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Call for Nominations: Pediatrics is seeking an Associate Editor for Case Reports. Please see the job description and the application.

pediatrics
November 2004, VOLUME114 /ISSUE 5

# Injury Risk of Nonpowder Guns

1. Danielle Laraque, MD,
2. and Committee on Injury, Violence, and Poison Prevention

## Abstract

Nonpowder guns (ball-bearing [BB] guns, pellet guns, air rifles, paintball guns) continue to cause serious injuries to children and adolescents. The muzzle velocity of these guns can range from approximately 150 ft/second to 1200 ft/second (the muzzle velocities of traditional firearm pistols are 750 ft/second to 1450 ft/second). Both low- and high-velocity nonpowder guns are associated with serious injuries, and fatalities can result from high-velocity guns. A persisting problem is the lack of medical recognition of the severity of injuries that can result from these guns, including penetration of the eye, skin, internal organs, and bone. Nationally, in 2000, there were an estimated 21840 (coefficient of variation: 0.0821) injuries related to nonpowder guns, with approximately 4% resulting in hospitalization. Between 1990 and 2000, the US Consumer Product Safety Commission reported 39 nonpowder gun–related deaths, of which 32 were children younger than 15 years. The introduction of high-powered air rifles in the 1970s has been associated with approximately 4 deaths per year. The advent of war games and the use of paintball guns have resulted in a number of reports of injuries, especially to the eye. Injuries associated with nonpowder guns should receive prompt medical management similar to the management of firearm-related injuries, and nonpowder guns should never be characterized as toys.

• nonpowder guns
• BB guns
• pellet guns
• air rifles
• paintball guns

## BACKGROUND

A traditional firearm gun is one that launches a projectile (ie, a bullet) by using the energy generated by burning of gunpowder. Nonpowder guns utilize the power of compressed air to launch a projectile. Nonpowder guns can be classified by the type of projectile they fire, the propulsion mechanism, or the type of barrel.14 The type of projectile can be lead, brass, steel, copper, or, most recently, a paintball. Paintballs are small gelatin projectiles that are 17 mm in diameter, filled with nontoxic, water-soluble paint, and intended to explode on contact with an object.5,6 This type of projectile is used in war games designed to mark the player with paint when he or she is hit. Air guns have been used since the 16th century7,8 in warfare and to kill game as large as deer.

The caliber of a projectile refers to its diameter and is measured in hundredths of an inch or millimeters. The caliber affects how much energy the projectile acquires before leaving the muzzle, or the end of the barrel. Tight-fitting missiles, those with little discrepancy between the diameter of the projectile and that of the muzzle, lead to higher velocities. In older nonpowder guns, the projectile was smaller than the barrel size, leading to dissipating of compressed air and an inefficient, low-velocity gun. Technical modifications of these guns have resulted in higher-velocity weapons.9 Standard pellet guns fire a pellet or spherical ball bearing (BB) with a diameter of less than 0.18 in (4.57 mm). Pellets have several designs, such as wad cutter, sharp pointed, round nosed, and hollow point. Each is suited for a different purpose. Hollow points are used for hunting, and the pellet's diameter increases on impact to cause maximum damage. Ballistic studies have shown that a larger caliber pellet will penetrate the body (eg, skin, bone) at lower velocities because of its increased mass. Skin penetration can be achieved, for example, at a velocity of approximately 331 ft/second with a 0.177-caliber pellet but at 245 ft/second with a 0.22-caliber pellet. Ocular penetration can occur at velocities as low as 130 ft/second.7,10 Polishing steel pellets with a plastic skirt increases velocity, accuracy, and range and is designed to increase penetration. Typically, high-velocity guns are classified as those with muzzle velocities higher than 350 ft/second (D. Tinsworth, MS, US Consumer Product Safety Commission [CPSC], written communication, November 26, 2001).7,9,1113

Projectiles can be fired by 3 propulsion mechanisms. The spring-piston type is a powerful spring that is cocked manually and released, driving the piston that shoots a stream of air. Use of the spring-piston can result in muzzle velocities between 250 and 350 ft/second. The carbon dioxide–powered gun uses a gas cartridge to generate a propulsive force that can produce muzzle velocities of 350 to 450 ft/second. Muzzle velocities ranging, on average, from 300 to 950 ft/second can be generated depending on the number of times the weapon is pumped, although velocities in excess of 1200 ft/second have been reported in the literature. This range of velocity overlaps velocities reached by traditional firearm pistols that have muzzle velocities from 750 ft/second to 1450 ft/second.1,3,9

The longer the gun barrel is, the higher the velocity. Gun barrels can be smooth or rifled. Rifled weapons produce a spin in the projectile, giving it more stability in flight. Dieseling of the barrel is achieved when oil placed in the barrel is combusted by the heat generated from friction, leading to an explosion. This is used to increase the speed of the projectile. Piggybacking entails simultaneously loading 2 pellets into the firing chamber, increasing the momentum and energy of the missile.9 A “zip gun” is a modified gun using homemade powder ammunition.8 These modifications of nonpowder guns can result in increased ability of these guns to cause serious injury, not unlike traditional powder guns.

## EXPOSURE AND INJURY PROFILE

The CPSC estimates that there are approximately 3.2 million nonpowder guns sold yearly.1214 Nonpowder guns are sold in many department stores, including toy stores.9 Eighty percent have muzzle velocities over 350 ft/second, and 50% have muzzle velocities between 500 and 930 ft/second. In 2000, the National Electronic Injury Surveillance System (NEISS), operated by the CPSC, collected information from a nationally representative sample of 100 US hospital emergency departments that included information on nonpowder gun injuries.

According to data from the Centers for Disease Control and Prevention (http://webapp.cdc.gov/sasweb/ncipc/nfirates.html and www.cdc.gov/ncipc/wisqars/nonfatal/datasources.htm) and the CPSC,12 in 2000 the overall nonfatal age-adjusted rate of injury from BB or pellet guns was 7.71 per 100000 population. In 2000, there were an estimated 21840 (coefficient of variation: 0.0821) nonpowder gun–related injuries treated in emergency departments (D. Tinsworth, MS, CPSC, written communication, November 26, 2001). Of these, 2% occurred in children 0 to 4 years of age; 49% occurred in children 5 to 14 years of age; 33% occurred in those 15 to 24 years of age; and the balance occurred in adults 25 years and older. Approximately 12% of injuries were to the eye; 24% were to the head and neck, excluding the eye; 63% were to extremities; and 1% were to other body areas. With the exception of the age group of 0 to 4 years, most victims were males. Sixty-six percent of injuries were diagnosed as either foreign-body lodgments or puncture wounds. There were no clear seasonal variations in the injury incidence. Nguyen et al15 provided a review of trends in BB or pellet gun injuries in children and adolescents in the United States from 1985 to 1999 derived from a special study using the NEISS, which focused specifically on injuries associated with penetrating gunshot wounds. On the basis of data from this study, in 1999 an estimated 14313 (95% confidence interval: 12025-16601) children and adolescents had BB or pellet gun–related injuries.

Many articles have been written detailing the clinical manifestations of children injured by nonpowder guns.310,1631 Some striking observations have been made. Lawrence3 reported on a series of 10 fatalities, 1 of which was a shot through the medial canthus of the eye in a 6-year-old. The weapon was a carbon dioxide–powered BB pistol. Bond et al10 described 16 children, 57% of whom required intraoperative treatment, and 19% of whom required other invasive procedures such as arteriogram or ventriculostomy. Thoracic injuries were associated with high morbidity and mortality when penetration of the chest wall occurred. Abdominal wounds were frequently associated with visceral injury and multiple perforations, usually of the small bowel. Peritoneal penetration was associated with a more than 80% chance of visceral injury. Transtracheal and brain injuries were also reported. These authors warn that the wound itself may seem trivial, but if not appreciated for their potential for tissue disruption, nonpowder gun injuries to the head, chest, and abdomen may have catastrophic results. They also note that the pellets from air guns have a propensity to embolize if the missile enters a blood vessel or the heart. The light weight of air gun pellets allows the missile to be swept by the blood flow more readily than heavier, higher-energy projectiles. Friedman et al7 report that the potential seriousness of pneumatic weapon injury is frequently underestimated. These authors concluded that injuries from air guns should be treated in a manner similar to those from low-velocity powder firearms. Bratton et al4 reviewed the clinical course of 101 children injured by nonpowder guns between 1988 and 1996 from Cincinnati, Ohio; Kansas City, Missouri; and Seattle, Washington. The case fatality rate for intracranial injuries was 30%, and 56% of patients required at least 1 surgical procedure. Amirjamshidi et al16 noted that air-gun pellet injuries are rare but catastrophic, with entrance usually through the orbit or the neck and the entry wound being so small that it may be disregarded on physical examination in the emergency department. They concluded that early recognition and correct management of possible complications is important to improve outcomes. Bhattacharyya et al17 reported on 42 children admitted to a level I pediatric trauma center for air-gun injuries over a 7-year period (1988-1995). They had a mean hospital stay of 7 days (range: 1-136 days) and a mean injury severity score of 8.3. Half of the children underwent operative procedures, and 38% had serious long-term disability. They concluded that these guns are not toys but are weapons, and injuries related to their use should be evaluated and managed in a similar fashion to powder-weapon injuries. These findings were similar to those of Walsh et al.18 In 1 study, the predominant risk factors for ocular injury from an air gun were lack of adult supervision, use of the gun for a purpose other than target practice, and being at a friend's home or yard.20 Hearing loss has been reported from nonpowder gun use,23 and suicides from the use of air guns have been documented.8,28 Concurrent use of alcohol has been noted also and probably contributes to misuse.

