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More AP Stat help
November 28th 2009, 06:01 PM #1
Nov 2009
More AP Stat help
1. A consumer organization estimates that 29% of new cars have a cosmetic defect such as a scratch or a dent when they are delivered to car dealers. This same organization believes that 7% have a
functional defects---something that does not work properly---and that 2% of new cars have both kinds of problems.
a) If you buy a new car, what's the probability that it has some kind of defect?
b) What's the probability it has a cosmetic defect but no functional defect?
c) If you notice a dent on a new car, what's the probability it has a functional defect?
d) Are the two kinds of defects disjoint events? Explain.
No. You can have both kinds of defects on one car.
e) Do you think the two kinds of defects are independent events? Explain.
Yes. Having one type of defect does not mean you will have the other type.
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Just out: Relativistic Quantum Chemistry - The Fundamental Theory of Molecular Science
The latest news from academia, regulators research labs and other things of interest
Posted: February 12, 2009
Just out: Relativistic Quantum Chemistry - The Fundamental Theory of Molecular
(Nanowerk News) John Wiley and Sons Ltd's new report "Relativistic Quantum Chemistry:
The Fundamental Theory of Molecular Science" will be published on March 3, 2009.
Written by two researchers in the field, this book is a reference to explain the
principles and fundamentals in a self-contained, complete and consistent way. Much
attention is paid to the didactical value, with the chapters interconnected and based
on each other. From the contents:
Relativistic Theory of a Free Electron: Dirac's Equation
Dirac Theory of a Single Electron in a Central Potential
Many-Electron Theory I: Quantum Electrodynamics
Many-Electron Theory II: Dirac-Hartree-Fock Theory
Elimination of the Small Component
Unitary Transformation Schemes
Relativistic Density Functional Theory
Physical Observables and Molecular Properties
Interpretive Approach to Relativistic Quantum Chemistry
From beginning to end, the authors deduce all the concepts and rules, such that
readers are able to understand the fundamentals and principles behind the theory.
Essential reading for theoretical chemists and physicists. | {"url":"http://www.nanowerk.com/news/newsid=9259.php","timestamp":"2014-04-17T12:56:15Z","content_type":null,"content_length":"32712","record_id":"<urn:uuid:2fbec1b6-19e4-4f37-9dac-9785de342280>","cc-path":"CC-MAIN-2014-15/segments/1397609530131.27/warc/CC-MAIN-20140416005210-00651-ip-10-147-4-33.ec2.internal.warc.gz"} |
What is WAVE SPEED MEASURED IN UNITS?
What units are used to measure the speed of a wave? Meters/second What are the units used to measure speed?, sometimes called "velocity". Others have mentioned certain PARTS of the units used to
measure speed
What units are used to measure the speed of a wave? Meters/second. What is the standard unit of measurement for wave speed? meters per second. Standard unit for the speed of light?
Please vote if the answer you were given helped you or not, thats the best way to improve our algorithm. You can also submit an answer or search documents about what is wave speed measured in units.
Frequency is the number of occurrences of a repeating event per unit time. ... A traditional unit of measure used with rotating mechanical devices is revolutions per minute, ... media in which the
wave speed is independent of frequency), ...
Speed is a measure of how fast something is moving. It is calculated by dividing the distance travelled by the time taken. speed of a wave measures how far one particular crest (or trough) moves in a
given amount of time.
Assuming a sinusoidal wave moving at a fixed wave speed, ... Wavelength is a measure of the distance between repetitions of a shape feature such as peaks, ... "bursts" of wave action where each wave
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Look at the equation lambda*f = c That reads wavelength times frequency gives speed. Speed has units of m/s Wavelength (units of meters for this ... In sound it's measured in htz. Like a sound wave
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Light is measured in many units. ... Or the spectroscopic shift can be converted into the speed of the receeding object, and measured either in kilometers per second, or (because on a galactic scale,
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Science Question;What is the Frequency Of sound wave? What unit is it measured in? I need to no for my science work .. Best Answer Asker's Choice . ... Speed of sound wave in air= Frequency of sound
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One theme of this unit has been that "a wave is a disturbance moving through a medium." There are two distinct objects in this phrase ... If a sound wave (speed = 340 m/s) returns to the camera 0.150
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The nature of a wave was discussed in Lesson 1 of this unit. ... Period, being a time, is measured in units of time such as seconds, hours, days or years. ... Frequency and speed are distinctly
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Wavelength can be defined as the distance between the crests of a wave (most commonly a sound or electromagnetic wave). ... What is the units of measure for wavelength? Meters, usually nanometers (10
^-9). Leave a Comment. Please login to add comments.
Peak 5 or 8 second gust speed (m/s) measured during the eight-minute ... The direction from which the wind waves at the wind wave period (WWPD) are coming. The units are degrees ... (APHA 1980).
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Frequency, wavelength, amplitude and wave speed. You need to know about these quantities used in waves - what they mean, the symbols used for them and the units used to measure them.
What unit is wave frequency measured ChaCha Answer: The standard unit for frequency is the hertz (Hz).
What is the symbol units of wave speed? ChaCha Answer: There is no symbol unit for a wave's speed. It will depend upon how you are me...
When describing waves that have a constant speed, such as sound or light waves, the frequency is inversely proportional to the wavelength. ... Hertz (Hz) is a unit of measure that denotes the number
of times an event occurs in one second.
Frequency is measured by the units of measurement known as Hertz denoted by the symbol Hz ... What is a wave measured in a frequency? ... angular frequency ω (also referred to by the terms angular
speed, radial frequency, circular frequency, orbital frequency, radian frequency, ...
Amplitude is measured in the amount of force applied over an area. The most common unit of measurement of force applied to an area for acoustic study is the Newtons per square meter ... A more
meaningful reference has been developed to measure the average amplitude of a wave over time, ...
How the wavelength of wave is measured? My idea of electromagnetic wave is there is an invisible line traveling through space moving up and down like a sine wave we see on CRO. ... which represent
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Wave speed is the speed at which a wave travels and an important concept in the wave unit covered in Physics. Video explanation on wave speed and example problems. Toggle navigation. math. Algebra;
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units. Speed and velocity are both measured using the same units. The SI unit of distance and displacement is the meter. ... stadium wave 16.447: 59.211: fastest human: ice skating (Jeremy
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... which are massless particles each traveling in a wave-like pattern and moving at the speed of light. Each photon contains a certain amount ... each with a different set of physical units? After
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What units are used to measure wavelength? Welcome New User! ( Create Account ... What units do you measure speed? Get paid for participating. Ask questions ... Wavelength of radio waves can be
measured in metres or centimetres for microwaves.
... (i.e. sound waves or beats) per second. It was named after scientist Heinrich Hertz. Ask a Question ... What Is The Unit Frequency Is Measured By Elaboration? Top Related ... angular frequency ω
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Frequencies are measured in hertz (Hz), ... The relationship between frequency and wavelength is that for a given speed, as the frequency increases, the wavelength decreases. ...
It is standard SI unit to measure intensity of sound or radiation. ... The speed of sound is the distance travelled during a unit of time by a sound wave propagating through an elastic medium. In dry
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Use our free online units converter for wave number. the calculator home page. Home ... and v p is the phase velocity of the wave. The phase velocity is the speed at which one phase of the wave ...
which is to say there are more cycles when we have a higher frequency for the same measured ...
Wavelength frequency conversion sound equation formula lambda audio radio acoustics acoustic sound typical waves wave length measure speed of sound formula for frequency light waves cycle duration
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(scalar) quantity: speed (formula) variable: r units of measure: m/sec ... unit of measure: relationships: length: scalar: L ...
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Wind speed units & wind directions Wind speed Calculator (Enter value in any field and click "Calculate")
The electrical signals can also be used to measure the characteristics of the sound, such as amplitude and frequency. Similarly, hydrophones convert sound in water into electrical signals that can be
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... Throughout this discussion, we measure angles in units of radians. Recall that 2π radians are equivalent to 360°, ... because ω is a rotational speed, it is measured in units of radians per
second. ... c is a constant of proportionality. In some cases, c is the speed of the wave.
If you didn't find what you were looking for you can always try Google Search
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2010 Hysteresis Controller and Delta Modulator- Two Viable Schemes for Current Controlled Voltage Source Inverter Reddy, B V; Chitti Babu, B
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2010 Image Compression Using Discrete Tchebichef Transform Algorithm Senapati, R K; Pati, U C; Mahapatra, K K
2008 Image Encryption by Novel Cryptosystem Using Matrix Transformation Acharya, B; Patra, S K; Panda, G
2008 Image Encryption Using Self-Invertible Key Matrix of Hill Cipher Algorithm Panigrahy, S K; Acharya, B; Jena, D
2005 Image segmentation Using Parallel Tabu Search Algorithm and MRF Model Nanda, P K; Patra, D
2006 Image Segmentation Using Thresholding and Genetic Algorithm Kanungo, P; Nanda, P K; Samal, U C
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2008 Implementation of Taguchi Design for Erosion of Fiber-Reinforced Polyester Composite Systems with SiC Filler Patnaik, A; Satapathy, Alok; Mahapatra, S; Dash, R R
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29-Aug-2009 An Improved Dynamic Response of Voltage Source Inverter using Novel Hysteresis Dead Band Current Controller Chitti Babu, B; Reddy, B V | {"url":"http://dspace.nitrkl.ac.in/dspace/browse-title?starts_with=I","timestamp":"2014-04-16T10:46:22Z","content_type":null,"content_length":"17315","record_id":"<urn:uuid:a73ddf7f-f61f-4238-9463-73af1117fc53>","cc-path":"CC-MAIN-2014-15/segments/1397609523265.25/warc/CC-MAIN-20140416005203-00156-ip-10-147-4-33.ec2.internal.warc.gz"} |
Hangman 1
Re: Hangman 1
_ _ _ M _ _ _ _
Re: Hangman 1
Is there a big T in there?
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Hangman 1
Re: Hangman 1
_ _ _ M _ T _ _
Re: Hangman 1
Is there a very large A in there?
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Hangman 1
Re: Hangman 1
There has to be an R in there.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Hangman 1
: Hangman 1
_ _ _ M _ TR_
Re: Hangman 1
The word is Geometry!
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Hangman 1
_ _ _ _ _ _ _ _ _ _ _ _
Re: Hangman 1
Is there an E in there?
The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
Re: Hangman 1
Is there a T in there?
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Hangman 1
Re: Hangman 1
T _ _ _ _ _ E _ _ _ _ _
Re: Hangman 1
An N?
The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
Re: Hangman 1
How about an M? That is surely in there.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Hangman 1
T _ _ m _ _ E _ _ _ _ N
Re: Hangman 1
The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
Re: Hangman 1
Re: Hangman 1
You capitalized the T, is this a name?
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Hangman 1
Re: Hangman 1
Thomas Edison
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Hangman 1
what the......
_ _ _ _ _ _ _
Re: Hangman 1
Is there a T?
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Hangman 1
_ _ _ _ _ t _
Re: Hangman 1
Can I get an R?
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Hangman 1
No R | {"url":"http://www.mathisfunforum.com/viewtopic.php?pid=241435","timestamp":"2014-04-18T03:12:55Z","content_type":null,"content_length":"32394","record_id":"<urn:uuid:23141eea-d4cd-47f1-b429-98e935cb6c2f>","cc-path":"CC-MAIN-2014-15/segments/1398223202774.3/warc/CC-MAIN-20140423032002-00348-ip-10-147-4-33.ec2.internal.warc.gz"} |
Difference Between Odds Ratio and Relative Risk
Odds Ratio Vs Relative Risk
When two groups are under study or observation, you can use two measures to describe the comparative likelihood of an event happening. These two measures are the odds ratio and relative risk. Both
are two different statistical concepts, although so much related to each other.
Relative risk (RR) is simply the probability or relationship of two events. Let’s say A is event 1 and B is event 2. One can get the RR by dividing B from A or A/B. This is exactly how experts come
up with popular lines like ‘Habitual alcoholic beverage drinkers are 2-4 times more at risk of developing liver problems than non-alcoholic beverage drinkers!’ This means that the likelihood of
variable A which is the risk of developing liver disease for habitual alcoholic beverage drinkers is relative to the same exact risk being talked about for variable B which includes the non-alcoholic
beverage drinkers. In this regard, if you belong to group B and that you are only 10% at risk for dying then it must be true that those from group A are 20-40% more at risk of dying.
The other measure ‘“ odds ratio (OR) is a term that already speaks of what it describes. Instead of using pure percentages (like in RR), OR uses ratio of odds. Take note, OR explains ‘odds’ not in
its colloquial definition (i.e. chance) but rather on its statistical definition which is the probability of an event over (divided by) the probability of a certain event not happening.
A good example is the tossing of a coin. When you happen to land the coin with its tails up 60% of the time (obviously it lands with heads 40% of the time), the odds of tails in your case is 60/40=
1.5 (1.5 times more likely to get tails than heads). But ordinarily, there’s really a 50 percent chance of landing on either heads or tails. So the odds are 50/50=1. So the question is on how likely
this event will not happen compared to it happening. The straightforward answer is that you are just equally likely to get either way. In written formula, with A being the likelihood for group 1
while B being the likelihood for group 2, the formula to get the OR is [A/(1-A)]/[B/(1-B)].
So if the probability of having liver disease among habitual alcoholic beverage drinkers is 20% and among non-alcoholic beverage drinkers is 2% the OR will be = [20%/(1-20%)] / [2%/(2-1%/)]=12.25 and
the RR of having liver disease when drinking alcoholic beverages will be = 20%/2%=10.
The RR and OR often have close results, but in some other situations they have very far numerical values most especially if the risk of occurrence is really very high to begin with. This scenario
gives a high OR while the RR is kept at a minimum.
1. The RR is much simpler to interpret and is most likely consistent with everyone’s intuition. It is the risk of a situation relative (in relation) to exposure. The formula is A/B.
2. OR is a bit more complicated and uses the formula [A/(1-A)]/[B/(1-B)].
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Everyday Situations - Lesh
Lesh, R. (1985). Processes, skills, and abilities needed to use mathematics in everyday situations. Education and Urban Society, 17(4), 439-446.
Lesh criticizes traditional textbooks and teaching methods, saying that their one-step problems rarely exercise students' skills, and that they do not reflect real-life mathematical situations.
Lesh believes that there should be as much--or more--emphasis in the mathematics classroom on understanding mathematical concepts and possible mathematical relationships as on accurate
computation. Students should be taught to recognize situations in which their mathematical skills can be utilized.
Mathematizable situations in the classroom, such as wallpapering a wall or balancing a class budget, cause students to engage in multiple mathematics processes and to learn how mathematical
concepts are related to one another in a useful and meaningful way. Such experiences also require students to talk and think about mathematics with one another and with the teacher.
Direct Quotes:
"A goal of the project was to identify important processes, skills, and understandings that are needed by students to use mathematical ideas in everyday situations. A substantial part of the
project consisted of students working together in small groups on problems that might reasonably occur in the normal lives of the students and their families: balancing a checkbook, planning a
vacation within a budget, wallpapering a room, estimating distances using a map, and so on" (p. 439).
"Getting a collection of isolated concepts in a youngster's head (e.g., measurement, addition, multiplication, decimals, proportional reasoning, fractions, negative numbers) does not guarantee
that these ideas will be organized and related to one another in some useful way; it does not guarantee that situations will be recognized in which the ideas are useful or that they will be
retrievable when they are needed" (pp. 439-440).
"The 'back to basics' movement that is currently influencing many of the nation's schools is often uninformed and misdirected. Results of National Assessment Tests, for example, show that 'Johnny
can add; computation with whole numbers is far from a lost art' (Carpenter et al., 1975: 457). In fact, although there are plenty of students who do poorly on computation problems, results from
most large-scale testing programs show that there are fewer of them today than at any time in the past. Today's youngsters run into difficulty in making inferences, solving problems, evaluating
the reasonableness of results, using references to 'look up' what they need to know, and so on. It is the complex skills, not the basic skills that are deteriorating. What we need is to get back
to complexity, where thinking is required in addition to simply knowing some isolated fact or procedure. In realistic situations in which mathematics is used, question asking, information
gathering, and trial-answer evaluating are often more important than simple answer giving. Real problems usually require more than simple one-step solutions" (p. 441).
