp. surjective) functions?

oM 3+o 7188560547018782648E "ap

Write \( \exp \) as a group homomorphism.



\[ \exp: {\mathbb{G}}_a({\mathbb{R}}) \to {\mathbb{G}}_m({\mathbb{R}}) .\]

unsorted

Write \( \exp \) as a group homomorphism.

4K3o3 4956675517703271057E "ap

Write \( \log \) as a group homomorphism. What are the domain and codomain?



\[ \log: {\mathbb{G}}_m({\mathbb{R}}) \to {\mathbb{G}}_a({\mathbb{R}}) .\]

unsorted

Write \( \log \) as a group homomorphism. What are the domain and codomain?

7I 3-} 7432670093357131937E "ap

True or false: \( \log(a+b) = \log(a) \log(b) \)



False

unsorted

True or false: \( \log(a+b) = \log(a) \log(b) \)

True or false: \( \log(a+b) = \log(a) + \log(b) \)



False

unsorted

True or false: \( \log(a+b) = \log(a) + \log(b) \)

oE1 221497566005716913E "ap

True or false: \( \log(a-b) = \log(a) / \log(b) \)



False: take \( b=1 \), this would force dividing by zero.

unsorted

True or false: \( \log(a-b) = \log(a) / \log(b) \)

C3-3 2756556640468010131E "ap

True or false: \( f^{-1}(A) \setminus f^{-1}(B) =_? f^{-1}(A\setminus B) \)



Yes. For forward direction, only an inclusion.

misc

True or false: \( f^{-1}(A) \setminus f^{-1}(B) =_? f^{-1}(A\setminus B) \)

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abcdefg



abasdsad \( y=mx+b \) asdasd \( \left\{{a, b}\right\} \).

asdasd \[ Y \coloneqq\left\{{a{~\mathrel{\Big\vert}~}bx=c}\right\}^{-1} .\]

definition

abcdefg

pw 3QK 4096065150833095935E "ap

Reduced scheme



A scheme \( X \) is reduced \( \iff {\mathcal{O}}_X(U) \) is a reduced ring for every open \( U \subset X \).

definitions

Reduced scheme

4u 31s 8745705504294304862E "ap

Comparison between maximal and prime ideals



?

commutative algebra

Comparison between maximal and prime ideals

Qs 3Y 1205135199777000530E "ap

Scheme definition of a variety



A variety is a finite type separated schemes over a field

algebraic geometry

Scheme definition of a variety

$q 3?E 3366358972416414862E "ap

Riemann-Roch Formula



\[ l(D)-l(K-D)=\operatorname{deg}(D)-p+1 \]

formula

Riemann-Roch Formula

Lo 3C 1959419667671991816E "ap

Kunneth (nice case)



\[ H_{k}(X\times Y;F) \cong \bigoplus _{{i+j=k}}H_{i}(X;F)\otimes H_{j}(Y;F) .\]

definition

Kunneth (nice case)

~m 3Es 8231976990502320698E "ap

Cohomology in terms of homology (nice case)



\[ H^i(X; \mathbb{Z}) = F(H_i(X; \mathbb{Z})) \times T(H_{i-1}(X; \mathbb{Z})) .\]

definition

Cohomology in terms of homology (nice case)

Ok 3K 7204969767132728497E "ap

\[ H_* \mathbb{RP}^2 \]



\[ {\mathbb{RP}}^2 = [\mathbb{Z}, \mathbb{Z}_2, 0, 0, 0, 0\rightarrow ] .\]

definition

\[ H_* \mathbb{RP}^2 \]

)i 3_/ 3453512937499526928E "ap

Ext Table



\[ \begin{array}{c|c|c|c} Ext & Z_m & Z & Q \\\hline Z_n & Z_d & Z_n & 0 \\\hline Z & 0 & 0 & 0 \\\hline Q & 0 & A_p/Q & 0 \end{array} \]

math

Ext Table

g 3C/ 9215437362333138607E "ap

Tor Table



\[ \begin{array}{c|c|c|c} Tor & Z_m & Z & Q \\\hline Z_n & Z_d & 0 & 0 \\\hline Z & 0 & 0 & 0 \\\hline Q & 0 & 0 & 0 \end{array} \]

math

Tor Table

e 3E/ 1844611241615072697E "ap

Hom Table



\[ \begin{array}{c|c|c|c} Hom & Z_m & Z & Q \\\hline Z_n & Z_d & 0 & 0 \\\hline Z & Z_m & Z & Q \\\hline Q & 0 & 0 & Q \end{array} \]

math

Hom Table

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