Paintballs used in war games are a relatively new phenomenon.5,6,21,25 Semiautomatic and fully automatic paintball pistols are available for purchase. The sport typically involves a team, designated fields, and referees used to ensure fair and “safe” play. Players should be 18 years and older, but younger adolescents are allowed to play with the consent of their parent(s). Private games also occur. The players often wear camouflage, and start-up costs for the weapon, goggles, and paintballs total $100 to$150. The paintballs consist of spherical shells filled with sorbitol, glycol, and food dye. The propulsion mechanism is usually a carbon dioxide canister, and muzzle velocities between 60 and 250 ft/second can be achieved. Given the size of the projectile, the resulting injuries are nonpenetrating. Locally manufactured paintballs are harder than the more expensive imported varieties and may be responsible for the severity of injuries reported in the United States. Importantly, the increasing popularity of war games has been associated with a number of reports in the literature of ocular injuries. These injuries have included but are not limited to hyphemas, commotio retinae, glaucoma, cataracts, choroidal rupture, corneal abrasion, conjunctival laceration, dislocation of the crystalline lens, macular hole, and retinal detachments.5,6,25,32,33 The visual outcome for many of these injuries is poor. Injuries have occurred even with eye-protective devices (EPDs), but no case of a player injured while properly wearing an EPD meeting the current American Society for Testing and Materials (ASTM) standards has been reported.6 Some players have sustained injuries to the eye when they have removed their goggles because of fogging. Current antifog inserts with a polycarbonate lens with a urethane-based hydroscopic coating can help prevent fogging. The current ASTM specifications do not involve testing EPDs for their ability to resist fogging but do require manufacturers to attach a warning to EPDs without antifog treatment noting that fogging may occur and recommending the use of an antifog solution. There have been no reported deaths directly related to paintballs, but the CPSC issued a warning on March 24, 2004, because of its investigation of 2 deaths caused by carbon dioxide canisters flying off paintball guns (www.cpsc.gov/CPSCPUB/PREREL/prhtml04/04105.html).

Before 1972, only 2 nonpowder gun–related fatalities were reported in the literature. However, between 1972 and 1982, the decade after the introduction of high-powered air rifles, 10 more fatalities were recorded by the CPSC.10 The number of deaths per year has increased since then. From 1990 to 2000, the CPSC reported 39 nonpowder gun–related deaths, of which 32 were children younger than 15 years, with an average of 4 deaths per year.11 The highest number of deaths occurred in 1989, 1990, and 1991.15 The trends in nonpowder gun fatalities and nonfatal injuries parallel the epidemic of firearm-related injuries and deaths of the past 2 decades.34

## SAFETY STANDARDS

An ASTM voluntary safety standard was originally published in 1978. The current edition was published in December 1992 (ASTM F589, Standard Consumer Safety Specification for Non-Powder Guns).35 This standard contains performance requirements to ensure the proper functioning of these products as well as provisions to address instructions, labeling, and marketing. The guns are general-purpose guns not classified as precision, adult, or training guns. For higher-power guns, the minimum labeled age is 16 years, and the potential for serious injury or death is indicated. For lower-power guns, the minimum labeled age is 10 years, and the risk of serious injury, particularly to the eye, is indicated, but not the risk of death. In the pediatric literature as early as 1984, Christoffel et al36,37 pointed to the dangers of nonpowder guns, noting that they were loosely regulated and could be legally purchased by young adolescents in most jurisdictions. They also emphasized the inadequacy of voluntary standards and proposed stricter regulations.

From June 1 through July 31, 1994, the CPSC conducted a limited study using follow-up telephone investigation of the circumstances of 55 cases of nonpowder gun–related injuries reported to the NEISS. Percentages provided in the analysis were based on national estimates projected from the 55 cases for the 2 months of the survey. Additionally, information on deaths was obtained from the CPSC Death Certificate, In-Depth Investigation, and Injury or Potential Injury Incident files for the period of January 1, 1985, through September 1, 1994. Information on 37 deaths was included (D. Tinsworth, MS, CPSC, written communication, November 26, 2001). Respondents were victims or parents or guardians of victims. The injury epidemiology mirrored that described for the 2000 data, with most injuries occurring in males younger than 16 years. Individuals who fired the guns ranged in age from 8 to 32 years, with most reported to be younger than 16 years. When the gun operator was younger than 16 years, there was no one 18 years or older present at the time of the incident in more than two thirds of the cases. The gun was most often reported to be a rifle and received as a gift, and two thirds were high-powered guns. As reported by the respondents, 51% of the hazard patterns were unintentional shootings, with victims coming into the line of fire during practice, discharges during loading of the gun, or incidents in which the gun “accidentally fired.” Fourteen percent were of unknown intent, 7% were intentional shootings, and 28% involved a gun thought to be unloaded, guns that discharged unexpectedly, ammunition that ricocheted, or fingers that became pinched in gun components. Ninety percent of those who died were younger than 16 years. Most fatalities resulted from wounds to the head or chest.

The CPSC report concluded that the effectiveness of age-specific warnings on packaging and instructions needed additional study and that adult supervision was often lacking. It was unclear from the data whether product modification to reduce hazards would be effective.

## LEGISLATIVE EFFORTS

Almost 30 states have regulations, ordinances, or laws covering nonpowder guns.9,38 Two of the strongest are in New York City and New York State. In New York City, air rifles and BB guns are prohibited, and licenses are not available. In New York State, no purchase or unsupervised use by someone younger than 16 years is permitted, and adult supervision is required at a shooting range or when hunting for someone of this age. Florida also has a strong law similar to the New York State law. In Florida, it is a second-degree misdemeanor for a minor younger than 16 years to use a BB gun, air gun, or gas-operated gun unless an adult is supervising the possession and the minor's parent has consented to such possession.

However, much variability exists, with some states regarding nonpowder guns as firearms and others not.15 Some state laws do not address nonpowder guns at all. Many authors have called for restrictions in sales and use of nonpowder guns, especially in light of the technologic advances that have resulted in much more powerful and dangerous weapons that are capable of killing and maiming.9,19,31,39

## SUMMARY/CONCLUSIONS

This technical report is focused mainly on the potential for injury and death associated with the use of nonpowder guns. Although some comments have been made on risk factors for injury, this report is not meant to be an exhaustive review of the behavioral risk factors for injury, nor does the report summarize the literature regarding the psychological implications or effects of the use of these weapons. The data presented do allow the following conclusions:

• Nonpowder guns pose a serious risk of injury, permanent disability, and even death.

• Since the 1980s, the use of high-powered air rifles has been associated with approximately 4 deaths per year.

• The range of muzzle velocities for nonpowder guns overlaps velocities reached by traditional firearms.

• Data suggest that lack of supervision and unstructured use may be risk factors contributing to the incidence of injury from nonpowder guns.

• EPDs can be useful in decreasing, but not fully eliminating, the incidence of ocular injuries associated with paintball use.

• Injuries associated with nonpowder guns should receive prompt medical management similar to the management of firearm-related injuries.

• Nonpowder guns (BB guns, pellet guns, air rifles, paintball guns) are weapons and should never be characterized as toys.

## Committee on Injury, Violence, and Poison Prevention, 2002–2003

Marilyn J. Bull, MD, Chairperson

Phyllis Agran, MD, MPH

M. Denise Dowd, MD, MPH

Victor Garcia, MD

H. Garry Gardner, MD

Gary A. Smith, MD, DrPH

Milton Tenenbein, MD

Jeffrey C. Weiss, MD

Joseph L. Wright, MD, MPH

## Liaisons

Ruth A. Brenner, MD

National Institute of Child Health and Human Development

Stephanie Bryn, MPH

Health Resources and Services Administration/Maternal and Child Health Bureau

Richard A. Schieber, MD, MPH

Centers for Disease Control and Prevention

Alexander (Sandy) Sinclair

Deborah Tinsworth

US Consumer Product Safety Commission

Lynne J. Warda, MD

## Staff

Rebecca Levin-Goodman, MPH

## Footnotes

• The guidance in this report does not indicate an exclusive course of treatment or serve as a standard of medical care. Variations, taking into account individual circumstances, may be appropriate.

• All policy statements from the American Academy of Pediatrics automatically expire 5 years after publication unless reaffirmed, revised, or retired at or before that time.

BB, ball bearingCPSC, US Consumer Product Safety CommissionNEISS, National Electronic Injury Surveillance SystemEPD, eye-protective deviceASTM, American Society for Testing and Materials


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Computer Science

# Graphs: Level 5 Challenges

Do you like carrots?

Given this $$100 \times 38$$ maze with two exits, find out the number of steps (left, right, up and down), including the step out of the maze, from the worst point inside, i.e, the point from which it takes the longest to get out.

The maze is presented in an obvious format. Absence of | or - represent absence of walls. Walls on the boundaries which are absent are exits.

To clarify, the title image is a joke. It has got nothing to do with the maze the problem links to.

Example

The worst point of the following $$3 \times 5$$ maze is the lower left corner. It takes 9 steps to get out from that point

 1 2 3 4 5 6 7 +-+-+-+-+-+ | | +-+ +-+ + + | | | | + +-+-+ + + | | | +-+ +-+-+-+

The king of the desert, and Link's arch nemesis, Ganondorf, has learned of the hero's rampant collecting of Heart Pieces, and that hardly aligns with his evil designs. Determined to stop the hero from progressing, the evil king has entered the Lost Woods himself, to collect Heart Pieces before Link can get them.

This link (1) contains a zip file, with a single file in the Input Format specified for this collection of problems, of 100 mazes.

• Link and Ganondorf begin at different locations within the wood, Link as an 'L', and Ganondorf as a 'G'.
• Link gets to move first, taking a single step, or remaining still if he chooses. If he touches an uncollected Heart Piece, it becomes his, permanently.
• Ganondorf moves next, taking a single step, or remaining still if he chooses. If he touches an uncollected Heart Piece, it becomes his, permanently, and Link can never collect it.
• This process repeats until Link cannot collect any more Heart Pieces.
• Naturally, Ganondorf is a perfect logician, and will choose his steps optimally so as to minimize the number of Heart Pieces Link is able to collect.

Let $$S$$ be the sum of the score for each maze. We define the score to be the index of the maze (1-based) multiplied by the maximum number of Heart Pieces Link can collect despite Ganondorf's interference.

Find $$\boxed{S}$$ .

Example:

Input:

4
5 5
##.##
..G..
H###H
.###.
..L..
5 5
##L##
.....
H###H
.###.
..G..
5 5
##.##
..G..
H#.#H
.#.#.
..L..
7 4
##L.G##
H.#.#.H
#.....#
H.###.H

Output:

13

Output Explained:

In the first maze, Ganondorf can indefinitely mimic Link's movements left and right until he becomes adjacent to a Heart Piece, which he can then collect before Link can, and then go fetch the second one before Link as well. Though Link can prolong the opening indefinitely, he will never collect a single Heart Piece either way, so the score for this maze is $$0$$.