[On a problem about individuals' overall performances in sports events, in which multiple pieces of information were given, including scores on running and jumping and qualitative comments from
the coaches, the students] "seemed driven to do some messy calculation with the numbers, sensible or not. In fact, most of the students were quite skillful at the arithmetic of *numbers*, but
many real problems involve the arithmetic of *quantities* -- a skill that mathematics textbook authors tend to assume is covered in science, and science textbook authors tend to assume is covered
in mathematics" (p. 444).
"In problems that require multi-step solution procedures, it is important to *plan* what you are going to do before doing it; *monitor* what you are doing *while* you are doing it; and *evaluate*
the sensibility of your results. These skills are seldom practiced in simple one-step textbook problems... The students who participated in our project did improve in their ability to deal with
problems [like the one described above], which we believe are similar to the kinds they will meet in everyday situations, job situations, or later mathematics courses. To some extent, our
students improved because we worked with them individually on some specific skills: graphing, measuring, estimating, and so forth. They probably also got better on problems that require planning,
organizing, and recording simply because our problems required them to practice and use these skills. Further, they were forced to reorganize their mathematical ideas because our problems usually
involved more than a single concept. The more organized character of their knowledge should benefit them greatly in future mathematics courses" (p. 445).
Carpenter, T. P., T. G. Coburn, R. E. Kays, and J. W. Wilson (1975). Results and implications of the NAEP mathematics assessment: secondary school. Mathematics Teacher, 68, 453-470.
Summary by Maria Ong | {"url":"http://mathforum.org/sarah/Discussion.Sessions/Lesh.html","timestamp":"2014-04-19T22:37:47Z","content_type":null,"content_length":"8135","record_id":"<urn:uuid:1da95fdf-add8-4eb8-92c3-f84180245862>","cc-path":"CC-MAIN-2014-15/segments/1398223202457.0/warc/CC-MAIN-20140423032002-00650-ip-10-147-4-33.ec2.internal.warc.gz"} |
Haskell Code by HsColour
{- |
A design study about how to design signal processors
that adapt to a common sample rate.
I simplified "Synthesizer.Inference.DesignStudy.Arrow" to this module
which uses only Applicative functors.
module Synthesizer.Inference.DesignStudy.Applicative where
import Data.List (intersect)
import Control.Applicative (Applicative(..), liftA3, )
data Rates = Rates [Int] | Any deriving Show
-- it is a Reader monad with context processing
data Processor a = P Rates (Rates -> a)
intersectRates :: Rates -> Rates -> Rates
intersectRates Any y = y
intersectRates x Any = x
intersectRates (Rates xs) (Rates ys) = Rates $ intersect xs ys
instance Functor Processor where
fmap f (P r f0) = P r (f . f0)
instance Applicative Processor where
pure x = P Any (const x)
(P r0 f0) <*> (P r1 f1) =
P (intersectRates r0 r1) (\r -> f0 r (f1 r))
runProcessor :: Processor a -> a
runProcessor (P r f) = f r
-- test processors
processor1, processor2, processor3 :: Processor Rates
processor1 = P (Rates [44100, 48000]) id
processor2 = P Any id
processor3 = P (Rates [48000]) id
process :: Processor (Rates, Rates, Rates)
process = liftA3 (,,) processor1 processor2 processor3
test :: (Rates, Rates, Rates)
test = runProcessor process | {"url":"http://hackage.haskell.org/package/synthesizer-0.0.3/docs/src/Synthesizer-Inference-DesignStudy-Applicative.html","timestamp":"2014-04-21T13:16:29Z","content_type":null,"content_length":"9350","record_id":"<urn:uuid:d2d8bad7-047a-4e39-ad40-08c329a7a08f>","cc-path":"CC-MAIN-2014-15/segments/1397609539776.45/warc/CC-MAIN-20140416005219-00583-ip-10-147-4-33.ec2.internal.warc.gz"} |
Physics Forums - View Single Post - light geodesic path
Quote by
pervect, mine text is red:
You'll need to clarify a bit. It sounds like you are asking for the null geodesics (the paths that light follow) given the metric of space-time (g_ij). not just g_ij However, you need to specify the
velocity as well as its position to calculate the geodesic. exactly, so I said: on step k, we're in x_i, with 'velocity' dx_i (=dt,i=4) ...if I'm understanding it correctly, because while geodesic
deviation will cause light beams to "curve", that curvature is equivalent ot an acceleration, and it won't affect the velocity of light (which after all is always equal to 'c' locally). that is, in
tangent euclidean coordinates, but x_i I am speaking about, are distant static observer ones
(there must be simple iterative solution) | {"url":"http://www.physicsforums.com/showpost.php?p=418890&postcount=3","timestamp":"2014-04-16T13:56:03Z","content_type":null,"content_length":"8312","record_id":"<urn:uuid:5402c702-08cb-44da-bd84-fa42110fc16c>","cc-path":"CC-MAIN-2014-15/segments/1397609523429.20/warc/CC-MAIN-20140416005203-00509-ip-10-147-4-33.ec2.internal.warc.gz"} |
Books have been written on the "definition" of probability. We shall merely note two properties: (a) statistical independence (events must be completely unrelated), and (b) the law of large numbers.
This says that if p[1] is the probability of getting an event in Class 1 and we observe that N[1] out of N events are in Class 1, then we have
A common example of direct probability in physics is that in which one has exact knowledge of a final-state wave function (or probability density). One such case is that in which we know in advance
the angular distribution f (x), where x = cosx[1] in an interval x[1] is Nf (x[1])x[1], where N, the total number of scattered particles, is a very large number. Note that the function f(x) is
normalized to unity:
As physicists, we call such a function a distribution function. Mathematicians call it a probability density function. Note that an element of probability, dp, is | {"url":"http://ned.ipac.caltech.edu/level5/Sept01/Orear/Orear3.html","timestamp":"2014-04-17T18:23:09Z","content_type":null,"content_length":"2906","record_id":"<urn:uuid:2ec8297f-617d-4a59-b09b-87696f2e1bfc>","cc-path":"CC-MAIN-2014-15/segments/1398223205375.6/warc/CC-MAIN-20140423032005-00565-ip-10-147-4-33.ec2.internal.warc.gz"} |
Symmetry and duality in normal basis multiplication
- J. SYMB. COMP , 2000
"... ..."
, 1993
"... Interest in normal bases over finite fields stems both from mathematical theory and practical applications. There has been a lot of literature dealing with various properties of normal bases
(for finite fields and for Galois extension of arbitrary fields). The advantage of using normal bases to repr ..."
Cited by 9 (0 self)
Add to MetaCart
Interest in normal bases over finite fields stems both from mathematical theory and practical applications. There has been a lot of literature dealing with various properties of normal bases (for
finite fields and for Galois extension of arbitrary fields). The advantage of using normal bases to represent finite fields was noted by Hensel in 1888. With the introduction of optimal normal bases,
large finite fields, that can be used in secure and e#cient implementation of several cryptosystems, have recently been realized in hardware. The present thesis studies various theoretical and
practical aspects of normal bases in finite fields. We first give some characterizations of normal bases. Then by using linear algebra, we prove that F q n has a basis over F q such that any element
in F q represented in this basis generates a normal basis if and only if some groups of coordinates are not simultaneously zero. We show how to construct an irreducible polynomial of degree 2 n with
linearly i... | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=1386151","timestamp":"2014-04-18T17:03:32Z","content_type":null,"content_length":"14395","record_id":"<urn:uuid:dbfc8da6-d8b5-469f-8f32-fa814c03bdc4>","cc-path":"CC-MAIN-2014-15/segments/1397609533957.14/warc/CC-MAIN-20140416005213-00566-ip-10-147-4-33.ec2.internal.warc.gz"} |
Magnetic field generated by current in semicircular loop at a point on axis
So, the parametric representation of a point on the semi-circle would be (0, bcos(t), bsin(t)) where b is the radius of the semi-circle.
The vector R is just [d, 0, 0] where d is the distance on the axis of the point
and then the e is the unit vector from R-r
But what's dr? And where does the switch to cylindrical coord come in? | {"url":"http://www.physicsforums.com/showthread.php?t=386396","timestamp":"2014-04-19T04:35:59Z","content_type":null,"content_length":"32519","record_id":"<urn:uuid:94a09b38-552b-4e0e-80d0-76a5d5b737f5>","cc-path":"CC-MAIN-2014-15/segments/1397609535775.35/warc/CC-MAIN-20140416005215-00211-ip-10-147-4-33.ec2.internal.warc.gz"} |
The Universe Adventure
Consequences of the Big Bang
Using the wealth of empirical information from redshift surveys, particle accelerator experiments, and detailed studies of galactic evolution and CMB anisotropy, scientists can predict what types of
conditions must have been present in the early Universe. Combining these parameters with the theoretical framework provided by General Relativity, cosmologists have generated a powerful model capable
of describing both the history and fate of the Universe.
Satellites, such as WMAP, provide scientists with important information about the history and fate of the Universe.
The Age of the Universe
We can estimate the age of the Universe by uncovering the ages of some of the cosmic bodies in the Universe.
• Earth: Using radioactive dating, we have discovered that the approximate age of Earth is 4.2 billion years. So, the Universe must be older than that.
• Stars: We can observe many stars at different ages. We can deduce from this that the oldest stars formed 10 to 12 billion years ago. So, the Universe must be older than that.
Cosmologists estimate the Universe to be 13.7 billion years old. How did they arrive at that?
Accurately Determining the Age of the Universe
Analyzing data from supernovae is one technique scientists use to estimate the age of the Universe.
Cosmologists get a more accurate estimate for the age of the Universe by analyzing the Universe's expansion rate. By studying the history of the expansion rate using redshift data from distant
galaxies and supernovae, we can project the expansion of space back to the beginning of time: the Big Bang. Running the expansion model backwards in this way tells us that the Universe is roughly
13.7 billion years old, by our most accurate estimates. | {"url":"http://universeadventure.org/big_bang/conseq-ageofuniv.htm","timestamp":"2014-04-19T01:51:24Z","content_type":null,"content_length":"8655","record_id":"<urn:uuid:dd29c54f-5c54-4b9c-af72-b96cd675fc81>","cc-path":"CC-MAIN-2014-15/segments/1397609535745.0/warc/CC-MAIN-20140416005215-00432-ip-10-147-4-33.ec2.internal.warc.gz"} |
Interest Rate Problem
November 2nd 2009, 11:58 AM #1
Oct 2009
Interest Rate Problem
If I had $3000 dollars compounded annually for 5 years, what would my interest rate be?
So far I have:
A(1) = 3000( 1 + r/1)^((1)(5))
Then I got:
ln3000 + 5ln( 1+r )
But, now I am unsure on how to go on from this point. Any help would be greatly appreciated! Thanks so much!!
Follow Math Help Forum on Facebook and Google+ | {"url":"http://mathhelpforum.com/calculus/111987-interest-rate-problem.html","timestamp":"2014-04-19T01:09:41Z","content_type":null,"content_length":"28725","record_id":"<urn:uuid:bbb86e5c-86a6-455b-a9f1-dfd434d55c02>","cc-path":"CC-MAIN-2014-15/segments/1397609535535.6/warc/CC-MAIN-20140416005215-00474-ip-10-147-4-33.ec2.internal.warc.gz"} |
Is this notation valid ?
December 5th 2010, 09:32 PM
Is this notation valid ?
Hi all,
let $\mathbb{W}_t$ be a subset of $\mathbb{W}$ for any $t$ (its elements are defined by $t$). If I want to say that if I performed union of every set $\mathbb{W}_t$ with $1 \leqslant t \leqslant
k$, I would obtain $\mathbb{W}$, is it valid notation to write :
$\displaystyle \mathbb{W} = \bigcup_{t=1}^{k} \mathbb{W}_t$
I don't want to use the heavy $\mathbb{W} = \mathbb{W}_1 \cup \mathbb{W}_2 \cup \dots \cup \mathbb{W}_k$ notation, is the shortcut above valid and understandable ?
Thanks all :)
December 6th 2010, 12:09 AM
Hi all,
let $\mathbb{W}_t$ be a subset of $\mathbb{W}$ for any $t$ (its elements are defined by $t$). If I want to say that if I performed union of every set $\mathbb{W}_t$ with $1 \leqslant t \leqslant
k$, I would obtain $\mathbb{W}$, is it valid notation to write :
$\displaystyle \mathbb{W} = \bigcup_{t=1}^{k} \mathbb{W}_t$
I don't want to use the heavy $\mathbb{W} = \mathbb{W}_1 \cup \mathbb{W}_2 \cup \dots \cup \mathbb{W}_k$ notation, is the shortcut above valid and understandable ?
Thanks all :)
Yes. Not only is it valid and understandable, that is the standard notation for what you are trying to do.
December 6th 2010, 12:14 AM
Yes, this notation is fine. As a minor point, the previous discussion should make it clear that t is an integer, not a real.
December 6th 2010, 12:19 AM
Yes, it is an integer, I often forget to add obvious stuff like this although I know I shouldn't :(
Thanks for your replies !! (Nod)
December 6th 2010, 04:11 AM
Rather than say "defined by t", I would say instead "t is the index".
Otherwise, from a practical, everyday working point of view, this is accepted notation, and is commonly found in mathematics, even among logicians and set theorists.
However, from a more strict, albeit pedantic, point of view, the notation is flawed: 'W' is being used in two different and incompatible ways. First it's used to symbolize a certain function,
then it is used to symbolize a particular subset of the union of the range of said function.
Usually, for purposes of mathematical communication, there is no harm in such usage, but from a more strictly formal point of view, the usage is flawed in the way I mentioned.
December 6th 2010, 05:13 AM
Sorry I am confused by the above post what function is 'W' being used to symbolise?
December 6th 2010, 09:32 AM
The function whose domain is some set of indices, of which 1 through k are members, and whose value is Wt for each t in the domain.
December 6th 2010, 11:34 AM
[QUOTE=Bacterius;592350]Hi all,
If I want to say that if I performed union of every set $\mathbb{W}_t$ with $1 \leqslant t \leqslant k$, I would obtain $\mathbb{W}$, is it valid notation to write :
is this what you are talking about? (sorry I am still a bit confused)
December 6th 2010, 06:35 PM
Thanks Moeblee for the detailed reply, I appreciate it :)
Hmmmm, $\mathbb{W}$ would be a non-empty set of natural integers (for instance), and $\mathbb{W}_t$ represents a subset of $\mathbb{W}$, whose elements are chosen as a function of $t \in \mathbb
{N}$. A crude example would be :
Originally Posted by Example
$1 \leq t \leq 3$, and $\mathbb{W} = \{4, 11, 15, 16, 71\}$
$\mathbb{W}_t = \{4, 11, 15\}$ if $t = 1$, $\mathbb{W}_t = \{15, 71\}$ if $t = 2$ and $\mathbb{W}_t = \{4, 16\}$ if $t = 3$.
And I just wanted to know if it was correct (syntaxically speaking) to write :
$\displaystyle \bigcup_{t=1}^{3} \mathbb{W}_t = \mathbb{W}_1 \cup \mathbb{W}_2 \cup \mathbb{W}_3 = \mathbb{W}$
For any $k \in \mathbb{N}$ - for the example $k = 3$.
December 6th 2010, 06:53 PM
Yes, that's all correct. And your use of 'W' in that way is well understood among mathematicians including set theorists. My only point though is that from an extremely technical point of view,
the symbol 'W' should not be used both for the function itself and the union that you mentioned. But, again, that is only from a very technical point of view that is not of concern in ordinary,
everyday mathematical communication.
