Some Thoughts on Dynamical Systems

I’ve resurrected this post from my draft graveyard after chatting with Chas Leichner about the lightly related notion of domain theory, and the interaction between computation and topos theory.

What is fiber bundle dynamics?

A fiber bundle expresses global phenomenon in terms of the output of local data.
Analogously, a geometric multi-scale modeling technique, aptly named fiber bundle dynamics, expresses macroscale data in terms of the output of microscale models.

For example: if the microscale model is molecular dynamics and the macroscale model is continuum hydrodynamics, then this formula is the Irving-Kirkwood formula that expresses stress in terms of the atomistic data from molecular dynamics. If the microscale model is replaced by Brownian dynamics, then this link is replaced by Kramer’s expression, etc.​

“…in concurrent coupling methods, one does not compute the constitutive relation within the full range of these variables – only the values that actually occur in the simulation are needed, and these might be a very small subset of the entire range.”
Ren, Seamless Multiscale Modeling of Complex Fluids via Fiber Bundle Dynamics

It’s interesting to think of (1) dependent types as fibrations and (2) a multi-scale model as a generalized transition system. Continue reading Some Thoughts on Dynamical Systems

A Unifying Language

Mathematics is a huge subject.

Category theory is one area of mathematics dedicated to exploring the commonality of structure between different branches of mathematics.

Categorical language allows us to ascend a layer of abstraction, and recognize the obvious underlying principles that guide seemingly unrelated concepts. Generality facilitates connections.

What is a category?

A category $C$ consists of:

  1. a class of objects $Ob(C)$
  2. For every ordered pair of objects $X$ and $Y$, a set $C(X,Y)$ of morphisms with domain $X$ and range $Y$ [$C(X,Y)$ is possibly empty] Note: $f \in C(X,Y)$  $\equiv$  $f : X\rightarrow Y$   $\equiv$  $X \overset{f}{\rightarrow} Y$.
  3. For every object an identity morphism $Id_x \in C(X,X)$.
  4. A composition law $$C(X,Y) \times C(Y,Z) \rightarrow C(X,Z)$$ $$(g,f) \rightarrow f\cdot g$$

The concept of composition follows naturally from the definition of path equivalence in graph theory: Two paths with the same source as destination are equal. Screenshot from 2014-08-11 00:10:36   Additionally, categories must satisfy the laws of associativity and identity.

Category Laws

Categories must obey 2 laws:

  1. Composition must be associative:Screenshot from 2014-08-11 00:09:55
  2.  Every object $a$ in $C$ has a morphism $id_a$ which is equivalent to a loop in graph theory. 431px-Self-loopThe identity morphism $id_a$ connects the object $a$ to itself, $$id_a: a \rightarrow a$$.

Screenshot from 2014-08-11 00:10:15

Two paths are equal if the source and destination of the paths are equal. With this in mind, we can represent the category laws of identity and associativity with diagrams.

Follow the arrows, recall that we write function composition backward! If we traverse $f$ then $g$, it is the convention to write $g \circ f$

Screenshot from 2014-08-11 10:06:43                                     Screenshot from 2014-08-11 10:06:54

Examples:

Groups, together with group homomorphisms, form a category (we will discuss these next lecture for those who have not danced with abstract algebra).

Each of the natural numbers is a category:

Screenshot from 2014-08-18 16:34:05

Categories have some nice properties

Any property which can be expressed in terms of (category, objects, morphism, and composition):

  • Dual: $D$ is $C$ with reversed morphisms
  • Initial: $Z \in obj(C)$ s.t. $\forall Y \in obj(C)$, #$hom(Z,Y) =1$. In other words: an object is initial if there exists a unique morphism from that object to any other object in $C$.
  • Terminal: $T \in obj(C)$ s.t. $T$ is initial in the dual of $C$
  • Functor: Structure preserving mapping between categories

Homomorphims Between Categories: What the func is a functor?

Screenshot from 2014-08-11 00:24:22

An example of associating morphisms from $C$ to $D$ with the functor $F(C)$.

Screenshot from 2014-08-11 00:24:59

A Reflection on the Unfication of Familiar Concepts

A few categories you have likely encountered before without recognizing it:

  • Set (sets and functions)
  • Vec (vectorial spaces and linear transformations)
  • Top (topological spaces and continuous maps)
  • Grp (groups and homomorphisms) — we will be discussing these next lecture for those who have not danced with abstract algebra.
  • Ab (abelian groups and homomorphisms)
  • Hask (Haskell types and functions)
  • Cat (categories and functors)

A categorical key to highlight some of the relationships between structure preserving maps:

Untitled picture

Here are some more advanced examples for those who feel groovy and algebraic:

  • $R$-Mod (R-modules and homomorphisms)
  • $Gr_R$ ($\mathbb{Z}$-graded $R$-modules and graded $R$-module homomorphisms)

arrows

Sources

Introduction to Haskell

Respect Our Work.

Already, criticism has be thrown at me for deciding to focus on pure mathematics. I am often told (with the best intentions) that I should go back to doing applied math/engineering.  I’m guessing that other people who’ve decided to study theoretical subjects suffer from a similar lack of respect.

