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• So the doctor lied, get over it. Stop whining, blaming. It’s YOUR body that’s killing you, nobody else’s fault.

Maybe he’s imperfect, this doctor, gave you the wrong drug, didn’t get you into a clinical trial, could have found something if he’d looked harder or at all. Big deal. Everybody’s imperfect.

Hindsight is twenty-twenty. Do you have a medical degree, eight years of residency? Then don’t second guess. Do as you are told and hope for the best.

You don’t pay half your salary for insurance, due to people who won’t accept responsibility. People like you.

Think you can do better? Good luck, but your time is limited. Better hope they take your insurance. They won’t.

I’m only looking out for you here, ignoring your disloyalty because I like you.

You’re one of my best customers. And I know you’re confused, don’t realize what you are saying.

Now stop fretting, take your medicine, do as I say. Be a mensch and die before six. I have a dinner.

• You are unseemly.

There is no room for unseemly things in my world.

You do not exist in my world.

You exist in this world.

It is not my world, and is unseemly.

My world must be seemly.

If you were to enter my world it would become unseemly.

I must remove you from the unseemly world.

Then my world will remain seemly.

• There is a hidden art, newest of the new, hippest of the hip. There are no places you can see it, no artists who can make it or, if made, none who can keep it. Found art is so passe. The new thing is Lost art, and my collection is vast.

• I see a world of broken dreams, mangled forms they failed to take, gadflies on a soul not mine, insouciant songs, dabbled in colors they never knew.

This unmoving maelstrom no seed of hope can hope to pierce. Fly elsewhere perfidious pollen, seek not to sully this confection with the grating melody of youth.

Why do I inveigh against a cast too great, an everchanging litany of misnamed misdeeds, clever devices of artless authors, unoriginal not ineffective.

The parade is distant touching me, but not upon me. Disgust alone breeds no tears, a quiescent enigma.

Why must I watch, moved unmoving, a relic of morality, another’s hysteric, sounding the hours til morn in somber peals of silence.

• There are four warnings which must be given to any man, that he may pass through this life unscathed.

What? How would I know what they are. Do I look unscathed? Are you saying my father was smarter than me, possessed this critical wisdom but neglected to pass it on? That’s insulting; I probably should scathe you.

When eating a tangerine, there may still be seeds, but they are small and can only break your teeth if you too are small. That is the reason you should decide to be big.

I’ll admit it, I don’t like your face. I may rearrange it. Do you think my dad is a pervert, the type of freak who puts his hands on little kids’ heads, teaches them things?

If you fall backward, you won’t see where you land. It is not likely that the person in whose arms you end up is the person for you. Best to fall forward and know whom you are falling for. Breaking your nose is a small price.

Do you always go around insulting the people who hate you? There are so many of us, where do you find the time. My dad was one. He hated you. He told me so. He said I should hurt you if ever I can.

We all want a child who is like us, so it is best to have yourself as a child. There is no law which prevents this, so if it does not happen that must be your fault.

You look like a no-can-do kind of guy, the sort who picks fights with people by not picking fights with them. Well, I’m itching for a can-do fight with a no-can-do sort.

If you are called upon to perform a blind taste test, it is best to lie. One or the other will be insulted, and you never should insult a large corporation. They are bigger than you and hate losing. Instead say that you love them both equally and unconditionally and have no taste.

Why would my dad tell you these things, but not me? He loved me, nurtured me, ate my brothers. Why would he do that but not tell me how to live my life? Why would I tell my dad these things instead of me. You’d think you already would know better and have told me them first.

• There is something not right about the house. It is too tall or too thin or the walls are at improbable angles.

Nothing seems as it should, and nobody who enters is ever seen leaving.

You can hear them go in, then a cry, some clattering, a groan. Silence.

The neighbors say the house always was there. Some say the Germans built it, some the English.

You never can tell whether people really vanished. Maybe they left by an attic, or a basement.

Perhaps there is a back door and they rejoined the crowd in front, pretending dismay at their own disappearance.

I believe each of us will enter the house one day. It is possible that some of us already have, but do not remember.

I was mistaken, that is not a crowd in front. It is a queue, and I am next.

• Tick Tock says the clock. Tock Tick say the sick.

Turn back your hands, they beg in vain. They are digging your graves, it laughs insane.

We’ll smash your dial, they threaten meekly. I have another, it chortles obscenely.

Tick Tock says the clock. Nobody hears or speaks.