December 6th 2010, 06:57 PM
MoeBlee I was replying to "hmmmm" :p (that was awkward, no pun intended)
Thanks for clarifying though, I'll make good use of it :)
December 6th 2010, 11:33 PM
yeah i wasnt confused by the notation i was asking moeblee for clarification of the point about W being used in two ways because I didnt understand sorry i think I may have just made the thread a
bit confusing but thanks anyway | {"url":"http://mathhelpforum.com/discrete-math/165432-notation-valid-print.html","timestamp":"2014-04-18T07:14:39Z","content_type":null,"content_length":"17208","record_id":"<urn:uuid:8fd5aaca-0468-4f5e-9c55-5c5dd907a9f8>","cc-path":"CC-MAIN-2014-15/segments/1397609532573.41/warc/CC-MAIN-20140416005212-00262-ip-10-147-4-33.ec2.internal.warc.gz"} |
Accidental release source terms
Environmental Health
Accidental release source terms
December 18, 2013, 2:53 pm
Accidental release source terms are the mathematical equations that quantify the flow rate at which accidental releases of air pollutants into the ambient environment can occur at industrial
facilities such as petroleum refineries, natural gas processing plants, petrochemical and chemical plants, oil and gas transportation pipelines, and many other industrial facilities. Accidental
releases in such facilities may occur through acts of nature (e.g., floods, hurricanes or earthquakes), operational errors, faulty design or inadequate maintenance.
Governmental regulations in a many countries require that the probability of such accidental releases be analyzed and their quantitative impact upon the environment and human health be determined so
that mitigating steps can be planned and implemented.
There are a number of mathematical calculation methods for determining the flow rate at which gaseous and liquid pollutants might be released from various types of accidents. Such calculation methods
are referred to as source terms, and this article on accidental release source terms explains some of the calculation methods used for determining the mass flow rate at which gaseous pollutants may
be accidentally released. Given those mass flow rates, air pollution dispersion modeling studies can then be performed.
Accidental release of a pressurized gas
When gas stored under pressure in a closed vessel is discharged to the atmosphere through a hole or other opening, the gas velocity through that opening may be choked or it may be non-choked. Choked
flow (also referred to as critical flow) is a limiting or maximum condition at which the gas velocity has attained the speed of sound in the gas.
Choked flow occurs when the ratio of the absolute upstream pressure to the absolute downstream pressure is equal to or greater than:
where k is the specific heat ratio of the discharged gas (sometimes called the isentropic expansion factor and sometimes denoted as γ, the Greek letter "gamma").
For many gases, k ranges from about 1.09 to about 1.41, and therefore the expression in equation (1) ranges from 1.7 to about 1.9, which means that choked velocity usually occurs when the absolute
upstream vessel pressure is at least 1.7 to 1.9 times as high as the absolute downstream pressure. In the case of a leak to the ambient atmosphere, the downstream pressure is the atmospheric
When the gas velocity is choked, the equation for the mass flow rate in SI units is:
where the terms are defined as stated below. If the upstream gas density, ρu is not known directly, then it is useful to eliminate it by using the ideal gas law corrected for the real gas
compressibility Z:
For the above equations, it is important to note that although the gas velocity reaches a maximum and becomes choked, the mass flow rate is not choked. The mass flow rate can still be increased if
the upstream pressure is increased or the temperature is decreased.
Whenever the ratio of the absolute upstream pressure to the absolute downstream pressure is less than in expression (1) above, then the gas velocity is non-choked and the equation for mass flow rate
or this equivalent form:
ṁ = mass flow rate, kilograms (kg) per second
C = discharge coefficient, dimensionless (usually about 0.72)
A = discharge hole area, square meters
k = cp ÷ cv = specific heat ratio of the gas
cp = specific heat capacity of the gas at constant pressure
cv = specific heat capacity of the gas at constant volume
ρu = real gas upstream density, kg per cubic meter = (M Pu) ÷ (Z R Tu)
Pu = absolute upstream pressure, Pa
Pd = absolute downstream pressure, Pa
M = the gas molecular mass, kg/kmol (also known as the molecular weight)
R = the universal gas law constant = 8314.472 Pa·m3 ÷ (kmol·K)
Tu = absolute upstream gas temperature, K
Z = the gas compressibility factor at Pu and Tu, dimensionless
The above equations calculate the initial instantaneous mass flow rate for the pressure and temperature existing in the source vessel when a release first occurs. The initial instantaneous flow rate
from a leak in a pressurized gas system or vessel is much higher than the average flow rate during the overall release period because the pressure and flow rate decrease with time as the system or
vessel empties. Calculating the flow rate versus time since the initiation of the leak is much more complicated, but more accurate. A comparison between two methods for performing such calculations
is available online.
The technical literature can be confusing because many authors do not explain whether they are using the universal gas law constant R which applies to any ideal gas or whether they are using Rs which
only applies to a specific individual gas. The relationship between the two is Rs = R/M.
• The above equations are for a real gas.
• For an ideal gas, Z = 1 and ρ is the ideal gas density.
• kmol = 1000 mol
Evaporation of a non-boiling liquid pool
Three different methods of calculating the rate of evaporation from a non-boiling liquid pool are presented in this section. The results obtained by the three methods are somewhat different.
The U.S. EPA method
The following equations are for predicting the rate at which liquid evaporates from the surface of a pool of liquid which is at or near the ambient temperature. The equations were developed by the
U.S. Environmental Protection Agency (U.S. EPA) using units which were a mixture of metric usage and United States usage.
The non-metric units have been converted to metric units for this presentation.
E = ( 0.1268 ÷ T) u0.78 M0.667 A P
E = evaporation rate, kg/min
u = windspeed just above the pool liquid surface, m/s
M = pool liquid molecular mass, dimensionless
A = surface area of the pool liquid, m2
P = vapor pressure of the pool liquid at the pool temperature, kPa
T = pool liquid absolute temperature, K
The U.S. EPA also defined the pool depth as 0.01 meter (m), i.e., one centimeter, so that the surface area of the pool liquid could be calculated as:
A = (pool volume, in m3) ÷ (0.01)
The U.S. Air Force method
The following equations are for predicting the rate at which liquid evaporates from the surface of a pool of liquid which is at or near the ambient temperature. The equations were derived from field
tests performed by the U.S. Air Force with pools of liquid hydrazine.
E = (4.161 × 10– 5) u0.75 TF M (PS ÷ PH)
E = evaporation flux, (kg/min) / m2 of pool surface
u = windspeed just above the liquid surface, m/s
TA = absolute ambient temperature, K
TF = pool liquid temperature correction factor, dimensionless (see equations (a) and (b) below)
TP = pool liquid temperature, °C
M = pool liquid molecular weight, g/mol
PS = pool liquid vapor pressure at ambient temperature, mmHg
PH = hydrazine vapor pressure at ambient temperature, mmHg (see equation (c) below)
(a) If TP = 0 °C or less, then TF = 1.0
(b) If TP > 0 °C, then TF = 1.0 + 0.0043 TP2
(c) PH = 760 exp[ 65.3319 − ( 7245.2 ÷ TA ) − ( 8.22 ln TA ) + ( 6.1557 × 10– 3 ) TA ]
Note: The function ln x is the natural logarithm (base e) of x, and the function exp x is e (approximately 2.7183) raised to the power of x.
Stiver and Mackay's method
The following equations are for predicting the rate at which liquid evaporates from the surface of a pool of liquid which is at or near the ambient temperature. The equations were developed by Warren
Stiver and Dennis Mackay of the Chemical Engineering Department at the University of Toronto.
E = k P M ÷ (R TA)
E = evaporation flux, (kg/s)/m2 of pool surface
k = mass transfer coefficient, m/s (which is taken to be 0.002 u)
TA = absolute ambient temperature, K
M = pool liquid molecular weight, g/mol
P = pool liquid vapor pressure at ambient temperature, Pa
R = the universal gas law constant of 8314.472 Pa·m3 ÷ (kmol·K)
u = windspeed just above the liquid surface, m/s
Evaporation of boiling, cold liquid pool
The following equation is for predicting the rate at which liquid evaporates from the surface of a pool of cold liquid (i.e., at a liquid temperature of about 0 °C or less).
E = ( 0.0001 ) ( 7.7026 − 0.0288 B ) ( M ) e– 0.0077B – 0.1376
E = evaporation flux, (kg/min)/m2 of pool surface
B = pool liquid boiling point at atmospheric pressure, °C
M = pool liquid molecular weight, g/mol
e = 2.7183, the base of the natural logarithm
Adiabatic flash of liquified gas release
Liquified gases such as ammonia or chlorine are often stored in cylinders or vessels at ambient temperatures and pressures well above atmospheric pressure. When such a liquified gas is released into
the ambient atmosphere, the resultant reduction of pressure causes some of the liquified gas to vaporize immediately. This is commonly referred to as "adiabatic flashing" and the following equation,
derived from a simple heat balance, is used to predict how much of the liquified gas is vaporized:
X = 100 ( HuL – HdL ) ÷ ( HdV – HdL )
X = weight percent vaporized
HuL = upstream liquid enthalpy at upstream temperature and pressure, J/kg
HdV = flashed vapor enthalpy at downstream pressure and corresponding saturation temperature, J/kg
HdL = residual liquid enthalpy at downstream pressure and corresponding saturation temperature, J/kg
If the enthalpy data required for the above equation is unavailable, then the following equation may be used:
X = 100 × cp (Tu – Td) ÷ Hv
X = weight percent vaporized
cp = liquid specific heat at upstream temperature and pressure, J/(kg · °C)
Tu = upstream liquid temperature, °C
Td = liquid saturation temperature corresponding to the downstream pressure, °C
Hv = liquid heat of vaporization at downstream pressure and corresponding saturation temperature, J/kg
Note: The words "upstream" and "downstream" refer to before and after the liquid passes through the release opening.
• Editors: D.W. Green and R.H. Perry (1984), Perry's Chemical Engineers' Handbook, 6th Edition, McGraw Hill, ISBN 0-07-049479-7.
• Handbook of Chemical Hazard Analysis Procedures (Appendix B), Federal Emergency Management Agency, U.S. Dept. of Transportation, and U.S. Environmental Protection Agency, 1989. Available online
at Handbook of Chemical Hazard Analysis, Appendix B. Scroll down to page 391 of 520 PDF pages. This handbook also provides the references below:
□ J.H. Clewell (1983), A Simple Method For Estimating the Source Strength Of Spills Of Toxic Liquids, Energy Systems Laboratory, ESL-TR-83-03.
□ G. Ille and C. Springer (1978), The Evaporation And Dispersion Of Hydrazine Propellants From Ground Spill, Environmental Engineering Development Office, CEEDO 712-78-30.
□ J.P. Kahler, R.C. Curry and R.A. Kandler (1980), Calculating Toxic Corridors, Air Force Weather Service, AWS TR-80/003.
• Risk Management Program Guidance For Offsite Consequence Analysis, U.S. EPA publication EPA-550-B-99-009, April 1999. Available online at Guidance for Offsite Consequence Analysis (Appendix D:
Equation D-1 in Section D.2.3 and Equation D-7 in Section D.6)
• Methods For The Calculation Of Physical Effects Due To Releases Of Hazardous Substances (Liquids and Gases), CPR 14E (Yellow Book), Chapter 2, pp. 2.67 - 2.68, The Netherlands Organization Of
Applied Scientific Research (TNO), 2005. Available for free registration and download here
• Calculating Accidental Release Rates From Pressurized Gas Systems From the www.air-dispersion.com website.
• W. Stiver and D. Mackay (1983), A Spill Hazard Ranking System For Chemicals, Proceedings of the Technical Seminar on Chemical Spills, Toronto, Canada, pp. 261-266 .
• W. Stiver and D. Mackay (1983), Evaporation Rates of Chemical Spills, Proceedings of the Technical Seminar on Chemical Spills, Toronto, Canada, pp. 1-8.
Beychok, M. (2013). Accidental release source terms. Retrieved from http://www.eoearth.org/view/article/51cbf2057896bb431f6a78d3 | {"url":"http://www.eoearth.org/view/article/51cbf2057896bb431f6a78d3/?topic=51cbfc78f702fc2ba8129e7d","timestamp":"2014-04-19T22:35:35Z","content_type":null,"content_length":"81832","record_id":"<urn:uuid:7855a9f8-f536-41d0-885b-7c8937f05da3>","cc-path":"CC-MAIN-2014-15/segments/1397609537754.12/warc/CC-MAIN-20140416005217-00171-ip-10-147-4-33.ec2.internal.warc.gz"} |
Re: Where can I find numerical codes for all LATEX Mathematical symbols
From: Will Robertson <wspr81@gmail.com> Date: Fri, 18 Feb 2011 09:40:03 +1030 Message-Id: <9CD376DD-76D1-47EB-8339-DD34A988A5AC@gmail.com> Cc: saf sied <saf_itpro@yahoo.com>, "www-math@w3.org" <
www-math@w3.org> To: David Carlisle <davidc@nag.co.uk>
(Sent quickly. Please excuse brevity.)
On 18/02/2011, at 8:58 AM, David Carlisle <davidc@nag.co.uk> wrote:
> There is some information on latex mappings in the mathml and entity spec sources in the file unicode.xml which may be downloaded from
> http://www.w3.org/2003/entities/2007xml/
> but probably the latex unicode-math pakgae has the most up to date and maintained information:
> http://www.tex.ac.uk/tex-archive/macros/latex/contrib/unicode-math/unimath-symbols.pdf
Note that these are almost entirely defined through the work of the STIX project, so they should be comprehensive, but note
1. The table is mostly a superset of what's available in LaTeX + amsmath + others, but if a font has symbols that didn't make it into unicode yet (e.g., the mnsymbol package) those symbols cannot be listed.
2. Some different font packages in LaTeX will give different names to the same symbol.
3. You can only use all of the symbols defined above if you use unicode-math, which only runs on LaTeX with the XeTeX or LuaTeX engine; hence it's a little experimental at this stage.
Received on Thursday, 17 February 2011 23:07:58 GMT
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Math Forum: Problem of the Week
On the brick "Women's Walkway" from the intersection of 33rd and Chestnut to the intersection of 34th and Walnut in Philadelphia, I became fascinated with the pattern of the bricks.
Here is an area that I'm filling with the pattern:
I thought it was economical how some bricks were cut in half and some of the halves were also cut in half to neatly fit to the edges.
If the top faces of the uncut bricks are 2 units by 1 unit,
1. how many bricks would it take to cover the full area of the pattern shown above?
2. what is the area?
[Note: If you are interested, there is a web page about the Women's Walkway project.] | {"url":"http://mathforum.org/pd/process/walkway.html","timestamp":"2014-04-19T08:23:16Z","content_type":null,"content_length":"2673","record_id":"<urn:uuid:9116445e-b545-465d-af25-24d004bab99c>","cc-path":"CC-MAIN-2014-15/segments/1398223203841.5/warc/CC-MAIN-20140423032003-00597-ip-10-147-4-33.ec2.internal.warc.gz"} |
Cryptology ePrint Archive: Report 2006/020
Scrambling Adversarial Errors Using Few Random Bits, Optimal Information Reconciliation, and Better Private CodesAdam SmithAbstract: When communicating over a noisy channel, it is typically much
easier to deal with random, independent errors with a known distribution than with adversarial errors. This paper looks at how one can use schemes designed for random errors in an adversarial
context, at the cost of relatively few additional random bits and without using unproven computational assumptions.
The basic approach is to permute the positions of a bit string using a permutation drawn from a $t$-wise independent family, where $t=o(n)$. This leads to two new results:
1. We construct *computationally efficient* information reconciliation protocols correcting $pn$ adversarial binary Hamming errors with optimal communication and entropy loss $n(h(p)+o(1))$ bits,
where $n$ is the length of the strings and $h()$ is the binary entropy function. Information reconciliation protocols are important tools for dealing with noisy secrets in cryptography; they are also
used to synchronize remote copies of large files.
