## Monday, March 05, 2007

### Mathematical singularity : f(x)=1/x

http://www.knuckletattoos.com/singularity/

In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability. See singularity theory for general discussion of the geometric theory, which only covers some aspects.

For example, the function

$f(x)=\frac{1}{x}$

on the real line has a singularity at x = 0, where it seems to "explode" to ±∞ and is not defined. The function g(x) = |x| (see absolute value) also has a singularity at x = 0, since it isn't differentiable there. Similarly, the graph defined by y2 = x also has a singularity at (0,0), this time because it has a "corner" (vertical tangent) at that point.

The algebraic set defined by y2 = x2 in the (x, y) coordinate system has a singularity (singular point) at (0, 0) because it does not admit a tangent there.

## Complex analysis

In complex analysis, there are four kinds of singularity, to be described below. Suppose U is an open subset of the complex numbers C, a is an element of U, and f is a holomorphic function defined on U \ {a}.

• The point a is a removable singularity of f if there exists a holomorphic function g defined on all of U such that f(z) = g(z) for all z in U − {a}.
• The point a is a pole of f if there exists a holomorphic function g defined on U and a natural number n such that f(z) = g(z) / (za)n for all z in U − {a}.

These three types of singularities are isolated. The fourth type is branch points; they require a more verbose definition, see branch point.

## From the point of view of dynamics

A finite-time singularity occurs when a kinematic variable increases towards infinity at a finite time. An example would be the bouncing motion of an inelastic ball on a plane. If idealized motion is considered, in which the same fraction of kinetic energy is lost on each bounce, the frequency of bounces becomes infinite as the ball comes to rest in a finite time. Other examples of finite-time singularities include Euler's disk and the Painlevé paradox.

## Algebraic geometry and commutative algebra

See main article singular point

In algebraic geometry and commutative algebra, a singularity is a prime ideal whose localization is not a regular local ring (alternately a scheme (mathematics) with a stalk that is not a regular local ring). For example, y2x3 = 0 defines an isolated singular point (at the cusp) x = y = 0. The ring in question is given by

$C[x,y] / (y^2 - x^3) \cong C[t^2, t^3].$

The maximal ideal of the localization at (t2,t3) is a height one local ring generated by two elements and thus not regular.

## Singular matrices

In linear algebra a square matrix is said to be singular when it is not invertible, that is when its determinant is zero.

## Singular value decomposition

Singular value decomposition (SVD) is a method of factorizing matrices. A non-negative real number σ is a singular value for M if and only if there exist normalized vectors u in Km and v in Kn such that

$Mv = \sigma u \,\mbox{ and } M^*u = \sigma v. \,\!$

The vectors u and v are called left-singular and right-singular vectors for σ, respectively. The factorisation is

$M = U\Sigma V^* \,\!$

where diagonal entries of Σ are equal to the singular values of M. The columns of U and V are left- resp. right-singular vectors for the corresponding singular values. It is widely used in statistics where it is used as a technique for solving linear least squares problems and is related to principal components analysis.

# How To Live In A Simulation

by Robin Hanson
6/26/2001, First version 3/14/2001
If you might be living in a simulation then all else equal you should care less about others, live more for today, make your world look more likely to become rich, expect to and try more to particpate in pivotal events, be more entertaining and praiseworthy, and keep the famous people around you happier and more interested in you.

People love to pretend, and to watch others pretending. From story-telling to plays to movies to virtual reality, we keep getting better at making people feel like they are watching imagined places and events. We also keep getting better at role-playing, i.e., creating enviroments where several people can see what happens when they all pretend they are different people in another time and place. Eventually such role-playing simulations may get so good that people will often forget that it is just a simulation.

This brings us to the intriguing premise of many recent movies, including The Matrix, 13th Floor, Truman Show, and Dark City: what if people in the future create role-playing simulations where the people in it do not know that it is a simulation? This premise naturally leads to a premise even more thought-provoking: future people might create simulations of a world much like our world. If so, how sure can each of us now be that we are not now living in a such a role-playing simulation?