In the second maze, Link and Ganondorf are in opposite positions as the first maze, with Link set back one space to compensate for first-turn advantage. Despite this, Link cannot employ the same strategy Ganondorf did - if he did, Ganondorf could simply sit still indefinitely, and Link would never collect a Heart Piece, which is Ganondorf's optimal end result. Link must therefore sacrifice one Heart Piece to Ganondorf by going after the other one, making the score for this maze $$1$$ times the index.

In the third maze, the situation is similar to the first maze, but now Link has a new avenue of approach. Though the path down the middle is longer towards each Heart Piece than the paths down the sides, by going up the middle Link forces Ganondorf to choose a single Heart Piece to defend, sacrificing the other to Link, making the score for this maze $$1$$ times the index.

In the fourth maze, Link is equidistant from all four Heart Pieces, same as Ganondorf, so with first turn advantage, he can get at least one. However, no matter which path Ganondorf takes, Link will also be able to get a second Heart Piece, but not a third, if he reacts properly to Ganondorf's movements. The score for this maze is thus $$2$$ times the index.

$$1 \times 0 + 2 \times 1 + 3 \times 1 + 4 \times 2 = 13$$

Details:

• Link will always appear once per maze, as an 'L'.
• Ganondorf will always appear once per maze, as a 'G'.
• There will be at most five Heart Pieces per maze, as 'H'.
• Open ground is given as '.'.
• Impenetrable trees are given as '#'.
• No other characters will be present in each maze.
• Ganondorf plays optimally, with the explicit goal of minimizing the number of Heart Pieces Link collects. Both characters have perfect information at all times.
• Link and Ganondorf can freely pass through each other.
• $$1 \leq W \times H \leq 30$$ in all mazes.

(1) - This link currently points to my Dropbox account, since there is currently no way to upload non-image content to Brilliant. Admin, feel free to delete this footnote if this is changed.

###### Image Credit: http://zelda.wikia.com/wiki/Ganondorf

How many pair of nodes $$(u,v)$$ in the following graph are there such that the shortest path between $$u$$ and $$v$$ is length $$2$$?

Details and Assumptions

×

format: plain


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# The Trigonometric Functions by Stefan Waner and Steven R. Costenoble

## This Section: 2. The Six Trigonometric Functions

2. The Six Trigonometric Functions

The two basic trigonometric functions are: sine (which we have already studied), and cosine. By taking ratios and reciprocals of these functions, we obtain four other functions, called tangent, secant, cosecant, and cotangent.

Cosine

Let us go back to the bicycle introduced in the preceding section, and recall that the sine of $t, \sin t,$ was defined as the $y-$coordinate of a marker on the wheel. The cosine of $t,$ denoted by $\cos t,$ is defined in almost the same way, except that this time, we use the $x-$coordinates of the marker on the wheel. (See the figure.)

$\cos t$ is defined as the $x-$coordinate of the point $P$ shown.

First notice that the coordinates of the point $P$ in the above diagram are $(\cos t, \sin t),$ and that the distance from $P$ to the origin is $1$ unit. From the distance formula in Chapter 8 of Calculus Applied to the Real World or Chapter 15 of Finite Mathematics and Calculus Applied to the Real World we have:

Square of the distance from $P$ to $(0, 0) = 1$
$(\sin t)^2 + (\cos t)^2 = 1$

We often write this as

$\sin^2t + \cos^2t = 1,$

ant so we have found a relationship between the sine and cosine function.

 Fundamental Trigonometric Identity $\sin^2t + \cos^2t = 1$

Let us now turn attention to the graph of the cosine function. The graph, as you might expect, is almost identical to that of the sine function, except for a "phase shift" (see the figure).

This gives the following new pair of identities.

 Further Relationships Between Sine and Cosine The cosine curve is obtained from the sine curve by shifting it to the left a distance of $\pi/2.$ Conversely, we can obtain the sine curve from the cosine curve by shifting it $\pi/2$ units to the right. $\cos t = \sin(t + \pi/2)$ $\sin t = \cos(t - \pi/2)$ Alternative formulation We can also obtain the \cosine curve by first inverting the sine curve vertically (replace $t$ by $-t$) and then shifting to the right a distance of $\pi/2.$ This gives us two alternative formulas (which are easier to remember) $\cos t = \sin(\pi/2 - t)$ $\sin t = \cos(\pi/2 - t)$

Question
Since we can model the cosine function with a sine function, who needs the cosine function anyway?

Technically, that is correct; we don't need the cosine function and we can get by with the sine function by itself. On the other hand, it is convenient to have the cosine function around, since it starts at its highest point, rather than zero.

 Modeling with the Cosine Function (General Cosine Curve) Note that the basepoint is at the higher point of the curve. All the constants have the same meaning as for the general sine curve: $A$ is the amplitude (the height of each peak above the baseline) $C$ is the vertical offset (height of the baseline) $P$ is the period or wavelength (the length of each cycle) $ω$ is the angular frequency, given by $ω = 2\pi/P$ $α$ is the phase shift (the horizontal offset of the basepoint; where the curve reaches its maximum)

Example 1 Cash Flows into Stock Funds

The annual cash flow into stock funds (measured as a percentage of total assets) has fluctuated in cycles of approximately 40 years since 1955, when it was at a high point. The highs were roughly $+15%$ of total assets, while lows were roughly $-10%$ of total assets.*

(a) Model this cash flow with a cosine function of the time $t$ in years with $t = 0$ representing 1955.
(b) Convert the answer in part (a) to a sine function model.

* Source: Investment Company Institute/The New York Times, February 2, 1997. p. F8.

Solution

(a) Cosine modeling is similar to sine modeling: We are seeking a function of the form

$P(t) = A \cos[ω(t-α)] + C.$

Amplitude $A$ and Vertical Offset $C:$
Since the cash flow fluctuates between $- 10%$ and $+15%,$ we see that $A = 12.5,$ and $C = 2.5.$

Period $P$:
This is given as $P = 40.$

Angular Frequency $ω:$
This is given by the formula

$ω = 2\pi/P = 2\pi/40 = \pi/20 \approx 0.157.$

Phase Shift $α:$
The basepoint is at the high point of the curve, and we are told that cash flow was at its high at $t = 0.$ Therefore, the basepoint occurs at $t = 0,$ and so $α = 0.$

Putting this together gives

$P(t) = A \cos[ω(t-α)] + C$
$= 12.5\cos(0.157t) + 2.5,$

where t is time in years.

(b) To convert between a sine and cosine model, we can use the relations given above. Here, let us use the formula

$\cos x = \sin(x + \pi/2).$

Therefore,

$P(t) = 12.5\cos(0.157t) + 2.5$
$= 12.5\sin(0.157t + \pi/2) + 2.5.$

The Other Trigonometric Functions

As we said above, we can take ratios and reciprocals of sine and cosine to obtain four new functions. Here they are.

Tangent, Cotangent, Secant, and Cosecant
 $\tan x = \frac{\sin x}{\cos x}$ tangent $\cotan x = \frac{\cos x}{\sin x} = \frac{1}{\tan x}$ cotangent $\sec x = \frac{1}{\cos x}$ secant $\cosec x = \frac{1}{\sin x}$ cosecant

Example 2

Use technology to graph the curve $y = \sec x$ for $-2\pi ≤ x ≤ 2\pi$

Solution

Since

$\sec x = 1/\cos x,$

we can enter this function as

	$Y_1 = 1/\cos(x).$

To set the window, let us use $-2\pi ≤ x ≤ 2\pi,$ and $-7 ≤ y ≤ 7.$ Here is the graph we obtain.

Question
What are the vertical lines doing here?

Since we defined the secant function as $\sec x = 1/\cos x,$ we know that it is not defined when the denominator is zero. That is, when

$\cos x = 0.$

Consulting the graph of $\cos x,$ we find that this occurs when $x = ±\pi/2, ±3\pi/2, ±5\pi/2,...$

Therefore, these values are not in the domain of the secant function. Further, as $x$ approaches these values, $\sec x$ becomes very large numerically, but changes sign when we cross these values, causing the graphing calculator to make sudden jumps from large negative values of y to large positive values. Thus, the vertical lines are asymptotes.

If you have studied the section on limits in Chapter 3 of Calculus Applied to the Real World, or Chapter 10 of Finite Mathematics and Calculus Applied to the Real World, you will recognize this phenomenon in terms of limits; For instance,

$x$$\pi/2^{-} \sec x = ∞ x$$\pi/2^{+}$ $\sec x = -∞$

Before we go on...

Here are the graphs of all four of these functions. You might try to reproduce them and think about the asymptotes

$\tan x = \sin x/\cos x$

$\cotan x = \cos x/\sin x$

$\sec x = 1/\cos x$

$\cosec x = 1/\sin x$

The Trig Functions as Ratios in a Right Triangle

Let us go back to the figure that defines the sine and cosine, but this time, let us think of these two quantities as lengths of sides of a right triangle:

We are also thinking of the quantity t as a measure of the angle shown rather than the length of an arc. Looking at the figure, we find that

 $\sin t =$ length of side opposite the angle $t = \frac{\text{opposite}}{1} = \frac{\text{opposite}}{\text{\text{hypotenuse}}}$ $\cos t =$ length of side adjacent to the angle $t = \frac{\text{adjacent}}{1} = \frac{\text{adjacent}}{\text{\text{hypotenuse}}}$
 $\tan t = \frac{\sin t}{\cos t} = \frac{\text{opposite}}{\text{adjacent}}$

This gives us the following six formulas

The Trigonometric Functions as Ratios in a Right Triangle

 Defining Formula Ratio in Right Triangle $\sin t = y-$coordinate of point $P$ $\sin t = \frac{\text{opposite}}{\text{hypotenuse}}$ $\cos t = x-$coordinate of point $P$ $\cos t = \frac{\text{adjacent}}{\text{\text{hypotenuse}}}$ $\tan t = \frac{\sin t}{\cos t}$ $\tan t = \frac{\text{opposite}}{\text{adjacent}}$ $\cotan t = \frac{\cos t}{\sin t}$ $\cotan t = \frac{\text{adjacent}}{\text{opposite}}$ $\sec t = \frac{1}{\cos t}$ $\sec t = \frac{\text{\text{hypotenuse}}}{\text{adjacent}}$ $\cosec t = \frac{1}{\sin t}$ $\cosec t = \frac{\text{hypotenuse}}{\text{opposite}}$

We would welcome comments and suggestions for improving this resource.