2. We improve the randomness complexity (key length) of efficiently decodable capacity-approaching "private codes" from $\Theta(n\log n)$ to $n+o(n)$.
We also present a simplified proof of an existential result on private codes due to Langberg (FOCS '04).
Category / Keywords: cryptographic protocols / Information reconciliation, fuzzy cryptography, error-correcting codes, private codes, information theory, derandomization, combinatorial cryptography
Date: received 17 Jan 2006, last revised 23 Jan 2006Contact author: adam smith at weizmann ac ilAvailable format(s): Postscript (PS) | Compressed Postscript (PS.GZ) | PDF | BibTeX Citation Version:
20060123:135618 (All versions of this report) Discussion forum: Show discussion | Start new discussion[ Cryptology ePrint archive ] | {"url":"http://eprint.iacr.org/2006/020/20060123:135618","timestamp":"2014-04-16T19:13:25Z","content_type":null,"content_length":"3430","record_id":"<urn:uuid:d554ab1b-dd94-4d9d-80f7-deb360428a86>","cc-path":"CC-MAIN-2014-15/segments/1397609524644.38/warc/CC-MAIN-20140416005204-00231-ip-10-147-4-33.ec2.internal.warc.gz"} |
Csáki, Endre - Alfréd Rényi Mathematical Institute of the Hungarian Academy of Sciences
• Strong limit theorems for a simple random walk on the 2-dimensional comb
• Heavy points of a d-dimensional simple random walk
• Strong approximations of threedimensional Wiener sausages AKI 1 and Yueyun HU 2
• Almost sure limit theorems for the maximum of stationary Gaussian sequences
• C2 Z2 x
• Frequently visited sets for random walks Endre Csaki # Antonia Foldes + Pal Revesz # Jay Rosen # Zhan Shi
• Publ. Math. Debrecen Manuscript (June 15, 2009)
• On Vervaat and Vervaat-error type processes for partial sums and renewals
• On the behavior of random walk around heavy points Endre Cski 1
• Random walk local time approximated by a Wiener sheet combined with an independent Brownian motion
• ON THE NUMBER OF CUTPOINTS OF THE TRANSIENT NEAREST NEIGHBOR
• On the increments of the principal value of Brownian local time
• On the local time of the asymmetric Bernoulli walk Dedicated to Professor Sndor Csrg on his sixtieth birthday
• Publ. Math. Debrecen Manuscript (June 15, 2009)
• Strong approximations of three-dimensional Wiener sausages Endre CSAKI1 and Yueyun HU2
• X0 = 0, X1, X2, . . . Ei := P(Xn+1 = i + 1 | Xn = i) = 1 -P(Xn+1 = i -1 | Xn = i)
• Lengths and heights of random walk Endre Cs aki 1+ and Yueyun Hu 2
• A joint functional law for the Wiener process and principal value
• ! #"%$ &$'( ) 0 $'132 4658789A@CB&D8EFHGPIRQTSVU EFXW`Y E@aE@3DTb QdcfeHgihqp F 5sr p I
• AN VINCZE (1912{1999) AND HIS CONTRIBUTION TO
• Maximal Local Time of a d-dimensional Simple Random Walk on Subsets.
• Lengths and heights of random walk Endre Csaki1 and Yueyun Hu2
• Strong limit theorems for a simple random walk on the 2-dimensional comb
• Large void zones and occupation times for coalescing random walks
• Strong approximations of three-dimensional Wiener sausages Endre CS'AKI1 and Yueyun HU2
• On Vervaat and Vervaat-error type processes for partial sums and renewals
• Pointwise and uniform asymptotics of the Vervaat error process
• Boundary Crossings and the Distribution Function of the
• On a class of additive functionals of two-dimensional Brownian motion and random walk
• On the ranked excursion heights of a Kiefer process by
• On the local time of random walk on the 2-dimensional comb
• Joint asymptotic behavior of local and occupation times of random walk in higher dimension
• On the ranked excursion heights of a Kiefer process AKI 1 and Yueyun HU 2
• Strong approximations of additive functionals of a planar Brownian motion
• Frequently visited sets for random walks Endre Csaki
• Long excursions of a random walk Endre Cs'aki*, P'al R'ev'esz* and Zhan Shi
• On the joint asymptotic behaviours of ranked heights of Brownian excursions
• On the increments of the principal value of Brownian local time
• Increment sizes of the principal value of Brownian local time
• A universal result in almost sure central limit theory Istv'an Berkes*, Endre Cs'aki
• Maximal Local Time of a d-dimensional Simple Random Walk on Subsets.
• Almost sure limit theorems for the maximum of stationary Gaussian sequences
• ISTVA'N VINCZE (1912-1999) AND HIS CONTRIBUTION TO
• On the increments of the principal value of Brownian local time
• Large void zones and occupation times for coalescing random walks
• Fields Institute Communications Volume 00, 0000
• A joint functional law for the Wiener process and principal value
• Almost Sure Limit Theorems for Sums and Maxima from the Domain of Geometric Partial Attraction of Semistable Laws
• Fractional Brownian motions as "higher-order" fractional derivatives of Brownian local times
• A strong invariance principle for two-dimensional random walk in random scenery
• Fields Institute Communications Volume 00, 0000
• Favourite sites, favourite values and jumping sizes for random walk and Brownian motion
• Strong approximations of additive functionals of a planar Brownian motion
• Asymptotic properties of ranked heights in Brownian excursions
• Frequently visited sets for random walks Endre Cs'aki* Ant'onia F"oldesy P'al R'ev'esz* Jay Rosenz Zhan Shi
• Heavy points of a d-dimensional simple random walk
• Lengths and heights of random walk Endre Cs'aki1y and Yueyun Hu2
• Strong limit theorems for anisotropic random walks on Z 2 Endre Cski 1
• Z2 Z2 | {"url":"http://www.osti.gov/eprints/topicpages/documents/starturl/01/483.html","timestamp":"2014-04-20T12:37:28Z","content_type":null,"content_length":"16605","record_id":"<urn:uuid:7b7dc9ea-ea82-4378-ba46-16cf81217914>","cc-path":"CC-MAIN-2014-15/segments/1397609538423.10/warc/CC-MAIN-20140416005218-00565-ip-10-147-4-33.ec2.internal.warc.gz"} |
Linear stability, transient growth and the role of viscosity stratification in compressible Couette flow
Malik, M and Dey, J and Alam, Meheboob (2008) Linear stability, transient growth and the role of viscosity stratification in compressible Couette flow. In: Int. Conf. Aerospace Science and
Technology, NAL, Bangalore.
Full text not available from this repository.
Linear stability and the nonmodal transient energy growth in compressible plane Couette flow are investigated for two prototype mean flows: (a) the uniform shear flow with constant viscosity, and (b)
the nonuniform shear flow with stratified viscosity. Both mean flows are linearly unstable for a range of supersonic Mach numbers (M). For a given M, the critical Reynolds number (Re) is
significantly smaller for the uniform shear flow than its nonuniform shear counterpart; for a given Re, the dominant instability (over all streamwise wave numbers, α) of each mean flow belongs to
different modes for a range of supersonic M. An analysis of perturbation energy reveals that the instability is primarily caused by an excess transfer of energy from mean flow to perturbations. It is
shown that the energy transfer from mean flow occurs close to the moving top wall for “mode I” instability, whereas it occurs in the bulk of the flow domain for “mode II.” For the nonmodal transient
growth analysis, it is shown that the maximum temporal amplification of perturbation energy, Gmax, and the corresponding time scale are significantly larger for the uniform shear case compared to
those for its nonuniform counterpart. For α=0, the linear stability operator can be partitioned into L∼L̅ +Re2 Lp, and the Re-dependent operator Lp is shown to have a negligibly small contribution to
perturbation energy which is responsible for the validity of the well-known quadratic-scaling law in uniform shear flow: G(t∕Re)∼Re2. In contrast, the dominance of Lp is responsible for the
invalidity of this scaling law in nonuniform shear flow. An inviscid reduced model, based on Ellingsen-Palm-type solution, has been shown to capture all salient features of transient energy growth of
full viscous problem. For both modal and nonmodal instability, it is shown that the viscosity stratification of the underlying mean flow would lead to a delayed transition in compressible Couette
Actions (login required) | {"url":"http://eprints.iisc.ernet.in/40973/","timestamp":"2014-04-17T13:26:25Z","content_type":null,"content_length":"22565","record_id":"<urn:uuid:59f98cfb-4849-4ab8-b3b5-39f039c2d0c9>","cc-path":"CC-MAIN-2014-15/segments/1397609530131.27/warc/CC-MAIN-20140416005210-00184-ip-10-147-4-33.ec2.internal.warc.gz"} |
Shannon and Maxine work in the same building and leave work
Question Stats:
68%31% (05:08)based on 47 sessions
alexpavlos wrote:
Shannon and Maxine work in the same building and leave work at the same time. Shannon lives due north of work and Maxine lives due south. The distance between Maxine's house and Shannon's house is 60
miles. If they both drive home at the rate 2R miles per hour, Maxine arrives home 40 minutes after Shannon. If Maxine rider her bike home at the rate of R per hour and Shannon still drives at a rate
of 2R miles per hour, Shannon arrives home 2 hours before Maxine. How far does maxine live from work?
A. 20
B. 34
C. 38
D. 40
E. 46
Here is the ratios approach to the problem:
Shannon drives at the speed of 2R in both the cases so she takes the same time. In the first case Maxine reaches home 40 mins after Shannon. In the second case, Maxine reaches 2 hrs after Shannon.
Why did Maxine take 1 hr 20 mins extra in the second case? Because she drove at half the speed.
Speed1: Speed 2 = 2:1
Time 1: Time 2 = 1:2 ( since distance stays the same)
The difference between Time1 and Time 2 is 1 hr 20 mins = 80 mins. So Time 1 must be 1hr 20 mins i.e. time taken by Maxine when she drives at speed 2R. Time taken by Shannon must be 1 hr 20 mins - 40
mins = 40 mins (because she reaches 40 mins early)
When their speeds were same in the first case,
Time taken by Maxine : Time taken by Shannon = 80 mins :40 mins = 2:1
Distance traveled by Maxine : Distance traveled by Shannon = 2:1
Total distance is 60 miles so Maxine lives 40 miles away and Shannon lives 20 miles away from office.
You mentioned time difference between time1 and time 2 is 80 mins... How dod you take the value for time1 as 80 mins?
Please explain the basis. | {"url":"http://gmatclub.com/forum/shannon-and-maxine-work-in-the-same-building-and-leave-work-132625.html?fl=similar","timestamp":"2014-04-16T04:24:55Z","content_type":null,"content_length":"185341","record_id":"<urn:uuid:756ae90d-e50f-4ddd-9b59-606647bd40bc>","cc-path":"CC-MAIN-2014-15/segments/1397609521512.15/warc/CC-MAIN-20140416005201-00138-ip-10-147-4-33.ec2.internal.warc.gz"} |
Proving hyperbolic identity Calculus Beyond Question
Hi thomas49th!
Originally Posted by thomas49th
Errr sorry I don't follow :S. I've only just started hyperbolics today and havn't used an imaginary numbers with them yet?
Thanks :)
In that case, all you need to know is that "hyperbolic" trig functions cosh sinh tanh sech coth and cosech work
the same as ordinary trig functions (for example, sinh(2x) = 2sinhx coshx), but occasionally you get a + instead of a minus (or vice versa) … I think only when you have
But, to be on the safe side, use
's method! | {"url":"http://www.allquests.com/question/4070040/Proving-hyperbolic-identity.html","timestamp":"2014-04-20T23:27:03Z","content_type":null,"content_length":"28150","record_id":"<urn:uuid:6a3b94ec-4589-4315-8028-6992da738467>","cc-path":"CC-MAIN-2014-15/segments/1397609539337.22/warc/CC-MAIN-20140416005219-00214-ip-10-147-4-33.ec2.internal.warc.gz"} |
Guttenberg, NJ Algebra 1 Tutor
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Maximal number of maximal subgroups
up vote 14 down vote favorite
Let $G$ be a finite group. I want to find an upper bound on the number of the maximal subgroups. My questions is does it possible to prove that the number of maximal subgroups of any finite group $G$
is at most $|G|^{100}$?
One can easily find that any subgroup is generated by at most $\log|G|$ elements thus the number of subgroups(in particular maximal subgroups) is at most $|G|^{\log|G|}$. Does it possible to improve
this upper bound. For an abelian group the number of maximal subgroups is at most $|G|$ and in fact I do not know any example where the number of maximal subgroups is more than $|G|$.
I am almost sure that I am not the first who is asking this question I would like to know if the answer to this question is known or either this is a hard question.
There's a load of good stuff on this here: aimath.org/news/wallsconjecture/wall.conjecture.pdf – Nick Gill Dec 7 '12 at 15:46
add comment
2 Answers
active oldest votes
Another extended comment. The best bound on the number of subgroups is $|G|^{(1/4+o(1))\log_2 |G|}$ proved in http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.100.913&rep=
rep1&type=pdf by Borovik, Pyber and Shalev.
For the number of maximal solvable subgroups they get $|G|^c$ for some constant. They don't estimate c but conjecture it is 1.
up vote 14 down Added. The survey László Pyber, Asymptotic results for simple groups and some applications. Groups and computation, II (New Brunswick, NJ, 1995), 309–327, DIMACS Ser. Discrete Math.
vote accepted Theoret. Comput. Sci., 28, Amer. Math. Soc., Providence, RI, 1997 claims there is a bound of the form $|G|^c$ for any group G on maximal subgroups but c is not described. It is
conjectured c=1 will do.
Added. Jesse in the comments below cites a more recent and better bound of $c|G|^{3/2}$. I am making this CW to not get credit for his answer.
6 In the paper Martin W. Liebecka, Laszlo Pyberb, Aner Shalev: On a conjecture of G.E. Wall. Journal of Algebra, 317, no.1, p. 184-197 (2007) they give an upper bound of $c|G|^{3/2}
$ for some absolute constant $c$. – Jesse Peterson Aug 15 '12 at 17:33
1 @Jesse, that should be an answer. – Benjamin Steinberg Aug 15 '12 at 17:54
3 There's no need to make this a CW. I learned of Wall's conjecture from Feng Xu who has considered generalizations for subfactors in a couple of papers. He asked a related question
about this in April: mathoverflow.net/questions/94553/… – Jesse Peterson Aug 15 '12 at 18:59
1 NB There's a typo in the reference - that should be Martin Liebeck. – HJRW Aug 15 '12 at 20:51
1 There is a book called subgroup lattices. Google it. – Benjamin Steinberg Aug 16 '12 at 13:46
show 5 more comments
The document I linked to above is sufficiently striking as to warrant an answer of its own. I hope it complements the community wiki above.
As mentioned above the relevant conjecture in this area is due to Wall:
Conjecture The number of maximal subgroups of a finite group $G$ is less than the order of $G$.
This has been the subject of much study with the landmark work (until recently) being the above-cited work of Liebeck, Pyber and Shalev. In addition to the result mentioned above they
show that the conjecture is true if the group $G$ is simple, up to a finite number of exceptions.
Now a quote from the linked document is relevant:
This largely directed attention to composite groups, where Wall in his original paper had at least shown the conjecture to be true for finite solvable groups. The key remaining cases
were known to be semidirect products of a vector space V with a nearly simple finite group G acting faithfully and irreducibly on it.
up vote 8
down vote It turns out that in this case Wall's conjecture implies some bounds on the cohomology groups $H^1(G,V)$. And, as the document relates, examples have now been found which violate these
bounds. In particular, Wall's conjecture does not hold.
In light of this development, the bound $C|G|^{3/2}$ mentioned above, also due to Liebeck, Pyber and Shalev, assumes greater importance. Although, as the linked document mentions, it is
likely that the value $3/2$ can be reduced a great deal.
