Editing User:Anisomorphism

Warning: You are not logged in. Your IP address will be publicly visible if you make any edits. If you log in or create an account, your edits will be attributed to your username, along with other benefits.

The edit can be undone. Please check the comparison below to verify that this is what you want to do, and then publish the changes below to finish undoing the edit.

Latest revision Your text
Line 1: Line 1:
<div class="nav-container">
Β  Β  <div class="nav-button">
[[read]]
Β  Β  </div>
Β  Β  <div class="nav-button">
[[Mechanics (Book)]]
Β  Β  </div>
</div>
I do math
I do math


= Algebraic Geometry of Computing =
Finite state machines appear in a variety of instantiations: mechanical, electronic, fluidic. The physical mechanisms involved necessitate that the design is described by differential equations, but ultimately the manipulation of abstracted "logical" states is the final goal. Thus we can describe the architecture of a general finite state machine with <math> \mathbb{Z}/2\mathbb{Z} </math> algebra (or other finite rings too).
=== Gates ===
Typically you will see a logic gate defined by its values on all combinations of inputs as a "truth table":
{| class="wikitable" style="margin:auto"
|+ AND
|-
! x !! y !! x AND y = AND(x,y)
|-
| 0 || 0 || 0
|-
| 0 || 1 || 0
|-
| 1 || 0 || 0
|-
| 1 || 1 || 1
|}
And statements written with logical connectives: <math> (x\and y)\or z = OR(AND(x,y),z) </math>
<br>
Along with distributive laws: <math> (x\and y)\or z = (x\or z)\and(y\or z) </math>, <math> (x\or y)\and z = (x\and z)\or(y\and z) </math>
<br>
De Morgan's laws: <math> \neg(x\and y) = (\neg x)\or (\neg y) </math>, <math> \neg(x\or y) = (\neg x)\and (\neg y) </math>
<br>
All of which apply to more complicated sentences rather than just individual variables. These laws along with commutative and associative laws are sufficient to evaluate and simplify any general logical expression, however we contend that this is the wrong language for computing and makes other important aspects - the dynamics and algebra - obscure.
<br>
There is one thing we can extract from logical connectives before moving on. The disjunctive normal form allows us to read truth tables and directly translate them into connective formulae which we can use later. Let us look at a different example which will help us escape the artificiality of AND and OR.
{| class="wikitable" style="margin:auto"
|+ XOR
|-
! x !! y !! XOR(x,y)
|-
| 0 || 0 || 0
|-
| 0 || 1 || 1
|-
| 1 || 0 || 1
|-
| 1 || 1 || 0
|}
XOR is only "true" or 1 when x or y but not both, are 1. Disjunctive normal form says that we can view the x, y entries as unary operators which return the input with no change, combine these as given on the lines which evaluate to 1, and take the OR of all of them for the total connective form of the truth table. Here is the third line: <math> x\and\neg y</math>.
<br> The total is:
<math>
\begin{align*}
&(x\and\neg y)\or (\neg x\and y) \\
=&(x\or(\neg x\and y))\and(\neg y\or(\neg x\and y)) \\
=&(x\or\neg x)\and (x\or y)\and(\neg y\or\neg x)\and(\neg y\or y) \\
=&(x\or y)\and (\neg y\or\neg x) \\
=&(x\or y)\and\neg (y\and x)
\end{align*}
</math>
<br>
This process can be viewed as a sum of "elementary functions" which are only 1 on one line each, and building a general function/table.
= Read prototype =
[[File:Linmechfieldsfolds.jpg|thumb|alt=Linear algebra, Mechanics, Relativity and Fields, Differential Geometry|The starter pack to physics and differential geometry]]
[[File:Linmechfieldsfolds.jpg|thumb|alt=Linear algebra, Mechanics, Relativity and Fields, Differential Geometry|The starter pack to physics and differential geometry]]




Our point of view is that the texts typically used in physics and especially mathematics degree tracks are window dressing for the real job of being a mathematical physicist or even an engineer. Excellent texts meet a certain standard we set here; Texts should be concise to respect the reader's time and occupations, interdisciplinary, at least relating mathematical tools between areas of mathematics:
Our point of view is that the texts typically used in physics and especially mathematics degree tracks are window dressing for the real job of being a mathematical physicist. Excellent texts meet a certain standard we set here; Texts should be concise to respect the reader's time and occupations, interdisciplinary, at least relating mathematical tools between areas of mathematics:
* Lang's algebra text contains examples and applications in geometry and number theory throughout
* Lang's algebra text contains examples and applications in geometry and number theory throughout
* Vaisman emphasizes the typically algebro-geometric method of sheaves in a differential geometry setting and to develop the theory of multiple sorts of manifolds
* Vaisman emphasizes the typically algebro-geometric method of sheaves in a differential geometry setting and to develop the theory of multiple sorts of manifolds
Line 75: Line 10:
* The algebraic topology texts are not "pure" either - focusing on applications to differential or algebraic geometry, and many more. Β 
* The algebraic topology texts are not "pure" either - focusing on applications to differential or algebraic geometry, and many more. Β 


Thus, the structure of this book list will be centered around core topics in theoretical physics which are already given direct connection to technology and reality, and the mathematics that follows from the theory rather than simply chasing popular formalisms. Future iterations will make an effort to connect with more computational content, such as that seen in representation theory or Olver's text on applications of Lie groups. This pertains to our last criterion that there should be some elementary aspects in a text - showing the translation of the abstract machinery into basic computations to make the relationships with other areas even more transparent.
Thus, the structure of this book list will be centered around core topics in theoretical physics which are already given direct connection to technology and reality, and the mathematics that follows from the theory rather than simply chasing popular formalisms. Future iterations will make an effort to connect with more computational content, such as that seen in representation theory or Olver's text on applications of Lie groups.