Mail us at:
 Stefan Waner (matszw@hofstra.edu) Steven R. Costenoble (matsrc@hofstra.edu)

Last Updated: March, 1997


text:
# Are there any planned experiments to intentionally create black holes in the lab for study?

And if they did what kind of machine would be making them? And what would they study?

• – Kyle Kanos Jun 11 '15 at 2:57
• ^Hilarious, this was under a lot of speculation when LHC was ran for the first time. :) – ritvik1512 Jun 11 '15 at 3:22
• You may be interested in the book by Volovik and colleagues `Artifical Black Holes'. – Arthur Suvorov Jun 11 '15 at 4:00
• But there are dumbholes. en.wikipedia.org/wiki/Sonic_black_hole – Paul Jun 11 '15 at 4:01

It is worth stressing that black holes are predictions of classical General Relativity models. Our experimental data have established that the underlying level of nature is quantum mechanical. There is a large body of research on quantizing gravity and unifying the three forces studied with particle physics experiments with the gravitational force. String theory is successful in doing that , unification of all forces and quantization of gravity, although a validated model has not yet appeared, as there are thousands of possible solutions of the landscape of string theories.

String theory introduces at least six extra dimensions which in the prevailing model proposals are curled to very small size, of the order of postulated string dimensions , (10^-33 cm) since they are unobservable in our four dimensional frame where we live and study nature. Phenomenologists exist so that they can propose measurable predictions from models, and there exist models, called large extra dimension models, which do give predictions for micro black holes that decay immediately , at the current energies of the LHC. These were the models which created a fuss about the LHC being dangerous, the hoi polloi not understanding the necessity of immediate decay to usual particles for such entities..

The experiments have looked for the signatures of these micro black holes and have set limits for these models. The conclusion is that no signs of large extra dimensions have been seen up to now, but the search will go on with the new data coming now from the LHC at double the energy.

The smallest possible black hole that could be observed would be one with a mass on the scale of the Planck mass (~ 22 µg) and a radius on the scale of the Planck length (really small).

Thermodynamics makes it impractical to pack multiple particles into such a small space, so the best bet would be to accelerate elementary particles to have relativistic mass greater than the Planck mass, and smash them together.

A machine to accomplish that would look like the LHC, but much, much bigger. The LHC accelerates protons up to 7 TeV. Plugging that energy into $E = mc^2$, we get m = 1.25 × 10-14 µg, which is 15 orders of magnitude (a quadrillion times) too small.

It's likely that LHC-style particle accelerator technology doesn't really scale that far, so "looks like the LHC" might be a stretch.

• … I'm not a particle physicist, but I think the individual quarks would need to reach half the Planck mass, not just the protons. The cross-section for the reaction would increase with energy. Since the particles are point masses with no ability to collide, the important thing is collecting a certain amount of mass-energy within the prospective event horizon. – Blackbody Blacklight Jun 11 '15 at 4:03

In 2008 Ulf Leonhardt of University of St. Andrews UK led the first research group to make an artificial black hole. Its purpose was to look for experimental evidence of "Hawking Radition," photons and neutrinos emitted by black holes. "Hawking Radiation" couldn't be detected from astrophysical black holes because the signal would be overwhelmed by the CMB radiation. These artificial black holes were created using metamaterials that mimic astrophysical black hole behavior without needing large gravities. Artificial black holes have also been made by Chinese and American research groups.

http://www.technologyreview.com/view/415718/artificial-black-hole-created-in-chinese-lab/

http://physicsworld.com/cws/article/news/2008/mar/06/artificial-black-hole-created-in-lab

If we calculate the schwarzschild radius (and below is a really convenient link if you don't want to calculate yourself.

http://physics.unl.edu/~klee/flash_astro/bhole_sim010.swf

And if we look at the size of an atomic nucleus: 1.6 fm (1fm= $10^{−15}$ m=0.000000000000001 m) for a proton in light hydrogen to about 15 fm for the heaviest atoms, such as uranium.

Now, I realize this is oversimplified, but even so, a black hole the size of a proton would need to have a mass of roughly 500 million tons (if I got my math right). Any black hole they might potentially make at CERN would need to be far far far far smaller than a proton - as in, super duper tiny and whether or not that is possible depends on the laws of physics of the very very very small more than it depends on what they want to do.

Are super tiny quantum black holes dangerous? Well, there's a lot of good and probobly quite a few bad articles on that very subject because of the attention it got, but the short answer is no, they would disintegrate in a stream of particles, like everything else that's created in CERN - assuming they could be created at all.

If CERN was able to create a black hole (they probobly can't), but if they could, then, they probobly would because it's very unlikely that it would be dangerous and because it would be a new discovery. We like new discoveries. We learn stuff. It provides insight into the early universe and the laws of physics for the extreme and the very very very tiny.

More powerful particle interactions happen by cosmic rays and super fast moving particles crashing into our atmosphere every day than happen in CERN. CERN is doing nothing that doesn't happen already. All they're doing is getting a much closer look at it.

• What does the "1.6 fm (10−15 m)" mean? (Is fm femtometers, and m meters?) – Volker Siegel Jun 11 '15 at 6:46
• I copy/pasted from the article - my formatting skills are poor, so apologies for lack of clarity, but yes. 1 FM = 10(-15) meters. The wiki link explains it with formatting. – userLTK Jun 11 '15 at 6:55

## protected by Qmechanic♦Jun 12 '15 at 6:01

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text:
# Population Variance under Interval Uncertainty: A New Algorithm

Population Variance under Interval Uncertainty: A New Algorithm In statistical analysis of measurement results, it is often beneficial to compute the range V of the population variance $$v=\frac{1}{2}\cdot\sum^{n}_{i=1}(x_{i}-E)^{2}\,(where E=\frac{1}{n}\sum^{n}_{i=1}x_{i})$$ when we only know the intervals $$[{\tilde x}_{i}-\Delta_{i},{\tilde x}_{i}+\Delta_{i}]$$ of possible values of the xi. In general, this problem is NP-hard; a polynomialtime algorithm is known for the case when the measurements are sufficiently accurate, i.e., when $$|{\tilde x}_{i}-{\tilde x}_{j}|\geq\frac{\Delta_{i}}{n}+\frac{\Delta_{j}}{n}$$ for all $$i\neq j.$$ In this paper, we show that we can efficiently compute V under a weaker (and more general) condition $$|{\tilde x}_{i}-{\tilde x}_{j}|\geq\frac{|\Delta_{i}-\Delta_{j}|}{n}$$ . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Reliable Computing Springer Journals

# Population Variance under Interval Uncertainty: A New Algorithm

, Volume 12 (4) – Aug 1, 2006
8 pages

/lp/springer_journal/population-variance-under-interval-uncertainty-a-new-algorithm-eNpxLHpOXq
Publisher
Springer Netherlands
Subject
Mathematics; Numeric Computing; Mathematical Modeling and Industrial Mathematics; Approximations and Expansions; Computational Mathematics and Numerical Analysis
ISSN
1385-3139
eISSN
1573-1340
D.O.I.
10.1007/s11155-006-9001-x
Publisher site
See Article on Publisher Site

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# EQUATIONS OF GEODESICS IN A TWO-DIMENSIONAL FINSLER SPACE WITH A GENERALIZED KROPINA METRIC

• Park, Hong-Suh (Department of Mathematics Yeungnam University) ;
• Lee, Il-Yong (Department of Mathematics Kyungsung University)
• Published : 2000.05.01
• 48 3

#### Abstract

The geodesic equation in a two-dimensional Finsler space is given by the differential equation of the Weierstrass form. In the present paper, we express the differential equations of geodesics in a two-dimensional Finsler space with a generalized Kropina metric.

#### Keywords

Finsler space;geodesic;($\alpha,\beta$)-metric;Weierstrass form;generalized Kropina metric;special ($\alpha,\beta$)-metric

#### References

1. The Theory of Sprays and Finsler spaces with Applications in Physics and Biolgy P.L. Antonelli;R.S. Ingarden;M. Matsumoto
2. Tensor N.S. v.56 On a Finsler space with a special (α,β)-metric H.S. Park;E.S. Choi
3. Revue Roumanie de Pures et Appliquees Equations of geodesics in two- dimensional Finsler spaces with (α,β)-metric M. Matsumoto;H.S. Park
4. Open System and Inform. Dynamics v.3 Every path space of dimension two is projectvely related to a Finsler space
5. J. Hokkaido Univ. Education. Sect.ⅡA v.46 On Finsler spaces with (α,β)-metric. Regularity, geodesics and the main scalars M. Kitayama;M. Azuma;M. Matsumoto
6. J. Korean Math. Soc. v.10 On Landsberg spaces of two dimensions with (α,β)-metric M. Hashinguchi;S. Hojo;M. Matsumoto
7. Mathl. Comput. Medeling v.20 Geodesics of two-dimensional Filnsler spaces
8. Reports on Mathmetical physics v.31 Theory of Finsler spaces with (α,β)-metric M. Matsumoto
9. Publ. Math. Debrecen v.51 On Finsler spaces of Douglas type, Generalization of the notion of Berward space S. Bacso;M. Matsumoto


text:
## 27.2 Relative glueing

The following lemma is relevant in case we are trying to construct a scheme $X$ over $S$, and we already know how to construct the restriction of $X$ to the affine opens of $S$. The actual result is completely general and works in the setting of (locally) ringed spaces, although our proof is written in the language of schemes.

Lemma 27.2.1. Let $S$ be a scheme. Let $\mathcal{B}$ be a basis for the topology of $S$. Suppose given the following data:

1. For every $U \in \mathcal{B}$ a scheme $f_ U : X_ U \to U$ over $U$.

2. For $U, V \in \mathcal{B}$ with $V \subset U$ a morphism $\rho ^ U_ V : X_ V \to X_ U$ over $U$.

Assume that

1. each $\rho ^ U_ V$ induces an isomorphism $X_ V \to f_ U^{-1}(V)$ of schemes over $V$,

2. whenever $W, V, U \in \mathcal{B}$, with $W \subset V \subset U$ we have $\rho ^ U_ W = \rho ^ U_ V \circ \rho ^ V_ W$.

