Math Stories
posted: 01-Aug-2025 & updated: 05-Aug-2025
NotebookLM Podcast
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\newcommand{\frobmap}[2]{\varphi_{#1,{#2}}} % %\newcommand{\algfontmode}{} % %\ifdefined\algfontmode %\newcommand\mathalgfont[1]{\mathcal{#1}} %\newcommand\mathcalfont[1]{\mathscr{#1}} %\else \newcommand\mathalgfont[1]{\mathscr{#1}} \newcommand\mathcalfont[1]{\mathcal{#1}} %\fi % %\def\DeltaSirDir{yes} %\newcommand\sdirletter[2]{\ifthenelse{\equal{\DeltaSirDir}{yes}}{\ensuremath{\Delta #1}}{\ensuremath{#2}}} \newcommand{\sdirletter}[2]{\Delta #1} \newcommand{\sdirlbd}{\sdirletter{\lambda}{\Delta \lambda}} \newcommand{\sdir}{\sdirletter{x}{v}} \newcommand{\seqk}[2]{#1^{(#2)}} \newcommand{\seqscr}[3]{\seq{#1}_{#2}^{#3}} \newcommand{\xseqk}[1]{\seqk{x}{#1}} \newcommand{\sdirk}[1]{\seqk{\sdir}{#1}} \newcommand{\sdiry}{\sdirletter{y}{\Delta y}} \newcommand{\slen}{t} \newcommand{\slenk}[1]{\seqk{\slen}{#1}} \newcommand{\ntsdir}{\sdir_\mathrm{nt}} \newcommand{\pdsdir}{\sdir_\mathrm{pd}} \newcommand{\sdirnu}{\sdirletter{\nu}{w}} \newcommand{\pdsdirnu}{\sdirnu_\mathrm{pd}} \newcommand{\pdsdiry}{\sdiry_\mathrm{pd}} \newcommand\pdsdirlbd{\sdirlbd_\mathrm{pd}} % \newcommand{\normal}{\mathcalfont{N}} % \newcommand{\algk}[1]{\mathalgfont{#1}} \newcommand{\collk}[1]{\mathcalfont{#1}} \newcommand{\classk}[1]{\collk{#1}} \newcommand{\indexedcol}[1]{\{#1\}} \newcommand{\rel}{\mathbf{R}} \newcommand{\relxy}[2]{#1\;\rel\;{#2}} \newcommand{\innerp}[2]{\langle{#1},{#2}\rangle} \newcommand{\innerpt}[2]{\left\langle{#1},{#2}\right\rangle} \newcommand{\closure}[1]{\overline{#1}} \newcommand{\support}{\mathbf{support}} \newcommand{\set}[2]{\{#1|#2\}} \newcommand{\metrics}[2]{\langle {#1}, {#2}\rangle} \newcommand{\interior}[1]{#1^\circ} \newcommand{\topol}[1]{\mathfrak{#1}} \newcommand{\topos}[2]{\langle {#1}, \topol{#2}\rangle} % topological space % \newcommand{\alg}{\algk{A}} \newcommand{\algB}{\algk{B}} \newcommand{\algF}{\algk{F}} \newcommand{\algR}{\algk{R}} \newcommand{\algX}{\algk{X}} \newcommand{\algY}{\algk{Y}} % \newcommand\coll{\collk{C}} 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Math Stories
Fundamental Theorems
Fundamental theorem of arithmetic
integer $n\geq2$ can be factored uniquely into products of primes,
i.e.,
exist distinct primes, $p_1$, …, $p_k$, and $e_1,\ldots, e_k\in\naturals$
such that
$$
n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k}
$$
Fundamental theorem of algebra
every non-constant single-variable polynomial with complex coefficients
has at least one complex root,
or equivalently,
(the field of complex numbers) is algebraically closed,
or equivalently,
every non-zero, single-variable, degree $n$ polynomial with complex coefficients has,
counted with multiplicity, exactly $n$ complex roots.
- the fundamental theorem of algebra, also called d'Alembert's theorem or the d'Alembert–Gauss theorem
- despite its name, not fundamental for modern algebra; named when algebra was synonymous with the theory of equations
Fundamental theorem of calculus
- first fundamental theorem of calculus - for continuous real-valued function $f:[a,b]\to\reals$, function $F:[a,b]\to\reals$ defined by $ F(x) = \int_a^x f(t) dt $ is uniformly continuous on $[a,b]$ and differentiable on open interval $(a,b)$ and $$ F'(x) = f(x) $$ for all $x\in(a,b)$, hence $F$ is antiderivative of $f$
- second fundamental theorem of calculus or Newton-Leibniz theorem - for real-valued function $f:[a,b]\to\reals$ and continuous function $F:[a,b]\to\reals$ which is antiderivative of $f$ in $(a,b)$, i.e. $ F'(x) = f(x) $, if $f$ is Riemann integrable on $[a,b]$, then $$ \int_a^b f(x) dx = F(b) - F(a) $$
Fundamental theorem of calculus for line integrals
line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve,
i.e.,
if $\varphi: X \to \reals$ is differentiable function and $\gamma$ is curve in $X\subset \reals$
which
starts at point $p\in\reals^n$
and
ends at point $q\in\reals^n$,
then
$$
\int_\gamma \nabla \varphi(x)^T dx = \varphi(q) - \varphi(p)
$$
- generalization of the second fundamental theorem of calculus of Fundamental theorem of calculus
Fundamental theorem of cyclic groups
every subgroup of a cyclic group is cyclic;
moreover,
for finite cyclic group of order $n$,
every subgroup's order is a divisor of $n$,
and exists exactly one subgroup for each divisor.
