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Month: September 2018

Beyond Cookbook Mathematics, Part 1

Beyond Cookbook Mathematics, Part 1

This post is due to the requests of several independent engineers and programmers. They expressed disappointment at their mathematics education and its failure to impart a deeper understanding of the formulas and algorithms they were taught to use. 

This also reflects my observations of teaching university mathematics over the years. I started as a TA (and frequent substitute lecturer) in 2008, and have taught all levels of calculus, basic statistics, and advanced undergraduate statistics thus far. I’ve certainly noticed even since 2008 a de-emphasis on proofs and “why” in favor of more examples, applications, and formulas in general. In fact, proofs were passed over in lectures in calculus courses meant for general engineers because “there wasn’t enough time”, or it was not considered something engineers needed to know. 

This attitude is actually fairly recent. Many of the older calculus books in my library were written for engineers in an undergraduate program, and these books are quite proof-heavy. Some examples are F. Hildebrand’s Advanced Calculus for Engineers (1949), Tom Apostol’s Calculus (Volumes 1 and 2), and R. Courant’s Differential and Integral Calculus (1938), to name a few. For a more modern text that still has some proof treatment, the 10th edition of Calculus: One and Several Variables, by Salas et al is a good resource. I used this one both as a student at Georgia Tech and an instructor.1. However, when I taught at University of Texas at Arlington as a grad student from 2013-2015 and then Foothill College in early 2018, the texts they chose to use was woefully inadequate to suit a college-level calculus course; proofs were nonexistent, and reasoning was thrown away in favor of contrived examples. The course was designed to steer engineers away from any proofs or thorough reasoning, showing the experience is somewhat widespread.2

I do not want to discuss the reasons for this departure; this isn’t an education site. What is important now is to discuss how to satisfy the desire of an engineer or other highly technical person using various mathematical topics/formulas to develop a deeper understanding of what he or she is doing. It wouldn’t be particularly helpful to simply suggest acquiring books and reading the proofs. 

The best thing an education can give is the ability to teach oneself through developing methods of logical thought and creativity. There are books on formal logic and mathematical proofs circulating, but they can be a bit daunting to start with, as they are quite abstract and discuss mathematical logic and proof theory in general.3What I’ll give here can be perhaps considered a friendly introduction as to how to begin using mathematical proofs to facilitate understanding of material. 

Start with what you know

Most of you who are reading this have likely taken a calculus course or two, and probably some basic linear algebra/matrix theory (especially if you’re in computer science). You’ve been exposed to limits, differentiation, continuity, determinants, and linear maps. You know more math than you think. We’ll use this material to begin learning how to deconstruct mathematical statements and arguments in order to understand how the pieces all fit together. (It’s really not unlike diagramming sentences.)

Pick a topic you know well. That way we’re not trying to introduce new material and learn how to read proofs at the same time. Dust off your old calculus book, or differential equations book. 

Definitions

Definitions are the most important thing in mathematics, and perhaps the most ignored by those using it. “Continuous” has a meaning. “Differentiable” has a meaning. Spending time to really understand the definition of a mathematical term will provide an unshakable foundation. 

Example: Let’s take something visual: the degree of a vertex in a graph. 

You might see this definition of degree of a vertex:

Def. 1: The degree of a vertex v in a graph G is the number of edges connected to v

Here’s a graph from one of my research articles. I realize this is a digraph, but we can still say that every node except the first is an end-vertex.

This is fairly intuitive and straightforward. One way to go about understanding this definition is to find other equivalent ways to express it. For example, we know that if there’s an edge sticking out of a vertex v, there must be something on the other end of the edge. (Graphs don’t allow dangling edges.) Thus, we might reframe this definition as 

Def. 2: The degree of a vertex v is the number of vertices adjacent (connected to) v

If we go one step further and collect all the vertices adjacent to v into a set or bucket, and name that bucket the neighborhood of v, we can write one more equivalent definition of the degree of a vertex.

Def. 3: The degree of a vertex v is the number of vertices in (or cardinality of) the neighborhood of v. (Formally, mathematicians would say that the number of elements in a set is the cardinality of the set.)

Notice what we’ve done here. We’ve taken one simple definition and expressed it three equivalent ways. Each one of these gives us a slightly different facet of what the degree of a vertex is. We can look at it from the perspective of edges or vertices.

Let’s take this definition and use it in another definition.

An end-vertex or pendant vertex in a graph is a vertex of degree 1. 

New definition using our previously defined degree. But what does it really tell us here? Can we visualize this? If a vertex only has degree 1, then we know that only one edge sticks out of that vertex. Equivalently, we can also say that it has only one neighbor vertex adjacent to it. The size of its neighborhood is 1. We now can picture an end-vertex quite nicely. 