Then there exists a morphism $f : X \to S$ of schemes and isomorphisms $i_ U : f^{-1}(U) \to X_ U$ over $U \in \mathcal{B}$ such that for $V, U \in \mathcal{B}$ with $V \subset U$ the composition

$\xymatrix{ X_ V \ar[r]^{i_ V^{-1}} & f^{-1}(V) \ar[rr]^{inclusion} & & f^{-1}(U) \ar[r]^{i_ U} & X_ U }$

is the morphism $\rho ^ U_ V$. Moreover $X$ is unique up to unique isomorphism over $S$.

Proof. To prove this we will use Schemes, Lemma 26.15.4. First we define a contravariant functor $F$ from the category of schemes to the category of sets. Namely, for a scheme $T$ we set

$F(T) = \left\{ \begin{matrix} (g, \{ h_ U\} _{U \in \mathcal{B}}), \ g : T \to S, \ h_ U : g^{-1}(U) \to X_ U, \\ f_ U \circ h_ U = g|_{g^{-1}(U)}, \ h_ U|_{g^{-1}(V)} = \rho ^ U_ V \circ h_ V \ \forall \ V, U \in \mathcal{B}, V \subset U \end{matrix} \right\} .$

The restriction mapping $F(T) \to F(T')$ given a morphism $T' \to T$ is just gotten by composition. For any $W \in \mathcal{B}$ we consider the subfunctor $F_ W \subset F$ consisting of those systems $(g, \{ h_ U\} )$ such that $g(T) \subset W$.

First we show $F$ satisfies the sheaf property for the Zariski topology. Suppose that $T$ is a scheme, $T = \bigcup V_ i$ is an open covering, and $\xi _ i \in F(V_ i)$ is an element such that $\xi _ i|_{V_ i \cap V_ j} = \xi _ j|_{V_ i \cap V_ j}$. Say $\xi _ i = (g_ i, \{ h_{i, U}\} )$. Then we immediately see that the morphisms $g_ i$ glue to a unique global morphism $g : T \to S$. Moreover, it is clear that $g^{-1}(U) = \bigcup g_ i^{-1}(U)$. Hence the morphisms $h_{i, U} : g_ i^{-1}(U) \to X_ U$ glue to a unique morphism $h_ U : g^{-1}(U) \to X_ U$. It is easy to verify that the system $(g, \{ h_ U\} )$ is an element of $F(T)$. Hence $F$ satisfies the sheaf property for the Zariski topology.

Next we verify that each $F_ W$, $W \in \mathcal{B}$ is representable. Namely, we claim that the transformation of functors

$F_ W \longrightarrow \mathop{\mathrm{Mor}}\nolimits (-, X_ W), \ (g, \{ h_ U\} ) \longmapsto h_ W$

is an isomorphism. To see this suppose that $T$ is a scheme and $\alpha : T \to X_ W$ is a morphism. Set $g = f_ W \circ \alpha$. For any $U \in \mathcal{B}$ such that $U \subset W$ we can define $h_ U : g^{-1}(U) \to X_ U$ be the composition $(\rho ^ W_ U)^{-1} \circ \alpha |_{g^{-1}(U)}$. This works because the image $\alpha (g^{-1}(U))$ is contained in $f_ W^{-1}(U)$ and condition (a) of the lemma. It is clear that $f_ U \circ h_ U = g|_{g^{-1}(U)}$ for such a $U$. Moreover, if also $V \in \mathcal{B}$ and $V \subset U \subset W$, then $\rho ^ U_ V \circ h_ V = h_ U|_{g^{-1}(V)}$ by property (b) of the lemma. We still have to define $h_ U$ for an arbitrary element $U \in \mathcal{B}$. Since $\mathcal{B}$ is a basis for the topology on $S$ we can find an open covering $U \cap W = \bigcup U_ i$ with $U_ i \in \mathcal{B}$. Since $g$ maps into $W$ we have $g^{-1}(U) = g^{-1}(U \cap W) = \bigcup g^{-1}(U_ i)$. Consider the morphisms $h_ i = \rho ^ U_{U_ i} \circ h_{U_ i} : g^{-1}(U_ i) \to X_ U$. It is a simple matter to use condition (b) of the lemma to prove that $h_ i|_{g^{-1}(U_ i) \cap g^{-1}(U_ j)} = h_ j|_{g^{-1}(U_ i) \cap g^{-1}(U_ j)}$. Hence these morphisms glue to give the desired morphism $h_ U : g^{-1}(U) \to X_ U$. We omit the (easy) verification that the system $(g, \{ h_ U\} )$ is an element of $F_ W(T)$ which maps to $\alpha$ under the displayed arrow above.

Next, we verify each $F_ W \subset F$ is representable by open immersions. This is clear from the definitions.

Finally we have to verify the collection $(F_ W)_{W \in \mathcal{B}}$ covers $F$. This is clear by construction and the fact that $\mathcal{B}$ is a basis for the topology of $S$.

Let $X$ be a scheme representing the functor $F$. Let $(f, \{ i_ U\} ) \in F(X)$ be a “universal family”. Since each $F_ W$ is representable by $X_ W$ (via the morphism of functors displayed above) we see that $i_ W : f^{-1}(W) \to X_ W$ is an isomorphism as desired. The lemma is proved. $\square$

Lemma 27.2.2. Let $S$ be a scheme. Let $\mathcal{B}$ be a basis for the topology of $S$. Suppose given the following data:

1. For every $U \in \mathcal{B}$ a scheme $f_ U : X_ U \to U$ over $U$.

2. For every $U \in \mathcal{B}$ a quasi-coherent sheaf $\mathcal{F}_ U$ over $X_ U$.

3. For every pair $U, V \in \mathcal{B}$ such that $V \subset U$ a morphism $\rho ^ U_ V : X_ V \to X_ U$.

4. For every pair $U, V \in \mathcal{B}$ such that $V \subset U$ a morphism $\theta ^ U_ V : (\rho ^ U_ V)^*\mathcal{F}_ U \to \mathcal{F}_ V$.

Assume that

1. each $\rho ^ U_ V$ induces an isomorphism $X_ V \to f_ U^{-1}(V)$ of schemes over $V$,

2. each $\theta ^ U_ V$ is an isomorphism,

3. whenever $W, V, U \in \mathcal{B}$, with $W \subset V \subset U$ we have $\rho ^ U_ W = \rho ^ U_ V \circ \rho ^ V_ W$,

4. whenever $W, V, U \in \mathcal{B}$, with $W \subset V \subset U$ we have $\theta ^ U_ W = \theta ^ V_ W \circ (\rho ^ V_ W)^*\theta ^ U_ V$.

Then there exists a morphism of schemes $f : X \to S$ together with a quasi-coherent sheaf $\mathcal{F}$ on $X$ and isomorphisms $i_ U : f^{-1}(U) \to X_ U$ and $\theta _ U : i_ U^*\mathcal{F}_ U \to \mathcal{F}|_{f^{-1}(U)}$ over $U \in \mathcal{B}$ such that for $V, U \in \mathcal{B}$ with $V \subset U$ the composition

$\xymatrix{ X_ V \ar[r]^{i_ V^{-1}} & f^{-1}(V) \ar[rr]^{inclusion} & & f^{-1}(U) \ar[r]^{i_ U} & X_ U }$

is the morphism $\rho ^ U_ V$, and the composition

27.2.2.1
$$\label{constructions-equation-glue} (\rho ^ U_ V)^*\mathcal{F}_ U = (i_ V^{-1})^*((i_ U^*\mathcal{F}_ U)|_{f^{-1}(V)}) \xrightarrow {\theta _ U|_{f^{-1}(V)}} (i_ V^{-1})^*(\mathcal{F}|_{f^{-1}(V)}) \xrightarrow {\theta _ V^{-1}} \mathcal{F}_ V$$

is equal to $\theta ^ U_ V$. Moreover $(X, \mathcal{F})$ is unique up to unique isomorphism over $S$.

Proof. By Lemma 27.2.1 we get the scheme $X$ over $S$ and the isomorphisms $i_ U$. Set $\mathcal{F}'_ U = i_ U^*\mathcal{F}_ U$ for $U \in \mathcal{B}$. This is a quasi-coherent $\mathcal{O}_{f^{-1}(U)}$-module. The maps

$\mathcal{F}'_ U|_{f^{-1}(V)} = i_ U^*\mathcal{F}_ U|_{f^{-1}(V)} = i_ V^*(\rho ^ U_ V)^*\mathcal{F}_ U \xrightarrow {i_ V^*\theta ^ U_ V} i_ V^*\mathcal{F}_ V = \mathcal{F}'_ V$

define isomorphisms $(\theta ')^ U_ V : \mathcal{F}'_ U|_{f^{-1}(V)} \to \mathcal{F}'_ V$ whenever $V \subset U$ are elements of $\mathcal{B}$. Condition (d) says exactly that this is compatible in case we have a triple of elements $W \subset V \subset U$ of $\mathcal{B}$. This allows us to get well defined isomorphisms

$\varphi _{12} : \mathcal{F}'_{U_1}|_{f^{-1}(U_1 \cap U_2)} \longrightarrow \mathcal{F}'_{U_2}|_{f^{-1}(U_1 \cap U_2)}$

whenever $U_1, U_2 \in \mathcal{B}$ by covering the intersection $U_1 \cap U_2 = \bigcup V_ j$ by elements $V_ j$ of $\mathcal{B}$ and taking

$\varphi _{12}|_{V_ j} = \left((\theta ')^{U_2}_{V_ j}\right)^{-1} \circ (\theta ')^{U_1}_{V_ j}.$

We omit the verification that these maps do indeed glue to a $\varphi _{12}$ and we omit the verification of the cocycle condition of a glueing datum for sheaves (as in Sheaves, Section 6.33). By Sheaves, Lemma 6.33.2 we get our $\mathcal{F}$ on $X$. We omit the verification of (27.2.2.1). $\square$

Remark 27.2.3. There is a functoriality property for the constructions explained in Lemmas 27.2.1 and 27.2.2. Namely, suppose given two collections of data $(f_ U : X_ U \to U, \rho ^ U_ V)$ and $(g_ U : Y_ U \to U, \sigma ^ U_ V)$ as in Lemma 27.2.1. Suppose for every $U \in \mathcal{B}$ given a morphism $h_ U : X_ U \to Y_ U$ over $U$ compatible with the restrictions $\rho ^ U_ V$ and $\sigma ^ U_ V$. Functoriality means that this gives rise to a morphism of schemes $h : X \to Y$ over $S$ restricting back to the morphisms $h_ U$, where $f : X \to S$ is obtained from the datum $(f_ U : X_ U \to U, \rho ^ U_ V)$ and $g : Y \to S$ is obtained from the datum $(g_ U : Y_ U \to U, \sigma ^ U_ V)$.

