- Topological Space
$X$ : The surface of the Earth. - Open Subsets
$U_i$ : These represent countries or regions on Earth, and$X$ is covered by these$n$ open subsets,$\left\{U_1, U_2, … ,U_n\right\}$ .
- For each open subset
$U_i$ (a country or region), you have a set$\mathcal{F}(U_i)$ , which is called the set of sections over$U_i$ . - The elements of
$\mathcal{F}(U_i)$ are specific functions or pieces of data defined over the region$U_i$ . E.g.:-
$\mathcal{F}(U_i)_1$ could be a function representing the temperature distribution in the country$U_i$ . -
$\mathcal{F}(U_i)_2$ could be a function representing the population density in$U_i$ . And so on for other types of data.
-
In theory,
The cardinality of the sets
-
$\mathcal{F}(U_i)$ for a country$U_i$ might be $\mathcal{F}(U_i)=\left\{ftemp, fdensity, felevation, … \right\}$, where each$f$ is a function or piece of data over that region. -
$\mathcal{F}(U_j)$ for another country$U_j$ might only include a subset of those functions, say just $\mathcal{F}(U_j)=\left\{ftemp, felevation\right\}, if population density hasn’t been measured there.
These differences reflect the real-world scenario where different regions might have different data available or might be described by different sets of functions. The key idea of sheaves is to handle these variations and allow for consistent “gluing” of this local information into a global picture.
\bigskip
Each region
An algebra over a field
-
$(A, +)$ is an abelian group (under addition), -
$(A, ⋅)$ is a vector space over$F$ , -
$(A, ×)$ is a ring, - and the operations are compatible, satisfying distributive laws.
Algebraic theory in classical universal algebra (or strictly, a presentation of an algebraic theory) consists of:
- operation symbols of specified arities
- equations
E.g. The (usual presentation of the) theory of groups:
- operation symbols of specified arities:
- an operation symbol
$1$ of arity$0$ - an operation symbol
$\blank\inv$ of arity$1$ - an operation symbol
$⋅$ of arity$2$ - equations: “the usual equations”
Let
Consider sets
This set contains all pairs
Let
That means, if for any two vectors
\begin{enumerate} \item addition: \begin{equation} \begin{split} f(\vec{u} + \vec{v}) = f(\vec{u}) + f(\vec{v}) \end{split} \end{equation}
\item scalar multiplication, i.e. homogeneity of degree 1: \begin{equation} f(c ⋅ \vec{u}) = c ⋅ f(\vec{u}) \end{equation} \end{enumerate}
By the associativity of the addition
\begin{equation} f(c_1 ⋅ \vec{u_1} + \dotsb + c_n ⋅ \vec{u_n}) = c_1 ⋅ f(\vec{u_1}) + \dotsb + c_n ⋅ f(\vec{u_n}) \end{equation}
If
Linearity meaning: can be drawn as a line.
Linear map in abstract algebra - module homomorphism
Linear map in category theory - morphism in the category of Modules over a given Ring
Function
For
Generalization of matrices to \(n\)-dimensional space. Also \(n\)-dimensional data containers with descriptions of the valid linear transformations between tensors.
See \href{https://youtu.be/tpL95Sd7zT0}{Jim Fowler: Tensor products}
0-dimensional tensor, i.e. a single number. E.g:
\begin{equation}
\begin{matrix}
1
\end{matrix}
\end{equation}
Scalar (dot) product: takes two vectors, returns a single number
1-dimensional tensor. An arrow with a length and orientation. E.g.:
\begin{equation}
\begin{bmatrix}
1 \
2
\end{bmatrix}
\end{equation}
Can represent a quantity with magnitude and direction (e.g. area):
- orientation is perpendicular to the area
- length is proportional to the amount the area
Vector (cross) product: takes two vectors
2-dimensional tensor. E.g.:
\begin{equation}
\begin{bmatrix}
1 & 2 \
3 & 4
\end{bmatrix}
\end{equation}
Matrix product (multiplication): takes two matrices, returns a matrix. Number of columns in the first matrix must be equal to the number of rows in the second matrix
E.g. 3-dimensional tensor. E.g:
\begin{equation}
\begin{bmatrix}
[ 1 & 2 ] & [1 & 0] \
[ 3 & 3 ] & [3 & 4]
\end{bmatrix}
\end{equation}
??? Tensor product
TODO
What is needed to understand
Where leads understanding of
What’s the next step? Why to study
Pure Strategy - set of decisions made with certitude.
Mixed Strategy - distribution of probabilities over some set of pure strategies.
Each player gives best response to the others. Nobody has an incentive to deviate from their actions if an equilibrum is played.
