Math is a liberal arts major

True Tales From IT Land

2011-05-21T17:09:00.000-07:00

Before I joined a financial services company I was a systems programmer, but for the last 15 years I've been doing a variety of jobs in what's known as "IT" (Information Technology). During those 15 years some goofy things occurred which I will share with you reader.

The Never Ending Elevation
No, this doesn't have anything to do with an erection lasting longer than four hours. An elevation is the introduction of new software into the production environment. Think of it as software having to get a job following college. Well this particular elevation involved the company's Intranet application with some major new features: single-signon with Windows and personalization of the home page.

Note: These two are interdependent. The single-signon allowed us to identify the user and this permitted the user's preferences to be stored and recalled.

This release had been delayed for several months because the project was dependent on another team to deliver the single-signon solution. When this feature was finally ready we scheduled the elevation to start on Friday evening and finish on Saturday. From the start things did not go well. There were configuration changes that could only be tested in the production environment because it was deemed too costly to replicate the hardware. By Saturday evening it was apparent that the elevation would continue into Sunday.

This was right at the time of my life when I was suffering from severe lower back pain. The stress of a major elevation (BTW I was the Technical Lead responsible for the elevation) was compounding my physical discomfort as I followed the proceedings by an extended conference call.

By Sunday morning we had the configuration right so that the single-signon part was working. Testing of the home page preferences was giving mixed results, however. Sometimes it worked and sometimes it did not. What was particularly strange was that the tests always worked in our test environment.

We finally isolated the problem to the database -- the production database was not behaving like the test database. It was time to get the on-call DBA on the line. This was my job and I spent four hours on the phone with the guy. He couldn't resolve the problem but I couldn't let him go because my boss wasn't going to let us go home until the problem was solved.

Finally the DBA called one of his colleagues and learned that a patch was needed to the production database driver. The patch had been applied everywhere else, but not to production. Finally the patch was applied, and everything worked. The elevation had taken around 40 hours to complete. 40 hours of pure hell.

The Urge to Kill
For some reason as Technical Lead I was expected to know every operational aspect of the Intranet. In other words when there was any operational problem, I had to diagnose it and fix it. The common term for this procedure was to find the "root cause" even if this meant tracking down the cosmic particle that flipped a bit on the memory chip of the server.

Remember I said that our Intranet had single-signon? This means that if you signed on to your Windows computer you did not have to signon to the Intranet application as it already recognized your login. This was no small feat because our Intranet ran on Unix servers with all kinds of crazy network devices in the picture.

For 99.999% of the time the single-signon worked perfectly but there were occasions when out of the clear blue, the home page would present a login form. We had seen this during testing and knew that the cause was some process consuming an unexpected amount of CPU such that the server could not complete the Kerberos protocol within a given number of milliseconds.

Usually this problem wasn't noticed and everything went back to normal. But one day operations noticed it and filed a trouble ticket. I was assigned to resolve the trouble ticket. Now our Intranet involves at least a half dozen servers not to mention specialized network devices. The problem was sporadic and did not follow a pattern. Here's your haystack; have fun finding the needle.

To make matters worse I had to report status via a daily teleconference called the DSR (Daily Status Report). The DSR is designed to improve the overall operations at the firm by the relentless pursuit of "root cause." Actually it's a means to pillory Tech Leads who have no idea why the software occasionally misbehaves.

Anyway, day after day I had to report no progress. Then suddenly someone from operations was on the call and said that he had traced the problem to one of the servers that was used to store employee pictures. He further stated that the problem occurred when a certain script was run by the database group. He arranged for a rep from that area to attend the DSR the next day.

The next day a manager from the database group was on the teleconference and was confronted with the miscreant script. He said "we don't have to run that script." The case was closed without the slightest apology to me or the rest of my team who had spent hours trying to diagnose it.

The Computer is in Control Here

My last tale is one of the most mysterious that I've ever had in almost 40 of computer work. Without going into tremendous detail I'll just say that I made a configuration change in the way one server authenticated to another server using stored credential (username/password) data.

To improve operations I needed to change the credentials the server used. First I tested this on my desktop by running the first server there. The credentials change worked fine.
Next I proceeded to the production environment. To ensure it worked as I made the change I monitored the second server's log file. If the new credentials were rejected it would show immediately in the logs, and I would be forced to back out the change. But the change worked and I could see it working in the log files.

About 30 minutes passed and the next thing I know my colleague tells me that he's seeing authentication errors on the first server. So I go directly to the credentials file that I had changed and see that it had reverted my change and is now using the previous username but no password. This is truly bizarre and marks the first time I have seen the system spontaneously reject a configuration change made by a human.

A Tenuous Analogy

2011-03-12T14:37:00.000-08:00

Let me offer you three words -- inaccessible, hardship, and beauty -- and ask you if there's anything that can tie these words together. Based on my experience there are two seemingly unrelated topics that come to mind: Antarctica and higher mathematics. I was introduced to the latter as an undergraduate math major over forty years ago and again as a graduate student just a few years ago. As for Antarctica I learned about the heroic age of exploration in the 1980's, and had the fortune to visit the continent in 2010.

Inaccessible - unreachable, remote, unattainable.

Many people consider higher mathematics as inaccessible. Certainly you need an aptitude for it, but it does exist. In 2008 over 1,360 Phds* were awarded in mathematics in the United States. The interior of Antarctica is nearly inaccessible. You have to be well-prepared and well-motivated to go there. However, around 1,000 people winter in Antarctica every year including 50 at the Amundsen-Scott South Pole Station.

Hardship - a condition that is difficult to endure; suffering; deprivation.

There's no doubt that the exploration of Antarctica involved hardship. Consider the tragic ending to Robert Falcon Scott's expedition to the South Pole in 1912. I can't cite a case of death while exploring mathematics, but certainly there have been nervous breakdowns and possibly suicides.

The one thing math researchers and explorers have in common is the intense need to know the unknown. Apsley Cherry-Garrard, a member of Scott's support team, had this to say about polar exploration:

Polar exploration is at once the cleanest and most isolated way of having a bad time which has been devised... There are many reasons which send men to the Poles, and the Intellectual Force uses them all. But the desire for knowledge for its own sake is the one which really counts and there is no field for the collection of knowledge which at the present time can be compared to the Antarctic. Exploration is the physical expression of the Intellectual Passion.

I think you can substitute "mathematic exploration" for "polar exploration" and maintain the meaning of the quotation.

Beauty - the quality present in a thing or person that gives intense pleasure or deep satisfaction to the mind, whether arising from sensory manifestations, a meaningful design or pattern, or something else.

I can attest to the beauty of the Antarctic Peninsula. The snowscapes are gorgeous; the icebergs are magnificent. Many mathematicians, scientists, and engineers laud the beauty of mathematical forms that contain deep symmetry and internal elegance.

Here are mathematician Bertrand Russell's words on this kind of beauty:

Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet
sublimely pure, and capable of a stern perfection such as only the greatest art
can show.

Here substitute "icebergs" for "mathematics" and again the meaning is preserved.

It would be silly to suggest that a heroic polar explorer would make a good mathematician or vice-versa. I count myself as neither an explorer nor a mathematician, yet I am fascinated by both species. The best I can do is explore words, make an analogy, and hope the analogy works.

*Phd's awarded in the United States in mathematics and statistics in 2008.

xxxxxx

My Polar Plunge

2010-12-11T13:38:00.001-08:00

"This is stupid," I thought. I was standing in my cabin wearing a bathing suit and a thick terry cloth robe with Ocean Nova embroidered on it. Ocean Nova was the cruise ship that had taken me to Port Lockroy (64^o50'S 63^o30'W) in the Antarctic Peninsula. What I thought was stupid was taking the polar plunge into the icy waters just for the right to brag about it later in life.

I walked down to the deck where we usually exit the ship to board the Zodiac boats that had been taking us to the shore for excursions. I found more than a few of my fellow passengers were willing to take the plunge. Plus the staff were prepared to give each plunger a shot of brandy afterwards. "Okay, I'm doing this," I said to myself.

I was the fifth in line. The process consisted of putting on a life vest and tying a pull line to it. The first plunger suited up and then jumped. A cheer went up from the onlookers. Now standing sans bathrobe in approximated 35^o F temperature I start to feel the cold. Plunger No. 2 goes next. A splash followed by a cheer. Same for plunger No. 3. Then while plunger No. 4 was suiting up there was an unexpected "kersplash." The ship's head chef, a real daredevil, had just jumped off the top deck of the ship without life jacket or safety line. "What a jerk," I thought later. "Just think if the guy got injured -- no food for the passengers."

They fished out the chef and it was back to the process. There goes No. 4 and someone points out that the plungers are not looking at the cameraman taking pictures from the Zodiac at water level. "I will not going to disappoint the cameraman," I resolved.

Now it's my turn. The fricking life vest is freezing and my teeth are chattering big-time. I can't wait to get this over with. I look directly at the cameraman and make a silly face. At least that's what I tell people when they see the pictures. I leap.

"Jesus H. Christ it's cold," I tell myself as I sink below the surface. I feel them pulling me right out of the frigid water. I get some water in my mouth and it's salty alright. I'm so eager to get out that I slam my left heel into the gang plank with a force that leaves it hurting for weeks afterwards.

Okay, I survived. I throw down the brandy and it warms me ever so slightly. I'm shivering full throttle now despite the fact that I have the Ocean Nova robe back on. I hurry back to my room feeling warmer the drier I get. I dry off completely, put on clothes, and lie in my bunk knowing that I did it. The battle's over, the war is won.

Facts:
1) 10 minutes in frigid waters is considered life-threatening.
2) Lynne Cox, world champion open-water swimmer, swam 1.2 mi. off the Antarctic Penisula for 25 minutes.

Trip to the End of the World

2010-11-25T05:02:00.000-08:00

It's when I travel that I feel most in the hands of God. Today I'm embarking on the longest physical journey of my life. In the next 26 hours I will travel 6,500 miles from Philadelphia, PA to Ushuaia, Argentina on the island of Tierra del Fuego.