• Does the silent man have nothing to say or is he tired of speaking truth to the wind?

It is puzzling there are so many, each shouting fragmented thought. Is it unthinkable to think?

Countless shadows cross, merge into impossible and obscene shapes. Where does one begin and the other end, how does one find oneself in such a mess?

Shadows once stood apart, sentinels of solitude and the undemarcated loss of time.

Is the shadow of a shadow truth? Perhaps it casts an object or is bound to some profane idempotence.

It may be that the shadows of shadows are other than things. Or perhaps they are not meant to be seen, the penumbra of things better forgotten, or things which never were. The absence of absence is not presence.

It is telling that a shadow has no bound, in maturity its potency without measure. No form contains such potential, this only shadow can achieve.

Even as it fades, a haze of dissolution, shadow mocks us. I am infinite for a moment, it laughs, but you … you will be for many moments, small and weak. Do you not envy me?

The shadow is haughty, but not grasping. It is wiser than we. One could easily confound parent with progeny. Foolish father, it says, will you never grow down?

It may seem that shadow is the little that passes through us, lessened by what we take, tasted and discarded. How very wrong.

It is defined by the light we did not block, what we could not consume, all the being and essence we could not comprehend. This is no small thing. How much greater than us shadow must be.

The light burns twice as bright, the unlight unburns even stronger. There is a difference between eternity and a heartbeat, man and his gods, however equal they may seem in the moment.

Light mesmerizes and charms, shadow dances at its behest. Only when the light fades, do we see that all the world is made of shadow, that shadow broke off the smallest piece of itself to give us some light.

• There was a bitter old woman who scowled at me in passing.

What cause do you have to be bitter? I asked her. It seems unfair to be bitter without a cause.

Do I now need a reason to be bitter?  Who are you to demand this of me?

It is wrong to scowl at passerby, I insisted. I know this because I am a passerby and you scowled at me.

A child presumes to lecture me,  she laughed. That is why I am bitter.

I smiled. You are laughing, so you must not be so bitter after all.

No, I am twice as bitter now because you made a bitter old woman laugh.

• When you are at home in the ordinary chaos of things coming and going, it is easy to ignore a fly. This can be a mistake.

There are flies, and there are flies. Pay close attention to the shape of the wings, the striations, the abdominal patina. These may be the give-away, the sign that this fly, out of billions, is a killer. It is the anathema, the 1943 copper penny, the brown recluse.

This does not mean it necessarily will go out of its way to kill you. It may be busy or lazy or simply not in the mood. It may bide its time until your child is asleep or it may decide you altogether unworthy of the effort. Then again, it may not.

You won’t know you are dead until some time has passed. This fly looks almost identical to any other and seems innocuous. Perhaps if you hadn’t shooed it or tried to swat it or made eye contact or failed to offer it a lucrative compensation package it simply would have gone away. But it did not, and the fault probably is yours.

Of course, you may not have recognized the fly, thought it ordinary, harmless. That is no excuse. If anything, it is insulting.

There are 230 visual characteristics that can be used to identify a fly. Killer and ordinary flies differ in only one of these, and nobody is sure which. Even the most renowned expert has little chance of telling. But perhaps you can do better, since you care, since you’re the one who will die.

There’s no certainty, only statistics. Find a way to bend these in your favor and perhaps you will live another day. Avoid the fly, run from it. Sometimes ignoring it can help; if there is no such thing it cannot hurt you.

Why should there be this fly? It has no right to exist, to threaten you and your child! But it does, and if you encounter it perhaps you can seduce it, persuade it to find somebody else – somebody less important, somebody less you (or your child). This rarely works, or perhaps you are that somebody else.

The sad truth remains: the fly is out there, unrelenting, buzzing, waiting. You must accept that it will kill you. If it is indeed a killer. Does it want your death or just some sugar?

Once the fly has bitten you, you will die. The fatality rate is 100%. Sometimes it is quick and painless, other times it can last for decades, culminating in one of many lingering, debilitating conditions. The symptoms are indistinguishable from ordinary illness, it is probably best not to bother with a doctor.

Save your money for a quality tomb. Finding a good place to spend eternity is difficult. Do you think there is room left in heaven or hell? Real estate is in high demand, you’ll likely end up stuck in your grave. Be certain it’s a nice one. Most important, make sure there are no holes, or a fly may get in.