Hopefully, this post will be helpful for you, my friends. Perhaps, to send to your families, engineers, etc. to save you time when you are sick of answering the same questions — laced with misunderstanding. Perhaps, to find consolation and recognize that you are not alone.

This post (admittedly written with the muses of frustration and maths evangelism) can be generalized to theoretical endeavors of any flavor. 

Why do you want to study theoretical science/math?

Image Source
Image Source

Why do I want to study pure math? The same reason that the scientist is providing above.

There is an itch in minds of the analytic and aesthetically inclined that mathematics scratches like no other.

I encourage you to read A Mathematician’s Apology and What is it Like to Understand Advanced Mathematics?.

What if you aren’t good at math?

It doesn’t matter if you’re good at mathematics or not if you enjoy doing it and have people willing to help you learn.

You’ve picked something that you enjoy doing, and you’re rolling with it.
Thereafter, if you do it for long enough, you’ll become good at it.

Don’t you want to help people?

Because of its abstractness, mathematics is universal in a sense that other fields of human thought are not. It finds useful applications in business, industry, music, historical scholarship, politics, sports, medicine, agriculture, engineering, and the social and natural sciences. The relationship between mathematics and the other fields of basic and applied science is especially strong.

Source

Here’s an exploration of Which Mathematical Ideas Have Done The Most To Change History?.

How can math make you a better person?

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Math provides you with powerful intuitions that can improve your daily life:

0. Maths develops the logical mindset, assiduous and aesthetic values needed to implement regular concise conversation and clear explanation. It teaches us to not accept hand-wavey proofs, but to derive them ourselves from first principles.

1. The ability to analyze a problem as a structure, work it out step by step and solve it generally translates between fields, allowing one trained in maths to easily move to other puzzle-based disciplines. For example, a maths person can transfer their skillset to computer programming by seeing code and programming languages as a collection of structures and rules governing their interactions.

2. An understanding of maths allows us to appreciate our environment by observing patterns and connections between objects that we wouldn’t otherwise see. After studying a field such as Model Theory, you begin to see deep connections (across fields of study) in the abstractions of seemingly unrelated concepts. Limiting yourself to the current ideas of one discipline makes original research unnecessarily difficult. I find that most original research is really just connecting past ideas into a new idea that is more than the sum of its parts.

3. Some visually oriented mathematicians overlay representations of abstract concepts on their environments as they go about daily life. This visual overlay of abstract stimuli increases the ability to appreciate things in their own right and builds visual intuition. However, overlaying mathematics on life differs significantly from making a mathematical model of your surroundings: the latter is far more concrete. Overlaying connections on stochastic stimuli is an enrichment to perception, whereas modeling stimuli is separate from it.

Visual intuition is important for research on the leading edge of science and mathematics (especially in geometry). Formalisms come after a topic is well understood.

4. The ability to describe mental pictures in a generalized symbolic fashion is incredibly powerful. The symbolic language of mathematics allows us to represent concepts generally without extraneous information, which permits each viewer to perceive the equations according to which mode of translation best suits their thought processes.

At this point, I’ll cease indulging my evangelism and go back my studies. 

College, Would You Like Fries With That?

I enjoy songwriting and have recently gotten into playing Third-Wave ska on my trombone.The basic premise of ska is to play an incredibly upbeat tune with shocking/macabre lyrics.

Here are the lyrics to a silly ska song I wrote in class. Note that I’m happy with my university experience; this song presents the point that university isn’t for everyone.

College? Would you like fries with that? by Catherine Ray

You’ll struggle through your classes, and they’ll let you out of school
Clutching a degree you’ve wasted your life to get, you fool
Up next, the corporate overlords will laugh and watch you squirm
They’ll lead you on in interviews, in the end you’ll just be spurned
Run with your tail between your legs,
maybe McDonald’s will accept your unskilled naivete.
Life has revealed its disguise
Yell, “I wish I’d Realized!”

Writing insightful term papers on the Third Reich,
you never thought you’d be one of those guys who quit their work on strike
First you paid thousands just to go to lectures and to class
All of that money and time, for 9 to 5, for a chair and a cubicle, to sit on your ass
The next 20 years consist of watching TV and arduous jobs, paying off your student loans
Working over time for mortgage, so you can pay for your worthless home
After work, go to the mirror.
Just look yourself in the eyes,
Yell. “I wish I’d realized!”

The goal of college isn’t to simply earn a scroll and a title
Get something concrete from it – a job, a pay raise, a promotion
Look back now – Clearly you’ll see, your only profit was debt and negative emotion.
Listen up, my friend. Getting some degree
won’t cement your name in most minds, surroundings, nor history
Education is awesome for your outlook on life, real life experience is the fastest way.
The assumption that education must be IRL – Get used to living in meatspace
Be an autodidact, it can’t be all that hard.
Get out your laptop, a pen and some flashcards
Learn a language, free eBooks, a chapter, or if you’re lazy, a word a day
All along there was Wikipedia, free online courses, to your dismay
It’s too late for you – unless, you’re a chrononaut
Life ain’t something you can revise.
Yell. “I wish I’d realized!”