I'm on my way to the Antarctic Peninsula. Why Antarctica? It's a place I've always want to visit since I saw the series "The Last Place on Earth" which recounted the race between Amundsen and Scott to be the first humans to the South Pole. Their story was full of hardship and deprivation, victory and the cruelest form of defeat. I appreciated their thirst for adventure and I wanted to see what they saw even it's in the form a comfort-filled expedition on a modern cruise ship.

The trip to Ushuaia consisted of three flights: Philadelphia to Atlanta (2.25 hrs.), Atlanta to Buenos Aires (10 hrs.), Buenos Aires to Ushuaia (3.5 hrs.). When I arrived in Buenos Aires it was morning and I had to wait three hours between flights. I was surprised how warm it was as I walked between terminal buildings. I sunned myself outside the domestic terminal because the terminal lobby was small and noisy.

I had a window seat on the flight to the city that calls itself the southernmost city in the world. As we approached the island I could see distinct waterways that separated the small but rugged mountains that filled the many small islands. These mountains were sporadically covered with snow as if they were decorated by a confectioner. The plane passed directly over the airport which looked like a field strip for small aircraft. But then the plane circled and dropped altitude and I realized that the small strip (Malvinas Argentinas International Airport) was our destination. In a minute we had landed at Ushuaia, my port of call for Antarctica.

At Home and at Work with a Kill-A-Watt

2010-05-17T18:17:00.001-07:00

This is a Kill-A-Watt. This is a Kill-A-Watt in action.

It's a specialized device that measures instantaneous energy consumption. I used it at home to measure power consumption from a Verizon set-top box under three conditions: 1) the box "off" 2) the box tuning a channel 3) the box tuning a channel with a 40" LCD TV on. I was surprised that with the box nominally off, it was drawing around 28W of energy. While tuning a channel it fluctuated around 28.5W. Finally with the TV on, the power consumption jumped to around 128W total.

So I called Verizon technical support to find out if I could turn the set-top box off overnight. They recommended not doing that. Why? During the night they occasionally distribute software updates to the unit. If the unit is off during an update, it may not be able to get it later resulting in a support call. Now that's 28W per hour on a 24 x 7 basis. That amounts to 245 KW per year times two since I have two set-top boxes in my house. That amounts to over 6% of my home's energy consumption last year!

I repeated a similar experiment at work attempting to measure my PC's energy consumption under these conditions: 1) the PC "asleep" in the morning with the monitor powered off 2) the PC "asleep" but displaying the login screen 3) the PC and monitor in use. I came up with these numbers: 108W, 175W, 190W.

I was surprised by the results of these experiments. The Verizon policy and the Vanguard policy concerning not powering-off the device were the same. Vanguard's policy revolves around the need to distribute software to PCs. The TIP Green Team, of which I am a member, has been told that Tech Ops is looking into powering off PCs remotely when there are no software distributions scheduled. If only Verizon would do the same for me!

The Four Greatest Mathematicians

2010-01-18T11:15:00.000-08:00

"History of Mathematics" was my final course in the master's program. I took it because I thought it would be fun, informative, and perhaps easy. The textbook we used was not the dreadnought A History of Mathematics by Victor Katz, but the sloop Journey Through Genius by William Dunham. Both books emphasize the development of mathematics from the perspective of Western civilization, but let's quickly thank the Arabian scholars who preserved the works of the ancient Greeks from annihilation. It was those Greeks who got the ball rolling with demonstrative mathematics around the sixth century B.C. It was the Arabs in the twelveth century who translated and expounded on the Greeks, and brought their works back to Europe.

During the course I learned the top four mathematicians of all time:

Archimedes (287 - 212 B.C.)
Newton (1642 - 1727)
Euler (1707 -1783)
Gauss (1777 - 1855)

I have taken the liberty of writing a short obituary for each of these towering figures.

Archimedes was a true self-starter who liked to apply math to everyday problems. At an early age he invented a water pump called the Archimedes screw which is still in use today. He exploited the techniques of "method of exhaustion" and double "reductio ad absurdum" to prove many theorems regarding familiar two and three dimensional figures. In order to save his city Syracuse from ruination by the Romans he devised innovative war machines to repel the overwhelming army.

Newton was perhaps the greatest self-learner the world has ever known. In his early twenties he spent two years in rustic isolation during which he: discovered the generalized binomial theorem; invented differential and integral calculus; recognized the universal gravitation as the key mechanism of the solar system; and developed insights into the nature of light by refracting it through a piece of glass called a prism. Having little interest in publishing his world-shattering findings, he turned his attention to alchemy for almost 40 years, but was persuaded to publish something by Edmund Haley. The result was a physics text called Philosophiae Naturalis Principia Mathematica which is considered the greatest contribution to science ever made by one man.

Euler was the most prolific publisher of pure and applied mathematics in history. His mathematical contributions ranged over number theory, calculus of variations, graph theory, complex analysis, and differential equations. He applied math to acoustics, engineering, mechanics, astronomy, and optics. The publication of his complete works was started in 1911, and the end is not yet in sight. Originally planned for 72 volumes, the discovery of new works pushed the project to an estimated 100 volumes. In order to aid his work in number theory he memorized the first 100 prime numbers, their squares, their cubes, all the way to their sixth powers. As a child he memorized the entire Aeneid and could recite it flawlessly late in life.

Gauss was a child prodigy who at the age of 17 invented a technique for inscribing a regular 17 sided polygon within a circle. This was a ruler and compass construction that stunned the math world since no one since classical times thought such a construction was possible. For his Phd. thesis he proved the Fundamental Theorem of Algebra, but being a perfectionist (more about this later) he improved upon the proof three times in subsequent years. His superlative text on number theory established modular arithmetic as the fundamental tool for its study. He tired of pure mathematics and turned his skills to scientific endeavors. In order to predict the position of the asteroid Ceres, he invented the technique of least squares and error theory in general. He mapped the earth's magnetic field, and along with Weber invented an early form of the telegraph. His perfectionist nature made him reluctant to publish until every detail of proof was beyond criticism. One biographer claimed that his unpublished work would have advanced mathematics by 50 years.

The most shocking thing I learned from this course is that all of the math I learned in high school, college, and graduate school only brought me up to around the year 1900. Seems that the twentieth century and beyond is reserved for Phd. students.

WHYY recording session (11/11/2009)

2009-11-12T17:09:00.000-08:00

I submitted a "This I Believe" essay to the Philadelphia public radio station WHYY, and was lucky enough to have my submission accepted. It was about math and though I thought it was a longshot, I knew it was unusual enough to attract attention. And yes, it was selected (broadcast on 11/13/2009). So this blog is about my adventure to the WHYY radio station to record it.

I left from Villanova station since I had a math class that very evening. In fact the building in the background is Mendel Hall where I have all my math classes.

No use in getting nervous so to relax on the train I'm playing Sudoku on my iTouch.

I exit the train at Market St. East and emerge from the station at 10th and Arch Sts. In this shot you can see the entrance to Chinatown and that storefront to the left below the traffic light is the Tracadero. The Troc is now a rock music venue and my son Tony has played at the balcony stage.

Where is the WHYY studio? Well it's across the street from the Consitution Center.

Here's the entrance.

And here's the reception desk. The receptionist didn't want her picture taken and is trying to hide in the picture. Why do people do that?

The station person who selected my essay was running behind schedule. She took me right to this studio where I snapped this picture. There's a special clock to the right of the screen that tells the radio people exactly how much time they have. It looks like the doomsday clock.

I was hoping to get some photos of my host, Elisabeth Perez-Luna, and myself but there was no time. You would see me in the photo below, but I had to take the picture myself.

After 30 minutes of reciting this essay over and over again I was beat. I tried my damnedest to put on a "radio voice." Ms. Perez-Luna had me make some alterations to the essay and she recorded it all using ProTools. I can't say what version is going to end up on the air after editing.

Goodbye WHYY, it was quite an experience.

Note: My essay was broadcast on Friday, November 20 at 5:45 pm. The station failed to give me any advanced notice so I missed it. You can find the essay and a podcast here.

Is There Anything as Simple/Complex as a Group?

2009-07-23T17:42:00.000-07:00

My summer 2009 session course was Group Theory. What is a group?

A group is an ordered pair (G, ·) such that G is a set, · is an associative binary operator on G, and $ e Î G such that:

If a Î G the a · e =a,

If a Î G, then $ a^-1
Î G such that a · a^-1 = e.

What could be simpler? A group is a set of objects and some kind of operation that has straightforward properties. Groups are all around us and come in all shapes (2D, 3D) and sizes (finite, infinite). Examples of finite groups are:
1) the symmetries of a square and rotations through various axes and
2) the set {0,1,2,3,4,5,6,7} with modulo 8 addition.

Examples of infinite groups are:
3) the set of integers and addition
4) the set of rational numbers – {0} and multiplication.

Okay we have sets of objects with an interesting behavior. Can we find any more of them? Well how about taking a subset of them and seeing if they are also interesting. Viola, consider the subgroup.

A subgroup of a group G is a subset which is a group under the operation in G. More precisely, a subgroup of a group (G, ·) is a group (H, *) such that H is a subset of G and * is the restriction of · to H x H. This means that multiplication is the same in H as it is in G.

Looking for subgroups in the examples above:
1) the horizontal and vertical rotations of a square is a subgroup of all the symmetries
2) {0,2,4,6} with modulo 8 addition is a subgroup of Z₈ above
3) the set of multiples of 3 is a subgroup of integers and addition
4) the set of rational numbers with even denominators – {0} is a subgroup.

Really simple stuff, right? Well let's define two terms, a theorem, another term, and another theorem and reconsider.

Definition: The normalizer of N(S) of S in G is defined by N(S) = {x Î G xS = Sx}.

Definition: If S and S' are subsets of group G, then S is conjugate to S' iff
$ x Î G such that S' = x^-1S x.

Theorem: Conjugacy is an equivalence relations.

Definition: The conjugate class on a subset S of a group G is the set Cl(S) of subsets S' of G which are conjugate to S.

Theorem: If S is a subset of a group G then [G:N(S)] = o(Cl(S)).