A related scenario is the holodeck of the television show Star Trek Next Generation. The holodeck offers computer-generated environments which allow real people to role-play not only with each other, but also with sophisticated computer-simulated people. Today, computer-simulated humans contain only a pale shadow of the complexity and sophistication of real humans. But eventually, if we continute to make better simulations at lower cost, at least some of our simulated humans may be as sophisticated as real humans. In a holodeck, a simulated person might not realize that they were simulated. So the question arises: how sure can we each be that we are not a simulated person in a future holodeck simulation?

Obviously we cannot now be sure that we are not living in a simulation. The more likely our descendants are to be rich, long-lasting, and interested in simulating us, the more simulations of people like us we should expect there to be on average, relative to real people like us. And so the more we expect our descendants to be rich like this, the more we should expect that we are in fact living in a simulation [Bostom 2001].

Now if we, like those characters in recent movies, discovered specific clues in the world around us suggesting that we do in fact live in a simulation, we would of course consider those clues carefully to see what they say about how we should live our lives. But in the absense of such specific clues, many observers have assumed that no implications follow from the mere possibility that we might live in a simulation.

This is not quite right, however. In general, your decisions should be based on a weighted average over the different possible worlds you might live in. If you assign a non-zero subjective probability to the possibility that your descendants will create sophisticated simulations which include people (real or simulated) like us ignorant of their status, then you should assign a non-zero subjective probability to the possibility that you now live in such a simulation. So to the extent that there are consequences of your actions which are different in a simulated world, and you care about these consequences, a non-zero probability of simulation should influence your decisions. The higher the probability you live in a simulation, the more influence that possibility should have on your decision.

Let us therefore consider in more detail how your decisions should be influenced by realizing that you might live in a simulation. To do this we will need to say what we mean by "should", and we will need some way to estimate what sorts of simulations we might live in, if we live in a simulation.

In this paper we will take "should" to simply mean satisfying the usual sort of human preferences. These include wanting to live longer and avoid pain, wanting this to be true of other people, wanting to be thought well of, and wanting to have influence. To guess at the the kinds of historical simulations our descendants may create, we will primarily reason by analogy from the kinds of simulations people now like to create, and their reasons for creating them. The future mixture of simulation types and reasons will of course diverge from today's mixture in ways that we do not anticipate. But this approach at least gives us a rough guide to action.

We expect our descendants to run historial simulations for several different kinds of reasons. First, some historical simulations will be run for academic or intellectual interest, in order to learn more about what actually happened in the past, or about how history would have changed if conditions had changed. Other historical simulations, however, perhaps the vast majority, will be created for their story-telling and entertainment value. For example, someone might ask their "holodeck" to let them play a famous movie actor at a party at the turn of the millenium.

If every simulation is an exact faithful reproduction of all of human history, including the indefinite future, the only decision implications are for those who care about influencing "real" history, or care about being thought of well by "real" people. People who care about these things should realize that such influence is harder than it seems. After all, if this is a simulation, the only way to influence the real world is to somehow influence whoever is observing this simulation.

Simulating events in full detail can be enourmously costly, however. Therefore most computer simulations today vary the detail at which they simulate various events. For example, a vibrating airplane wing is usually simulated in finer detail at places where it bends more, or where air currents near it change more. In general, the level of detail appropriate for any one place depends on how much more expensive it is to produce such detail, and on how influential larger errors there are in producing errors in the final results of interest. Since it is harder to vary the simulation detail in role-playing simulations containing real people, these simulations tend to have some boundaries in space and time at which the simulation ends.

The possibility that we living in a limited simulation with varying detail offers many more implications for how we should live our lives. Consider, for example, a computer simulation of a party at the turn of the millenium created to allow a particular future guest to participate. This simulation might be planned to last only one night, and at the start be limited to the people in the party building, and perhaps a few people visible from that building. If the future guest decided to leave the party and wander the city, the simulated people at the party might be erased, to be replaced by simulated people which populate the street where the party-goer walks.

If you knew that you were a simulated person in this party simulation, and you wanted to live as long as possible, you might want to discourage anyone from leaving the party. If the simulation might end early were the future guest to become bored, you might also want to make sure everyone had a good time. And your motivation to save for retirement, or to help the poor in Ethiopia, might be muted by realizing that in your simulation you will never retire and there is no Ethiopia.

Of course you know nothing as specific as that you are in a millenial party simulation for the benefit of the short guy in the red hair. But to the extent that, on average, the implications of the other possible simulations tend to push your decisions in the same direction, you can draw decision implications from the possibility that you live in a simulation.