Example: Close windows to make air conditioning working:
Everybody just gives up without trying to convince others to close the window.
Example: Party organisation - follow the majority:
Majority joins - those skipping are penalized by “missed something”. \
Majority skips - those joining are penalized by “booring”.
Nash Equilibrum TODOs:
- Write action profiles for everyone (the matrix).
- Calculate optimal mixed strategies for everyone in order to get Nash Equilibrum.
- Calculate maxmin strategy and maxmin value (i.e. when the other guys do max harm to the i-th guy).
Whenever all agents agree on ordering of outcomes the social welfare function selects that ordering.
Independence of Irrelevant Alternatives:
If the selected ordering between two outcomes depends only on the relative
ordering they are given by the agents.
Dictator:
Single agent whose preferencies always determine the social ordering.
Arrows Theorem:
Any social welfare function that is pareto efficient and independent of
irrelevant alternatives is dictatorial.
Market transition
Dragan Djuric: Clojure on GPU \
Bayadera (Bayesian): very fast \
Bayesian is hard to compute, multi model, many dimensional problem, complex hyperspace \
Markov Chain Monte Carlo simulations (MCMC): difficult to parallelize \
JAGS/Stan (state-of-the-art bayesian C++ tools)
Linear Complementarity formulation
Support Enumeration Method
Every even integer
3Blue1Brown: Visualizing the Riemann hypothesis and analytic continuation
The real part of every non-trivial zero of the Zeta function
Or: \
All the nontrivial zeroes of the analytic continuation of the Riemann zeta
function
Every simply connected, closed 3-manifold is homeomorfic to the 3-sphere (Donuts)
Every problem whose solution can be quickly verified by a computer can also be quickly solved by a computer.
With
There is no set with cardinality strictly between the cardinalities of integers and real numbers. Notes: R surj P(N) (Power series - Mocninovy rad)
Every positive integer
Barber paradox is derived from Russell’s paradox.
\begin{tabbing}
Rule \hspace{7em} \= Expression
Difference \>
See \href{https://youtu.be/M8xlOm2wPAA}{Bayes’ Theorem applied to disease diagnosis} on YouTube.
\begin{table}[H]
\begin{tabular}{|l|l|l|l|l|l|}
\hline
\multicolumn{2}{|l|}{Objective Health} & \multicolumn{2}{l|}{Test result} & Outcome & Event
- Generall test correctness: 0.09 + 0.63 = 0.72 (i.e. proper results for ill + proper results for healthy persons)
- Just guessing “everybody’s healthy” gives 90% “generall test correctness” because the test is wrong only for ill patients and they make up 10% of the population.
;; +-- test positive 0.9: 0.1 * 0.9 = 0.09
;; |
;; +----- ill 0.1 --+
;; | |
;; | +-- test negative 0.1: 0.1 * 0.1 = 0.01
;; ---+
;; | +-- test positive 0.3: 0.9 * 0.3 = 0.27
;; | |
;; +-- healthy 0.9 --+
;; |
;; +-- test negative 0.7: 0.9 * 0.7 = 0.63
;; test negative, i.e. says "you're healthy" and the patient is really
;; ill (has the condition)
(/ 0.01 (+ 0.01 0.63)) = 0.015625
;; test positive, i.e. says "you're ill" and the patient is really ill (has
;; the condition)
(/ 0.09 (+ 0.09 0.27)) = 0.25
;; test negative, i.e. says "you're healthy" and the patient is really
;; health (doesn't have the condition)
(/ 0.63 (+ 0.01 0.63)) = 0.984375
;; test posivite, i.e. says "you're ill" and the patient is really
;; healthy (doesn't have the condition)
(/ 0.27 (+ 0.09 0.27)) = 0.75If event
Example: The probability it was cloudy this morning, given that it rained in the
afternoon.
Higher dimensional analogues for studying loops = (alternative to) Homotopy
groups
Simplices: analogs of triangles in higher dimensions
Loops around sphere: captuers 2-dimensional hole in the sphere
HoTT foundational framework; notions of paths in a space; equality and quivalence.
Easier translation of mathematical proofs to a programming language of proof assistants (than before).
Identity is equivalent to equivalence, in particular: equivalent types are identical.
For all types
- There’s a function
$UA: (A ≅ B) → (A = B)$ such that from a proof equivalence of$A ≅ B$ it constructs a proof of equality$A = B$ . Moreover a proof equivalence of$A ≅ B$ is equivalent to a proof of equality$A = B$ . I.e.$(A ≅ B) ≅ (A = B)$ . - it allows to create a homotopy calculus w/o introduction of differential variety and even w/o an introduction of real numbers
Entier Relativ i.e. Set of Integers