Oh no, now we're in the deep end of the pool and we don't know how to swim. This is typical of Group Theory (and higher math in general). Define something that's pretty simple and prove some stuff about it. Then define a new abstraction on top of that and prove some more theorems. Then define
some more stuff on top of THAT... You get the picture.

In this manner group theory leads to these kinds of groups: cyclic, abelian, free, solvable, divisible, decomposable, p-groups, supersolvable, M groups, etc. How can something so simple now be so complex?

P.S. the text was "Group Theory" by W. R. Scott which I would avoid like the plaque. It has very, very few examples and I think examples are needed to wrap your head around all of the abstractions being thrown at you.

Crypto Applet

2009-07-20T17:06:00.000-07:00

If you want to use the crypto applet right away click
here.

Introduction

My introduction to Cryptography was "MAT 8790: Cryptography" during Villanova's Summer Session 2007. The text for this course was Cryptography: Theory and Practice by Douglas Stinson. During this course I found myself writing Java programs to explore the algorithms discussed and to sharpen my programming skills. By the end of the course I had a couple of the simple cipher systems implemented in a "uniform" way. By uniform I mean that the APIs for the cipher system adhered to a Java interface and could be invoked in the same fashion. To those familiar with object oriented programming concepts, I had implemented a Strategy pattern for Stinson's cryptosystems.

The project lay dormant until I saw it as an opportunity to release it as an educational tool, and seek credit for an independent study. Thus was born the Crypto Applet project.

To make it educational, I tried to implement several simple ciphers that a high school student could absorb without difficulty: Shift, Affine, and Substitution. Hopefully the student's interest would be piqued to the point that she may investigate the RSA and ElGamal ciphers, and the mathematics behind them. The applet's demo pane permits the student to interact with the ciphers by entering parameters for the encryption key and various plaintext strings to encrypt with the cipher. The plaintext and ciphertext values can be displayed "alphabetically," as per Stinson, or numerically. A numerical display shows the inner workings of the encryption algorithm more clearly than the alphabetical display.

In the course of the project I learned a great deal about the cipher algorithms, the Java math library, the Java swing library, and HTML coding. I will share some insights by visiting each cipher and discuss what I learned doing the implementation.

I also released all source code under the Academic Free License v.3.0. I choose this one from the many Open Source license choices, because AFL has an academic orientation. Feel free to download the source into the Eclipse IDE and read the source code.

Program Technology and Architecture

The applet consists of two jar files: 1) the crypto "engine" (crypto.jar) 2) the Graphical User Interface (GUI) written in Java Swing (cryptogui.jar). I first considered writing this as a pure web application (servlets, jsp's, etc.) but was discouraged by the fact that I would have to find a host that would run the web application. Villanova's IT department (UNIT) does not provide this service to students; however, they do provide content space with few restrictions. This made the choice of Swing/applet for me, although at the start I did not know Swing programming. I also knew that any user of the application would have to have the Java 5.0 run-time system loaded into their browser program. This imposes a barrier to potential users who would have to upgrade to this level of the Java run-time.

The crypto engine can be run standalone from the command line but it doesn't support user interaction. My first step in providing a user interface was not to implement a GUI in Swing. I had developed the engine with pre-defined encryption key parameters and needed to extend the engine with parameter setting routines. So I implemented a command line interface (cryptoui.jar) to the engine wherein I could perfect the entry and validation of parameters and exception handling. Since this runs from the command line, it is not part of the applet, but it was a worthwhile stepping stone. When the command line version was working properly I undertook the more complex Swing programming. This resulted in the applet version.

Here is the HTML for the applet invocation:

<applet code="ciphergui.CipherApplet.class" name="Crypto Applet"

archive="crypto.jar,cryptogui.jar"
width=620 height=450 codebase="crypto">

<param name="bgcolor" value="ffffff">

<param name="fontcolor" value="000000">

Your browser is not Java enabled.

</applet>

As you can see there's not a lot to it, however, I have to make several points.

The HTML page with the above code references a nested folder called "crypto." This is where the two jar files reside on the server. The entry point for the applet is the CipherApplet class which must be referenced by package name.

The applet displays both html (Introduction, Theory panes) and text (code panes) within Swing components. The html and text content can be stored on the server and referenced as external to the Java code. This meant however, that if someone wanted to run the applet on their own server, say after making an enhancement, they had to replicate the content from the original server. To avoid this, I embedded the html and text within the jar file itself and used Java APIs to load it. Below is an example of how the Intro pane is displayed:


JEditorPane htmlCode = new JEditorPane();
htmlCode.setContentType("text/html");
htmlCode.setEditable(false);
try {
 InputStreamReader br = new InputStreamReader(getClass().getResourceAsStream("html/intro.html"));
 htmlCode.read(br, null);
 br.close();
 } catch (FileNotFoundException e1) {
     e1.printStackTrace();
 } catch (IOException e1) {
     e1.printStackTrace();
}

The critical API is getResourceAsString which reads from a folder nesting within the jar file itself.
I will have more to say about Java Swing programming later.

On the Difficulty of Naming Things

I've been a software engineer my entire career and will attest to what I believe is the most difficult task – inventing good names during programming. This is especially true for object-oriented programming where a good design for a class is that fact that the class does one and only one thing. Now this project is small enough that I feel that I did adhere at least to this design principle. However, I will admit that I've been inconsistent in my naming.

I treat "crypto" and "cipher" interchangeably throughout the program. I think cipher predominates, but the package names and project names still say "crypto." Please forgive my inconsistency.

Also I wish I had named "CryptoUI " as "CryptoCLI" where CLI means "Command Line Interface." As it stands "CryptoUI" maybe easily confused with "CryptoGUI."

Converting Letters to Numbers

There are two utility classes ConvertAlphabet and ConvertTrigramAlphabet whose job is to translate between alphabetic representation and numeric representation.

ConvertAlphabet is used by the simple ciphers which operate on values in the limited range of 0 to 25.

ConvertTrigramAlphabet is used by RSA and ElGamal ciphers which operate on larger values. It is imperative that these ciphers convert groups of letters (three letters to a trigram) before the ciphers operate on them. If you simply let RSA and ElGamal operate on individual values in the range 0 to 25 the encrypted values are just a one-for-one scrambling of the plaintext value to cipher value. The numeric values must be large in absolute value and for this reason a trigram mapping is used. As explained in the applet's Introduction page the triplet "dog" is mapped to the numeric value 2398 as follows:

d	=>	3*(26)²	=	2028
o	=>	14*(26)¹	=	364
g	=>	6*(26)⁰	=	+ 6
				2398

I also had to deal with the fact that the entire plaintext value must be a multiple of three. I had the choice of truncated an extra character or two, or padding the string with a character or two. I chose the latter using a 'b' (think 'blank') for the pad character.

I noticed in the course of the project that the complex algorithms produce encrypted values that are larger than any input or output values. This is shown in the values related to the string "dog" where the encrypted value is not a three, but a four character string, "BLND."

This forced me to re-implement the routine int2Trigram routine. The old and new versions are shown.


/**
 * Convert an int to a trigram value. Example: 3*26^2 + 14*26 + 6 = 2398 =>
 * "dog"
 *
 * @deprecated
 */
private static String int2Trigram_old(int val) {
   int i0, i1, i2;
   i0 = val / 676; // 676 = 26^2
   val = val - (i0 * 676);
   i1 = val / 26;
   val = val - (i1 * 26);
   i2 = val;
   return "" + int2Char(i0) + int2Char(i1) + int2Char(i2);
}

/**
 * Convert an int to a trigram value. Example: 3*26^2 + 14*26 + 6 = 2398 => "dog"
 *
 * @param val numeric value to convert to a trigram.
 */
private static String int2Trigram(int val) {
   StringBuffer buf = new StringBuffer();
   int divisor = 26;
   int b = 0;
   int i;
   for (i=0; val>0  i<3; i++)
   {
       b = val % 26;
       buf.append((char) (b + (int) 'a'));
       val = val / divisor;
   }
   return buf.reverse().toString();
}

This proves another programming adage first espoused by the rock band Jethro Tull: "Nothing Is Easy."

Shift Cipher

The Shift cipher is the simplest of the ciphers. For this reason I chose this as the example in the introduction. The encryption algorithm is straightforward:


 public int[] encrypt(int[] ba) {
  int len = ba.length;
  int[] cipher_ba = new int[len];
  for (int i=0; i < len; i++)
  {
      cipher_ba[i] = (ba[i] + SHIFT) % 26;
  }
  return cipher_ba;
 }

Programming Notes:

The parameter ba contains the numeric values for the plaintext to be encyrpted.
A new array cipher_ba holds the numeric values following the encryption.
'%' is the modulus operator in Java.

The decryption algorithm is almost as straightforward; however, Java performs signed arithmetic on int (integer) values. So the result of the modulus operator may be a negative number.


 public int[] decrypt(int[] ba) {
  int len = ba.length;
  int mod;
  int[] plain_ba = new int[len];
  for (int i=0; i < len; i++)
  {
      mod =(ba[i] - SHIFT) % 26;
  //  Java modular arithmetic is signed so convert negative residues.
      mod = mod < 0 ? mod+26 : mod;
      plain_ba[i] = mod;
  }
  return plain_ba;
 }

Programming Notes:

The conditional operator, ?: , acts like an in-line if statement. Following '?' is the true value; following ':' is the false value.

Affine Cipher

The Affine cipher is a little more sophisticated than the Shift cipher. I won't present the encryption/decryption algorithm, but will comment on the parameter entry. Parameter entry is made in the GUI part of the applet. The Affine cipher just validates that A, the multiplicative factor, is relatively prime to 26. And that B, the shift factor, is between 0 and 25 inclusive.