• Good news for all of you who’ve been asking me to make my writing available! I’m posting a short work at least once, and sometimes twice, a week. They will be a mix of older and newer ones, and hopefully the pipeline will remain well-populated as I continue to write and edit. For the most part, the pieces are either poems or the type of very very short fiction/vignette that seems to be unclassifiable in today’s literary market.

These are the same sort of pieces (and sometimes the same actual pieces) that go into my chapbooks and short-work books, such as the forthcoming The Man Who Stands in Line. I intend to occasionally post full-length short stories as well, though for me “full-length” still can be quite short. Given the chaos of my schedule, it is less than likely I will adhere to any consistent timing. But as a rough rule of thumb, I will aim to post works on Sundays and (if two works that week) Wednesdays.

All posted content is free to view for anyone, and subject to the Creative Commons License described on the website.

If you would like to be kept apprised of new postings, please subscribe to my RSS feed.

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• I run three meetup groups in the Boston area, so I decided to include links to them from this site.

1.  Cambridge and Boston Friendly Writing Group:  This is an open group loosely based off a great group I used to attend in NY (that one was run by Zack Hample).  The idea is that we write for a little over an hour and then (optionally) share what we have written.  As with many meetup groups, there are lots of members but only a few attend.  Our meetings tend to be small and fun.  In terms of membership, we may now be the largest writing group in Boston!

2.  Cambridge Advanced Math Study Group:  This is a smaller group (admission is selective) which works its way through math textbooks.   Our focus so far has been Algebraic Topology and Differential Geometry.

1. Boston Area Research Projects Group: This is for people who want to discuss research projects outside their areas of expertise. Projects like the ones found on this site. Admission is selective.

If you feel that any of these are of interest to you (even if you’re just visiting the Boston area or here temporarily) feel free to join or apply; we’d love to have you!

• “The following is the text of a letter I sent to the NY Times questioning the FDA’s decision to ban Trans fats.  Whether or not one regards it as a good idea, the decision was based on a small amount of scientific evidence of recent vintage pointing to (optimistically) a reduction of 20K heart attacks per year.  On the other hand, there is a great deal of evidence that smoking directly causes 480K deaths per year.   The FDA’s priorities seem illogical.   Here is the letter itself.  It was not selected for publication:

To the Editor:

Re “F.D.A. Sets 2018 Deadline to Rid Foods of Trans Fats” by Sabrina Tavernise (June 16, 2015)

Whatever one’s opinion of banning Trans fats, the FDA’s priorities in doing so are puzzling.  According to the CDC, cigarette smoke is directly responsible for over 480,000 American deaths each year.  This is a far higher and more clearly attributable mortality rate than that from Trans fats.   If the FDA has the authority to ban Trans fats, then surely it has the authority to ban tobacco products.  If banning a ubiquitous component of our foods is deemed expedient, then surely it is even more expedient to ban a substance that lacks even a nominal claim to utility.    A nationwide prohibition against tobacco products is an obvious measure that would save at least 20 times as many American lives as the ban on Trans fats, yet nary a word from the FDA.”

• 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.

• 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.

• It often seems like the inflation figures touted by officials and economists have little connection with the real world.  There are a number of reasons for this, some technical and some political.  But there is a deeper problem than the means and motives for calculating any specific index.   The issue is that any aggregate number is likely to deviate significantly from one’s personal experience.  Each of us saves for different reasons and spends in different ways.  Without taking these specific choices into account, we cannot accurately represent or protect against the inflation that we individually encounter.  This paper elaborates on this idea and explains how each of us can identify the relevant components of inflation, and best hedge our savings.

• Taxes necessarily are unpopular. They represent an economic burden and do not yield obvious benefits. Though some make a show of embracing their civic duty, few voluntarily would undertake to do so if given a choice. The criminal penalties attached to evasion and the substantial efforts at enforcement are evidence of this. Nonetheless, there is a tie between one’s sense of social responsibility and the palatability of taxes. A perception that our sacrifice benefits ourselves, our loved ones, and society as a whole can mitigate the pain it causes. Conversely, if our hard earned money vanishes into an opaque hole of possible waste and corruption, resentment is engendered.