Using Definitions

Many applications rely on checking to see if a definition is satisfied. 

  • Is f(x) = \sin(x) continuous?
  • Do I have any end vertices in my network?

Here we are taking a specific example and looking to see if a definition is satisfied, typically because we know that (due to theorems) we get certain properties we either want (or maybe don’t want) if the definition is satisfied. 

For example, network engineers like to have resilient networks. By “resilient”, I mean that they’d like to be able to tolerate a link failure and still be able to send information anywhere on the network. Intuitively, it would be really bad if a particular link failure isolated a node/switch so that no information could reach it. 

Let’s try to frame that in mathematical terms. A network can be drawn as a graph, with circles representing nodes/switches/computers/whatever, and edges between nodes representing the physical links connecting them. We want to design a network so that a single link failure anywhere in the network will not isolate a node. 

We can mathematically represent a link failure by the removal of an edge. So we can take our graph representing our network, and start testing edges by deleting them to see if a node ends up isolated with no edges emanating from it. Or…we could return to a definition from earlier and think about this mathematically. 

If a single link failure isolates a particular node, then that means only one edge sticks out from it. That means that node has degree 1, by our first definition of degree. Thus, we may now conclude that this node also fits the definition of an end-vertex. 

Moving back to the practical space, we conclude: a vertex can only be isolated via a single-link failure if it’s an end-vertex. We can also write the statement the other way:  If a vertex is an end-vertex, then the deletion of its incident edge isolates it. 

Now, thanks to these definitions, and our understanding of them, we can find a way to spot all the end-vertices in a network. Since we have multiple ways of looking at this problem, we can find the way that suits us best.

Using definition 1, we can count the number of links from each node. Any node that has only one link connected to it is an end-vertex, and the failure of that link will isolate that node. 

Using definition 2, we can count the number of adjacent neighbors, especially if we have a forwarding table stored for each node. If any node only has one other node in its table, it’s an end-vertex, and the removal of the link connecting the two nodes would isolate our vertex. 

I used a fairly visual, practical definition and example here. Other definitions in mathematics can get fairly involved; I’ve spent hours simply picking apart a definition to understand it. But the strategy doesn’t change. The first step in developing a deeper understanding of mathematics is to pay attention to definitions–not just what they say, but what they mean. The next article will discuss how we use definitions to write theorems and understand proofs. 

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The Hathlor Classification System

The Hathlor Classification System

Many researchers have their own libraries, and The Math Citadel is no different. Both Jason and I have spent many hours buried in the shelves of bookstores new and used, the stacks of university library shelves, and the rows of books in public libraries across four states now. During this time, we’ve amassed our own collection of new, used, out-of-print, and rare math and engineering texts. Some we acquired as part of our formal studies, most are acquired tangentially, and sometimes haphazardly. 

As we’re both very interested in organization and cataloguing in general, we noticed the different methods libraries tend to use to shelve their vast collections, particularly the technical books. Most university libraries we’ve visited utilize the more modern Library of Congress classification system, while public libraries tend to be a mix of the Dewey Decimal System and the Library of Congress system, varying by library. 

What we ultimately realized in our time exploring these shelves is that both of these methods for cataloguing books are insufficient for our needs. One of our biggest desires in a classification system was a way to determine some of the content of the text in a database or card catalogue listing, without necessarily having to pull the book from the shelf (which may be in storage). One other shortcoming we noticed is that neither system accounts for a “continuum-style” classification. As an example, many abstract algebra texts contain chapters on number theory, but some do not. Perhaps two different abstract algebra texts have different focus topics; one may focus more heavily on group theory, while another spends much of its pages discussing advanced matrix theory. The titles do not always reflect this, nor do the classification codes used by either the Dewey Decimal System nor the Library of Congress system.

So, we invented our own. Born of months of discussion and trial, we finally settled on a new one that takes the best aspects of both original systems, and also expands to include additional information. We’ll begin by briefly describing the two original classification systems, then describe our new Hathlor system. Our system is customizable to suit anyone’s needs, and we’re excited to share a passion project of ours. 

The Dewey Decimal System

This system is the one most are familiar with from school libraries. It was created by Melvil Dewey in 1876, and introduced a relative index that allowed for library expansion. Wikipedia has a great account of the history and development of the system, so we’ll not dive too deeply here; we’ll merely describe it. The system separates books by discipline, ten major classes overall, each divided into ten sections. It’s a hierarchical classification system that endeavors to classify a book as specifically as possible by adding classes after the decimal points. 