One final interesting quote:
A conjecture of Aschbacker and Guralnick, not made at the conference... would now rise to be the main conjecture in maximal subgroup theory. (The conjecture states that it is the
number of conjugacy classes of maximal subgroups that is bounded, less than the number of conjugacy classes of elements in the group.)
Anyone interested should definitely read this document. Not only is it interesting mathematically, it's a very engaging account of how this recent breakthrough was achieved.
Your link was broken. I think I fixed it (that is it links to something now, though it's possible you intended to link to something else on that site). – Noah Snyder Dec 7 '12 at 22:26
Fantastic answer! I hadn't heard about this breakthrough, though like Jesse I've been interested in Wall's conjecture from the subfactor side. – Noah Snyder Dec 7 '12 at 22:36
@Noah, thanks for fixing the link (you got the right one) and for the positive feedback - appreciated. nick – Nick Gill Dec 10 '12 at 16:39
add comment
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Mathematics Concept & Skill Development Lecture Road Safety Messages for All: When walking on a road, when is it
Series: Webvideo consolidation of site lessons and www.whyslopes.com >> Work and Study Tips >> H Jigsaw puzzles and problem solving safer to be on the side allowing one to see oncoming traffic?
lesson ideas in preparation. Price to be Next: [H more - Routine to non-routine problem solving.] Previous: [G. Written
determined. work formats for developing and showing skill.] [1] [2] [3] [4] [5] [6] [7] [8] Play with this [unsigned]
[9] [10][11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
Bright Students: Top universities want you. While Complex Number Java Applet
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Algebra Word Problems
7 Complex Numbers Appetizer. Return to Page Top
If your interest is in solving algebra word problems at the high school level, I
What is or where is the square root of -1. With would recommend learning how to solve linear equations in several unknowns in an
rectangular and polar coordinates, see how to add, effortless fashion. High school students who can solve linear equations in one
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plane. The visual or geometric approach here known eliminated to formulate a single equation in one unknown quantity to solve. The
in various forms since the 1840s, demystifies the trick here is to draw or extract a single equation from the given information.
square root of -1 and the associated concept of But in most such words problems, it is easier to extract or draw from the given
"imaginary" numbers. Here complex number information several linear equations in several unknowns to solve. Each sentence
multiplication illustrates rotation and dilation in the word problem gives an equation in one or more unknowns or quantities. Now
operations in the plane. the algebraic way of writing and thinking can be used to eliminate variables and
to solve for the one or more quantities of interest in an effortless fashion.
The algebraic solution of linear equations involves the elimination of variables
Geometric Notions with Ruler & Compass to obtain say one equation in one unknown. This elimination process may be
Constructions better done and recorded with algebraic notation. Going directly to one equation
in one unknown to solve a problem requires more work to be done with words.
1 Initial Concepts & Terms 2 Angle, Vertex & Side www.whyslopes.com >> Work and Study Tips >> H Jigsaw puzzles and problem solving
Correspondence in Triangles 3 Triangle Isometry/ Next: [H more - Routine to non-routine problem solving.] Previous: [G. Written
Congruence 4 Side Side Side Method 5 Side Angle work formats for developing and showing skill.] [1] [2] [3] [4] [5] [6] [7] [8]
Side Method 6 Angle Bisection 7 Angle Side Angle [9] [10][11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
Method 8 Isoceles Triangles 9 Line Segment
Bisection 10 From point to line, Drop Return to Page Top
Perpendicular 11 How Side Side Side Fails 12 How
Side Angle Side Fails 13 How Angle Side Angle
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Integral Help (Check if I'm doing these right)
December 13th 2007, 11:44 AM #1
Integral Help (Check if I'm doing these right)
Use the form of the definition of the integral given in the Properties of Integrals to evaluate integrals below:
1. $\int\limits_1^4{(x^2+2x-5)dx}$
I then get:
$\int\limits_1^4{(x^2)dx} + \int\limits_1^4{(2x)dx} - \int\limits_1^4{(5)dx}$
I do the calculations and end up with
$\frac{n(n+1)(2n+1)}{n^3} (\frac{27}{6})$
Take up the limit as n -> infinite and get
$\int\limits_1^4{(x^2+2x-5)dx} = \frac{-1}{2}$
2. $\int\limits_0^5{(1+2x^3)}dx$
basically I did the same thing for this problem and end up with
$\int\limits_0^5{(1+2x^3)}dx = 1$
Here's another problem that I quite don't understand.
3. Find the Riemann sum for $f(x) = sinx$ , $0 \le x \le \frac{3pi}{2}$
with six terms, taking he saple points to be right endpoints. Give your answer correct to six decimal places.
There are several other parts to this particular question but I think I can manage to answer those if I find out how to do this part.
No, I am sorry to say, you are not doing them right.
What exactly are you trying to do?. I see a possible relationship to a Riemann sum there.
By the right endpoint method:
$\frac{27}{n^{3}}\sum_{k=1}^{n}k^{2}+\frac{36}{n^{2 }}\sum_{k=1}^{n}k-\frac{6}{n}\sum_{k=1}^{n}1$
Now, use the identities you learned. I see you used one in your post(the sum of the squares). Then, take the limit as $n\rightarrow{\infty}$.
You're shooting for 21, not -1/2.
Try the same technique on this one, if that's what you need to do.
No, I am sorry to say, you are not doing them right.
What exactly are you trying to do?. I see a possible relationship to a Riemann sum there.
By the right endpoint method:
$\frac{27}{n^{3}}\sum_{k=1}^{n}k^{2}+\frac{36}{n^{2 }}\sum_{k=1}^{n}k-\frac{6}{n}\sum_{k=1}^{n}1$
Now, use the identities you learned. I see you used one in your post(the sum of the squares). Then, take the limit as $n\rightarrow{\infty}$.
You're shooting for 21, not -1/2.
Try the same technique on this one, if that's what you need to do.
Hmm... I see what I have done wrong. Thank you for your help. I still need help with the last question though which I added while you were posting.
when using K, K = any number in that interval right?
Break sin(x) up into 6 partitions over the designated limits
That's the width of each rectangle. Use that in your sine to find the heighths of the rectangles. The area of a rectangle is width times height.
Add them up.
$sin(\frac{\pi}{4})+sin(\frac{\pi}{2})+......+sin(\ frac{3\pi}{2})$
Then multiply that by your width of $\frac{\pi}{4}$
Here's a graph:
Last edited by galactus; November 24th 2008 at 05:38 AM.
I'm having problems simplifying this:
$f(x_{i}) = \frac{5}{n} [1 + 2(1 + \frac{5i}{n})^3]$
any help?
$f(x_{i}) = \frac{5}{n} [1 + 2(1 + \frac{5i}{n})^3]$
$= \frac{5}{n} + \frac{10}{n}(1 + \frac{5i}{n})^3$
I get to that point but the cube, or shall I say the 1 +, is bothering me. Is there any way to simplify it or something?
This is my solution to the second one.
${\Delta}{x} = {\frac{5}{n}}$ and $x_i = \frac{5i}{n}$
$f(xi) = [ 1 + 2(\frac{5i}{n})^3] \frac{5}{n}$
I simplify and get
$= [\frac{15}{n} + \frac{625i^3}{n^4}$
Do the stuff with the sigma notation and get
$\frac{15}{n} + \frac{625}{2}(\frac{n(n+1)}{n4})^2$
I take the limit of that and get $\frac{625}{2}$
Looks like you need to add 5.
Then you get:
$\frac{1250}{n^{4}}\sum_{k=1}^{n}k^{3}+\frac{5}{n}\ sum_{k=1}^{n}1$
= $\frac{625}{n}+\frac{625}{2n^{2}}+\frac{635}{2}$
Now, see the limit?.
Looks like you need to add 5.
Then you get:
$\frac{1250}{n^{4}}\sum_{k=1}^{n}k^{3}+\frac{5}{n}\ sum_{k=1}^{n}1$
= $\frac{625}{n}+\frac{625}{2n^{2}}+\frac{635}{2}$
Now, see the limit?.
hmm. yeah I got to go over and make sure I didn't make the same mistake with other problems. Thanks
December 13th 2007, 12:08 PM #2
December 13th 2007, 12:11 PM #3
December 13th 2007, 12:57 PM #4
December 13th 2007, 03:30 PM #5
December 13th 2007, 04:05 PM #6
December 13th 2007, 04:36 PM #7
December 13th 2007, 04:49 PM #8
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The Soft Error Experts
The business of System Reliability : Defining the characteristics of our best target market with a simple cost model.
Supply chain managers have developed tools to monitor the supply of components for large-scale systems, including parameters like lead time, single source of supply, potential replacement parts,
price, INCOTERM in their analyses. They usually work with component engineers to gather the data needed for their management tools. Reliability has historically been factored into the design (life
duration) and the maintenance program mostly for deterministic ageing effect creating a slow drift in performance. Another type of reliability issue is unexpected, random failure like SEEs that can
occur at anytime, anywhere in the system, causing potential catastrophic failures. Because of the difficulty of modelling and predicting SEEs, they are not necessarily well analyzed as they are
sometimes like the needle in the haystack for large systems. When components show this type of reliability issues in the field, the information is supposed to be fed back to the supply chain’s
monitoring system, which organizes repairs and recalls, and design change request if needed. These operations often come at a large cost to the system vendor, both internal and as contract penalties.
A simplified cost estimation model of this risk and the associated costs can be represented below:
Total cost is:
C = C1 + P[1]*C3 + C4
Where: C1 represents the cost associated to development, implementation, fabrication, sales, etc. C3 represents the cost of repair or recall of the product. C4 represents the cost of maintenance.
Adding the capability of assessing and correcting SEEs before shipment drastically lowers the risk of failure in the field. It modifies the cost structure as shown below:
The total cost would then be:
C[R] = C1 + C2 + P[2]*C3 + C4 (2)
Where: C2 is the overall cost of improving the reliability (analysis, test, mitigation, ...). P2 is the probability of a failure given the result (and recommendations) of the reliability audit.
Statistically speaking, this probability follows Bayesian statistics.
The difference of cost between the two approaches would then be:
ΔC = C[R] – C = C2 + C3*(P[2] - P[1]) (3)
ΔC needs to be negative in order for reliability audit to make business sense and generate positive Return On Investment. Note that if reliability analysis is performed well, the repair cost C3
should be much lower than in the previous case. For the sake of simplicity, we'll assume that C3 is the same (worst case).
ΔC<0 nbsp="" o:p="">0>
The conditions for this to happen are clear from (3), and define our target markets and our offering:
· C3 is large. It corresponds to certain applications and industries. Our experience of cases where this cost can be prohibitive: aerospace, medical devices and cloud infrastructure
· P2 << P1: this condition is achieved mainly with accurate analysis tools, deep knowledge and expertise in the field and effective mitigation strategies
· C2 is as small as possible: C2 has two components, the cost of analysis and the cost of mitigation. C2 depends on the stage where the problem is audited, obviously the earlier the cheaper.
Therefore, for the target markets for which cost of failure is prohibitive, we have to bring enough expertise to significantly lower the probability of failure through accurate analysis and effective
mitigation, and our intervention should be as early as possible in the design phase. These statement are very helpful when defining our product portfolio and the type of data and engagement that
we'll be seeking.
if you find these thoughts interesting, or you'd like to react to this blog, let us know! Your comments are welcome as usual!
DAC 2012: Interview with EE Times
DAC was quite busy this year at the Moscone center in San Francisco.
It was a good way to test market demand for Soft Error solutions, or at least the interest of different industries about this reliability problem. Whereas many are aware of the problem, more than
ever see it as a concern and try to be proactive about it. Summary of concerned markets is shown in this snapshot taken from the graphics on our booth:
We've also been interviewed by EETimes online TV. You can see the it here.
The Australian Transportation Safety Board (ATSB) released on December 19^th 2011 the final report of investigations of two repeated nose dive incidents during a Qantas Airline Airbus A330 flight
from Singapore to Perth in October 2008 (Qantas Flight 72) resulting in an accident. The plane landed at a nearby airport after the incident which caused at least 110 injured passengers/crewmembers
and some damages to the inside of the aircraft.
Here’s the full report: http://www.atsb.gov.au/media/3532398/ao2008070.pdf
See section 3.6.6 page 143 for discussion about Single Event Effect and Appendix H.
The report is inconclusive about the root cause of the incident. Its origin occurred in an avionic equipment called the TLN101 ADIRU (Air Data Inertial Reference Unit).
Incorrect data for all the flight parameters were sent by this unit to the other avionics equipments, eventually creating a false angle of attack information misleading the central computer that
reacted with a quick nose down maneuver.
The report mentions that probably the wrong signal came from the CPU (Intel 80960MC) inside the ADIRU. Other chips interacting with the CPU and therefore potentially sending wrong signals are an ASIC
from AMIs, wait state RAM from Austin Semiconductor (64kx4) and RAM from Mosaic Semiconductor (128kx8).
When skimming through the report and especially the section about SEE, I had few thoughts:
The estimated SEU failure rate of the equipment is 1.96 e-5 per hour, or 19.600 FIT (note: none of the memories were protected by ECC or EDAC). At an altitude of 37.000 ft the neutron acceleration
factor compared to ground reference (NYC) is 83x (report data), therefore the equivalent FIT at ground level should be 236 FIT. The order of magnitude seems about right, even though I’d like to have
more data about the process node and total size of embedded memory. This FIT rate is just an estimate (from theory, not from test) and seems to take only memory SBU (Single Bit upset) into account.
The investigators couldn’t reproduce the symptoms through test. They focused mainly on neutron testing at 14MeV. I imagine this is because it was a source that they could access easily. Maybe a wider
neutron range up to hundreds of MeV (like white neutron spectrum at Los Alamos, TSL or Triumf) would have been more appropriate, especially to create MCU (Multi Cell Upsets). The report states that
the rate MCU/SBU is about 1%, so they didn’t investigate further. This depends on the process node! At latest technologies (40nm, 28nm) this ratio can be up to 40% on SRAM.
The components seemed to have been manufactured with older process nodes. But as such, did they check the effect of thermal neutron (Boron 10 was used in older technologies)? Of alpha particles
contamination of the package?
I believe that this report needs a little more details on the issue, a little more investigation to try to be more conclusive….Any thoughts and comments?
Every second the equivalent of 63 billion CDs of data transit through the world’s internet (source: Cisco). That’s 1.5ZB per year (1ZB= 270 B!). As of December 31st 2011, 2.3 billion persons are
using the internet, a 5.3X increase from the previous year! (source: internetworldstat.com). As lifestyle in almost every country in every continent is moving towards ubiquitous mobile lifestyle,
demand for remote mass storage and cloud computing capacity is rapidly increasing. These numbers are mindboggling and leave us to estimate the impact of failure of this infrastructure leading to
service disruption: we are not talking about thousands nor millions of users affected. We are talking about hundred of millions!
Obviously cloud services architectures involve heavy redundancies, mirror imaging of servers in different geographical location, disaster recovery procedures…Still, isn’t there some single point of
failure? As it is a well adopted fact that software can show bugs, viruses, worms…how about hardware? When firewalls, watchdogs and other software procedure are commonly put in place, aren’t we less
keen to accept hardware failure? Even trickier: what about hardware generated data corruption? The hardware shows no sign of failure, the software is not infected by viruses….still something’s not
Soft errors, even as being a small contributor of the overall reliability of systems, can still be the source of undetected failures that propagate to whole systems.
What are we doing to mitigate this problem, especially in cloud computing and data storage infrastructure?
Soft error in semiconductor devices is a reality and a threat to safety and reliability in micro electronic devices.
This tutorial gives you tools to assess your device level of risk and provide a 4 step solutions plan to prevent soft error . | {"url":"http://www.thesofterrorexperts.blogspot.com/","timestamp":"2014-04-18T00:12:53Z","content_type":null,"content_length":"78010","record_id":"<urn:uuid:73347cbd-48bd-433e-9c83-eb61283da38e>","cc-path":"CC-MAIN-2014-15/segments/1397609532374.24/warc/CC-MAIN-20140416005212-00134-ip-10-147-4-33.ec2.internal.warc.gz"} |
DCL, extended precision, and floating point?