The taxes paid by an individual represent a substantial sum to him, but a mere pittance to the government. If there is no accounting for this money, then it appears to have been squandered. This assumption is natural, as the government is known to be a notorious spendthrift. Nor does the publication of a voluminous, incomprehensible, and largely euphemistic budget lend transparency. Even if it were perfectly accurate, and every taxpayer troubled to read it, the human mind isn’t wired to accurately grasp the relationships between large numbers. Thirty thousand dollars in taxes is minuscule compared to a billion or ten billion or a hundred billion, and it makes little difference which of those quantities is involved. Therefore an effort to elicit confidence through a full disclosure of expenditures would be ill fated even if well intentioned. However it would serve to enforce accountability, and should be required in addition to any other measures employed. If nothing else, this would allow watchdog organizations to analyze government behavior and identify waste.

So how could we restore individual faith in the system of government expenditure? There is in fact a way to do so and encourage fiscal responsibility at the same time. Individuals like to know where their money went. A successful tactic of certain charities is to attach each donation to a specific child or benefit. A person feels more involved, is more likely to contribute, and is better satisfied with their contribution if it makes a tangible difference. We need to know that we aren’t wasting our money.

The pain of an involuntary contribution may be assuaged through a similar approach. It may even transform into pride. There will be individuals who remain resentful, just as there are those who do not donate to charity. And some people simply don’t like being forced to do anything. However the majority of taxpayers likely will feel better if they know precisely where their money went.

We propose that an exact disposition of each individual’s taxes be reported to him. At first glance, this may seem infeasible. Funds are drawn from pooled resources rather than attached to such specific revenue streams. However, what we suggest can be accomplished without any change in the way the government does business, and our reporting requirement would not prove onerous. The federal, state, and most local governments already meticulously account for expenses – even if they do not exhibit particular restraint in incurring them. They must do so for a variety of legal and regulatory reasons, and records generally exist even if not publicly available.

Individual tax contributions need only be linked to expenditures at the time of reporting, but this must be done consistently. To that end, expenses could be randomly matched with the taxes that paid for them. This could be done each February or March for the prior year. We simply require that each dollar of taxes collected be assigned to one and only one dollar spent and vice versa. If there is a surplus, then some taxpayers would receive an assignment of “surplus” and if there is a deficit then certain expenses will be assigned a non-tax source - such as borrowed money or a prior year’s surplus. If a taxpayer’s contribution has been marked as surplus, then his true assignment is deferred until such time as the surplus is spent (again using a lottery system for matching). If it covers a prior year’s deficit then it is matched against that year’s excess expenses. The point is that every dollar of taxpayer money eventually is matched against a real expense.

For example, one taxpayer’s report could read 10K toward the construction of 121 example plaza, New York'', or better still3K used for the purchase of air conditioning units, 5K for ductwork, and 2K for electrical routing for work done at XXX and billed to YYY contracting on ZZZ date. Work completed on AAA date.’’ An individual receiving such a report would feel a sense of participation, accountability, and meaningful sacrifice.

It may seem that few people would feel pride in defraying the cost of mundane items, but such an objection is misguided. These are real expenses and represent a more comprehensible and personal form of involvement than does a tiny fraction of an abstract budget. If an expense would appear wasteful, pointless, or excessive, then it is appropriate to question it.

What of the pacifist whose money goes toward weapons or the religious individual whose taxes pay for programs that contravene his beliefs? It may seem unfair to potentially violate a taxpayer’s conscience by assigning him an unpalatable expense. But no exceptions should be made. Their money is being spent in the manner described. Whether their contribution is diluted or dedicated, they live in a society that violates their ideals and they should vote accordingly.

It is our belief that a feeling of involvement in the operation of government, along with the requisite increase in transparency, would alleviate much of the general perception of irresponsibility, excess, and unaccountability. An individual may object to his relative contribution, but the means of its use would no longer be inscrutable. This could go a long way toward restoring faith in our government.

• In most real situations, we must make decisions based on partial information. We should neither allow this uncertainty to prevent action or pretend to perfect certainty in taking action. Yet in one area with a great impact on an individual’s freedom and well-being we do just that. Judges and juries are required to return an all-or-nothing verdict of guilt. They may not use their experience, intelligence, and judgment to render a level of confidence rather than a mere binary choice.

I propose adopting a sentencing mechanism based on a probabilistic assessment of guilt or innocence. This allows jurists to better express their certainty or lack thereof than does our traditional all-or-nothing verdict. The natural place to reflect such an imputed degree of guilt is in the sentencing phase. I discuss the implications of such a system as well as certain issues with implementation.

• The current popular notion of democracy is something to the effect of the will of the people is effected through voting.’’ Though this is a far cry from the original meaning of the word or its various incarnations through history, let’s take it as our working definition. It certainly reflects the basic approach taken in the United States. Though often confounded by the public mind with a vague cultural notion of freedom, it only conforms to this when taken together with certain other principles – such as explicit protections of individual liberties.