Wikipedia gives an example of a book on Finsler geometry:

  1. The book’s broadest class is 500-Natural Sciences and Mathematics 
  2. Next, it’s specifically a mathematics text (not physics or biology), which is the first section, so we are at 510
  3. Within mathematics, its topic is geometry, the 6th topic listen, thus 516
  4. Now we may move to decimal places to further classify as specifically as we like. This text is an analytic geometry, the third subtopic in geometry. Thus, we may now list the text as 516.3
  5. We wish to further classify, so among analytic geometries, this one discusses metric differential geometries, the 7th subdivision of analytic geometry. If the library classes the books to two decimal places, the code is 516.37
  6. We may even wish to subdivide metric differential geometry and move to three decimal places, finishing with 516.375–its final classification code.

Pros

This system allows very fine granularity and specificity in classification; one could decide to divide further and move to four, five, or more decimal places. It provides a total ordering, so shelving is sensical. 

Cons

One thing I personally don’t like about this system is the length the codes can reach without really offering tons more information. It breaks single topics down into very fine divisions if desired, but one would need to know the possible subdivisions to look at the code of a book and discern all that information. One other issue is that the book is restricted to one and only one class. Many mathematics and engineering texts don’t fit neatly into one class, division, or subdivision. A book on algebraic statistics would span abstract algebra and statistics, yet under the Dewey Decimal system, I’m not only forced to classify it under one or the other, but the “other” I don’t use disappears, and the text can get lost. A statistician might not always just browse the algebra shelf, yet this text might be of interest to him. 

Library of Congress

The LCC was developed by the Library of Congress, and is the most common system deployed in academic and research libraries in the United States. Herbert Putnam invented the system in 1897. For additional history, check here. This system has a larger number of broad classes, and uses letters to denote general classes and narrower topics. There are 21 broad classes, using all letters of the alphabet excepting I, O, W, X, and Y. Each broad class is divided into subclasses, though the number of subclasses may vary. For example, Class B (Philosophy, Psychology, and Religion) is divided into 15 subclasses, one of which is BV-Practical Theology. Class Q (Science) is divided into 12 subclasses, one of which is QA-Mathematics. 

Underneath these broad classes is a narrower topic of 4 digits, then a cutter number that represents the author, corporation, or title, and finally the last line notes the year of publication. 

As an example, from the University of Mississippi library

Title: Price Control under Fair Trade Legislation

Author: Ewald T. Grether

  1. This text falls under HF (Social Sciences → Commerce).
  2. The library codes it 5415, signaling Business → Marketing → General Works
  3. Next, the cutter number for Grether is .G67, and finally, the year of publication is 1939. 

In summary, the LCC call number would look like this:


HF
5415
.G67
1939


 

Pros:

There are more classes than the Dewey Decimal system.

Cons:

We thoroughly dislike this system for a number of reasons. The primary concern is that looking at the call number on the spine of the book tells very little useful information, and gives some information that is irrelevant to evaluation of the content, such as the cutter number and publication year. It also shares the same issue as the Dewey Decimal system in that multiple subjects contained in a text are lost by forcing the text into one class.

In both of these systems, it’s difficult to determine where some more multidisciplinary or multitopic texts might be found without knowing the exact name of the text. It would be impossible to use either of these systems to locate some text that discusses graph theory and algorithms, but also mentions applications to chemistry. It is with this motivation that we devised our new system.

The Hathlor Classification Codes

The main goal of the new system was to allow a user without intimate familiarity with the minutest details of the scheme, but with knowledge of the specific subjects, to discern information from the spine or given code of a text. Neither LCC nor DDC give this benefit. We describe our system and reasoning here. 

Hierarchical Yet Lateral

We have divided material in the library into a hierarchical set of groups. The largest group a text can be a member of is the Subject. Our library currently contains 5 subjects, each given a sensical one or two-letter code:

SubjectCode
ChemistryC
Computer ScienceCS
EngineeringE
MathematicsM
PhysicsP

Within each Subject are a number of Topics, where each topic has a three letter code. Again, these codes were designed sensibly, so that one can infer the topic from the three letter code (as opposed to needing to know that QA is Mathematics in the LCC system, for example). To illustrate, the table below shows how we divided Mathematics into ten Topics

TopicCode
Applied and Engineering Mathematics AEM
AlgebraALG
AnalysisANA
Differential/Integral EquationsDIE
Discrete MathematicDSC
FundamentalsFUN
GeometryGEO
Number TheoryNUT
Statistics/Probability TheorySPT
TopologyTOP

Finally, within each Topic are a varying number of Subtopics. Texts can contain many Subtopics, and in the spirit of the MAC address, we give a variable length binary code that indicates which Subtopics are contained. For example, the Topic of Fundamentals contains four Subtopics: Calculus in bit 1, Logic in bit 2, Set Theory in bit 3, and Trigonometry in bit 4. 

A general code will look like this:

Subject – Topic.XXXX

where the single-letter subject code is given, the three letter topic code is given, and the X is an indicator of whether or not a particular subtopic is contained in the work. Note that the number of X’s may vary by topic. 