Similarly, suppose given two collections of data $(f_ U : X_ U \to U, \mathcal{F}_ U, \rho ^ U_ V, \theta ^ U_ V)$ and $(g_ U : Y_ U \to U, \mathcal{G}_ U, \sigma ^ U_ V, \eta ^ U_ V)$ as in Lemma 27.2.2. Suppose for every $U \in \mathcal{B}$ given a morphism $h_ U : X_ U \to Y_ U$ over $U$ compatible with the restrictions $\rho ^ U_ V$ and $\sigma ^ U_ V$, and a morphism $\tau _ U : h_ U^*\mathcal{G}_ U \to \mathcal{F}_ U$ compatible with the maps $\theta ^ U_ V$ and $\eta ^ U_ V$. Functoriality means that these give rise to a morphism of schemes $h : X \to Y$ over $S$ restricting back to the morphisms $h_ U$, and a morphism $h^*\mathcal{G} \to \mathcal{F}$ restricting back to the maps $h_ U$ where $(f : X \to S, \mathcal{F})$ is obtained from the datum $(f_ U : X_ U \to U, \mathcal{F}_ U, \rho ^ U_ V, \theta ^ U_ V)$ and where $(g : Y \to S, \mathcal{G})$ is obtained from the datum $(g_ U : Y_ U \to U, \mathcal{G}_ U, \sigma ^ U_ V, \eta ^ U_ V)$.

We omit the verifications and we omit a suitable formulation of “equivalence of categories” between relative glueing data and relative objects.

## Comments (3)

Comment #2540 by Noah Olander on

In the last sentence of the statement of lemma 26.2.1 you have written "for $V \subset U \subset S$ affine open," when you should have written "for $V, U \in \mathcal{B}$ such that $V \subset U$"

Comment #2548 by Dario Weißmann on

In 26.2.1 and 26.2.2: I don't think the $X$ constructed is unique, just unique up to unique isomorphism. Noah Olander's comment should also apply to 26.2.2.

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## Statistical Science

### Permutation Methods: A Basis for Exact Inference

Michael D. Ernst

#### Abstract

The use of permutation methods for exact inference dates back to Fisher in 1935. Since then, the practicality of such methods has increased steadily with computing power. They can now easily be employed in many situations without concern for computing difficulties. We discuss the reasoning behind these methods and describe situations when they are exact and distribution-free. We illustrate their use in several examples.

#### Article information

Source
Statist. Sci., Volume 19, Number 4 (2004), 676-685.

Dates
First available in Project Euclid: 18 April 2005

https://projecteuclid.org/euclid.ss/1113832732

Digital Object Identifier
doi:10.1214/088342304000000396

Mathematical Reviews number (MathSciNet)
MR2185589

Zentralblatt MATH identifier
1100.62563

#### Citation

Ernst, Michael D. Permutation Methods: A Basis for Exact Inference. Statist. Sci. 19 (2004), no. 4, 676--685. doi:10.1214/088342304000000396. https://projecteuclid.org/euclid.ss/1113832732

#### References

• Bradley, J. V. (1968). Distribution-Free Statistical Tests. Prentice-Hall, Englewood Cliffs, NJ.
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• Dwass, M. (1957). Modified randomization tests for nonparametric hypotheses. Ann. Math. Statist. 28 181--187.
• Edgington, E. S. (1995). Randomization Tests, 3rd ed. Dekker, New York.
• Ernst, M. D. and Schucany, W. R. (1999). A class of permutation tests of bivariate interchangeability. J. Amer. Statist. Assoc. 94 273--284.
• Fisher, R. A. (1935). The Design of Experiments. Oliver and Boyd, Edinburgh.
• Fisher, R. A. (1936). The coefficient of racial likeness'' and the future of craniometry. J. Royal Anthropological Institute of Great Britain and Ireland 66 57--63.
• Garthwaite, P. H. (1996). Confidence intervals from randomization tests. Biometrics 52 1387--1393.
• Grogan, W. L., Jr. and Wirth, W. W. (1981). A new American genus of predaceous midges related to Palpomyia and Bezzia (Diptera: Ceratopogonidae). Proc. Biological Society of Washington 94 1279--1305.
• Higgins, J. J. (2004). An Introduction to Modern Nonparametric Statistics. Brooks/Cole, Pacific Grove, CA.
• Ihaka, R. and Gentleman, R. (1996). R: A language for data analysis and graphics. J. Comput. Graph. Statist. 5 299--314.
• Jöckel, K.-H. (1986). Finite sample properties and asymptotic efficiency of Monte Carlo tests. Ann. Statist. 14 336--347.
• Kennedy, P. E. and Cade, B. S. (1996). Randomization tests for multiple regression. Comm. Statist. Simulation Comput. 25 923--936.
• Lehmann, E. L. (1975). Nonparametrics: Statistical Methods Based on Ranks. Holden--Day, San Francisco.
• Manly, B. F. J. (1997). Randomization, Bootstrap and Monte Carlo Methods in Biology, 2nd ed. Chapman and Hall, London.
• Pitman, E. J. G. (1937a). Significance tests which may be applied to samples from any populations. J. Roy. Statist. Soc. Suppl. 4 119--130.
• Pitman, E. J. G. (1937b). Significance tests which may be applied to samples from any populations. II. The correlation coefficient test. J. Roy. Statist. Soc. Suppl. 4 225--232.
• Pitman, E. J. G. (1938). Significance tests which may be applied to samples from any populations. III. The analysis of variance test. Biometrika 29 322--335.


text:
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What are some good public contribution opportunities? (100\$ bounty) 2020-06-18T14:47:51.661Z
Gurkenglas's Shortform 2019-08-04T18:46:34.953Z
Implications of GPT-2 2019-02-18T10:57:04.720Z
What shape has mindspace? 2019-01-11T16:28:47.522Z
A simple approach to 5-and-10 2018-12-17T18:33:46.735Z
Quantum AI Goal 2018-06-08T16:55:22.610Z
Quantum AI Box 2018-06-08T16:20:24.962Z
A line of defense against unfriendly outcomes: Grover's Algorithm 2018-06-05T00:59:46.993Z

Comment by Gurkenglas on The Apprentice Thread · 2021-06-17T13:57:02.782Z · LW · GW

[APPRENTICE] Advanced math or AI alignment. I'm bad at getting homework done and good at grokking things quickly, so the day-to-day should look like pair programming or tutoring.

[MENTOR] See https://www.lesswrong.com/posts/MHqwi8kzwaWD8wEQc/would-you-like-me-to-debug-your-math. The first session has the highest leverage, but if my calendar doesn't end up booked (there is one slot in the next two weeks booked out of like 50), more time per person makes sense. My specialization is pattern-matching to correctly predict where a piece of math is going if it's good. When you science that art you get applied category theory.

Comment by Gurkenglas on Experiments with a random clock · 2021-06-14T06:32:48.096Z · LW · GW

Break the minute hand off your wristwatch. Maybe some of the hour hand too.

Comment by Gurkenglas on Would you like me to debug your math? · 2021-06-12T08:31:53.820Z · LW · GW

Seems like less of a market niche, but link it!

Comment by Gurkenglas on Would you like me to debug your math? · 2021-06-11T18:16:08.048Z · LW · GW

It's not my one trick, of course, but it illustrates my usefulness. It's more maintainable not just because it is shorter but also because it has decades of theory behind it. Drawing the connection unlocks inspiration from entire branches of math. And the speedups from standing on the shoulders of giants go far beyond the constant factors from vectorized instructions.

Comment by Gurkenglas on Finite Factored Sets: Orthogonality and Time · 2021-06-11T13:41:28.517Z · LW · GW

subpartitions

So you're doing category theory after all! :)

Comment by Gurkenglas on Would you like me to debug your math? · 2021-06-11T11:47:08.889Z · LW · GW

Sure, I'll try it. I don't expect to be an order-of-magnitude power multiplier in that case, though.

Comment by Gurkenglas on The reverse Goodhart problem · 2021-06-08T21:41:15.184Z · LW · GW

I wouldn't relate it to humans. In just about any basic machine learning setting, (train, test) has aspect 2. In fact, what you describe speaks in favor of modeling preferences using something other than utility functions, where aspect 3 is ruled out.

re your natural example, I would expect that as one shifts from 90% doing the best for the worst off + 10% attainable utility preservation to 100% the former, average welfare goes down.

Comment by Gurkenglas on Speculations against GPT-n writing alignment papers · 2021-06-08T12:59:00.867Z · LW · GW

Take into account that the AI that interprets needs not be the same as the network being interpreted.

Why do you think that a mere autocomplete engine could not do interpretability work? It has been demonstrated to write comments for code and code for specs.

Comment by Gurkenglas on Speculations against GPT-n writing alignment papers · 2021-06-08T12:55:45.465Z · LW · GW

The error correction needs to be present in the original network because I also do some of the converting network into english. The only reason I don't do everything myself is that it takes too long. The proportion can be higher at the topmost levels because there are less tasks there. The error correction doesn't let it completely ignore what I would do at the low levels because on the 1% I can compare its outputs to mine, so they need to at least superficially look similar.

If we find that there's a bunch of redundancy, we can check whether there is any way to cut it down that would suddenly identify a bunch of mesa optimization. So the mesaoptimizer would have to take over the network entirely or trick its bretheren as well as us.

Comment by Gurkenglas on Speculations against GPT-n writing alignment papers · 2021-06-07T22:57:17.811Z · LW · GW

Translating to a tree of natural language descriptions and back lets you

1. use different instances of GPT at each step, so that any coded messages would have to rely on the common training set rather than shared internals.
2. do a random subset of the tasks yourself, so that any subversion of the target system would have to be based on subtle biases, perhaps with a small number of outright lies that it hopes we don't implement manually, rather than the description being fabricated from whole cloth.

Having the thing write papers is merely an existence proof of embedded agency being irrelevant except for deconfusion.

Intelligent agents causally responsible for your existence.