This aside, let us consider the components necessary for democracy. To do so, we must make some supposition regarding the ability of an individual voter to render a decision. We assume that every voting individual, regardless of aptitude, is capable of determining their purpose in voting. We say purpose'' rather thancriterion’’ because we refer to a moral choice, what they hope to achieve by voting. This is far more basic and reliable than any specific set of issues or criteria. A person knows their value system, even if they can not or do not have the means of accurately expressing it. The desires to improve the country, foster religious tenets, create a certain type of society, support the weak, advance one’s own interest, protect a specific right, or promote cultural development cannot easily be manipulated or instilled. While it is possible to create a sense of urgency or attach specific issues or criteria to these values, one’s purpose itself is a reflection of that individual’s view of society and their relationship with it. To meaningfully participate in the democratic process, an individual must translate this purpose into particular votes in particular elections. Note that a purpose may embody a plurality of ideals rather than any specific one (such as in the examples above).

It is the function of democracy to proportionately reflect in our governance and society the individual purposes of the citizenry. A number of components are involved, any of whose absence undermines its ability to do so. While the consequent process may retain all the trappings of a democracy, it would not truly function as one. Though it could be argued that such imperfection is natural and speaks to the shortcomings of the participants rather than a failing of the institution itself, such a claim is misguided. Regardless of cause, if the people’s will is not accurately reflected then the society does not conform to our popular notion of a democracy. Whether another system would perform better is beyond our present consideration. We simply list certain key requirements for a democracy to function as we believe it should, and allow the reader to decide the extent to which our present society satisfies them.

Note that a particular government need not directly represents the interest of every citizen, but its formation and maintenance must meaningfully do so. In some loose sense this means that (1) the effect of a citizen is independent of who that citizen is, and (2) the opinion of a majority of citizens is reflected in the actions of the government. These are neither precise requirements nor ones satisfied in practice, particularly in representative democracies. However they reflect our vague cultural concept of democracy.

The following are the major components necessary for a democracy to function as we believe it should.

#### Choice

Once a voter has decided upon a set of positions that reflect their purpose, they must have a means of voting accordingly. There must be sufficient choice to allow an individual to embody those positions in their vote. Furthermore, the choice must be real. Marginal candidates with no chance of winning may be useful for registering an opinion, but they do not offer true participation in the election. If there are only two major candidates then the voter’s entire purpose must be reduced to a binary decision. Only if it happens to be reflected in one of the choices at hand would their view be expressible.

If there are two major candidates and they differ only on a few issues that are of no consequence to a particular individual, then that person cannot express his purpose by voting. For example if a voter feels very strongly about issue X, and both major candidates have the same opposing position on that issue, then he cannot make his will known in that election. It may be argued that the presence of small candidates serves exactly this purpose and that if sentiment is strong enough one could prevail. This is not born out by history. In a two party system, a voter is reduced to a binary choice between two bundled sets of positions. As a more extreme example, suppose there are several major issues and the candidates agree on one of them. Even if every single person in the country holds the opposite position on that issue, their will still cannot be effected through that election. If there were no other important issues, then one or the other candidate surely would take the popular position – or a third party candidate would do so and prevail. However in the presence of other issues, this need not be the case.

Finally, there must be some reason to believe that the actions of a candidate once elected will reflect their proclaimed positions. Otherwise, it will be years before the voter can penalize them. Without such an assurance – and history certainly does not offer it – a nominal choice may not be a real one. The people then acts the part of a general who cannot move his troops, however much he may threaten or cajole them.

#### Information

A well-intentioned individual must have a way of locating and obtaining information whose accuracy is not in question or, if uncertain in nature, is suitably qualified. Voters must have access to accurate and sufficient information. In order to translate their purpose into a vote, an individual must be able to determine the choices available and what they actually entail. Moreover, he must be able to determine the relative importance of different issues in effecting his purpose. Fear mongering, inaccurate statistics, and general misinformation could lead him to believe that a particular issue ‘X’ is of greater import than it truly is. Instead of focusing on other issues ‘Y’ and ‘Z’ which are more germaine to his purpose, he may believe that dealing with issue ‘X’ is the most important step toward it. Similarly, if the views of candidates are obfuscated or misrepresented or the significance of events is disproportionately represented, an accurate translation of his purpose into a vote may be denied a person. Even a perfectly rational and capable voter cannot make a suitable decision in the absence of information or in the presence of inaccurate information. This said, not every vehicle should be expected to provide such information. If a person prefers to listen to a news station that reports with a particular bias, that is not the fault of the information provider – unless it does so subtly and pretends otherwise.