For example, if not many simulations last through all of human history, then the chance that your world will end soon is higher than it would be if you were not living in a simulation. So all else equal you should care less about the future of yourself and of humanity, and live more for today. This remains true even if you are highly uncertain of exactly how long the typical simulation lasts.

Also, in general the behavior of many people far from the simulated people of interest might be randomly generated based on statistics from previous simulations, or come from "cached" records of previous simulated people. Some "people" in a crowd simulation might even be run by very simple programs that have them wiggle and mumble "peas and carrots" like extras supposedly did once in movie crowd scenes. Assuming you don't care as much about these fake simulated people, then all else equal you shouldn't care as much about how your actions affect the rest of the world.

If simulations tend to be ended when enough people in them become confident enough that they live in a simulation, then if you want to live long you might want to prevent this from happening. However, assuming the actual history of our descendants included many people who said it was likely that they were living in a simulation, then it should be all right if a similar number of people in a simulated world say this. It might even be a problem if too few people said this.

More generally, if our descendants tend to be more likely to simulate a world the more similar that world is to the actual world of their recorded history, then if you want your world to continue, all else equal you want it to look more like what their history recorded. And since their history eventually resulted in very rich and powerful descendants, all else equal you want the world you live in to look like that will happen.

If our descendants tend to be more interested in simulating "pivotal" people and events from their history, then you should raise your estimate of the chances that the events and people around you will be considered pivotal to your descendants. You should also try to encourage this to happen, as it will make the simulators less likely to drop you from their simulation, or to end that simulation. If you can identify an especially interesting event around you, you might also try to prevent it from ending, as the simulation might end soon after the event does.

If our descendants prefer their simulations to be entertaining, all else equal, then you should want you and the events around you to be entertaining as well, all else equal. "All the world's a stage, and the people merely players." Of course what is regarded as entertaining does vary somewhat across time and cultures, and our distant descendants' tastes will likely vary from ours as well. So one should emphasize widely shared features of entertaining stories. Be funny, outrageous, violent, sexy, strange, pathetic, heroic, ... in a word "dramatic." Being a martyr might even be a good thing for you, if that makes your story so compelling that other descendants will also want to simulation you.

If our descendants sometimes play parts in their simulations, if they are more likely to play more famous people, and if they tend to end simulations when they are not enjoying themselves, then you should take care to keep famous people happy, or at least interested. And if they are more likely to keep in their simulation the people they find more interesting, then you should try to stay personally interesting to the famous people around you.

If our descendants like to play a moral God with their simulations, punishing and rewarding people in the simulation based on how they lived their lives, you might do well to live what they will consider praiseworthy lives. Of course you'll have to figure out the common features of morality in descendants who are willing to play God. (It would seem inconsistent of them to greatly emphasize humility, for example. Inconsistency and morality are hardly strangers though.)

In sum, if your descendants might make simulations of lives like yours, then you might be living in a simulation. And while you probably cannot learn much detail about the specific reasons for and the nature of the simulation you live in, you can draw general conclusions by making analogies to the types and reasons of simulations today. If you might be living in a simulation then all else equal it seems that you should care less about others, live more for today, make your world look likely to become eventually rich, expect to and try to participate in pivotal events, be entertaining and praiseworthy, and keep the famous people around you happy and interested in you.

Nick Bostrom, "Are You Living In A Simulation?" http://www.nickbostrom.com/sim/simulation.doc, March 2001.

This page was immediately inspired by reading a paper by Nick Bostrom, and more indirectly inspired by movies such as The Matrix and Thirteenth Floor. Thanks to Nick Bostrom, Lee Corbin, Tyler Cowen, Hal Finney, Chris Hibbert, Mark Walker, and an anonymous referee for comments.

http://malfunctionjunction.net/?p=322

### Boston Police Blow Up Suspicious Looking Man

Friday, March 2 12:00 AM ET

# Boston Police Blow Up Suspicious Looking Man

By Brian Briggs

Boston, MA – There were more tense moments today after Boston Police were forced to blow up a suspicious looking man near a lamp post.

An alert city worker called in the man after noticing that he had been leaning against the lamp post for more than ten minutes.