The Question is:
According to DCL Help, symbol data types available are basically just string
and integer.
Is there any way to use a longword or double type in order to calculate numbers
that are greater than 2x32.
Do I need to code my own program to do the calculations and output the numbers.
And is there any way to represent floating point with DCL?
The Answer is :
DCL supports longword integers and strings. Longwords are 32 bits.
There is no DCL-provided mechanism to calculate with non-integer
values, nor with values that exceed a longword.
You will want to code your own program, as DCL is not particularly
suited to this task. In theory, you could code DCL routines to
provide multiple-precision arithmetic, but the results would be
neither pretty nor speedy.
DCL is appropriate for file and string manipulations, but if you want
to do numbers, please use a compiler or a calculator -- freeware or
shareware executable images that provide for calculations from within
DCL procedures have been referenced in newsgroups in the past. | {"url":"http://h71000.www7.hp.com/wizard/wiz_5462.html","timestamp":"2014-04-20T05:43:31Z","content_type":null,"content_length":"6703","record_id":"<urn:uuid:4d1cbc7b-a8fd-44b9-b1b7-eec929ee19ce>","cc-path":"CC-MAIN-2014-15/segments/1397609538022.19/warc/CC-MAIN-20140416005218-00488-ip-10-147-4-33.ec2.internal.warc.gz"} |
Distinct equivalent classes the null set
December 30th 2009, 02:33 AM
Distinct equivalent classes the null set
Let R be an equivalence relation on a set S. Let E and F be two distinct equivalence classes of R. Prove that E and F = null set.
Do i show that itys transitive im a bit stuck. Help much appreciated thanks.
December 30th 2009, 02:43 AM
Do you mean $E \cap F = \emptyset$?
Assume there exists an element in $E \cap F$, call it $x$. Then you can apply transitivity as every element of $E$ is equivalent to $x$, as is every element of $F$. $e \sim x$ and $x \sim f \
Rightarrow e \sim f \Rightarrow E = F$, a contradiction.
December 30th 2009, 02:45 AM
Do you mean $E \cap F = \emptyset$?
Assume there exists an element in $E \cap F$, call it $x$. Then you can apply transitivity as every element of $E$ is equivalent to $x$, as is every element of $F$. $e \sim x$ and $x \sim f \
Rightarrow e \sim f \Rightarrow E = F$, a contradiction.
Yep thats what i meant thanks.
January 5th 2010, 10:21 AM
Do you mean $E \cap F = \emptyset$?
Assume there exists an element in $E \cap F$, call it $x$. Then you can apply transitivity as every element of $E$ is equivalent to $x$, as is every element of $F$. $e \sim x$ and $x \sim f \
Rightarrow e \sim f \Rightarrow E = F$, a contradiction.
acually im a bit confused how is this a contradiction, it doesnt show its in the null set, just that e is in f?
January 5th 2010, 10:50 AM
You assumed that there is some element in $E\cap F$. As Swlabr showed, this implies $E=F$, however you took E,F to be distinct equivalence classes, ie. $E eq F$ - so this is a contradiction.
January 5th 2010, 12:47 PM
More of a forward-knowledge looking back approach (since you need to do this problem to prove what I'm about to say), but a relation $R$ on $S$ induces a partition $\pi$ of $S$ where the blocks
are the equivalence classes. If that is a definition in your book the answer follows immediately. | {"url":"http://mathhelpforum.com/advanced-algebra/121965-distinct-equivalent-classes-null-set-print.html","timestamp":"2014-04-19T04:52:40Z","content_type":null,"content_length":"14120","record_id":"<urn:uuid:29c999ee-39ac-4c5e-b785-ab5e7f9d6355>","cc-path":"CC-MAIN-2014-15/segments/1398223206118.10/warc/CC-MAIN-20140423032006-00228-ip-10-147-4-33.ec2.internal.warc.gz"} |
Segmentation by grouping junctions
Results 1 - 10 of 82
- IEEE Transactions on Pattern Analysis and Machine Intelligence , 2001
"... In this paper we address the problem of minimizing a large class of energy functions that occur in early vision. The major restriction is that the energy function’s smoothness term must only
involve pairs of pixels. We propose two algorithms that use graph cuts to compute a local minimum even when v ..."
Cited by 1384 (52 self)
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In this paper we address the problem of minimizing a large class of energy functions that occur in early vision. The major restriction is that the energy function’s smoothness term must only involve
pairs of pixels. We propose two algorithms that use graph cuts to compute a local minimum even when very large moves are allowed. The first move we consider is an α-βswap: for a pair of labels α, β,
this move exchanges the labels between an arbitrary set of pixels labeled α and another arbitrary set labeled β. Our first algorithm generates a labeling such that there is no swap move that
decreases the energy. The second move we consider is an α-expansion: for a label α, this move assigns an arbitrary set of pixels the label α. Our second
- IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE , 2001
"... After [10, 15, 12, 2, 4] minimum cut/maximum flow algorithms on graphs emerged as an increasingly useful tool for exact or approximate energy minimization in low-level vision. The combinatorial
optimization literature provides many min-cut/max-flow algorithms with different polynomial time compl ..."
Cited by 794 (48 self)
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After [10, 15, 12, 2, 4] minimum cut/maximum flow algorithms on graphs emerged as an increasingly useful tool for exact or approximate energy minimization in low-level vision. The combinatorial
optimization literature provides many min-cut/max-flow algorithms with different polynomial time complexity. Their practical efficiency, however, has to date been studied mainly outside the scope of
computer vision. The goal of this paper
- IEEE Transactions on Pattern Analysis and Machine Intelligence , 2004
"... Abstract—In the last few years, several new algorithms based on graph cuts have been developed to solve energy minimization problems in computer vision. Each of these techniques constructs a
graph such that the minimum cut on the graph also minimizes the energy. Yet, because these graph construction ..."
Cited by 699 (21 self)
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Abstract—In the last few years, several new algorithms based on graph cuts have been developed to solve energy minimization problems in computer vision. Each of these techniques constructs a graph
such that the minimum cut on the graph also minimizes the energy. Yet, because these graph constructions are complex and highly specific to a particular energy function, graph cuts have seen limited
application to date. In this paper, we give a characterization of the energy functions that can be minimized by graph cuts. Our results are restricted to functions of binary variables. However, our
work generalizes many previous constructions and is easily applicable to vision problems that involve large numbers of labels, such as stereo, motion, image restoration, and scene reconstruction. We
give a precise characterization of what energy functions can be minimized using graph cuts, among the energy functions that can be written as a sum of terms containing three or fewer binary
variables. We also provide a general-purpose construction to minimize such an energy function. Finally, we give a necessary condition for any energy function of binary variables to be minimized by
graph cuts. Researchers who are considering the use of graph cuts to optimize a particular energy function can use our results to determine if this is possible and then follow our construction to
create the appropriate graph. A software implementation is freely available.
, 2001
"... In this paper we describe a new technique for general purpose interactive segmentation of N-dimensional images. The user marks certain pixels as “object” or “background” to provide hard
constraints for segmentation. Additional soft constraints incorporate both boundary and region information. Graph ..."
Cited by 657 (14 self)
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In this paper we describe a new technique for general purpose interactive segmentation of N-dimensional images. The user marks certain pixels as “object” or “background” to provide hard constraints
for segmentation. Additional soft constraints incorporate both boundary and region information. Graph cuts are used to find the globally optimal segmentation of the N-dimensional image. The obtained
solution gives the best balance of boundary and region properties among all segmentations satisfying the constraints. The topology of our segmentation is unrestricted and both “object” and
“background” segments may consist of sev-eral isolatedparts. Some experimental results are presented in the context ofphotohideo editing and medical image seg-mentation. We also demonstrate an
interesting Gestalt example. A fast implementation of our segmentation method is possible via a new mar-$ow algorithm in [2].
- IJCV , 2003
"... In this paper we present a statistical framework for modeling the appearance of objects. Our work is motivated by the pictorial structure models introduced by Fischler and Elschlager. The basic
idea is to model an object by a collection of parts arranged in a deformable configuration. The appearance ..."
Cited by 524 (15 self)
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In this paper we present a statistical framework for modeling the appearance of objects. Our work is motivated by the pictorial structure models introduced by Fischler and Elschlager. The basic idea
is to model an object by a collection of parts arranged in a deformable configuration. The appearance of each part is modeled separately, and the deformable configuration is represented by
spring-like connections between pairs of parts. These models allow for qualitative descriptions of visual appearance, and are suitable for generic recognition problems. We use these models to address
the problem of detecting an object in an image as well as the problem of learning an object model from training examples, and present efficient algorithms for both these problems. We demonstrate the
techniques by learning models that represent faces and human bodies and using the resulting models to locate the corresponding objects in novel images.
"... Several new algorithms for visual correspondence based on graph cuts [7, 14, 17] have recently been developed. While these methods give very strong results in practice, they do not handle
occlusions properly. Specifically, they treat the two input images asymmetrically, and they do not ensure that a ..."
Cited by 266 (11 self)
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Several new algorithms for visual correspondence based on graph cuts [7, 14, 17] have recently been developed. While these methods give very strong results in practice, they do not handle occlusions
properly. Specifically, they treat the two input images asymmetrically, and they do not ensure that a pixel corresponds to at most one pixel in the other image. In this paper, we present a new method
which properly addresses occlusions, while preserving the advantages of graph cut algorithms. We give experimental results for stereo as well as motion, which demonstrate that our method performs
well both at detecting occlusions and computing disparities.
- In Proc. Int. Conf. Computer Vision , 2003
"... We propose a principled account on multiclass spectral clustering. Given a discrete clustering formulation, we first solve a relaxed continuous optimization problem by eigendecomposition. We
clarify the role of eigenvectors as a generator of all optimal solutions through orthonormal transforms. We t ..."
Cited by 181 (7 self)
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We propose a principled account on multiclass spectral clustering. Given a discrete clustering formulation, we first solve a relaxed continuous optimization problem by eigendecomposition. We clarify
the role of eigenvectors as a generator of all optimal solutions through orthonormal transforms. We then solve an optimal discretization problem, which seeks a discrete solution closest to the
continuous optima. The discretization is efficiently computed in an iterative fashion using singular value decomposition and nonmaximum suppression. The resulting discrete solutions are nearly
global-optimal. Our method is robust to random initialization and converges faster than other clustering methods. Experiments on real image segmentation are reported. optima consist not only of the
eigenvectors, but of a whole family spanned by the eigenvectors through orthonormal transforms. The goal is to find the right orthonormal transform that leads to a discretization. ˜X normalize
- In IEEE Conference on Computer Vision and Pattern Recognition , 1998
"... Markov Random Fields (MRF’s) can be used for a wide variety of vision problems. In this paper we focus on MRF’s with two-valued clique potentials, which form a generalized Potts model. We show
that the maximum a posteriori estimate of such an MRF can be obtained by solving a multiway minimum cut pro ..."
Cited by 166 (22 self)
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Markov Random Fields (MRF’s) can be used for a wide variety of vision problems. In this paper we focus on MRF’s with two-valued clique potentials, which form a generalized Potts model. We show that
the maximum a posteriori estimate of such an MRF can be obtained by solving a multiway minimum cut problem on a graph. We develop efficient algorithms for computing good approximations to the minimum
multiway cut. The visual correspondence problem can be formulated as an MRF in our framework; this yields quite promising results on real data with ground truth. We also apply our techniques to MRF’s
with linear clique potentials. 1
- IN IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE , 1999
"... In a traditional classification problem, we wish to assign one of k labels (or classes) to each of n objects, in a way that is consistent with some observed data that we have about the problem.
An active line of research in this area is concerned with classification when one has information about pa ..."
Cited by 161 (2 self)
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In a traditional classification problem, we wish to assign one of k labels (or classes) to each of n objects, in a way that is consistent with some observed data that we have about the problem. An
active line of research in this area is concerned with classification when one has information about pairwise relationships among the objects to be classified; this issue is one of the principal
motivations for the framework of Markov random fields, and it arises in areas such as image processing, biometry, and document analysis. In its most basic form, this style of analysis seeks a
classification that optimizes a combinatorial function consisting of assignment costs --- based on the individual choice of label we make for each object --- and separation costs --- based on the
pair of choices we make for two "related" objects. We formulate a general classification problem of this type, the metric labeling problem; we show that it contains as special cases a number of
standard classification f...
- Proc. IEEE Computer Vision and Pattern Recognition Conf. , 2000
"... A pictorial structure is a collection of parts arranged in a deformable configuration. Each part is represented using a simple appearance model and the deformable configuration is represented by
spring-like connections between pairs of parts. While pictorial structures were introduced a number of ye ..."
Cited by 161 (9 self)
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A pictorial structure is a collection of parts arranged in a deformable configuration. Each part is represented using a simple appearance model and the deformable configuration is represented by
spring-like connections between pairs of parts. While pictorial structures were introduced a number of years ago, they have not been broadly applied to matching and recognition problems. This has
been due in part to the computational difficulty of matching pictorial structures to images. In this paper we present an efficient algorithm for finding the best global match of a pictorial structure
to an image. The running time of the algorithm is optimal and it it takes only a few seconds to match a model with ve to ten parts. With this improved algorithm, pictorial structures provide a
practical and powerful framework for qualitative descriptions of objects and scenes, and are suitable for many generic image recognition problems. We illustrate the approach using simple models of a
person and a car. | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=16115","timestamp":"2014-04-21T13:15:05Z","content_type":null,"content_length":"39605","record_id":"<urn:uuid:e205fe8b-766f-48e1-b95c-9a3d06455cab>","cc-path":"CC-MAIN-2014-15/segments/1397609539776.45/warc/CC-MAIN-20140416005219-00433-ip-10-147-4-33.ec2.internal.warc.gz"} |
Deformation theory of representations of an algebraic group
up vote 22 down vote favorite
For an algebraic group G and a representation V, I think it's a standard result (but I don't have a reference) that
• the obstruction to deforming V as a representation of G is an element of H^2(G,V⊗V^*)
• if the obstruction is zero, isomorphism classes of deformations are parameterized by H^1(G,V⊗V^*)
• automorphisms of a given deformation (as a deformation of V; i.e. restricting to the identity modulo your square-zero ideal) are parameterized by H^0(G,V⊗V^*)
where the H^i refer to standard group cohomology (derived functors of invariants). The analogous statement, where the algebraic group G is replaced by a Lie algebra g and group cohomology is replaced
by Lie algebra cohomology, is true, but the only proof I know is a big calculation. I started running the calculation for the case of an algebraic group, and it looks like it works, but it's a mess.
Surely there's a long exact sequence out there, or some homological algebra cleverness, that proves this result cleanly. Does anybody know how to do this, or have a reference for these results? This
feels like an application of cotangent complex ninjitsu, but I guess that's true about all deformation problems.
While I'm at it, I'd also like to prove that the obstruction, isoclass, and automorphism spaces of deformations of G as a group are H^3(G,Ad), H^2(G,Ad), and H^1(G,Ad), respectively. Again, I can
prove the Lie algebra analogues of these results by an unenlightening calculation.
Background: What's a deformation? Why do I care?
I may as well explain exactly what I mean by "a deformation" and why I care about them. Last things first, why do I care? The idea is to study the moduli space of representations, which essentially
means understanding how representations of a group behave in families. That is, given a representation V of G, what possible representations could appear "nearby" in a family of representations
parameterized by, say, a curve? The appropriate formalization of "nearby" is to consider families over a local ring. If you're thinking of a representation as a matrix for every element of the group,
you should imagine that I want to replace every matrix entry (which is a number) by a power series whose constant term is the original entry, in such a way that the matrices still compose correctly.
It's useful to look "even more locally" by considering families over complete local rings (think: now I just take formal power series, ignoring convergence issues). This is a limit of families over
Artin rings (think: truncated power series, where I set x^n=0 for large enough n).