#### Aptitude

A voter must have the intelligence, critical reasoning, motivation, and general wherewithal to seek out accurate information, detect propaganda or advertising, and make an informed decision. Their perceived interest must coincide with their true interest, and their purpose be accurately represented in the choice they make. It may seem that we are advocating the disenfrachisement of a segment of the population, individuals who – while failing to meet some high standard – have valid purposes of their own which they too have the right to express. This is not the case, nor is our standard artificial. We are merely identifying a necessary ingredient, not endorsing a particular path of action. Moreover, the argument that they would be deprived of a right is a specious one. Such individuals \emph{are} disenfranchised, whether or not they physically vote. They lack the ability to accurately express their purpose, and easily are misled, confused, or manipulated. At best they introduce noise, at worst their votes may systematically be exploited. A blind person may have a valid destination, but they cannot drive there.

#### Access

Voters must be willing and able to participate. They cannot be blocked by bureaucratic, economic, legal, or practical obstacles – especially in a way that introduces a selection bias. Their votes must be accurately tallied and their decision implemented.

#### Structure

Not only must the structure of the democratic process treat all voters equally, their de facto influence must be equal. Depending on the nature of the voting system, certain participants may have no real influence even if the system as a whole treats them symmetrically. A simple example would be a nation consisting of four states with blocks of 3, 3, 2, and 1 votes, where each block must vote as a unit. Regardless of the pattern of voting, citizens in the state with a single vote can never affect the outcome. If that vote is flipped, the majority always remains unchanged. This particular topic is addressed in another paper.

There certainly are many other technical and procedural requirements. However those listed above are critical components that directly determine a voter’s ability to express their will through the democratic process. In their absence, voters could be thwarted, manipulated, misled, or confused. The purpose of democracy isn’t to tally votes, but to register the will of the people. Without the choice and tools to express this will, the people can have nothing meaningful to register.

• We often hear of cases that offend our sense of fairness – excessive sentences, minor crimes that are punished more severely than serious crimes, or two equivalent crimes that are punished very differently. Rather than attempt to solve a politically and legally intractable problem, we ask a more theoretical question: whether an individual can assign sentences in a way that seems reasonable and consistent to him.  Our system is a means of doing so.   We offer a simple algorithmic method that could be used by an individual or review board to ensure that sentences meet a common-sense standard of consistency and proportionality.

We intend to offer a less mathematical and more legally-oriented version of this article in the near future.

• 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.

• We should require that every bill be ratified by a second vote, one year after its original passage. It goes into effect as normal, but automatically expires if not ratified at the appropriate time.

Sometimes foolish legislation is passed in the heat of the moment or due to short term pressures. Perhap there is an approaching election, or the media has flamed popular hysteria over some issue, or there is a demand for immediate action with no time for proper deliberation, or an important bill is held hostage to factional concerns, or legislators are falling all over one another to respond with a knee jerk reaction to some event. There are many reasons why thoughtful consideration may succumb to the influences of the moment. The consequences of such legislation can be real and long lasting. Law enforcement resources may be diverted or rights suppressed or onerous demands made on businesses. It is true that legislation may be repealed, but this requires an active effort. The same forces that induced the original legislation, though weakened by time, may threaten to damage anyone who takes the initiative to rectify it.

Here is a simple proposal that could address this problem: Every piece of legislation should be voted on a second time, one year after its original passage. This vote would serve to ratify it. By making this mandatory, the burden of attempted repeal is not placed on any individual. Rather, legislators need simply change their vote. This is less likely to create a fresh political tempest, the issue’s emotional fury long spent. When an act is passed, it goes into effect as normal. However one year from that date, it must be ratified or it will expire. Obviously this should only apply to bills for which such ratification is meaningful; there would be no point in revoting on the prior year’s budget after the money has been spent. By requiring a ratification vote, legislators are given time to breath, sit back, and consider the ramifications of a particular piece of legislation. The intervening year also may provide some flavor of its real effect. A similar approach could be used at all levels of government.