To validate A, and to find its inverse modulus 26, I pre-calculated all of the numbers that are relatively prime to 26 and their inverses. Show below is the validation.


 private final int[] validA = { 0, 1, 0, 3, 0, 5, 0, 7, 0, 9, 0, 11, 0, 0,
                                0, 15, 0, 17, 0, 19, 0, 21, 0, 23, 0, 25 };
 private final int[] validAInverse = { 0, 1, 0, 9, 0, 21, 0, 15, 0, 3, 0,
                               19, 0, 0, 0, 07, 0, 23, 0, 11, 0, 5, 0, 17, 0, 25 };
/**
 * Validate and set the multiplicative value for the cipher.
 *
 * @param valA is the candidate multiplicative value.  It must be relatively prime to 26.
 * @throws IllegalArgumentException.
 */
 public void setA(int valA) throws IllegalArgumentException
 {
    if (valA < 0  valA > 25)
    {
        throw new IllegalArgumentException("A value must be in [0..25]");
    }
    if (validA[valA] == 0)
    {
        throw new IllegalArgumentException("A value must be relatively prime to 26");
    }
    A = valA;
    A_INVERSE = validAInverse[valA];
}

I could have used the Euclidean Algorithm to determine if A is relatively prime to 26 and another algorithm to find its inverse, but a table lookup works fine. This demonstrates the classic time/space tradeoff that one encounters in computer programming frequently.

Substitution Cipher

The reader should be familiar with Substitution cipher as it is the familiar cryptogram puzzle found in newspapers. The only tricky part of its implementation was inputting the permutation that is to be used as the encryption key. "That's simple," I thought, "just have the user type the 26 letters of the alphabet in random order and use that as the permutation."

Well, I wanted to give feedback while the user was typing so my first implementation was to display a label after the input field showing the letters that had not been entered. This works fine if the user works methodically from left to right, but as I tested the implementation, I found that I sometimes used the mouse to reposition the cursor. This wrecked havoc (as did use of CTRL-C, and CTRL-V) on the implementation as I soon discovered that I had to implement a word processor in order to echo the unused characters accurately. So in the second implementation, I waited until the entire permutation is entered before validating it.

A good way to verify the implementation is to enter a permutation (just following the keyboard left to right from top to bottom) and then enter the alphabet as the string to encrypt. You will see something like this.

RSA Cipher

RSA is the first cipher that introduces high order mathematics. The algorithm requires two large prime numbers to be entered by the user. Validation of the primes and RSA algorithm implementation are made ridiculously simple by the Java BigInteger library. When I discovered the methods of this library I realized that it was designed with RSA in mind. The methods I use for validation, encryption, and decryption are:

isProbablePrime
multiply
modInverse
modPow

The encryption and decryption algorithms are each implemented in only 12 lines of code!

El Gamal Cipher

When I first encountered the El Gamal cipher I dismissed it for commercial purposes because, by design, it doubles the length of the ciphertext value. Since ciphers are intended for communication purposes, this means the transmission must be twice the length, say of an RSA ciphertext. After working with El Gamal I now see that this negative can be viewed as a positive since the use of a random variable makes the cipher that more difficult to decrypt.

In addition to the encryption keys there is a random value, called K, that is used by the encryption/decryption algorithm. In my first implementation I had the user enter this value and use it repeatedly. Then I realized that I could get closer to a one-time pad by having this value represent the seed to a random number generator. This seed is given to the Java Random library and as each trigram is encrypted, a new K value is generated by the Random class.

Another interesting aspect is that a primitive element (root) of the private key, called ALPHA, must be validated. My implementation is simply to generate powers of the element modulus P until finally (and this is guaranteed) it is equal to 1:

If ALPHA is a primitive root then the power is one less than the prime P.


 public void setALPHA(int alpha) throws IllegalArgumentException
 {
//  Validate that alpha is a primitive element for A.
    BigInteger valAlpha = new BigInteger(Integer.toString(alpha));
 // Here we have alpha^1
    BigInteger valAlphaPower = valAlpha;
    int pow;
    for (pow = 2; pow < p; pow++) {
        valAlphaPower = valAlphaPower.multiply(valAlpha);
        BigInteger val = valAlphaPower.mod(P);
        if (val.equals(BigInteger.ONE)) {
           break;
        }
    }
    if (pow < p - 1) {
         throw new IllegalArgumentException("ALPHA must be a primitive root for P");           
    }
    ALPHA = alpha;
    BigInteger valBETA = (new BigInteger(Integer.toString(alpha))).modPow(A, P);
    BETA = valBETA.intValue();
 }

This implementation proves to be computationally intensive and pins the processor for a few seconds. See how long it takes on your machine with these values:

P = 31847

A = 7899

ALPHA =5

SEED=1234

I will have more to say about this computation in the Swing Programming section.

Swing Programming

As I mentioned I did not know any Swing programming at the beginning of this project. I did have experience developing GUI programs in Visual Basic so "event driven" programming was in my background.

I bought an eleven hundred page O'Reilly book titled Java Swing that was no help whatsoever since it was too low-level. I found the tutorials and sample code on Sun's website to be most useful. The tool that really saved the day was the Visual Editor plugin for Eclipse. This plugin is barely maintained by the Eclipse community, but it still works like a charm. You will have difficulty finding a version compatible with Ganymede.

The most challenging area of the GUI programming was the validation of the parameters with the goal of giving feedback to the user. Swing generates an event when the Enter key is pressed, the only event that the program initially responded to. I immediately discovered that I, myself, was using the Tab key to move between fields, so I decided to program a Tab event the same as an Enter event. Then the first time I observed another user (my son) interact with the program, he used the mouse to move from field to field! I never implemented the mouse event, but I did give feedback in the form of a check mark to indicate that the parameter has been validated. Hopefully, this conditions the user to use the correct input technique.

The very last parameter validation that I implemented was the primitive element for ElGamal encryption. I was hoping to find an efficient algorithm for this but ended up with a brute force enumeration of powers. This is disconcerting to the user because all of the other parameters are validated almost instantaneously (thanks again to BigInteger!).

I thought of changing the label for ALPHA entry to "WAIT...WAIT…" below:

 private void actionALPHAPerform() {
  String aVal = jTextALPHA.getText();
  String aError = null;
  int val = 0;
  try {
            val = Integer.parseInt(aVal);
            jLabelALPHA.setText("WAIT...WAIT");
            cipher.setALPHA(val);   
  } catch (NumberFormatException ex) {
      aError = "Please enter a number.";
  } catch (IllegalArgumentException ex) {
      aError = ex.getMessage();
  }
  if (null != aError) {
    jTextALPHA.selectAll();
    ShowParmValid.ShowParmInacceptable(jLabelALPHAError, aError);
  }
  else {
   ShowParmValid.ShowParmAcceptable(jLabelALPHAError);
   jTextSEED.requestFocus();
  }
}

This change had no effect whatsoever. Why? Some investigation led to this passage in the Swing documentation:

By the way, applets that use Swing components must be implemented as subclasses of JApplet, and components should be added to the JApplet content pane, rather than directly to the JApplet. As for any applet, you should never perform time-consuming initialization in the init() or start() method; instead, you should start a thread that performs the time-consuming task.

This passage indirectly hints at the need for starting a separate thread on which to call the setALPHA method. This I will leave as an enhancement to the reader; however, if you download the code and see incomprehensible thread creation logic in this area, than you can assume that I have risen to the challenge myself.

Finally I want to comment on the HTML displayed within the JEditorPane class. My first implementation was to display scanned images of the Cryptosystem descriptions from Stinson's book. I sought permission from the publisher Chapman & Hall/CRC but was told that these images could only be displayed on a password protected website. This was too restrictive so I translated the image content to HTML version 3.2 (HTML4 was not allowed due to the fact that JEditorPane implements only v. 3.2). Note that it is the applet that renders the HTML displayed on the screen, not your (hopefully more modern) browser.

To demonstrate what HTML v3.2 looks like here is some of the HTML for ElGamal theory:


<b>Cryptosystem: ElGamal Public-Key</b>
<br><br>
Let <font face="Georgia"><i>p</i></font>, be a prime such that the <body>Discrete Logarithm</b>
problem in <font face="Georgia">(Z<sub>p</sub><sup>*</sup>,⋅)</font> is infeasible,
and let α ∈ <font face="Georgia">Z<sub>p</sub><sup>*</sup></font> be a primitive element.
Let <font face="Georgia"><i>P = Z<sub>p</sub><sup>*</sup></i></font>,
<font face="Georgia"><i>C = Z<sub>p</sub><sup>*</sup> x Z<sub>p</sub><sup>*</sup></i></font>
, and define
<br><br>
<div align="center"> <font face="Georgia">
 <i>K</i> = {(p,α,a,β) : β ≡ α<sup>a</sup> (mod p)}</font>.
</div>
<br><br>
The values <font face="Georgia">p, α</font> and <font face="Georgia">β</font> are the public key,
and  <font face="Georgia">a</font> is the private key.

This is not the most readable and maintainable HTML, but it will do in a pinch.

Tools

I used the following tools to develop the Crypto Applet:

Eclipse (Ganymede version)

Visual Editor plugin (org.eclipse.ve)

Mathematica v.7 Student Edition

Microsoft FrontPage 2003 Ed.

I used Mathematica to check the program's calculations, especially when I received unexpected results.

FrontPage is the University approved tool for deploying content to the student's website.

Conclusion

I hope you enjoy the Crypto applet in its current form but I will immediately suggest these improvements:

More educational content

Addition of new ciphers

Conversion to a plugin architecture

The first two bullets are self-explanatory. A plugin architecture would permit the dynamic binding of new ciphers at run-time. You would prove yourself to be a first-class Java programmer by implementing this architecture.

Comments and feedback are welcome.

Bibliography

Castro, E. (2003). HTML for the World Wide Web (5th Edition). Berkeley: Peachpit Press.

Eckstein, R., Loy, M., & Wood, D. (1998). Java Swing. Sebastopol: O'Reilly & Associates.

Flanagan, D. (2005). Java in a Nutshell (5th Edition). Sebastopol: O'Reilly Media.

Flanagen, D. (2000). Java Examples in a Nutshell (2nd Edition). Sebastopol: O'Reilly & Associates.

Sierra, K., & Bates, B. (2005). Head First Java (2nd Edition). Sebastopol: O'Reilly Media.

Stinson, D. (2006). Cryptography: Theory and Practice (Third Edition). Boca Raton: Chapman & Hall/CRC.