So here's what I mean precisely. Suppose A and A' are Artin rings, where A' is a square-zero extension of A (i.e. we're given a surjection f:A'→A such that I:=ker(f) is a square-zero ideal in A'). A
representation of G over A is a free module V over A together with an action of G. A deformation of V to A' is a free module V' over A' with an action of G so that when I reduce V' modulo I (tensor
with A over A'), I get V (with the action I had before). An automorphism of a deformation V' of V as a deformation is an automorphism V'→V' whose reduction modulo I is the identity map on V. The
"obstruction to deforming" V is something somewhere which is zero if and only if a deformation exists.
I should add that the obstruction, isoclass, and automorphism spaces will of course depend on the ideal I. They should really be cohomology groups with coefficients in V⊗V^*⊗I, but I think it's
normal to omit the I in casual conversation.
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5 Answers
active oldest votes
A representation of G on a vector space V is a descent datum for V, viewed as a vector bundle over a point, to BG. That is, linear representations of G are "the same" as vector
bundles on BG. So the question is equivalent to the analogous question about deformations of vector bundles on BG. We could just as easily ask about deformations of vector bundles on
any space X.
Given a vector bundle V on X, consider the category of all first-order deformations of V. An object is a vector bundle over X', where X' is an infinitesimal thickening (in the
example, one may take X = BG x E where E is a local Artin ring and X' = BG x E' where E' is a square-zero extension whose ideal is isomorphic as a module to the residue field). A
morphism is a morphism of vector bundles on X' that induces the identity morphism on V over X.
If X is allowed to vary, this category varies contravariantly with X. Vector bundles satisfy fppf descent, so this forms a fppf stack over X.
This stack is very special: locally it has a section (fppf locally a deformation exists) and any two sections are locally isomorphic. It is therefore a gerbe. Moreover, the
up vote 21 down isomorphism group between any two deformations of V is canonically a torsor under the group End(V) (this is fun to check).
vote accepted
Gerbes banded by an abelian group H are classified by H^2(X,H) (this is also fun to check); the class is zero if and only if the gerbe has a section. If the gerbe has a section, the
isomorphism classes of sections form a torsor under H^1(X,H). The isomorphisms between any two sections form a torsor under H^0(X,H). (This implies that the automorphism group of any
section is H^0(X,H).)
In our case, H = End(V), so we obtain a class in H^2(X,End(V)) and if this class is zero, our gerbe has a section, i.e., a deformation exists. In this case, all deformations form a
torsor under H^1(X,End(V)), and the automorphism group of a deformation is H^0(X,End(V)).
All of the cohomology groups above are sheaf cohomology in the fppf topology. If you are using a different definition of group cohomology, there is still something to check.
$G$ was an algebraic group. Now what is meant by $BG$ ? – Wilberd van der Kallen Jan 19 '11 at 15:36
$BG$ is the stack of $G$-torsors (principal $G$-bundles). – Keerthi Madapusi Pera Jan 19 '11 at 16:13
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The statements about the group and Lie algebra in the question are special cases of a more general fact.
Namely, if $A$ is an associative algebra and $V$ an $A$-module, then obstructions to deformations of $V$ lie in the Hochschild cohomology group $HH^2(A,{\rm End}(V))$, freedom of
deformation in $HH^1(A,{\rm End}(V))$, and infinitesimal automorphisms in $HH^0(A,{\rm End}(V))$. This is rather easy to check using the bar complex.
up vote 16 Now, the statement for Lie algebras is the special case $A=U({\mathfrak g})$, recalling that for any $U({\mathfrak g})$-bimodule $M$,
down vote $$ HH^\ast (U({\mathfrak g}),M)=H^\ast({\mathfrak g},M_{ad}). $$
Similarly, for affine algebraic groups, it is the special case $A=O(G)^\ast$, where $O(G)$ is the coalgebra of regular functions, recalling that for any (algebraic) $G$-bimodule $M$, $$
HH^\ast(O(G)^\ast,M)=H^\ast(G,M_{ad}). $$
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Here's not a complete answer, but I think an enlightening trick. Deformations of V over the dual numbers are always in bijection with Ext^1(V,V) in any abelian category. The trick is
that if you have a deformation V', you have a long exact sequence:
Hom(V,V) -> Hom(V',V) -> Hom(V,V) -> Ext^1(V,V) -> Ext^1(V',V) -> Ext^1(V,V) -> Ext^2(V,V)
up vote 8 down You can see that the extension splits if and only if the image of the identity under the boundary map is trivial (using Baer sum, you can extend this trick to show that two extensions
vote are isomorphic if and only if the image of the identity is the same).
I think the obstruction in Ext^2(V,V) you had in mind is the image of that class under the next boundary map, by a similar argument.
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I will offer a sketch of an argument, and maybe someone who knows what a stack is can make it happen for real. There is probably a non-stacky deformation theory of commutative Hopf algebras,
but I don't know what it looks like.
Deforming G as a group should be the same as deforming BG as a plain old geometric object. Pulling back a point in BG along a cover by a point is very roughly taking a based loop space, and
the deformed loop space comes with the deformed composition law. Similarly, deforming a representation of G should be the same as deforming a sheaf on BG.
up vote I'm going to assume G is smooth. Then the tangent complex of BG mapping to a point is just the sheaf Ad, concentrated in degree 1. If we boldly assume that deformation theory of/on stacks
6 down works just like deformations of/on schemes, but maybe with some degree shifts, we should get the answers you want. For deforming G in particular, there is a canonical class in H^2(BG, Ad
vote [-1]) that classifies obstructions, and if that vanishes, H^1(BG, Ad[-1]) classifies deformations and H^0(BG, Ad[-1]) classifies automorphisms of a deformation. When deforming the sheaf V,
one usually sees the sheaf End(V) written as coefficients.
Olsson wrote a paper on deformations of representable morphisms of stacks, and while the morphism BG -> S isn't representable, one might benefit from asking the author for additional details
if one were, say, working in the same building as he.
I've heard this approach that "deforming BG as a stack is the same as deforming G as a group," and I like it, but I don't see why any deformation of BG must be of the form BG' for G' a
deformation of G. Also, I don't understand how to do deformations of representations, but I guess the thing I have to understand is deformation theory of coherent sheaves on a scheme. I'll
ask Martin about it when I get the chance. – Anton Geraschenko Oct 16 '09 at 5:07
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About what Anton said at the end about deformations of a group. Let $m_0$ be the standard multiplication. Then I want to consider a deformation of the form $m:(G \times \epsilon \mathfrak
{g}) \times (G \times \epsilon \mathfrak{g}) \to G \times \epsilon \mathfrak{g}$ where $m(g_1, g_2) = m_{0}(g_1,g_2) + \epsilon m_1 (g_1,g_2)$. When you write out the associativity condition
up vote $m\circ (m \times 1) = m \circ (1 \times m)$ it seems that you find that $(g_1,g_2) \mapsto (m_{1}(g_{1},g_{2}))(g_{1}g_{2})^{-1}$ is a group cohomology cocycle for G acting on $\mathfrak{g}
4 down $ by the adjoint representation. Now one has to identify $H^{2}(G,Ad)$ with $H^{2}(BG,Ad)$ (taking care of the topology somehow).
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Originally Posted by
F = ma will be how much impact (i.e. force) the mass will produce.
It is the increased kenetic energy a heavier mass will carry as compared to a lighter one given the same velocity that would have to be dissipated by the braking system, so:
K = 1/2 * (MV^2)
So given same velocity (V), the higher mass (M) will produce higher kenetic energy to be dissipated by the braking system.
Man all this talk brings me back to college physics, which I'm kinda rusty on now. Correct me if I'm wrong in any of this. And if you wanna get really technical, you could find out what the kinetic
friction is for your tires on dry asphalt (0.5 to 0.8 generally) and brake pads on rotors and factor that in. Oh, and don't forget wind resistance. Man, you could get really technical.
I'll try my best to explain things here. I'm going to assume that whoever reads this post has little to no knowledge of classical mechanics. For those of you who took a course or two in physics, no
offense. Just hang tight!
Aviography is pretty much correct in everything he says, but in this scenario we really don't have to factor in energy, whether it be kinetic or potential (in regards to classical mechanics). For
those of you who don't know, energy is a measurement of work equal to force times distance (W=F*x), measured in Joules (J). We could measure the amount of work done by the decelerating vehicle, but
it is really not necessary.
What we're really concerned with here is finding out the distance the vehicle will travel with the added weight compared to the vehicle without the added weight. In my scenario we will find out the
distance, and as an added bonus, we will find out the time it takes to decelerate at a constant rate from a given velocity to rest with and without the added weight.
If you know the formula for force, acceleration, and velocity, and you can manipulate the formulas to find out unknowns, you're pretty much in luck. Anyway, let's begin.
First off, let's start off by using the SI system (which is mostly in metric units). Let's assume you have a 1200 kg vehicle and plan on adding 100 kg in the trunk later to test your theory.
As you know, velocity (V) is equal to distance divided by time: V=x/t
In the SI system, distance (x) is measured in meters, and time is measured in seconds (s)
So let's assume you accelerate at a constant rate from 0 km/h to 100 km/h (62 mph) in 8 seconds. 100 kilometers per hour in meters per second is approximately 28.77 m/s. Therefore, after 8 seconds,
your final velocity is 28.77 m/s.
Acceleration (a) is equal to velocity divided by time: a=V/t
Acceleration can also be calculated by change in velocity divided by time: (final velocity - initial velocity)/t
The SI unit for acceleration is meters per second per second, or commonly phrased as meters per second squared (m/s^2). Our initial velocity was 0 km/h, or 0 m/s, and our final velocity is 28.77 m/s,
so if we plug our numbers into the acceleration formula, we get roughly 3.6 m/s^2 (28.77 m/s - 0 m/s)/8 s.
Force (F) is equal to mass multiplied by acceleration: F=m*a
The SI unit for mass is measured in kilograms, and the SI unit for force is measured in Newtons (N), named after Sir Issac Newton.
If you plug in 1200 kg for your mass and 3.6 m/s^2 for your acceleration, it would take 4320 N of force to accelerate your 1200 kg vehicle from 0 km/h to 100 km/h in 8 seconds.
Following me so far? Simple physics, and we're not factoring in friction or wind resistance.
Now here's a question: how much force would it take to decelerate your 1200 kg car from 100 km/h to 0 km/h in exactly 8 seconds?
The answer is 4320 Newtons.
Deceleration is merely negative acceleration. Technically, the answer could also be -4320 N depending on which directions you consider positive and negative. Basically you have to apply a certain
force to accelerate a mass, and you would have to apply an equal amount of force in the opposite direction to bring that mass back to rest (4320 N - 4320 N = 0 N).
To put this in terms of vehicle acceleration and deceleration, consider stepping on the gas pedal positive acceleration and hitting the brakes deceleration, or negative acceleration.
Now let's add 100 kg to your 1200 kg car to bring a total mass of 1300 kg.
Before you added the extra weight to your car, you accelerated at a rate of 3.6 m/s^2, and you decelerated at the rate of 3.6 m/s^2. Interpreting this, you hit the brakes with a certain amount of
force that caused you to decelerate at 3.6 m/s^2. In this next scenario, you're going to decelerate at the same rate of 3.6 m/s^2.
How much force would be generated if a 1300 kg mass were to be decelerated at a rate of 3.6 m/s^2? Assume acceleration to be positive in this scenario.
Applying the equation F=m*a, we get 4680 Newtons of force.
Interpreting this, before adding the extra weight, it only required 4320 N to stop your car, but after adding an extra 100 kg to your vehicle, it required 4680 N of force to stop the car.
Now think about what you would have to do to stop your car with more force. Figured it out yet? You'd have to press the brake pedal harder. If you haven't already figured it out yet, if your
acceleration remains constant, your vehicle is traveling the same distance regardless the mass of your vehicle. It just requires more force to reach that distance in a certain amount of time when the
vehicle is heavier.
If you're wondering "What distance? We didn't define any distance!" then let's backtrack.
We're assuming you accelerated at a constant rate, but don't worry about that. You traveled from 0 km/h to 100 km/h. 100 km/h is 28.77 m/s, right? You traveled for 8 seconds, right? So 28.77 meters
times 8 seconds is 230.16 meters (or approximately 755.12 feet).
Now your goal was to decrease your stopping distance by adding extra weight. Hey, it was a good theory, but let me explain some more. You were probably assuming you could use the same amount of brake
force (or less) to stop your moving vehicle with the added weight, right?
Remember 4320 Newtons? That's the amount of force required to stop your 1200 kg moving vehicle from 100 km/h to 0 km/h in 8 seconds. Now let's say you apply 4320 Newtons of force to stop your vehicle
with an extra 100 kg of added weight, or 1300 kg vehicle. What would your deceleration rate be?
Rearranging F=m*a to solve for a, you get a=F/m, or force divided by mass.
4320 N/1300 kg equals approximately 3.32 m/s^2.
So, now that we have figured out your deceleration rate, let's figure out how long it would take to stop your vehicle with 4320 N of force.
As you know, acceleration is equal to velocity divided by time (a=V/t). We know the acceleration (3.32 m/s^2) and velocity (28.77 m/s), but the unknown is time. Still following?
If you rearrange the formula for acceleration to find time, you get t=V/a. let's plug and chug.
28.77 m/s / 3.32 m/s^2 = 8.66 seconds approximately.
It's not a huge difference, but as you can see, it would take longer to stop a heavier mass with the same amount of force.
How much farther would you travel in 8.66 seconds?
Remember velocity is equal to distance divided by time (V=x/t), and if you rearrange the equation to solve for distance, you get x=V*t.
28.77 m/s * 8.66 s = 249.15 meters approximately (or 817.42 feet).
Without the added weight, you traveled a distance of 230.16 meters (or approximately 755.12 feet). The difference in distance is 18.99 meters, or 62.3 feet.
Average length of a vehicle (according to a study done in Portland, Oregon measuring 390 vehicles) is roughly 13.52 feet. 62.3 feet is roughly 4.61 car lengths. If you used the same amount of force
applied to the brakes on your vehicle with the 100 kg added weight as you did without the added weight, you would have traveled 4.61 extra car lengths to stop, assuming you traveled from 100 km/h to
0 km/h with an average deceleration rate of 3.32 m/s^2.
I hope this little physics lesson and scenario helps you in your decision on whether or not to add extra weight to your vehicle to help braking time and distance. There are many other factors that
would also affect your braking distance and time, including the coefficient of kinetic friction between your tires and the road (The coefficient of friction will vary due to material of tires and the
road and whether the road is wet, dry, or has sand, gravel, or snow on it.), the coefficient of static and kinetic friction between your brake pads and rotors, the force of air resistance, and
gravitational resistance due to an incline or decline (slope) of the road you are traveling on. Just remember that this scenario did not factor friction or external forces, and acceleration (and
deceleration) were assumed to be a constant rate.
As far as adding a few sandbags or bricks in your trunk, you will add more force against your rear axle (due to the force of gravity), and indirectly, your rear wheels. Since your car is front wheel
drive, the back axle is merely pulled whereas the front axle is pushed by the engine. More force will be required from the engine to pull the extra weight, meaning that your vehicle will accelerate
at a lower rate than normal in a certain gear at a certain RPM. If you want to accelerate at the same rate as you normally do, you could compensate by increasing your RPMs and applying more force, or
press down harder on your gas pedal, but you could potentially lose traction on your front wheels and spin out until you gain traction again. Think of this as towing and applying too much gas causes
you to spin out.
Next, suppose you take a hard left turn (without braking) at a given velocity without the added weight in the trunk. In this scenario you do not lose traction in your rear wheels and spin out. The
inertia of the vehicle wants to move forward, but your front axle is trying to force the vehicle to move left. The vehicle eventually curves to the left safely due to rotational inertia.