• My close friend and fellow writer Naturi Thomas has published her memoir “How to Die in Paris”. My personal friendship with her aside, Naturi is one of the most talented writers I have met. I personally dislike memoirs in general, but couldn’t put her’s down. I literally read it cover to cover on a cross-country flight. Having enjoyed her stories over the years, I always found her to be an amazingly captivating author. Her talent is brought to new heights in How to Die in Paris. Rather than playing for sympathy or evincing self-pity, she takes a sequence of terrible events (both from her childhood and time in paris) and weaves an entertaining narrative filled with wry commentary and humorous observations on the human condition. I highly recommend her book; it’s a really fun read.

How to Die in Paris Official Site

How to Die in Paris Facebook Page

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• In another post, I discussed the mathematical calculation of optical parameters for a configuration of stacked lenses and camera components. As is evident from the example worked out there, the procedure is somewhat tedious. Instead, it is better to spend twice the time writing a program to do it. Fortunately I already did this and offer it to you, gentle reader, to use and criticize. I expect no less than one rabid rant about some aspect that doesn’t pedantically conform to the IEEE standard. This is working code (and has been checked over and tested to some extent). I use it. However, it is not commercial grade and was not designed with either efficiency or robustness in mind. It is quick and dirty – but graciously so.

Think of this as a mine-shaft. You enter at your own risk and by grace of the owner. And if you fall, there won’t be non-stop human interest coverage on 20 TV channels as rescue workers try to extract you. That’s because you’re not a telegenic little kid and this is a metaphor. Rather, you will end up covered in numeric slime of dubious origin. But I still won’t care.

All this said, I do appreciate constructive criticism and suggestions. Please let me know about any bugs. I don’t plan to extensively maintain this program, but I will issue fixes for significant bugs.

The program I provide is a command line unix (including MacOS) utility. It should be quite portable, as no funky libraries are involved. The program can analyze a single user-specified configuration or scan over all possible configurations from an inventory file. In the latter case, it may restrict itself to configurations accessible using the included adapters or regardless of adapter. It also may apply a filter to limit the output to “interesting” cases such as very high magnification, very wide angle, or high telephoto.

The number of configurations can be quite large, particularly when many components are available, there are no constraints, and we account for the large number of focal/zoom choices for each given stack. For this reason, it is best to constrain scans to a few components in an inventory (by commenting out the components you don’t need). For example, if one has both 10 and 25mm extension tubes then try with only one. If this looks promising, restrict yourself to the components involved and uncomment the 25mm as well.

Either through the summary option or the use of a script to select out desirable configurations, the output may be analyzed and used for practical decisions. For example, if a 10x macro lens is needed and light isn’t an issue then a 1.4X telextender followed by a 200mm zoom followed by a reversed 28mm will do the trick. It will have a high f-stop, but if those components are already owned and we don’t need a low f-stop it may be far more cost-effective option than a dedicated ultra-macro lens (there aren’t any at 10X, but a 5X one is available).

For simple viewing of the results, I recommend the use of my “tless” utility. This isn’t a shameless plug. I wrote tless for myself, and I use it extensively.

• 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.

In the present piece, I examine the theory behind the determination of those properties for any particular setup.  Given a set of components (possibly reversed) and some readily available information about them and the camera, we deduce appropriate optical matrices, construct an effective matrix for the system, and extract the overall optical properties – such as focal length, nearest object distance, and maximum magnification.  We account for focal play and zoom ranges as needed.

The exposition is self-contained, although this is not a course on optics and I simply list basic results.  Rather, I focus on the application of matrix optics to real camera lenses.  I also include a detailed example of a calculation.

As far as I am aware, this is the only treatment of its kind.  Many articles discuss matrix methods or the practical aspects of reversing lenses for macro photography.  However, I have yet to come across a discussion of how to deduce the matrix for a camera lens and vice-versa.

After reading the piece, you may wonder whether it is worth the effort to perform such a calculation.  Wouldn’t it be easier to simply try the configurations?  To modify the common adage, a month on the computer can often save an hour in the lab.  The short answer is yes and no.  No I’m not an economist, why do you ask?

If you have a specific configuration in mind, then trying it is easier.  However, if you have a set of components and want to determine which of the hundreds of possible configurations are candidates for a given use (just because the calculation works, doesn’t mean the optical quality is decent), or which additional components one could buy to make best use of each dollar, or which adapter rings are needed, or what end of the focal ranges to use, then the calculation is helpful.  Do I recommend doing it by hand?  No.  I even used a perl script to generate the results for the example.  As mentioned, a freeware program to accomplish this task in a more robust manner will be forthcoming.  Think of the present piece as the technical manual for it.