Line 93: Line 28:
The '''Aspirational''' section contains some of the big ideas, which may be repeated from earlier but deserve emphasis. These are the triumphs of mathematics, peaks that everyone deserves to climb.
The '''Aspirational''' section contains some of the big ideas, which may be repeated from earlier but deserve emphasis. These are the triumphs of mathematics, peaks that everyone deserves to climb.


== Basic Mathematics ==
== Fill in Gaps ==
<div class="flex-container" style="clear: both;">
<div class="flex-container" style="clear: both;">
{{BookListing
{{BookListing
Line 100: Line 35:
| title = === Basic Mathematics ===
| title = === Basic Mathematics ===
| desc = Review of arithmetic, algebra, trigonometry, logic, and geometry by Serge Lang.
| desc = Review of arithmetic, algebra, trigonometry, logic, and geometry by Serge Lang.
}}
{{BookListing
| cover = Shilov Linear Algebra Cover.jpg
| link = Linear Algebra (Book)
| title = === Linear Algebra ===
| desc = Linear algebra of linear equations, maps, tensors, and geometry by Georgi Shilov.
}}
}}
{{BookListing
{{BookListing
Line 114: Line 43:
}}
}}
</div>
</div>
== Landau ==
== Royal Road to Differential Geometry and Physics ==
<div class="flex-container">
<div class="flex-container">
{{BookListing
| cover = Lawvere Sets for Mathematics Cover.jpg
| link = Sets for Mathematics (Book)
| title = === Sets for Mathematics ===
| desc = Categorical approach to set theory by F. William Lawvere.<br>
'''Backbone reference:'''
* [[{{FULLPAGENAME}}#Set Theory and Metric Spaces|Set Theory and Metric Spaces]]
* [[{{FULLPAGENAME}}#Foundations of Analysis|Foundations of Analysis]]
}}
{{BookListing
| cover = Shilov Linear Algebra Cover.jpg
| link = Linear Algebra (Book)
| title = === Linear Algebra ===
| desc = Linear algebra of linear equations, maps, tensors, and geometry by Georgi Shilov.
}}
{{BookListing
{{BookListing
| cover = Landau Course in Theoretical Physics V1 Cover.jpg
| cover = Landau Course in Theoretical Physics V1 Cover.jpg
Line 121: Line 65:
| title = === Mechanics ===
| title = === Mechanics ===
| desc = Classical mechanics of particles by Lev Landau.<br>
| desc = Classical mechanics of particles by Lev Landau.<br>
<div class="flex-container" style="clear: both;">
'''Prerequisite:'''
{{BookListing
* [[{{FULLPAGENAME}}#Calculus|Calculus]]
| cover = Mechmath.jpg
'''Backbone reference:'''
| link = Mechanics (Book)
* [[{{FULLPAGENAME}}#Ordinary Differential Equations|Ordinary Differential Equations]]
| title = === Applications ===
| desc = Symplectic geometry and other mathematical Structures of Classical Mechanics
}}
</div>
}}
}}
{{BookListing
{{BookListing
Line 135: Line 75:
| title = === The Classical Theory of Fields ===
| title = === The Classical Theory of Fields ===
| desc = Classical field theory of electromagnetism and general relativity by Lev Landau.<br>
| desc = Classical field theory of electromagnetism and general relativity by Lev Landau.<br>
<div class="flex-container" style="clear: both;">
'''Prerequisite:'''
{{BookListing
* [[{{FULLPAGENAME}}#Linear Algebra|Linear Algebra]]
| cover = Fieldsmath.jpg
| link = The Classical Theory of Fields (Book)
| title = === Applications ===
| desc = Differential/Riemannian geometry and other mathematical Structures in Relativistic Field Theory
}}
</div>
}}
}}
{{BookListing
| cover = Landau Quantum Mechanics.jpg
| link = Quantum Mechanics (Book)
| title = === Quantum Mechanics ===
| desc = Quantum Mechanics of particles, atoms, molecules by Landau and Lifshitz<br>
<div class="flex-container" style="clear: both;">
{{BookListing
| cover = Quantmath.jpg
| link = Quantum Mechanics (Book)
| title = === Applications ===
| desc = Generalized functions, spectra of self-adjoint operators, and other mathematical Structures in Quantum Mechanics
}}
</div>
}}
{{BookListing
| cover = Lawvere Sets for Mathematics Cover.jpg
| link = Sets for Mathematics (Book)
| title = === Sets for Mathematics ===
| desc = Categorical approach to set theory by F. William Lawvere.<br>
'''Backbone reference:'''
* [[{{FULLPAGENAME}}#Set Theory and Metric Spaces|Set Theory and Metric Spaces]]
* [[{{FULLPAGENAME}}#Foundations of Analysis|Foundations of Analysis]]
}}
{{BookListing
{{BookListing
| cover = Bishop Tensor Analysis Cover.jpg
| cover = Bishop Tensor Analysis Cover.jpg
Please note that all contributions to The Portal Wiki may be edited, altered, or removed by other contributors. If you do not want your writing to be edited mercilessly, then do not submit it here.
You are also promising us that you wrote this yourself, or copied it from a public domain or similar free resource (see The Portal:Copyrights for details). Do not submit copyrighted work without permission!
Cancel Editing help (opens in new window)

Template used on this page:

Retrieved from "https://theportal.wiki/wiki/User:Anisomorphism"