Jul 012015
 
 07/01/2015

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Especially in urban areas, two locations may be quite close geographically but difficult to travel between. I wondered whether one could create a map where, instead of physical distances, points are arranged according to some sort of travel-time between them. This would be useful for many purposes.

Unfortunately, such a mapping is mathematically impossible in general (for topological reasons). But so is a true map of the Earth, hence the need for Mercator or other projections. The first step in constructing a useful visualization is to define an appropriate Travel-Time metric function. Navigation systems frequently compute point-to-point values, but they are not bound by the need to maintain a consistent set of Travel Times between all points. That is our challenge — to construct a Travel Time metric.

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Jul 012015
 
 07/01/2015

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This is a scary piece in which I analyze precisely how many voters would be required to trigger a Constitutional Convention and ratify any amendments it proposes.  Because the 2/3 and 3/4 requirements in the Constitution refer to the number of States involved, the smaller States have a disproportionate effect.  In Congress, the House counterbalances this — but for a Constitutional Convention, there is no such check.

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Jan 312012
 
 01/31/2012

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Both information theory and statistical mechanics make rather cavalier use of a simple continuous version of the discrete entropy. Treatments often gloss over a number of subtleties in the definition of such a quantity, and this can lead to confusion. A proper continuous version of the discrete entropy is not easy to construct and may not exist. The differential entropy commonly bandied about actually is a discrete entropy in disguise, and possesses an implicit coarse-graining scale.

In this article, I review discrete entropy and probability densities, carefully analyze the continuous limit and issues encountered, and touch on several possible approaches. An enumeration of various axiomatic formulations also is provided. The piece is pedagogical and does not contain original research, though I offer a couple of my own thoughts on possible means of generalizing entropy.

While an acquaintance with probability and entropy is assumed, the discussion is fairly self contained and should be accessible to a broad audience.

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Aug 232011
 
 08/23/2011

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I like to play around with various configurations of camera lenses.  This partly is because I prefer to save money by using existing lenses where possible, and partly because I have a neurological condition (no doubt with some fancy name in the DSM-IV) that compels me to try to figure things out.  I spent 5 years at an institute because of this problem and eventually got dumped on the street with nothing but a PhD in my pocket.  So let this be a warning: keep your problem secret and don’t seek help.

A typical DSLR (or SLR) owner has a variety of lenses.  Stacking these in various ways can achieve interesting effects, simulate expensive lenses (which may internally be similar to such a stack), or obtain very high magnifications.  Using 3 or 4 lenses, a telextender, a closeup lens, and maybe some extension rings (along with whatever inexpensive adapter rings are needed), a wide variety of combinations can be constructed.  In another entry, I’ll offer a companion piece of freeware that enumerates the possible configurations and computes their optical properties.

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Aug 012011
 
 08/01/2011

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While exploring theoretical physics and computer science, I commonly encounter large sets whose cardinalities are of interest. Rather than endlessly recalculate these as needed, I would prefer to have a single reference which consolidates all of the salient results. To my knowledge such a work does not exist, so I decided to create it. Consider it a missing chapter on cardinality from Abramowitz and Steguin.

There are many excellent works on the rigorous development of cardinal theory, the more intricate aspects of the continuum hypothesis, and various axiomatic formulations of set theory. Rather than emphasize these, the present work attempts to summarize practical results in cardinal arithmetic as well as list the cardinalities of many common sets. No attempt at rigor or a systematic development is made. Instead, sufficient background is provided for a reader with a basic knowledge of sets to quickly find results they require. Proof sketches offer the salient aspects of derivations without the distraction of formal rigor. Where I perceive that pitfalls or confusion may arise (or where I encountered them myself), I have attempted clarification.

In addition, I included a discussion of infinite bases and integration from the standpoint of cardinality. These are topics that are of interest to me. Hopefully, others will find their mention useful as well.

If you detect any errors in my exposition, wish to offer suggestions for improvement, or know of any omitted references or proofs, I would be grateful for your comments.

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Apr 292008
 
 04/29/2008

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Have you ever wondered what really is meant by a “deciding vote” on the Supreme Court or a “swing State” in a presidential election? These terms are bandied about by the media, but their meaning isn’t obvious. After all, every vote is equal, isn’t it?  I decided to explore this question back in 2004 during the election year media bombardment.  What started as a simple inquiry quickly grew into a substantial project. The result was an article on the subject, which I feel codifies the desired understanding. The paper contains a rigorous mathematical framework for block voting systems (such as the electoral college), a definition of “influence”, and a statistical analysis of the majority of elections through 2004. The work is original, but not necessarily novel. Most if not all has probably been accomplished in the existing literature on voting theory. This said, it may be of interest to a technical individual interested in the subject. It is self-contained, complete, and written from the standpoint of a non-expert in the field. For those who wish to go further, my definition of “influence” is related to the concept of “voting power” in the literature (though I am unaware of any analogue to my statistical definition).

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Jun 011998
 
 06/01/1998

Once upon a time there was a physicist. He was productive and happy and dwelt in a land filled with improbably proportioned and overly cheerful forest creatures. Then a great famine of funding occurred and the dark forces of string theory took power and he was cast forth into the wild as a heretic. There he fought megalomaniacs and bureaucracies and had many grand adventures that appear strangely inconsistent on close inspection. The hero that emerged has the substance of legend.

But back to me. I experienced a similar situation as a young physicist, but in modern English and without the hero bit.   However, once upon a time I DID write physics papers. This is their story…
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