• 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.

• This is a memorial poem for my friend and mentor, Diana Festa. It appears in the 2011 edition of Mobius (only available in print).

Diana hosted a poetry group that I and a number of friends attended. Her encouragement, guidance, and cooking all were unsurpassed. In a world filled with selfishness, she was a rare island of kindness. Unfortunately she recently passed away, leaving the world a poorer place.

Mobius Poetry Magazine

• Over the years, I’ve found delimited text files to be an easy way to store or output small amounts of data. Unlike SQL databases, XML, or a variety of other formats, they are human readable. Many of my applications and scripts generate these text tables, as do countless other applications. Often there is a header row and a couple of columns that would best be kept fixed while scrolling. One way to view such files is to pull them into a spreadsheet, parse them, and then split the screen. This is slow and clumsy, and updates are inconvenient to process. Instead, I wanted an application like the unix utility ‘less’ but with an awareness of table columns. The main requirements were that it be lightweight (i.e. keep minimal content in memory and start quickly), parse a variety of text file formats, provide easy synchronized scrolling of columns and rows, and allow horizontal motion by columns. Strangely, no such utility existed. Even Emacs and vi don’t provide an easy solution. So I wrote my own unix terminal application. I tried to keep the key mappings as true to “less” (and hence vi) as possible. The code is based on ncurses and fairly portable. The project is hosted on Google Code and is open source.

• This is a poem of mine that was published in Caveat Lector.

• 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).

• Back in 2007, I started an Ezine with a few fellow writers. But then the literary powers that be realized the awesome potential of my poetry (a phrase very rarely heard) and decided to put an end to this. They forced me to return to wall street and earn lots of money. However, we put out one issue and I think it has some excellent stories in it.

Among the writers is published author Naturi Thomas. I must mention that the cover design is my own, lest you think it acquired at great expense from a seasoned professional.

My own contributions are also directly accessible:

• “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…

My research was in an area called Renormalization Group theory (for those familiar with the subject, that’s the “momentum-space” RG of Quantum Field Theory, rather than the position-space version commonly employed in Statistical Mechanics – although the two are closely related).

In simple terms, one could describe the state of modern physics (then and now) as centering around two major theories: the Standard Model of particle physics, which describes the microscopic behavior of the electromagnetic, weak, and strong forces, and General Relativity, which describes the large scale behavior of gravity. These theories explain all applicable evidence to date, and no prediction they make has been excluded by observation (though almost all our effort has focused on a particular class of experiment, so this may not be as impressive as it seems). In this sense, they are complete and correct. However, they are unsatisfactory.  Their shortcomings are embodied in two of the major problems of modern physics (then and now): the origin of the Standard Model and a unification of Quantum Field Theory with General Relativity (Quantum Field Theory itself is the unification of Quantum Mechanics with Special Relativity). My focus was on the former problem.  The Standard Model is not philosophically satisfying.   Besides the Higgs particle, which is a critical component but has yet to be discovered, there is a deeper issue. The Standard Model involves a large number of empirical inputs (about 21, depending on how you count them), such as the masses of leptons and quarks, various coupling constants, and so on. It also involves a specific non-trivial set of gauge groups, and doesn’t really unify the strong force and electro-weak force (which is a proper unification of the electromagnetic and weak forces). Instead, they’re just kind of slapped together. In this sense, it’s too arbitrary. We’d like to derive the entire thing from simple assumptions about the universe and maybe one energy scale. There have been various attempts at this. Our approach was to look for a “fixed point”. My studying which theories are consistent as we include higher and higher energies, we hoped to narrow the field from really really big to less really really big – where less really really big is 1. My thesis and papers were a first shot at this, using a simple version of Qunatum Field Theory called scalar field theory (which coincidentally is useful in it’s own right, as the Higgs particle is a scalar particle). We came up with some interesting results before the aforementioned cataclysms led to my exile into finance.

Unfortunately, because of the vagaries of copyright law I’m not allowed to include my actual papers. But I can include links. The papers were published in Physical Review D and Physical Review Letters. When you choose to build upon this Earth Shattering work, be sure to cite those. They also appeared on the LANL preprint server, which provides free access to their contents. Finally, my thesis itself is available. Anyone can view it, but only MIT community members can download or print it. Naturally, signed editions are worth well into 12 digits. So print and sign one right away.