What do you mean you can think of this, I told this to you :D

Comment by Gurkenglas on Tracey Davis and the Prisoner of Azkaban - Part 2 · 2021-06-07T09:27:59.796Z · LW · GW

Was the conversation in chapter one after this or is Tracey going back and forth between the two, as a Tracey does?

You, me, the Three Broomsticks, a bottle of fizzix and a nuclear missile?

Channelling Luna, I see.

Comment by Gurkenglas on SIA is basically just Bayesian updating on existence · 2021-06-04T17:32:39.677Z · LW · GW

If you try to model yourself as a uniformly randomly selected observer somewhere in the Solomonoff prior, that doesn't work because there isn't a uniform distribution on the infinite naturals. When you weight them by each universe's probability but allow a universe to specify the number of observers in it, you still diverge because the number goes up uncomputably fast while the probability goes only exponentially down. In the end, probabilities are what decision theories use to weight their utility expectations. Therefore I suggest we start from a definition of how much we care about what happens to each of the forever infinite copies of us throughout the multiverse. It is consistent to have 80% of the utility functions in one's aggregate say that whatever happens to the overwhelming majority of selves in universes of description length >1000, it can only vary utility by at most 0.2.

Comment by Gurkenglas on Finite Factored Sets · 2021-06-04T16:27:12.205Z · LW · GW

Call the index set of X IX. Call the partition into empty parts indexed by S 0S. We have 0 ⊣ I ⊣ D ⊣ ⊔ ⊣ T.

None of the our three adjunction strings can be extended further. Let's apply the construction that gave us histories at the other 5 ends. Niceness is implicit.
- The right construction of TS->X is the terminal S->S' with a TS'->X: The image of ⊔(TS->X).
- The left construction of X->0S is the initial S'->S with a X->0S': The image of I(X->0S).
- The left construction of B->FX is the initial X'->X with a B->FX': The image of ∨(B->FX).
- The right construction of Δ1•->S is the terminal •->• with a Δ1•->S: The image of Δ•(Δ1•->S).
- The left construction of S->Δ∅• is absurd, but can still be written as the image of Δ•(S->Δ∅•).
- The history of ∨B->X is the terminal B->B' with a ∨B'->X: Breaks the pattern! F(∨B->X) does not have the information to determine the history.

In fact, ⊔T, I0, ∨F, Δ•Δ1 and Δ•Δ∅ are all identity, only F∨ isn't.

Comment by Gurkenglas on Tracey Davis and the Prisoner of Azkaban - Part 1 · 2021-06-04T10:47:42.265Z · LW · GW

She left her wand on the bar and she didn't immediately lose? Moody must have expected her to have another wand, given that she left her wand on the bar theatrically.

Comment by Gurkenglas on Finite Factored Sets · 2021-06-03T20:44:11.192Z · LW · GW

Let 1 be the category with one object • and one morphism. Let Δx be the constant functor to x.

A set is a family of • called elements. A set morphism S->S' has a 1-morphism between each element of S and some element of S'. The 1-morphisms •->Δ•S correspond to the set morphisms Δ∅•->S. The 1-morphisms Δ•S->• correspond to the set morphisms S->Δ1•. We have Δ∅ ⊣ Δ• ⊣ Δ1.

Let 0 the empty category. • is the empty family. A 1-morphism has nothing to prove. There's no forgetful functor 1->0 so the buck stops here.

Comment by Gurkenglas on Finite Factored Sets · 2021-06-03T09:02:21.391Z · LW · GW

Let's try category theory.

A partition is a family of sets called parts. A partition morphism X->X' has a function from each part of X to some part of X'. It witnesses that X is finer than X'¹.

The underlying set of a partition is its disjoint union. Call the discrete partition of S DS. The functions S->⊔X correspond to the partition morphisms DS->X. Call the trivial partition of S TS. The functions ⊔X->S correspond to the partition morphisms X->TS. In terser notation, we have D ⊣ ⊔ ⊣ T.

A factorization is a family of partitions called factors. A factorization morphism B->B' has a partition morphism to each factor of B' from some factor of B.²

The underlying partition of a factorization is its common refinement. Call the trivial factorization of X FX.³ The partition morphisms X->∨B correspond to the factorization morphisms FX->B: We have F ⊣ ∨. The absence of "discrete factorizations" as a right adjoint to ∨ is where histories come from.

A history of ∨B->X is a nice⁴ B->H with a ∨H->X. The history of ∨B->X is its terminal history. Note that this also attempts to coarsen each factor. ∨B->X being weakly after ∨B->X' is witnessed by a nice ∨H->X' or equivalently H->H'. ∨B->X and ∨B->X' are orthogonal iff the pushout of B->H and B->H' is the empty factorization.

Translating "2b. Conditional Orthogonality" is taking a while (I think it's something with pushouts) so let's post this now. I'm also planning to generalize "family" to "diagram". Everyone's allowed to ask stupid questions, including basic category theory.

¹: Which includes that X might rule out some worlds.
²: Trying to avert the analogy break cost me ~60% of the time behind this comment.
⁴: Nice means that everything in sight commutes.

Comment by Gurkenglas on If You Want to Find Truth You Need to Step Into Cringe · 2021-06-01T19:59:27.207Z · LW · GW

Unattractiveness => Cringe

Or Cringe => Unattractiveness, or they have a common cause. The people in the video may be unattractive because the author wanted to convince the viewer that weeabooism correlates with low social status. When you listen to the linked song, the reasoning it brings forward for why the viewer should consider weeaboos cringe is that they believe to have higher social status than they do.

Comment by Gurkenglas on If individual performance is Pareto distributed, how should we reform education? · 2021-05-25T09:23:23.150Z · LW · GW

How do you measure performance, then? If you can only rank it, distributions mean nothing.

Comment by Gurkenglas on Don't feel bad about not knowing basic things · 2021-05-24T18:49:43.859Z · LW · GW

You can map across any monad, but not everything you can map across is a monad. Applicatives are in between.

Comment by Gurkenglas on Don't feel bad about not knowing basic things · 2021-05-24T08:13:34.030Z · LW · GW

In the middle are stuff in between.

What an Applicative is? :)

Comment by Gurkenglas on Are PS5 scalpers actually bad? · 2021-05-18T18:13:56.353Z · LW · GW

Couldn't producers just hold an auction and have the proceeds beyond the price they're allowed by public opinion to charge go to charity?

Comment by Gurkenglas on European Soylent alternatives · 2021-05-16T10:56:45.005Z · LW · GW

Is this up to date?

Comment by Gurkenglas on How to compute the probability you are flipping a trick coin · 2021-05-15T15:23:05.486Z · LW · GW

The exponential is because updates happen on a logarithmic scale. Do you have a simple variant of the problem in mind where we don't get exponentials? When I try to construct one, I have to start from "we don't get exponentials" and calculate how the probabilities of different hypotheses would have to converge over time.

Comment by Gurkenglas on How to compute the probability you are flipping a trick coin · 2021-05-15T06:47:43.237Z · LW · GW

2^-n is in fact the probability of a coin showing n heads. Where is the choice?

Comment by Gurkenglas on Agency in Conway’s Game of Life · 2021-05-13T06:55:43.694Z · LW · GW

I would say it all depends on whether there is a wall gadget which protects everything on one side from anything on the other side. (And don't forget the corner gadget.)

If so, cover the edges of the controlled portion in it, except for a "gate" gadget which is supposed to be a wall except openable and closable. (This is relatively easier since a width of 100 ought to be enough, and since we can stack 10000 of these in case one is broken through - rarely should chaos be able to reach through a 100x10000 rectangle.)

Wait 10^40 steps for the chaos to lose entropy. The structures that remain should be highly compressible in a CS sense, and made of a small number of natural gadget types. Send out ships that cover everything in gliders. This should resurrect the chaos temporarily, but decrease entropy further in the long run. Repeat 10^10 times, waiting 10^40 steps in between.

The rest should be a simple matter of carefully distinguishing the remaining natural gadgets with ship sensors to dismantle each. Programming a smiley face deployer that starts from an empty slate is a trivial matter.

If walls are constructible, there's no need for gates, and also one could allow a margin for error in the sensory ships: One could advance the walls after claiming some area, in case a rare encounter summons another era of chaos.

All this is less AGI than ordinary game AI - a bundle of programmed responses.

Comment by Gurkenglas on Is driving worth the risk? · 2021-05-11T07:44:03.246Z · LW · GW

I think making utility linear in years is a mistake. The remote possibility of finding a physics hack to control infinite matter in finite time does not curbstomp all other considerations, therefore the utility of that outcome is finite. I prefer 66% of BB(1000) years to 33% of BB(10000) years. I am uncertain about my preferences, but utility functions are not aggregated by taking the expectation.

The only authority on your preferences is yourself; but reasonable agents, when a hypothetical proves them dutch-bookable/incoherent, will become less certain about their preferences.

What this cashes out to is that you should calculate the value not of a year but of (an extra 1% chance of) making it to takeoff. (From what you would do in (perhaps physically impossible) hypotheticals.)

Comment by Gurkenglas on MikkW's Shortform · 2021-05-10T18:46:50.110Z · LW · GW

So a forager animal with no predators isn't free because it has to look for food?

Comment by Gurkenglas on Domain Theory and the Prisoner's Dilemma: FairBot · 2021-05-10T07:09:50.340Z · LW · GW

Yeah, enacting comes in at a higher level of interpretation than is yet considered here. The increasing levels of interpretation here are: Set theory or other math foundations; we consider sets of queries and beliefs and players with functions between them; we add porder and monotonicity; we specify proof engines and their properties like consistency; we define utilities, decision theories, and what makes some players better than others. (Category theory is good at keeping these separate.) I'd start talking about "enacting" when we define a decision theory like "Make the decision such that I can prove the best lower bound on utility.". What do you mean by deciding on a belief state? "Decision" is defined before I establish any causation from decisions to beliefs.

Oh, I thought you meant you didn't see why any two beliefs had an upper bound. My choice to make players monotonic comes from intuition that that's how the math is supposed to look. I'd define Query=P(Decision) as Decision->2 as well but that plainly makes no sense so I'm looking for the true posetty definition of Query, and "logical formulas" looks good so far. Switching back and forth sounds more like you want to do multiple decisions, one after the other. There's also a more grounded case to be made that your policy should become more certain as your knowledge does, do you see it?