Complex Numbers: More Real Than You Imagine

2009-05-05T17:11:00.000-07:00

My last course was titled Complex Analysis where we studied functions f(z) of complex variables z = x+ i*y. We methodically defined complex versions of familiar functions like sine, cosine, exp, and the like, and then studied their properties including differentiation and integration. If this sounds boring, it’s not. For example, we have Euler’s Identity ⅇ^ⅈπ=-1 which relates four of the primary constants of mathematics. The identity follows directly from Euler’s definition of ⅇ^ⅈθ=cosθ+ⅈsinθ to which he was led so that familiar properties like ⅇ^(x1+x2)=ⅇ^(x1)×ⅇ^(x2) worked for complex numbers as they did for reals.

Now what’s particularly fascinating about Complex Analysis is that is can be used to prove all kinds of results involving real variables. Let’s define the norm of a complex number as x + i y = (x^2 +y^2)^1/2 . There’s a property of complex numbers that says

Use this to define the multiplication of the sum of squares of two real numbers in terms of complex numbers (“≡” means “is defined as”). Example:
Here we have a result from number theory that states that the product of a sum of two squares is itself the sum of two squares. This is always true, although somewhat surprising to see the ease with which the calculation proceeds using complex numbers.Okay, you’re not impressed. How about the proof of the Fundamental Theorem of Algebra which states that if

is a non-constant polynomial of degree ≥ 1 with complex coefficients, then the equation

has a root in .
This theorem is impossible to prove using the real number system; however, with complex numbers it’s almost a one liner when you employ Liouville’s Theorem.

Another very strange phenomenon has to do with real integrals like:

This result cannot be derived by standard calculus. Instead it is a fall-out of the Cauchy Integral Theorem (CIT) and Cauchy Integral Formula (CIF). To get this result you have to find the right contour in the complex plane and perform complex integration on its boundary. One part of the contour will be the x-axis giving the real-valued integral on the left. The complex integrals are computed using either CIT or CIF and when the results are combined out pops a real number!

To really appreciate these wonders you must take the course. My wonderment leads me to a mathematical version Plato’s allegory of the cave. In The Math Cave the real numbers are the shadows that we see and take for reality. However, in truth, the complex numbers are the source of the shadows.

Rock's Ridiculous Sports Quiz

2009-02-12T08:03:00.000-08:00

Please enter you answers as a comment to the post.

1. Make a list of sports where the low score wins. Note that all timed sports are excluded e.g. running, swimming, auto racing ,etc. (10 point for each sport)

2. (Essay Question) Identify the sport where ambidexterity is most valuable. (40 points)

3. There are two kinds of sports – sports where the boundary line is “in bounds” and sports where the boundary line is “out of bounds.” Make a two column list of such sports with column A representing "in bounds" sports and column B representing "out of bounds" sports. (10 points for every complete row, i.e. correct answer in column A and column B)

4. Identify the sport whose time structure (hint: this is not the same as the length of the game) changes from high school to college to the pros. (15 points for each sport)

5. Consider the following sports: baseball, football, and soccer. Which sport gives the greatest number of combinatorial lineups given the restricted interchangeability of playing positions. For example, in baseball:
a. Any outfielder can play any outfield position
b. The first baseman only plays first base
c. The other three infielders can play any infield position (except first base)
d. The pitcher and catcher cannot exchange position

Given the assumptions above there are 36 lineups here. Your answer should state assumptions and the number of lineups. (20 points)

6. Name a sport which has a scoring system in which you can win more points than your opponent, win more games than your opponent, and still lose the match to your opponent? (10 points)

Give the specifics of losing a match such a match (more points than opponent and more games than opponent). Assume that every game of the match had the fewest possible points scored. (20 points)

7. List the size of the items below in increasing order of diameter: baseball, basketball, golf ball, hockey puck, tennis ball. (5 points for each correct answer in the list)

8. In baseball to “bat around” means that every player bats exactly once in a half inning. When this occurs what is the maximum # of runs that can be scored? The minimum # of runs? (Hint: the maximum is > 6 given an extraordinary set of circumstances for the home team). (5 points for the correct minimum number; 40 points for the correct maximum number)

9. Here’s another “there are two kinds of sports” questions. Sports that do not have an unlimited overtime play and those that do. As before make a two column list. Column A lists sports that have unlimited OT to settle the contest and column B lists sports that determine the victor by some kind of “shootout.” (10 points for every complete row)

Answers below (keep scrolling) but give it a shot before giving up.

1. Golf, cross-country, show jumping (equestrian sport)

2. This question is extremely hard to score because it is hard to pick the correct sport in the first place. Some sports by their nature are symmetrical in the sense that any asymmetry would be considered a distinct handicap. Take for example, swimming (or running). If the player were not ambidextrous he or she would probably be at a serious disadvantage. So is the answer swimming?

The question is loaded because you tend to think of sports that are “handed” (also “footed”). But even in sports that are “handed” you would not expect ambidexterity at all positions. Have you ever heard of an ambidextrous quarterback or an ambidextrous pitcher? I haven’t.

My preferred answer is soccer. Now soccer players are born “footed,” that is, they have a favored kicking leg, but through constant training the superior player does not show a preference for either leg. Such a player must be able to dribble, pass, and score with either leg when the opportunity arises.

3.
Out of Bounds In Bounds
Football Baseball
Basketball Soccer
Tennis* Tennis*
Relay Race
Long Jump

*When serving a player “foot faults” when he steps on the baseline before striking the ball. Once the ball is in play it is considered in bounds if it touches any part of the court boundary.

4. Basketball. High school: 4 quarters, college: 2 halves, pros: 4 quarters. Why it changes like this is a mystery to me.

5. Football. Non-interchangeable positions: Quarterback, Center, Middle Linebacker. Interchangeable positions: Guards and Tackles (24 combinations), Left and Right Linebacker (2 combinations), Free Safety and Cornerbacks (6 combinations). Total permutations: 288.

6. Tennis. Player A loses to Player B in a five set match 6-0, 6-6 (7-5), 6-0, , 6-6 (7-5), 6-6 (7-5). Every gave won by player A was a shutout (50-love) and every game won by player B was won by the minimum margin of two points (50-30). Thus player A won 30 games to player B’s 21 games (tiebreaker counts as one game). Player A won 167 points to Player B’s 105 points.

7.
Golf ball: 1.680 inches
Tennis ball: 2.575-2.700 inches
Baseball: 2.87 - 2.94 inches
Hockey puck: 3 inches
Basketball: 9.39 inches

8. Max = 9, min = 3. Most people would give 6 as the maximum # of runs that can be scored in a half inning with each player batting exactly once, however consider this situation. A team is down by 8 runs in the bottom of the ninth and makes a miraculous comeback scoring 9 runs without incurring a single out. Nine players have batted and the ninth has hit a walk off grand slam. The half inning ends not due to the third out being made, but because the game has ended.

9.
Unlimited OT Limited OT
Baseball Football
Basketball Soccer
Hockey * Hockey *

* In profession hockey during the regular season shootouts determine the games winner; however in the playoffs unlimited overtime is the norm.

Linear Algebra

2008-12-26T11:47:00.000-08:00

What do you say about Linear Algebra? You learn Gaussian elimination and vector spaces and you're done. Except if the textbook your teacher chooses insists that you approach LA from a geometric perspective. Then you must interpret systems of linear equations as hyperplanes and such craziness.

To summarize there are two ways of representing a subspace V Ì R^n:
1. V=Span(v₁,…,v_k) for appropriate vectors v₁,…,v_k. This is the generalization of our parametric representation of lines and planes in R³.

2. V= N(A) for an appropriate m x n matrix A. This is the representation by a homogeneous system of linear equations, in which we interpret V as the intersection of the m hyperplanes defined by A_i × x = 0.

That's it, enough said.

Advanced Calculus = Real Analysis

2008-12-24T11:35:00.000-08:00

Newton and Leibnitz are both credited with inventing calculus around the turn of the 18th century. A lay person might think that these geniuses both wrote a 300 page math textbook on the subject to gain fame -- wrong! They had the key ideas regarding rate of change (differentiation) and its opposite (integration). The ideas were so powerful that physicists, astronomers, and engineers started to use the new math to solve the pressing problems of the day. Everyone was happy except ensuing mathematicians who introduced functions and series that the undeveloped calculus could not address.

The Advanced Calculus course I just completed used the book "A Radical Approach to Real Analysis" to teach the foundations of calculus from an historical perspective. The book cites the work of Fourier, Cauchy, Abel, and Dirichlet (there were certainly others) as the 19th century mathematicians who laid the rigorous foundation for calculus so that strange new functions like the one below could be analyzed.

I perceived two central tools used for the area of mathematics called real analysis (sometimes called "the "epsilon/delta" game or "epsilonics") -- the Mean Value Theorem and Uniform Convergence Theorems. Every student of calculus should remember the Mean Value Theorem:

Let y = f(x) be a function with the following two properties: f(x) is continuous on the closed interval [a,b]; and f(x) is differentiable on the open interval (a,b). Then there exists at least one point c in the open interval (a,b) such that:

This is an incredibly powerful theorem that can be used to prove numerous results. Here's something I had to prove as a homework assignment:

The notion of Uniform Convergence and the theorems related to it really gave mathematicians a handle on functions defined as the sum of infinite series of functions (e.g. Fourier series). This topic is a little too complex to discuss here except to summarize that a function having the property of uniform convergence can be integrated term by term and potentially differentiated term by term (for this the sum of derivatives must also be uniformly convergent).

Generally if a function is not uniformly convergent it fails at some troublesome point. Consider the function:
on the interval [0,1]. If you take enough terms (increase the value of N) this function behaves like the zero function, f(x) = 0, for all points in the interval [0,1] except at 0. Here the function "blows up" as shown in this plot:

Simply put the function is not convergent at x = 0 (see the big hump just to the left of .2).

All of this analysis helped put calculus on a rigorous footing going into the 20th century and enabled the production of the calculus tomes (mine is almost 900 pages!) from which many of us studied. So even though calc students distain epsilon/delta arguments and the like, I salute the pioneers of calculus. Hail to Fourier, Cauchy, Abel, and Dirichlet!