Example: Strang’s Linear Algebra and its Applications is a basic linear algebra text covering fundamental matrix theory, Gaussian elimination, linear systems, orthogonal projections, determinants, eigenvalues and eigenvectors, and positive-definite matrices. Subtopics in Algebra are Abstract Algebra, Category Theory, Linear Algebra, and Matrix Theory, in that order (alphabetically). 

To code this book, we note that its subject is Mathematics, with topic Algebra, and containing linear algebra and matrix theory, but not Abstract Algebra, or Category Theory.

Thus, the Hathlor code for the book is 

M-ALG.0011


Cross-Topic Texts

Many texts in mathematics contain material that spans topics. Our codes reflect this, and may be extended to include those additional topics in descending order of importance or inclusion to the work, separating the Topic.Subtopic codes by a colon. One may do this as many times as necessary to encapsulate all appropriate topics. Thus, we see that now the code may take the form of 

Subject-PrimaryTopic.XXXX:SecondaryTopic.XX:…


Example: Apostol’s Linear Algebra: A First Course with Applications to Differential Equations is primarily a linear algebra text, but does have a significant discussion of differential equations. Thus, the subject is still clearly Mathematics, but now we have a primary topic of Algebra and a secondary topic of Differential/Integral Equations. Differential/Integral Equations has three subtopics: integral equations, ordinary differential equations, and partial differential equations. Apostol only addresses the second of these in his text. 

Thus, the Hathlor code for this book is 

M-ALG.0010:DIE.010


As a note, one can have as many topics indicated as necessary. Joyner’s Adventures in Group Theory is an engaging, multi-topic text that touches on many fundamental areas of mathematics, presenting abstract algebra matrix theory, logic, set theory, and a tiny bit of graph theory through the lens of recreational mathematics. Its Hathlor code is M-ALG.1001:FUN.0110:DSC.0100.

Final Generalization: Cross-Subject Texts

We also noticed in reading many technical works that authors may span entire subjects. Some texts in analysis focus only on the pure mathematics, and others discuss applications of analysis to physics or electrical engineering. The Hathlor codes account for this as well, giving the book a primary subject and as many subsequent subjects as necessary. One finishes the entire subject-topic-subtopic classification prior to moving onto the next, noting a new subject by the set <> of symbols. The final general Hathlor code thus takes on the following form:

PrimarySubject-PrimaryTopic.XXX:SecondaryTopic.XXXX:…<>SecondarySubject.PrimaryTopic.XXXX:…


Example: Hohn’s Applied Boolean Algebra discusses Boolean algebra and some of its applications, particularly in electrical engineering and relay circuits.The text is primarily a mathematical treatment, but it would be foolish not to note its electrical engineering motivations. Thus is spans the subjects of mathematics and engineering. Its Hathlor code is

M-DSC.1000<>E-ELE.1


Shelving: Simple Lexicographic Order

Shelving books by the Hathlor classification code is done in lexicographic order, moving left to right, maintaining a simple, unambiguous shelving system. Thus, all chemistry books are shelved before all computer science books. Within each subject, we move by alphabetical order by topic according to the three letter codes. All (Primary) Algebra (ALG) books come before all (primary) Analysis (ANA) books. Within each topic, we shelve by the binary indicators. Thus, M-ALG.1000 comes before M-ALG.01001. If the text contains secondary/tertiary topics, we file those after those with no subsequent topics, the same way that “can” precedes “candle” in a dictionary. The secondary topics are filed in alphabetical order, and so forth. The texts that span subjects then follow those that do not span subjects, still shelved by their primary subject. 

 

Why Reinvent the Wheel?

As mentioned before, we felt that the current systems in use both omit useful information regarding the content of the works and add extra information a user doesn’t typically care about, such as the LCC’s cutter number. In addition, a researcher or browser may simply have a general idea of the types of things he would like a text to contain, but neither the DDC nor the LCC provides a simple way to search for such things. Ours provides a way to search via a simple regular expression query, returning a set of texts previously unknown to the user that fit the subjects, topics, and subtopics he seeks, particularly books that contain all he seeks. 

Though neither of us have formally worked in a library, nor to either of us hold any formal degrees in library science, we both have a deep and abiding passion for reading, mathematics, organization, and classification. Beyond that, our collection now spans over 220 books, so we definitely needed a better way to shelve our library. We both think it’s a pity that very few technical libraries outside of universities exist anymore, particularly in the private sector. Company libraries for engineers and scientists such as the (highly underrated and unknown) library at NASA Ames Research Center should be cared for, revitalized, and restarted. We’re happy to help with that. 

 

–Rachel Traylor and Jason Hathcock

 

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