Comment by Gurkenglas on Domain Theory and the Prisoner's Dilemma: FairBot · 2021-05-09T22:34:35.925Z · LW · GW

The belief state diagram is upward closed because I included the inconsistent belief states. We could say that a one-query player "decides to defect" if his query is proven false. Then he will only decide on both decisions when his beliefs are inconsistent. Alternatively we could have a query for each primitive decision, inducing a monotone map from P({C,D}) to queries; or we could identify players with these monotone maps.

I didn't follow the bit about being modeled as oneself. Every definition of the belief space gives us a player space, yes? And once we specify some beliefs we have a tournament to examine, an interesting one if we happen to pipe the player's outputs into their inputs through some proof engines. Define enacted.

Comment by Gurkenglas on Domain Theory and the Prisoner's Dilemma: FairBot · 2021-05-09T16:46:30.487Z · LW · GW

In intuitionistic logic, "Cooperate iff either of C or ⊥ is provable." is equivalent to "Cooperate iff (C or ⊥) is provable.". So should we only consider players of form "Cooperate iff _ is provable.", where _ is some intuitionistic formula? Well...

Comment by Gurkenglas on Why are the websites of major companies so bad at core functionality? · 2021-05-09T14:34:21.274Z · LW · GW

I suppose for messy real-world tasks, you can't define distances objectively ahead of time. You could simply check a random 10 (x,f(x)) and choose how much to pay. In an ideal world, if they think you're being unfair they can stop working for you. In this world where giving someone a job is a favor, they could go to a judge to have your judgement checked.

Though if we're talking about AIs: You could have the AI output a probability distribution g(x)  over possible f(x) for each of the 100 x. Then for a random 10 x, you generate an f(x) and reward the AI according to how much probability it assigned to what you generated.

Comment by Gurkenglas on Why are the websites of major companies so bad at core functionality? · 2021-05-08T19:24:44.816Z · LW · GW

How would you goodhart this metric? To be clear, you want to map x to f(x), but this takes a second of your time. You pay them to map x to f(x), but they map x to g(x). After they're done mapping 100 x to g(x), you select a random 10 of those 100, spend 10 seconds to calculate the corresponding g(x)-f(x), and pay them more the smaller the absolute difference.

Comment by Gurkenglas on Why are the websites of major companies so bad at core functionality? · 2021-05-08T16:09:23.682Z · LW · GW

Couldn't you make them care by making their pay dependent on how well they predict what you would decide, as measured by you redoing the decision for a representative sample of tasks?

Comment by Gurkenglas on interpreting GPT: the logit lens · 2021-05-01T08:53:58.823Z · LW · GW

floating point underflow to simulate relu

Oh that's not good. Looks like we'd need a version of float that keeps track of an interval of possible floats (by the two floats at the end of the interval). Then we could simulate the behavior of infinite-precision floats so long as the network keeps the bounds tight, and we could train the network to keep the simulation in working order. Then we could see whether, in a network thus linear at small numbers, every visibly large effect has a visibly large cause.

By the way - have you seen what happens when you finetune GPT to reinforce this pattern that you're observing, that every entry of the table, not just the top right one, predicts an input token?

Comment by Gurkenglas on interpreting GPT: the logit lens · 2021-04-29T17:16:27.870Z · LW · GW

gelu has the same property

Actually, gelu is differentiable at 0, so it is linear on close-to-zero values.

Comment by Gurkenglas on When Should the Fire Alarm Go Off: A model for optimal thresholds · 2021-04-29T06:29:19.045Z · LW · GW

not having to pay  is effectively the same as gaining

No! If you're going to add/multiply something to your utility function for convenience, you have to do it for every action. When the building is on fire, deciding whether to turn on the sprinklers is a decision on whether to spend T and gain D, so V(TP)-V(FN) needs to be D-T.

Comment by Gurkenglas on What topics are on Dath Ilan's civics exam? · 2021-04-27T07:39:27.888Z · LW · GW

This is how you get Latin courses.

Comment by Gurkenglas on NTK/GP Models of Neural Nets Can't Learn Features · 2021-04-23T09:42:09.172Z · LW · GW

Can we therefore model fine-tuning as moving around in the parameter tangent space around the pre-trained network?

Comment by Gurkenglas on Löb's theorem simply shows that Peano arithmetic cannot prove its own soundness · 2021-04-22T12:17:37.579Z · LW · GW

We are trying to prove some statement p. When we're proving it for all pairs of real numbers x and y, the spell "Without loss of generality!" gives us the lemma "x<=y". When we're proving it for all natural numbers, the spell "Complete induction!" gives us the lemma "p holds for all smaller numbers". When we're working in PA, "Löb's Theorem!" gives us the lemma "Provable(p)".

Edit: And in general, math concepts are useful because of how they can be used. Memorize not things that are true, but things you can do. See this comment.

Comment by Gurkenglas on Löb's theorem simply shows that Peano arithmetic cannot prove its own soundness · 2021-04-22T12:08:16.021Z · LW · GW

Then apparently "PA can't prove its own soundness." is an even weaker true statement among the ones one might choose to remember :).

Comment by Gurkenglas on Löb's theorem simply shows that Peano arithmetic cannot prove its own soundness · 2021-04-22T11:26:35.128Z · LW · GW

Misunderstanding. I'm saying "In order to prove p, it suffices to prove □p→p.". Compare to complete induction.

Comment by Gurkenglas on Löb's theorem simply shows that Peano arithmetic cannot prove its own soundness · 2021-04-22T10:11:03.171Z · LW · GW

Yes, but 3 is a one-way street. You should remember the theorem,  not the corollary.

Comment by Gurkenglas on Löb's theorem simply shows that Peano arithmetic cannot prove its own soundness · 2021-04-22T09:32:49.023Z · LW · GW

If you told me to write down "PA can't prove its own soundness.", I would not write down Löb's theorem, I would write down "¬(PA ⊢ ∀p: Provable(p)→p)".

I would translate Löb's theorem as "In PA, when proving something, you get the lemma that it is provable for free.". Compare to "When proving a property for all natural numbers, you get the lemma that it holds for all smaller natural numbers for free."

Comment by Gurkenglas on Are there opportunities for small investors unavailable to big ones? · 2021-04-20T00:44:21.846Z · LW · GW

How does price discrimination serve the public? I got the impression it's the sort of drawback of markets that gets regulated away when it gets pronounced enough, like network effects.

Comment by Gurkenglas on Superrational Agents Kelly Bet Influence! · 2021-04-17T10:47:35.994Z · LW · GW

Suppose instead of a timeline with probabilistic events, the coalition experiences the full tree of all possible futures - but we translate everything to preserve behavior. Then beliefs encode which timelines each member cares about, and bets trade influence (governance tokens) between timelines.

Comment by Gurkenglas on Why has nuclear power been a flop? · 2021-04-16T19:57:18.432Z · LW · GW

a tax on each kilowatt-hour

Wouldn't this almost precisely incentivize approving anything immediately?

Comment by Gurkenglas on Computing Natural Abstractions: Linear Approximation · 2021-04-16T09:27:23.814Z · LW · GW

Y can consist of multiple variables, and then there would always be multiple ways, right? I thought by indirect you meant that the path between X and Y was longer than 1. If some third cause is directly upstream from both, then I suppose it wouldn't be uniquely defined whether changing X changes Y, since there could be directions in which to change the cause that change some subset of X and Y.

Comment by Gurkenglas on Computing Natural Abstractions: Linear Approximation · 2021-04-15T22:48:18.757Z · LW · GW

If matrix A maps each input vector of X to a vector of which the first entry corresponds to Y, subtracting multiples of the first row from every other row to make them orthogonal to the first row, then deleting the first row, would leave a matrix whose row space is the input vectors that keep Y at 0, and whose column space is the outputs thus still reachable. If you fix some distribution on the inputs of X (such as the normal distribution with a given covariance matrix), whether this is losslessly possible should be more interesting.

Comment by Gurkenglas on Computing Natural Abstractions: Linear Approximation · 2021-04-15T21:34:05.754Z · LW · GW

With differential geometry, there's probably a way to translate properties between points. And a way to analyze the geometry of the training distribution: Train the generator to be locally injective and give it an input space uniformly distributed on the unit circle, and whether it successfully trains tells you whether the training distribution has a cycle. Try different input topologies to nail down the distribution's topology. But just like J's rank tells you the dimension of the input distribution if you just give the generator enough numbers to work with, a powerful generator ought to tell you the entire topology in one training run...

If the generator's input distribution is uniform, Σ is diagonal, and the left SVD component of J is also the left (and transposed right) SVD component of JΣJᵀ. Is that useful?

Comment by Gurkenglas on Computing Natural Abstractions: Linear Approximation · 2021-04-15T19:52:37.748Z · LW · GW

What a small world - I was thinking up a very similar transparency tool since two weeks ago. The function f from inputs to activations-of-every-neuron isn't linear but it is differentiable, aka linear near (input space, not pixel coordinates!) an input. The jacobian J at an input x is exactly the cross-covariance matrix between a normal distribution 𝓝(x,Σ) and its image 𝓝(f(x),JΣJᵀ), right? Then if you can permute a submatrix of JΣJᵀ into a block-diagonal matrix, you've found two modules that work with different properties of x. If the user gives you two modules, you could find an input where they work with different properties, and then vary that input in ways that change activations in one module but not the other to show the user what each module does. And by something like counting the near-zero entries in the matrix, you could differentiably measure the network's modularity, then train it to be more modular.

Train a (GAN-)generator on the training inputs and attach it to the front of the network - now you know the input distribution is uniform, the (reciprocals of) singular values say the density of the output distribution in the direction of their singular vector, and the inputs you show the user are all in-distribution.

And I've thought this up shortly before learning terms like cross-covariance matrix, so please point out terms that describe parts of this. Or expand on it. Or run away with it, would be good to get scooped.


text:
A box has dimensions of 19 inches long, 1.7 feet wide, and 6 inches high. What is the volume of the box?

Jul 23, 2016

The volume is $193.8$ cubic inches

Explanation:

A box is a rectangular prism.

The volume is calculated by the formula
length x width x height

$V = l w h$

$l = 19$in
$w = 1.7$in
$h = 6$in

$\left(19\right) \left(1.7\right) \left(6\right) = V$

$193.8 = V$