Grad Math - Those Super Scandinavians

2008-07-27T05:25:00.000-07:00

This summer's (2008) graduate course was Mathematical Modeling. There was something suspicious about the enrollment -- the course was filled to capacity with 25 students. Reason? The teacher was considered an easy grader.

The course required a presentation on a math modeling topic of one's own choosing. At the very start of the course I chose the "floating license problem" which has to do with determining how many floating (concurrent) licenses to purchase for a community of users. The floating license model is one of many ways that commercial software is licensed. The idea is that you don't have to buy every user a license. You buy a few with the knowledge that at peak usage some users may have to wait to obtain a license to use the software.

Now this may sound like a modern problem because this form of licensing is at most 40 years old. However the problem is a classic one from queuing theory and was solved when the first telephone systems were being built in the early twentieth century. The research was published by Agner Erlang (1917) and Tore Engset (1918). Erlang was Danish; Engset was Norwegian.

I picked this topic for two reasons: to get credit at school and to get credit at work. At work my department manages licenses for software development tools and some of the licenses are of the floating variety. Since I had no idea where to start I did a literature search and came up with a fantastic hit in IEEE Transactions on Software Engineering -- "Determining the Proper Number and Price of Software Licenses" -- written by five Finns at the University of Turku in Finland. The theory and applications were all there in the paper.

Now my fellow students were content to regurgitate the papers on topics that they found interesting. I decided to go a mile farther and reproduce the results of the authors. In the paper they presented analytic results for different combinations of users and licenses. They also described a software simulation they performed to reinforce the analytic results.

So there you have it. I decided to implement a simulation using Mathematica. Mathematica is a symbolic math package that is partly responsible for bringing me back to higher mathematics. Here's what I had to go by (the "requirements" if you will) from the Finnish paper:

The generation of the requests is done separately for each user by reserving the given number of nonoverlapped intervals from the set of working hours. This is done by choosing the starting times randomly and then by checking whether the service intervals obtained in this way overlap. If there are overlapping intervals, the simulation generates a new set of intervals...

Here is my schedule generating function written in Mathmatica:


GenerateSchedule[p_, l_, u_]:= Module[{duplicates,randTimes,schedule,j},
duplicates = True;
While [duplicates == True,
 randTimes = RandomChoice[Range[1,p],RandomChoice[Range[l,u]]];
 duplicates = (Length[Union[randTimes]] < Length[randTimes]);
];
schedule = Table[0, {p}];
Do[schedule[[j]] = 1, {j, randTimes}];
schedule
]

The meaning of the parameters:
p - number of time units for the schedule
l - lower limit for the # requests during the time period
u - upper limit for the # requests during the time period

Here is a sample schedule for 10 minutes with 4 license requests:

minute	1	2	3	4	5	6	7	8	9	10
use	0	1	1	0	0	1	0	0	1	0

Back to class. On the last session of class everyone did their presentations. Most of the presentations did not involve the use of any math tools. A couple did use Microsoft Excel and one used Minitab. So I guess I won with Mathematica.

Graduate Seminar Topic - Gödel's Theorems

2008-04-17T18:21:00.000-07:00

I took this course because it was required and because I didn't like any of the other courses offered. The easiest way to describe the course is that it's an independent study but with a group of students. Everyone picks their own topic and is required to give four half-hour presentations and write four 10-page papers. The topic I picked was Gödel's incompleteness theorems. A topic from what some people call logic and others call metamathematics. It's a great subject that few math majors even know about. I knew about it from the classic book "Gödel, Escher, Bach : an Eternal Golden Braid" by Douglas Hofstadter which was given to me as a Christmas present when it was first published in 1979. Of course I didn't read it then. Heck I was single and who in their right mind reads a 700+ page technical book?

Now I must pick the main thrust of the rest of this posting. I could: 1) describe my presentations and try to teach you Gödel's results 2) give short reviews of the books that I read for the class 3) describe my experience of the class. The last choice might be amusing since the other students' topics ranged far and wide. Some topics were poorly chosen and equally poorly presented. A couple of the topics were very amusing like the construction of magic squares or the history of computational devices including finger counting to 100,000. I don't think that I can really explain the theorems before you get bored so I'll review my sources for the presentations and the papers.

"Gödel, Escher, Bach" or GEB is certainly a classic. Only a portion is about Gödel's theorems, of course. In general it's hard to say what this book is about because it's about so many different things. Many Escher drawings are reproduced and the text explains their significance in context of recursion, self-reference, and self-replication. Bach's fugues are analyzed in a similar way. And the book amusingly explains basic logic and Gödel's theorems. The author develops some fantastic analogies to give you different ways to understand what's going on. He compares the Gödel sentence to a vinyl record that's specifically designed to break the record player that will play it (nowadays, think of an mp3 file that's designed to break the player that will play it). And he invents amusing, yet deeply philosophical, dialogs between Achilles, the Tortoise, and the Crab. And let's not overlook the theme of Zen Buddhism that runs through the book.

My favorite takeaway from this book is a diagram that Hofstadter devised that explains Gödel's theorems visually. Here it is:

Gödel basically proved that a formal system designed to encompass arithmetic cannot prove all arithmetic truths. There are unreachable truths that are beyond the formal system. That's the left side of the picture. Since the book covers Eastern thought, or at least Zen thinking, he included the opposite of formal arithmetic on the right. In that system you start with negative axioms and prove the negation of theorems. But that side too is incomplete since it has unreachable falsehoods.

So now you do know Gödel's results. I bit the bullet and read the entire book (742 pages) and recommend it to anyone (who really, really likes math, logic, programming, or artificial intelligence).

My primary source was "A Profile of Mathematical Logic" by Harold DeLong. This book is a readable history of logic from the ancient Greeks up through the middle of the 20th century. DeLong gives a good treatment of Gödel's theorems adhering very closely to Gödel's research paper that was published in 1931. I "borrowed" the whole progression for my presentations/papers from this book. The scholarship behind this book is superb as the bibliography is 24 pages long. Highly recommended for the philosophy or math lover.

In 1958 a small paperback entitled "Gödel's Proof" was written by Nagel and Newman. This book is highly readable. If you had to read a book on Gödel this should be the one.

Next is a collection of scholarly papers on metamathematics titled "From Frege to Gödel: A Source Book of Mathematical Logic, 1879 – 1931." This collection includes two of Gödel's most famous papers translated into English. One proves that first order predicate calculus is complete. The other that formal arithmetic is incomplete.

My last source was a brand new book "An Introduction to Gödel's Theorems." This is a serious textbook designed for a one or two semester course leading up to the theorems. It is as much fun to read as a standard logic textbook. In fact I found a number of other books like this that were written for a course taught by the book's author.

About Kurt Gödel

To say that Gödel was a genius is an understatement. The originality of his first incompleteness proof is breathtaking. He has been called the greatest logician since Aristotle. Unfortunately he suffered from paranoia and hypochondria most of his adult life. In his 72nd year he literally starved himself to death fearing that his food was poisoned.

My Mathematical Biography

2008-01-17T17:11:00.000-08:00

I had to write this for a course I'm taking so I thought that I might as well share it with the world.

Currently I’m a first year graduate student in Villanova’s master’s degree mathematics program. A lot of my friends have asked me, “are you crazy? Why are you doing this?” To them I give a short reply, “I like math.” Besides liking math I have to catch up to the rest of my family. My two brothers have Phd degrees (not in math). My wife has two masters degree (certainly not in math). I feel like the dummy in the family holding just an MS degree in computer science (from Villanova, of course).

My interest in math was piqued by raising my two sons. One is now a sophomore at Tufts University. He never needed any help with his homework and went through his school district’s “gifted program.” As a gifted student in mathematics he was compelled to participate in MathCounts competition during middle school. I went to the first competition and saw the team do poorly due to lack of preparation. So I called the teacher in charge the next year and asked if I could assist coaching the team. He said “sure,” and by eighth grade the team placed 5th in Chester county, and one member of the team finished 6th individually.

I learned that having one very bright son does not guarantee two bright sons. My other son is not academically oriented. I tutored him a lot in math and had to review his text books closely in order to explain the concepts the way that I thought they were presented in class. I was amazed at how much new material is taught in geometry compared to what I learned many, many years ago. The same goes for Algebra 1 and 2 although the difference was not so dramatic in these subjects.

So I re-entered the field of mathematics without a definite goal in mind knowing that I received a random mathematics education as an undergraduate. I say random because I don’t have a good foundation in calculus (it’s a long story) and the only undergraduate courses that I really enjoyed were number theory and abstract algebra. I majored in math as an undergraduate because because math was one subject where there as absolutely no BS (I don’t mean Bachelor of Science here). I’m hoping to squeeze by on the required analysis courses having strengthened my calculus by taking the MIT Open Course Ware course in single variable calculus (using a great textbook by George Simmons).

I guess I may teach a few math courses since an early retirement looks feasible for me. I did try teaching computer science in the early eighties but I didn’t like it at the time. I do know that I enjoy being a student again.

Perhaps the real reason that I’m studying math again is to impress a college friend who is a bone-fide mathematician. His name is John P. D’Angelo and he’s a tenured professor at the University of Illinois at Urbana-Champaign. John is a legend in my mind, and I hope to see him again some day. Believe it or not, as an undergraduate his teachers would have him read their research papers to look for errors prior to submitting them for publication. He earned his Phd from Princeton University during the period when there was a phantom in Fine Hall. The phantom was named John Forbes Nash and John spoke with him on many occasions.

So if you know of any good employment opportunities for a “seasoned” worker with a double masters degree, please let me know.

Graduate Course in Number Theory

2007-12-22T18:49:00.000-08:00

I just completed my second graduate math course. The subject was number theory which is known as the "queen of mathematics." Before I describe the course I want to briefly mention that last summer I toured the old growth redwood forests in California. When I walked through the redwoods I felt like I was in an enchanted forest. I marveled at the towering sequoias which ranged in age from 1,500 to 300 years old. Each tree is a monument to nature herself and the forest has an eternal quality to it.