Officer Charlie O'Hara of the bomb squad said, "We got a report of a man loitering in a high-traffic area with a bulky coat and a backpack. We cleared the area, snuck up behind him, attached the explosives and detonated him."

Police later learned the man, Evan Johnson, was waiting for his girlfriend, Cindy Collins, who was getting a cappuccino at a nearby Starbucks. His backpack contained books and all that was under his coat was a "Wish you were beer" t-shirt.

Collins explained why Johnson was by the lamp post. "There was a long line at Starbucks, and Evan wanted to smoke so he waited outside for me. Next thing I know I look up and I see police blowing him up. I always told him that down jacket made him look puffy."

Mr. Johnson is only the latest scare to hit Boston. Last month a marketing stunt by a cartoon show paralyzed the city. Two weeks ago police shut down Boston's largest mall after a teen passed gas in an Abercrombie and Fitch. Just this week police blew up a traffic counting device.

"You can never be too careful," said O'Hara. "I'd blow up a hundred innocent people if I can prevent another 9/11."

### Article: 60,000 Marriages Broken by Iraq, Including Mine

Yet more evidence that the National Guard and the Department of Defense has no idea of the far reaching impacts and the implications of our nations 'War on Terror'.

By Stacy Bannerman, The Progressive. Posted March 5, 2007.

When one military wife got the news that her husband was coming home from Iraq, they didn't tell her he was going to bring the war back with him.

I was folding fliers for a high school workshop on nonviolence when my husband, a mortar platoon sergeant with the Army National Guard 81st Brigade, walked into my office and said, "I got the call."

We hadn't talked about the possibility of him being deployed for months, not since President Bush had declared, "Mission accomplished." But I knew exactly what he meant; I didn't know then what it would mean for us.

We weren't prepared, and neither was the Guard. The Guard sent him into harm's way without providing some of the basic equipment and materials, such as global positioning systems, night vision gear, and insect repellant, that he would rely on during his year-long tour of duty at LSA Anaconda, the most-attacked base in Iraq, as determined by the sheer number of incoming rockets and mortars, which averaged at least five per day.

Unlike active duty military, the National Guard had no functional family support system or services in place. While the Guard was scrambling to get it together, my husband was already gone, and I was alone, just months after we had moved to Seattle.

Twenty-four hours after Lorin boarded the plane for Iraq, I hung a blue star service flag -- denoting an immediate family member in combat -- in the front window. Then I closed the blinds, hoping to keep the harbingers of death at bay. They still got in, through the phone, the Internet, the newspaper, and the TV.

Each week, I heard of a friend's husband or son: wounded, maimed, shot, hit, hurt, burned, amputated, decapitated, detonated, dead. A glossary of pain. I checked icasualties.org all the time, cursing and crying as the numbers rose relentlessly, praying that Lorin wouldn't be next.

I got involved with Military Families Speak Out, which is exactly what the name suggests: an organization of people with loved ones in uniform who are adamantly opposed to the war in Iraq. We were breaking the military's traditional code of silence by publicly protesting this war, and the pushback was intense, particularly for military wives. I was ostracized by the women married to men in my husband's company, and my husband was reprimanded by his superior officers. I was an "unruly spouse," and Lorin could "expect adverse career consequences."

I thought being forced to serve in a war based on lies was itself an "adverse consequence." I said as much during an interview on Hardball with Chris Matthews, which just happened to be broadcast on the big-screen TV during lunchtime in the mess tent at Anaconda. Lorin didn't see it, but approximately 5,000 of the troops he was serving with did. He heard about it for weeks, but never asked me to stop. He had his own questions and concerns about Operation Iraqi Freedom.

During the run-up to the war, when 76 percent of Americans supported the invasion of Iraq, we protested in the streets of Spokane. But he was contractually bound and committed to his men. He clung to what he'd been briefed on regarding the Guard's mission in Iraq, which included building schools for kids.

Two months into his deployment, I got a call from him, and he said, choking up, that there was an "accident." Two Iraqi children were dead because he gave the order to fire a couple of mortar rounds. Several weeks later, he phoned again, his voice flat and emotionless, to tell me that the men he had dinner with the previous night had been killed by the same Iraqi soldiers that they were training six hours earlier.

Days went by without any communication -- anxious hours, restless nights. I swerved between anger and fear.