Fundamental theorem of equivalence relations
equivalence relation $\sim$ on set $X$ partitions $X$;
conversely,
corresponding to any partition of $X$,
exists equivalence relation $\sim$ on $X$
Fundamental theorem of finite abelian groups
every finite abelian group
can be expressed as direct sum of cyclic subgroups of prime-power order,
i.e.,
any finite abelian group $G$ is
isomorphic to direct sum of form
$$
\bigoplus_{i=1}^u \left(\integers/{k_i}\integers\right)
$$
in either of the following canonical ways
- numbers $k_1$, …, $k_u$ are powers of (not necessarily distinct) primes
- $k_1$ divides $k_2$, which divides $k_3$, and so on up to $k_u$
- also known as basis theorem for finite abelian groups
Fundamental theorem of finitely generated abelian groups
- Fundamental theorem of finitely generated abelian groups generalizes Fundamental theorem of finite abelian groups in two ways
- primary decomposition - every finitely generated abelian group is isomorphic to a direct sum of primary cyclic groups and infinite cyclic groups, i.e., every finitely generated abelian group $G$ is isomorphic to group of form $$ G = \integers^n \oplus \left(\integers/q_1 \integers\right) \oplus \cdots \oplus \left(\integers/q_t \integers\right) $$ where $n\geq0$ is rank, and numbers $q_1, \ldots, q_t$ are powers of (not necessarily distinct) prime numbers; in particular, $G$ is finite if and only if $n=0$, values of $n, q_1, \ldots, q_t$ are (up to rearranging indices) uniquely determined by $G$, i.e., exists one and only one way to represent $G$ as such decomposition
- invariant factor decomposition - can also write any finitely generated abelian group $G$ as direct sum of form $$ G = \integers^{n} \oplus \left(\integers /{k_{1}}\integers\right) \oplus \cdots \oplus \left(\integers /{k_{u}}\integers\right) $$ where $k_1$ divides $k_2$, which divides $k_3$ and so on up to $k_u$; again, rank $n$ and invariant factors $k_1, \ldots, k_u$ are uniquely determined by $G$ (here with a unique order); rank and sequence of invariant factors determine group up to isomorphism
Fundamental theorem for Galois theory
for finite Galois extension, $K/k$
- map $H \mapsto K^H$ induces isomorphism between set of subgroups of $G(K/k)$ & set of intermediate fields
- subgroup, $H$, of $G(K/k)$, is normal if and only if $K^H/k$ is Galois
- for normal subgroup, $H$, $\sigma\mapsto \restrict{\sigma}{K^H}$ induces isomorphism between $G(K/k)/H$ and $G(K^H/k)$
Fundamental theorem on homeomorphisms
for two groups $G$ and $H$ and
group homeomorphism $f:G\to H$,
normal subgroup $N\subset G$,
natural surjective homeomorphism $\varphi:G\to G/N$
if $N$ is subset of $\Ker{f}$,
exists unique homeomorphism $h:G/N\to H$
such that
$$
f = h \circ \varphi
$$
Fundamental theorem of ideal theory in number fields
every nonzero proper ideal in ring of integers
of number field admits unique factorization into
product of nonzero prime ideals;
in other words, every ring of integers of number field is Dedekind domain
Fundamental theorem of linear algebra
number of columns of matrix $M$ is
sume of rank of $M$ and nullity of $M$,
or equivalently,
dimension of domain of linear transformation $f$
is sum of rank of $f$ (dimension of image of $f$)
and
nullity of $f$ (dimension of kernel of $f$)
Fundamental theorem of linear programming
for linear program
$$
\begin{array}{ll}
\mbox{minimal} & c^Tx
\\
\mbox{subject to} & Ax \leq b
\end{array}
$$
if $P=\set{x\in\reals^n}{Ax \leq b}$
is bounded polyhedron (hence polytope)
and $x^\ast$ is optimal solution,
then $x^\ast$ is either extreme point (i.e., vertex) of $P$
or lies on some face of $P$
Fundamental theorem of symmetric polynomials
for every commutative ring $A$,
denote ring of symmetric polynomials in variables $X_1$, …, $X_n$
with coefficients in $A$
by $A[X_1,\ldots,X_n]^{S_n}$,
which is polynomial ringt in $n$ elementary symmetric polynomials $e_k(X_1,\ldots,X_n)$
for $k=1,\ldots,n$,
then
every symmetric polynomial $P(X_1,\ldots,X_n) \in A[X_1,\ldots,X_n]^{S_n}$
has unique representation
$$
P(X_1, \ldots, X_n) = Q(e_1(X_1,\ldots,X_n), \ldots, e_n(X_1,\ldots,X_n))
$$
for some polynomials $Q\in A[Y_1,\ldots,Y_n]$,
or equivalently,
ring homeomorphism
that sends $Y_k$ to $e_k(X_1,\ldots,X_n)$
for $k=1,\ldots,n$
defines
an isomorphism between
$A[Y_1,\ldots,Y_n]$
and
$A[X_1,\ldots,X_n]^{S_n}$
Duality
Dualities
-
duality
- “very pervasive and important concept in (modern) mathematics''
- “important general theme having manifestations in almost every area of mathematics''
-
dualities appear in many places in mathematics, e.g.
- dual of normed space is space of bounded linear functionals on the space (page~here)
- dual cones and dual norms are defined ( & )
- can define dual generalized inequalities using dual cones ()
- can find necessary and sufficient conditions for $K$-convexity using dual generalized inequalities ()
- duality can be observed even in fundamental theorem for Galois theory, i.e., $G(K/E) \leftrightarrow E$ & $H \leftrightarrow K^H$ ()
- exist dualities in continuous / discrete functions in time domain and continuous / discrete functions in frequency domain, i.e., as in Fourier Transformation
- However, never fascinated more than duality appearing in optimization, e.g.,