I found my number theory course was similar to visiting the ancient forest. We learned theorems from Euclid (e.g. the infinitude of prime numbers) and Diophantus that date back over 2,000 years. We learned venerable theorems from Fermat, Euler, and Gauss that date back over 300 years. Each theorem stands as a pillar of mathematical truth for all eternity.

I have to admit, though, that I was lost in the forest of number theory at times. At certain points I could understand individual theorems and their proofs, but for long stretches I stumbled along. The journey began with chapters on divisibility and congruences. These are straightforward topics that can be taught to bright high school students or the brightest middle school students. Fermat’s Little Theorem and Euler’s generalization were covered. Of course, the totient function f(n) was defined (the count of numbers relatively prime to n) and in a later chapter the remarkable formula was proven. Translated into English this states that the sum of the totient function ranging over all of the divisors of n is equal to n itself (futher translation not available).

Okay they were the easy chapters. Now we move onto quadratic reciprocity and quadratic forms. Our textbook An Introduction to the Theory of Numbers by Niven, Zuckerman, and Montgomery mentions that "Gauss discovered the quadratic reciprocity law just before his 18^th birthday. After a year of strenuous effort he found the first proof, in 1795, at the age of nineteen." I find this fact comforting. The world’s greatest mathematician had to struggle for a year to find a proof for a fact that he knew to be true. So it’s okay to struggle with math homework – that’s its purpose in life.

Binary quadratic forms have the form f(x,y) = ax²+bxy + cy². Mathematicians have studied BQFs intensely and have a complete understanding of them. I can’t say that I have a complete understanding but I did learn a technique based on BQFs that can be used for the following arithmetic parlor trick. Given the prime number 398417 find two numbers whose squares sum to 398417 (answer: 631² + 16²).

For some reason our teacher decided to skip Chapter 4 which has marvelous results like the Moebius inversion formula (don’t ask). So we continue with the chapter on Diophantine equations, the most famous of which is Fermat’s Last Theorem (there are no integer solutions for xⁿ+ yⁿ = zⁿ where n > 2). This leads to an area of mathematics I had never learned before called elliptic curves (these are not elliptical curves). The geometric analysis of these curves (chord and tangent method) yields additional solutions to the Diophantine equation on which they are based. In our class we visit the foothills of the mountains that Andrew Wiles conquered in his 1993 proof of Fermat’s Last Theorem.

We skip chapter 6 to get to the chapter on Continued Fractions. I’m in the embarrassing situation of having to tell people that I’m studying fractions in graduate mathematics, but it’s worth it. These fractions are small wonders and the basis for new insights into irrational numbers. I also learned of Pell’s equation and was able to fully understand an anecdote relating to the Indian mathematician Ramanujan. One of his friend’s gave him a puzzle to determine a house number that met certain conditions. Ramanujan was cooking at the time, but he was able to instantly state the result in general terms using a continued fraction based on Pell’s equation.

For the last two classes our teacher covered eclectic topics in multiplicative number theory. At the very least I learned what Dirichlet’s series are and the Reimann zeta function in particular. Oh to solve the Riemann hypothesis relating to the roots of the zeta function!

Note on textbook: It seems that the textbook by Niven, Zuckerman andMontgomery is widely used as a graduate text. It has survived the test of time having endured five editions since 1961. My belief is that it was a fit text in its early editions, but by the fifth edition it has grown flaccid and overweight. The authors give the most succinct proofs and even skip steps at times.

If I felt stultified by the topics at hand, the authors helped me achieve that state.

The Beebee Gun -Or- What Was I Thinking?

2007-11-22T18:20:00.000-08:00

"Dad, can I have a beebee gun?"

"No, you can't."

"Why not? Mike has a beebee gun."

"I don't care. Beebee guns are dangerous and you can't have one."

This conversation was repeated many times when my son Tony was in fifth grade. I'm usually one who can put his foot down and say "no" but Tony found a soft spot in my defenses. He promised to raise his grades if I got him a beebee gun.

"Well you just missed honor roll the last marking period. If you promise to try harder I'll get you a beebee gun."

"I promise."

So I purchased a beebee handgun and let Tony use it in our newly finished basement. Even though I insisted that he use some sort of backstop I started to find beebees embedded in the drywall. I thought "what he really needs is a huge backstop." So I went and bought three sheets of plywood and attached them together via hinges so that they can be bent at an angle to form a backstop, much like a dressing curtain.

The shooting continues and I start to find embedded beebees nowhere close to the backstop. I warn Tony that unless his aim improves the beebee gun is gone. He makes his usual promise and I let the matter drop.

The next thing I know he has several of his friends over and passes around the beebee gun. They must have had a first class shoot-out because I find beebees in the ceiling tiles, the drywall, in the flourescent light enclosures. I also find that the TV took a glancing hit that took a 1/4 inch chip out of the screen.

Okay, that's it, I confiscate the beebee gun (actual picture shown above). Tony and I now spend some quality time together digging out beebees, spackling the drywall, and collecting hundreds of beebees from the all nooks and crannies of the basement.

Now dear reader you must be saying that I'm a complete fool, and I can't deny it. I wanted some sway over Tony and I thought the beebee gun just might do the trick. Of course Tony did not make the honor roll the next marking period.

Even today, many years later, I come upon a stray beebee that escaped my determined effort to remove all evidence of my folly. And when I do, I hold it up to the light and say to myself, "What was I thinking?"

First Graduate Math Course

2007-07-26T17:08:00.000-07:00

This is my first post as a graduate math student having completed my first course at Villanova University. Since I was eager to start I took an eight week summer course where each session was 3.5 hours long.

It was intense. The first homework assignment took between 15 and 20 hours to complete even though I skipped problems that were too hard or too time-consuming. Fortunately, the homework subsided in later sessions and even the sessions near the end of the term let out early.

We used what must be "the" book on the subject - Cryptography Theory and Practice by Douglas Stinson. The breadth of mathematics covered in the book (and we covered only about 20% of the book) is amazing. Linear algebra, abstract algebra, probability, and above all number theory. Euler considered number theory to be the queen of mathematics. It is certainly the linchpin of modern cryptography.

My favorite parts to this course were the programming assignments and the study of the RSA public key cryptosystem. I wrote the programs in Java 6 and learned new language features in the process. I'd estimate that I wrote about 1,000 lines of code. My teacher commented that I wrote "nice programs." Damn straight. I've been programming for over 35 years.

RSA public key stuff is fascinating because it's pure number theory. To study it you dip into many number theory topics including primality (this means "is it a prime number") testing and factorization of huge numbers. Now wonder Rivest, Shamir, and Adleman won the Turing Award in 2002. What I don't understand is why the El Gamal cryptosystem became the basis for the Digital Signature Algorithm standard instead of RSA. RSA is just as good for this and it means you have to purchase only one package for your business.

Another thing I don't understand is the theory behind the Index Calculus algorithm for computing discrete logarithms, namely the change in the modulus from p to p -1 as you go from (1) to (2) below.

(1) a^x_j º p₁^e_1j p₂^e_2j ... p_B^e_Bj (mod p)

(2) x_j º e_1j log_a(p₁) + ... + e_Bj log_a(p_B) (mod p-1)

Kind reader. If you can explain, please post a comment with the explanation. My teacher claimed that it was a direct result from Fermat's Little theorem, but I still don't see it.

Dick Cheney Goes to Heaven

2007-03-05T16:46:00.000-08:00

It’s just like I heard it would be like. I see myself lying prostrate on the floor. I see people giving me CPR, others calling for help. I hear someone say “it’s too late, he’s gone.” “That’s okay,” I think. “I’m glad the chest pain is over.” Then I thought to myself, “They better give me a good, Reagan-level funeral” as the scene dissolves.

So here I am standing by a desk with a man who looks too busy to notice me. He looks like the late Senator Everett Dirksen, very distinguished, very authoritative. I admire his dark blue suit, his red tie, and especially his American flag lapel pin. I see a nameplate on the desk – Senator Peters.

“Hello,” I say in my deepest, most gravitas laden voice, hoping to command some respect from this lackey.

He puts down a paper, looks up, and smiles. “Welcome, Dick. We’ve been expecting you for some time now. Your resume is remarkable. You should have no problem getting in.”

“Getting in?” I ask. “Is there anything I have to do? Is there anything that I haven’t already done?”

“Not really, Dick, just a preliminary interview. That will be it.”

Senator Peters: “Your service to the American people is wonderful. You were Chief of Staff to President Ford. You represented your state in Congress for 10 years. You served as Secretary of Defense. You were Vice-President during one of the most trying periods of time in United States history.

DC: “Thank you sir. I’ve very proud of my service to America. If you didn’t know, I received the Presidential Medal for Freedom in 1991.”

Senator Peters: “But you never served in the Armed Forces, especially during the Vietnam War.”

DC: “Well, like many young men at the time, I was in college and sought a student deferral. After college, I received a family hardship deferral.”

Senator Peters: “If you never knew the terribleness of war firsthand, how do you feel about sending America’s best into harm’s way? Seems that you’ve done that twice in the Persian Gulf.”

DC: “I know it’s a tremendous responsibility, but I know what’s best for America.”

Senator Peters: It seems that you made up your mind about Iraq long before there was enough evidence to wage war against it.”

DC: “Saddam Hussein was a brutal dictator. Our intelligence reports stated that he had links to Al-Qaeda and was cultivating weapons of mass destruction. Anyone who disputes this just doesn’t know the facts.”

Senator Peters: “It even appears that you benefited from the war with your ties to the oil industry and the defense industry.”

DC: “Nothing could be farther from the truth. Those accusations typically come from my political opponents. I have not benefited for any activity that I believe is in the best interest of the United States of America.”

Senator Peters: “Well, as I said at the start Mr. Vice President, your resume is impeccable. Congratulations and welcome to heaven. Yours is the first door down the hall on the right. It’s the red door.”

I see the red door down the hall and walk toward it. It’s this easy then, entering heaven. Christ, I deserve it. With all of the great service I gave to the good old U, S, of A. Working for that idiot Ford, never reaching the top, and then working for idiot #2, Bush, for eight years. Thank god that they listened to me, and did what I told them, those fucking idiots.