His e-mails were sometimes delayed, or returned to him as undeliverable, with portions blacked out by military censors. The ones that got through asked for more homemade treats, baby wipes, batteries, movies, and magazines. One missive informed me about rockets landing next to the trailer where he slept ... while he was in bed. Another ended abruptly because he was under attack.

Finally, the phone rang with the news that my husband was coming home, after nearly a year in Iraq. They didn't tell me he'd bring the war with him.

He'd been back for almost two months, but he was still checking to see where his weapon was every time he got in a vehicle. He drove aggressively, talked aggressively, and sometimes I could swear that he was breathing aggressively. This was not the man I married, this hard-eyed, hyper-vigilant stranger who spent his nights watching the dozens of DVDs that he got from soldiers he served with in Iraq. He couldn't sleep, and missed the adrenaline surge of constant, imminent danger. The amateur videos of combat eased the ache of withdrawal from war, but did nothing to heal my soldier's heart.

At a conference on post-deployment care and services for soldiers and their families, a Marine Corps chaplain asked, "How do you know if you're an SOB? Your wife will tell you!"

Har-de-har-har-har. The remark got the predictable round of applause from the capacity crowd, which, with one exception, wasn't living with anyone who had recently returned from Iraq. I was that exception, and it infuriated me that this was a joke. The Pentagon's solution for the constant stress endured by those of us who felt bewildered and betrayed was: "Learn how to laugh." With help from the Pentagon's chief laughter instructor, families of National Guard members were learning to walk like a penguin, laugh like a lion, and blurt "ha, ha, hee, hee, and ho, ho."

Emotional isolation is one of the hallmarks of post-combat mental health problems. The National Guard didn't conduct follow-up mental health screening or evaluations of the men in my husband's company until they had been home for almost eight months. Nearly a year later, in August of 2006, my husband was informed of his results: Post-Traumatic Stress Disorder (PTSD). It was obvious that he was suffering, but when I brought it up, he parroted what the military told him: "Give it time."

Time wasn't a panacea for Jeffrey Lucey, Doug Barber, or the dozens of other Guard members and Reservists who have committed suicide after serving in Iraq. Time hasn't helped the hundreds of homeless Iraq War veterans wandering lost in the streets of what military families are assured is a deeply grateful nation. Time is most definitely not on our side.

My husband has served his time with the Guard. He's got more than twenty-three years of actual service, and almost twenty years of "good time" that qualifies him for retirement benefits.

But then he learned about a few loopholes. Now, if he serves as a member in good standing for 364 days in a year, instead of 365, that year isn't credited as time served toward his retirement. If he's deemed irreplaceable -- he's one of a handful of mortar platoon sergeants who've seen combat -- the Guard can retain him for several more years after his contract expires.

He is surprised by this, but I'm not. I no longer expect that the Department of Defense will keep its promises to the soldiers or their families. I don't pretend that the Pentagon will adhere to its policies. And I know from experience that "support the troops" is a slogan, and not a practice.

On January 11, 2007, the Pentagon discarded the time limit that prevented Guard members and Reservists from serving more than 24 total months on active duty for either the Iraq or Afghan wars. The Pentagon's announcement came in the wake of President Bush's decision to deploy an additional 21,500 troops to Iraq.

The escalation contradicts the advice of top U.S. military officials. Although the majority of Americans are opposed to the "surge," most members of Congress are reluctant to block the supplemental appropriations request that will fund it, claiming that they don't want to abandon the troops. Congress has abandoned the troops for nearly four years. It is the soldiers, their families, and the people of Iraq that pay the human costs. The tab so far: more than 3,000 dead U.S. troops, tens of thousands of wounded, over half a million Iraqi casualties, roughly 250,000 American servicemen and women struggling with PTSD, and almost 60,000 military marriages that have been broken by this war. Including mine.

It was hard to reconnect after more than a year apart, and the open wound of untreated PTSD made it virtually impossible. Lorin is still the best evidence I have of God's grace in this world, but we just couldn't find our way back together after the war came home.

Stacy Bannerman is the author of "When the War Came Home: The Inside Story of Reservists and the Families They Leave Behind." She is a member of Military Families Speak Out, www.mfso.org, and can be contacted at her website, stacybannerman.com.