I open the door and walk in. I notice that I’m wearing a warm-up suit and that I feel fine. No chest pain at all. I also notice that there’s a sweet smell of incense in the room and there’s strange, foreign music playing. Not exactly what I thought heaven would be like, but not bad either.

The room’s kind of dark and it takes my eyes a little time to adjust. To my surprise I can see a giant round bed fitted with a scarlet bedspread and silken sheets. Then I notice the mirror over the bed. This looks like a scene from playboy mansion. What next? Girls?

Oh my god, it is girls. Young, voluptuous girls enter from a side door. So many I can’t count. Each is dressed in silk gauze. Each is dark-eyed, with a dark complexion, though I wouldn’t call any of them “black.”

They file past me and every one either smiles at me, or purses her lips, or lifts her bosom with one of her hands. Christ, am I going to get laid here? Looks like this Wyoming cowboy is going to shoot off his pistol a few more times.

So I look over these girls. None of them says a word, but their body language speaks volumes. Half of them just sit on the floor with their legs open wide and grin at me. Some look young enough to be my grandchild. Others are older, but not nearly as old as my wife. Choices, choices.

Okay, I’ll take you as I point to teenage beauty. She squeals in delight, and I heard a sigh of disappointment from all of the others. This beautiful creature literally skips around me as we head to the round bed. By the time we reach the bed, she’s naked. All I have to do is drop pants on my jumpsuit. I’m amazed that at my age, I’m hard as a rock and ready to go.

So we’re both down on sheets, which are kind of slippery. I finger her and see that she’s wet as a mop. Since she’s ready, I’m ready. I’m going in and … oh my god that hurts. It feels like my dick was just cut off.

Next thing I know I’m sitting at the desk of Senator Peters again. He’s shaking his head.

Senator Peters: “Dick, you failed the interview.”

DC: “What? We had the interview. I did great.”

Senator Peters: “That was the preliminary interview. We knew that you were good with words, a veritable prevarication machine you might say. So we put you to the test, the silent interview. You failed just like the rest of them.”

“We can now call you on all of your bullshit. You were so good on earth that we had trouble telling the bullshit from the non-bullshit. You were so convincing that even from our vantage point, we couldn’t tell if you were sincere or not. Here, I'll just tell you what we've got on you.”

“You help found the ‘Project for the New American Century’ which advocated the U.S. use its military strength as an instrument of foreign policy.”

“You directed the Energy Task Force and published a report that stated it is in the US interest to manipulate oil rich countries to secure future oil supplies.”

“You were the Bush spokesman for the invasion of Iraq and the removal of Saddam Hussein stating repeatedly that:
- he was developing weapons of mass destruction
- he was an ally of Al-Qaeda
- he supported the September 11 attacks against the US.”

“You saw to it that Halliburton Corp. received $7B no-bid contracts following the US invasion of Iraq.”

“You, sir, are a warmonger and a war profiteer. Goodbye Mr. Cheney,” says Senator Peters. He pushes a button under his desk, and just like in the movies, a trapdoor opens under my seat, and I fall into darkness. And as I fall I feel it getting hotter and hotter. “Christ, what’s happening,” I think. And it keeps getting hotter and hotter.

The next thing I know I’m lying of the floor of a cell with harsh lights shining down on me. There’s a mustached man with army fatigues standing over me. The bastard kicks me with all of his might in the gut.

“Hello, Cheney. Welcome to hell,” Saddam says.

The Recess Posse

2007-01-02T16:19:00.000-08:00

This is a true story from my youth. I haven’t shared it before now.

In eighth grade all of the "good" boys, as deemed by the school nuns, were impressed into the school's safety patrol. So I was picked to be a safety.

We were required to conduct ourselves in a morally fit manner, and were given specific assignments. My assignment was particularly bad -- I was to oversee the first grade class during lunch recess in the rear of the school.

I hated this. I was separated from anyone close to my age, and I had to enforce the nun's rules concerning recess. First and foremost, the first graders were not allowed to run! Heaven forbid they might trip and fall on the asphalt. So for the first two weeks I was constantly chasing after the violators and making them do a time-out by standing against a chain link fence.

It didn't take long before I noticed that it was the same bunch of kids that I was punishing day after day. Since I was going crazy enforcing the silly rule, I decided to make my own rules. I rounded up all of the "bad" first graders and deputized them. Of course, they wanted to know what that meant, so I told them that they were on my side and when I gave the order, they had to run after any kid they saw running and bring him to me.

What a great idea this turned out to be! Not only did I give them a chance to exercise, but I had the most enthusiastic enforcement team you could imagine. If someone outside the posse started to run, the posse would collar him right away. It was so effective that after a while, to relieve boredom, members of the posse would just start running to see how long they could elude their friends.

It never occurred to me that if the nuns found out what I was doing, I would have been severely punished, maybe even suspended. As luck would have it, I received a promotion and changed assignments. This wasn't due to the taming of the running lust of first graders, but rather to the fact that the captain safety was kicked off the force. He was caught talking during morning prayers and was instantly dismissed by the eighth grade nun.

Everyone near the top of the ranks was bumped up one position. I became lieutenant safety and was assigned the bicycle rack at an entirely different area of the school.

My story was never told till now.

Bad Line Call

2006-12-30T11:32:00.000-08:00

Tuesday night is tennis night, and this one was just like any other tennis night, except that it was different. It was the last tennis night of the contract season, and I was playing my tennis nemesis, B. I expected to lose to B. that night; I just didn’t know how badly I would lose. I expected to lose because I always lose to him. It’s not that B. is in a totally different league than me, like he’s a tennis pro and I’m a rank amateur. We’re in the same league, and that’s what so frustrating about the fact that he always wins.

B. is a player who instinctively knows how to play the game. He knows how to hit all of the strokes. He knows when to come to the net, and when not to come to the net. He knows how to position himself when the ball travels to the opponent’s side of the court. His tennis instincts are as strong as a predatory lion’s instincts are when it comes to stalking an antelope.

If B. has a weakness it’s his lack of conditioning. Since our contract time is limited to one hour this isn’t a big factor. In fact B. compensates for it by his sheer will to win. Let me rephrase that – B. refuses to lose. He pushes himself to the limit whenever defeat is near. It doesn’t matter whether it’s losing a set, a game, or a single point. B. refuses to lose.

But something was different that night. We were tied 4-4 and I was holding my own. I had gotten to this score on occasion in the past, when B. would suddenly realize that he might lose. B. would focus with incredible intensity and his serves would blaze into the corners of the service box. If his service didn’t win the point, his groundstrokes would, always attacking my weak backhand until I either hit a short ball or a shallow lob, either of which was enough for him to put away the point. Of course, I knew all this and it would give me a case of the nerves just like an antelope gets when he sees the lion.

B. served in the ninth game. I could tell he was tired because he started to serve and volley. He wanted the points to be short like the quick kill by the lion going for the jugular. I just kept lobbing my returns and instead of sailing long, they were landing deep. This not only frustrated B., it also exhausted him as he sprinted back to make a return. With the game tied at 30 – 30 we start trading groundstrokes, moving each other from side to side, each hoping to hit a winner and end the point. B. hit a deep drive to my baseline. I couldn’t catch up to it, but I could watch it closely. I saw it land squarely on the line, and I immediately call it “out.” B. was too far away to challenge; he just bent over to catch his breath. On the next point he opened his forehand and the ball sailed out of bounds. I had just broken his serve.

I served out the last game. On set point B. was totally spent. After he returned service he came to the net to rattle me as he had done so many times before. I lobbed him and he didn’t even try to make a save. He just turned and watched the ball land in bounds. I had won my first set in 10 years.

But what about that bad line call? I knew it was the wrong call as soon as I made it. Since then I have thought long and hard about it. If he had won that point, he may have won the game and eventually the set. The lion feasts again. I thought of all of the injustices I had endured. Doesn’t it matter that B. foot faults frequently when he serves? Doesn’t it matter that B. enjoys trouncing me 6-0, 6-0 when I’m playing poorly? What about the times he gets his hot serve going and deliberately picks on my pathetic backhand return? And he keeps doing this through both sets of the contract hour. I know these are all rationalizations, serving me, the antelope. What would your call be?

The Handshake

2006-12-29T12:43:00.000-08:00

The plane had landed at Philadelphia International Airport and had rolled to a stop outside of the gate. I knew that this was the time to make my move. I stood up and leaned over the mammoth figure still reclining in his seat. He looked at me crossly. “Sir,” I said, “you are the greatest basketball player of all time.” I stuck my hand out forcing him to reciprocate. “Thanks man,” he mumbled as he shook my Lilliputian hand.

Now how can you sleep when the greatest basketball player of all time is sitting directly in front of you? I couldn’t. All I could think of was what he could still do on the court against the Sixers new big man, Darryl Dawkins. The Dipper would just eat him up, I thought, on offense and defense. But that’s a fantasy. Wilt will never set foot on a court again. Maybe I’m dreaming this. After all it’s the red eye from LA to Philly and you’re supposed to sleep.

What can you say about Wilt Chamberlain? Most sports fans know that he scored 100 points in a single professional game. But how about the 50+ points per game he averaged for an entire season? Or the record 55 rebounds he grabbed in a single game against Bill Russell. Or the only time a professional player scored a double-triple-double – 22 points, 25 rebounds, and 21 assists. These records are as impossible as an asthmatic climbing Everest without oxygen.

In his early days he was unstoppable. The NBA felt it had to at least slow Wilt down so they changed the rules. The three-second violation, widening the foul lane, and both offensive and defensive goaltending rules were instituted to throttle The Dipper. They didn’t.

Why am I thinking about this? I’m supposed to get some sleep. Don’t you think Wilt’s sleeping in his seat?

As I board the plane late in the evening, I’m surprised at the check-in counter. I have just received a free upgrade to first class. I enter the first class section and see that there are more seats than people. Now I can really spread out, relax, and sleep. But wait a minute, who is this towering figure now boarding at the last minute that everyone seems to know. My God it’s Wilt Chamberlain, and he’s sitting in the row in front of me! How like Wilt to hold up an entire flight until he’s good and ready to fly. After all he’s probably the only profession coach who failed to show for his team’s games. Now what should I say to him?