Now suppose you take the same turn at the same velocity with the added weight. You could potentially sling your rear end around and fishtail. Why? As you try to curve your vehicle to the left, more
force (I say force because in this scenario you are fighting wind resistance and friction from the tires and the surface.) and inertia is trying to keep the vehicle moving straight. Again, as you are
turning, rotational inertia comes into play. Imagine a tennis ball tied to a string, and you are swinging the ball in circles from the string with your hand. Where the string is being held in your
hand is the axis of the rotation. Your hand is a fixed axis. If you release the string, the ball will travel straight instead of curving. Now imagine swinging a bowling ball tied to a string. You
swing the ball around and let the string go again. Notice how the bowling ball requires more force to rotate it? Now let's apply rotational inertia to your car with the extra weight added.
Suppose your front axle is axis of rotation, the car itself is the string, and the extra weight in the trunk is the ball. More force is going to be required to rotate the vehicle around the left turn
with the added weight instead of making the turn without the weight. Why? Inertia wants to keep the car moving forward.
A certain amount of frictional force is applied between your tires and the road as you make a turn. Net force is the sum of all forces, and frictional force is considered negative since it is
fighting your positive force. In this scenario, F(net) = F(applied) - F(frictional). While your vehicle with the added weight is making the turn, if your net force is greater than 0 Newtons, chances
are you will lose traction and spin out in the direction your vehicle was traveling when frictional force was overpowered. This can be related to you letting go of the string attached to the ball.
Remember on snowy or rainy roads, less frictional force is being applied to your vehicle. Therefore, adding extra weight to the rear of your vehicle would be a valid cause to lose traction during a
curve or turn at a given velocity on less frictional roads. Adding snow tires would help increase friction between the tires and the road, though.
Anyway, like I said, I hope this helps! You guys chime in if you want! | {"url":"http://www.driveaccord.net/forums/showthread.php?t=55423","timestamp":"2014-04-20T05:42:59Z","content_type":null,"content_length":"166136","record_id":"<urn:uuid:8bcf34a6-bd6b-42c9-9a1d-49bb45d7cbf3>","cc-path":"CC-MAIN-2014-15/segments/1397609538022.19/warc/CC-MAIN-20140416005218-00485-ip-10-147-4-33.ec2.internal.warc.gz"} |
[Numpy-discussion] What's wrong with matrices?
Ed Schofield schofield at ftw.at
Sun Jul 9 01:35:29 CDT 2006
On 08/07/2006, at 10:22 PM, JJ wrote:
> 3) In some operations, especially range selection operations, a N,
> 1 matrix is
> turned into a 1,N matrix. This is confusing relative to matlab and
> problematic,
> in my view.
This sounds like a bug. Can you give any examples of this happening
with the latest release or SVN version?
> <snip>
> 5) If X is a 1,n matrix, then X.T should return a n,1 matrix, I
> think.
This should be the case. Could you post a code snippet that violates
Thanks for your feedback, JJ!
-- Ed
More information about the Numpy-discussion mailing list | {"url":"http://mail.scipy.org/pipermail/numpy-discussion/2006-July/009302.html","timestamp":"2014-04-19T15:49:37Z","content_type":null,"content_length":"3188","record_id":"<urn:uuid:ecaa5438-4c47-4d41-bdde-64af508a3fe2>","cc-path":"CC-MAIN-2014-15/segments/1397609537271.8/warc/CC-MAIN-20140416005217-00191-ip-10-147-4-33.ec2.internal.warc.gz"} |
degrees / radians
January 10th 2007, 12:43 AM
degrees / radians
Ok, I have some questions, these are all using radians
Question 1
Convert the following angles to radians giving your answers correct to 3SF.
a) 20 degrees
b) -72 degrees
c) 400 degrees
d) -140 degrees
e) 760 degrees
Question 2
An angle "p" subtends an arc of length 25cm in a circle of radius R cm.
The area of the sector POQ is 72cm (squared)
Forumulate two equations in r and "p" Find the values of r and "p"
Question 3
A cylindrical pipe of diameter 1.5m contains water to a depth of 0.9m.
a) find the cross sectional area of the water
b) if the water is flowing at a rate of 60 litres per second fin dthe speed of the water in m/s
I would really appreciate any help offered, these are the final pieces of some homework and i really need help, thank you
January 10th 2007, 04:41 AM
The relationship between degrees and radians is that $\pi$ rad = 180 degrees.
a) $\frac{20 \, degrees}{1} \cdot \frac{\pi \, rad}{180 \, degrees} \approx 0.349 \, rad$
The others are done the same way.
January 10th 2007, 05:10 AM
Hello, Kim!
You're expected to know these formulas.
In a circle of radius $r$ with a central angle of $\theta$ radians:
. . the length of arc is: . $s \:=\:r\theta$
. . the area of the sector is: . $A \:=\:\frac{1}{2}r^2\theta$
2) An angle $\theta$ subtends an arc of length 25cm in a circle of radius $R$ cm.
The area of the sector POQ is 72 cm².
(a) Formulate two equations in $R$ and $\theta.$
(b) Find the values of $R$ and $\theta.$
Part (a)
We know that the arc length is 25 cm.
. . So we have: . $R\theta\:=\:25$[1]
We know that the area of the sector is 72 cm².
. . So we have: . $\frac{1}{2}R^2\theta\:=\:72$[2]
Part (b)
Divide [2] by [1]: . $\frac{\frac{1}{2}R^2\theta}{R\theta} \:=\:\frac{72}{25}\quad\Rightarrow\quad\boxed{R \,=\,\frac{144}{25}\text{ cm}}$
Substitute into [1]: . $\frac{144}{25}\theta\:=\:25\quad\Rightarrow\quad\b oxed{\theta \,=\,\frac{625}{144}\text{ radians}}$ | {"url":"http://mathhelpforum.com/trigonometry/9790-degrees-radians-print.html","timestamp":"2014-04-21T16:49:07Z","content_type":null,"content_length":"9951","record_id":"<urn:uuid:68325ea9-dac8-4e12-a8ab-4a0dfe59e9d3>","cc-path":"CC-MAIN-2014-15/segments/1397609540626.47/warc/CC-MAIN-20140416005220-00610-ip-10-147-4-33.ec2.internal.warc.gz"} |
Short Circuit Capacity: Basic Calculations and Transformer Sizing
Need a transformer? Medium Voltage Cast Resin and Oil Immersed are available here
Short Circuit Capacity: Basic Calculations and Transformer Sizing
Copyright © 2001 Francis J. Martino
Short circuit capacity calculation is used for many applications: sizing of transformers,
selecting the interrupting capacity ratings of circuit breakers and fuses, determining if a
line reactor is required for use with a variable frequency drive, etc.
The purpose of the presentation is to gain a basic understanding of short circuit capacity.
The application example utilizes transformer sizing for motor loads.
Conductor impedances and their associated voltage drop are ignored not only to present a
simplified illustration, but also to provide a method of approximation by which a plant
engineer, electrician or production manager will be able to either evaluate a new application
or review an existing application problem and resolve the matter quickly.
Literature containing a detailed discussion of short circuit capacity calculations are
available within the electrical power transmission industry. [1]
The following calculations will determine the extra kVA capacity required for a three
phase transformer that is used to feed a single three phase motor that is started with
full voltage applied to its terminals, or, "across-the-line."
Two transformers will be discussed, the first having an unlimited short circuit kVA
capacity available at its primary terminals, and the second having a much lower input
short circuit capacity available.
kVA of a single phase transformer = V x A
kVA of a three phase transformer = V x A x 1.732, where 1.732 = the square root of 3.
The square root of 3 is introduced for the reason that, in a three phase system,
the phases are 120 degrees apart and, therefore, can not be added arithmetically.
They will add algebraically.
Transformer Connected To Utility Power Line
The first transformer is rated 1000 kVA, 480 secondary volts, 5.75% impedance.
Rated full load amp output of the transformer is
1000 kVA / (480 x 1.732) = 1203 amps
The 5.75% impedance rating indicates that 1203 amps will flow in the secondary if
the secondary is short circuited line to line and the primary voltage is raised from
zero volts to a point at which 5.75% of 480 volts, or, 27.6 volts, appears at the
secondary terminals. Therefore, the impedance (Z) of the transformer secondary may
now be calculated:
Z = V / I = 27.6 volts / 1203 amps = .02294 ohms
The transformer is connected directly to the utility power lines which we will
assume are capable of supplying the transformer with an unlimited short circuit
kVA capacity. The utility company will always determine and advise of the short
circuit capacity available at any facility upon request.
With unlimited short circuit kVA available from the utility, the short circuit
amperage capacity which the transformer can deliver from its secondary is
480 volts / .02294 = 20,924 amps
An alternative method of calculating short circuit capacity for the above
transformer is:
1203 amps x 100 / 5.75% = 1203 / .0575 = 20,922 amps
Another alternative is to consult a reference manual. Cutler- Hammer Consulting
Application Catalog, 12th Edition, gives the specifications for the above mentioned
transformer and the value of the short circuit capacity in Table A25 on page A-59.
The short circuit capacity is given as 20,900 amps.
Now we are ready to apply a motor to the terminals of the transformer secondary.
We must determine the voltage drop which will be caused by the motor inrush on
start. If the voltage remains within the rated voltage of the motor, then no oversizing
of the transformer is required.
Motors rated for 460 volts are for use with distribution systems that are rated at
480 volts. The rating system allows a twenty volt drop in the distribution system
which may occur along the feeder cables which connect the power transformer to
the load. The NEMA specification for a standard motor is that it requires the motor to
be capable of operating at plus or minus 10% of nameplate voltage. Therefore, the
voltage drop on inrush should not be allowed to drop below 460 volts less 10%,
or, 414 volts.
The transformer will be asked to supply power to a motor which has a full load amp
rating of 1203 amps, which will fully load the transformer. Therefore, we will rate
the motor at 460 V x 1203 A x 1.732, or, 958.5 kVA. We will assume that our motor
will have an inrush of 600% of its full load rating which will cause an inrush of
460 V x 1203 A x 600% x 1.732 = 5751 kVA
The voltage drop at the transformer terminals will be proportional to the motor load.
The voltage drop will be expressed as a percentage of the inrush motor load compared
to the maximum capability of the transformer. [2] The transformer has a maximum kVA
capacity at its short circuit capability, which is
480 V x 20,924 A x 1.732 = 17,395 kVA
The voltage drop on motor inrush will be
5751 kVA / 17,395 kVA = .331, or, 33.1%
The transformer output voltage will drop to 480 x .669, or, 321 volts. Thus, we can
see that the transformer is much too small to use a motor that has a full load rating
equal to the full load capacity of the transformer.
The transformer must be sized so that its short circuit capabilty is equal to or
greater than 5751 kVA times 10, or, 57,510 kVA in order to have a voltage drop of 10%
or less. Therefore, the short circuit amperage capacity of the transformer to be used
must be a minimum of
57,510 kVA / (480 V x 1.732) = 69176 amps
A typical 2500 kVA, 5.75% impedance transformer will have a short circuit capacity of
52,300 amps. The next highest standard size transformer at 3750 kVA will have a 6.5%
impedance and would have a short circuit output capability of 69,395 amps which will be
In the particular application discussed, the ratio of the selected standard size
transformer kVA to motor kVA is 3750 kVA / 958.5 kVA = 3.91. Thus the transformer
rating is 391% larger, or, nearly four times, the rating of the motor. Note the non-linear
effect of the impedance rating of the transformers on their short circuit capacities.
Transformer Connected To An Upstream Transformer
The second transformer we will examine will have a finite short circuit capacity
available at its primary rather than an unlimited capacity. We will assume that a facility
derives its power from the same 1000 kVA transformer mentioned above and that the
second transformer is connected directly to the terminals of the 1000 kVA transformer.
Thus, feeder cables between the two transformers are eliminated and the impedance of
cables are not taken into account. However, the smaller the motor leads, the less will be
both the short circuit capacity and the voltage delivered to the motor terminals.
The second transformer, which will have a 480 volt primary and a 480 volt secondary, will
be used to power a 20 HP, 3 phase, 460 volt motor which will be started at full voltage. The
motor will be the only load on the transformer.
Sales catalogs by various manufacturers will invariably recommend a "minimum
transformer kVA" of 21.6 for use with a 20 HP motor. The minimum transformer kVA ratings
are for use with multiple motors on a single transformer. A multiple motor configuration
will be discussed in the next section of this article.
The 21.6 kVA is calculated as follows:
480 volts x 26 nominal amps x 1.732 = 21.6 kVA
The transformer manufacturers will give a 20 HP motor a nominal full load amp rating
of 27 amps, thus allowing no extra capacity:
460 volts x 27 nominal amps x 1.732 = 21.5 kVA
One motor manufacturer has rated a 20 HP motor at 26 Full Load Amps, 460 VAC,
205 Locked Rotor Amps, 81% Power Factor. The motor will present a load of
460 volts x 26 amps x 1.732 = 20.7 kVA
The starting motor kVA load with inrush current will be
460 V x 205 A x 1.732 = 163.3 kVA
We will consider using a 30 kVA general purpose transformer to supply the 20 HP motor.
The transformer will have a nominal impedance of 2.7% and an ouptut of 36.1 amps at
480 volts. The short circuit current capacity that can be delivered to the 21.6 kVA
transformer by the upstream 1000 kVA transformer is 20,924 amps, or, 17,395 kVA.
The short circuit amperage capacity of a transfomer with a limited system short circuit
capacity available at its primary is:
transformer full load amps / (transformer impedance + upstream system impedance as seen
by the transformer)
upstream system impedance as seen by the transformer =
transformer kVA / available primary short circuit capacity kVA
36.1 amps / [2.7% + (30 kVA / 17,395 kVA)] =
36.1 / (2.7% + .0017%) = 36.1 / .0287 = 1258 short circuit amps
The transformer output voltage drop upon motor inrush will be:
motor inrush kVA / short circuit kVA =
163.3 kVA / (480 V x 1258 A x 1.732) = 163.3 kVA / 1046 kVA =
.156 = 15.6 %
A 30 kVA transformer rating is too small as the motor voltage drop will exceed 10%.
A 45 kVA transformer with a 2.4% impedance and an output of 54.1 amps at 480 volts
would have a short circuit capacity of 2034 amps. The voltage drop upon motor inrush
would be 9.66%.
For a single motor and transformer combination, one transformer manufacturer
recommends that the motor full load running current not exceed 65% of the transformer
full load amp rating. [3] Thus, for our 26 amp motor the transformer rating should be
a minimum of 40 amps, or, 33.3 kVA.
Multiple Motors On A Single Transformer
The minimum transformer kVA is given by transformer manufacturers so that a transformer
may be sized properly for multiple motors. If there are five motors on one transformer, add
the minimum kVA ratings and then add transformer capacity as necessary to accomodate the
inrush current of the largest motor.
The transformer thusly selected will be capable of running and starting all five motors
provided that only one motor is started at any one time. Additional capacity will be required
for motors starting simultaneously.
Also, if any motor is started more than once per hour, add 20% to that motor's minimum
kVA rating to compensate for heat losses within the transformer.
Motor Contribution to Short Circuit Capacity
When a fault condition occurs, power system voltage will drop dramatically. All motors
that are running at that time will not be able to sustain their running speed. As those motors
slow in speed, the stored energy within their fields will be discharged into the power line.
The nominal discharge of a motor will contribute to the fault a current equal to up to four
times its full load current.
With our 1000 kVA, 1203 amp transformer example given above, we will assume that all
1203 amps of load are from motors. The actual short circuit current will equal 20,924 amps
plus 400% of 1203 amps for a total of 25,736 short circuit amps.
When sizing the transformer for motor loads, the fault current contribution from the
motors will not be a consideration for sizing. However, the motor contribution must be
considered when sizing all branch circuit fuses and circuit breakers. The interrupting
capacity ratings of those devices must equal or exceed the total short circuit capacity
available at the point of application.
Motor contribution to short circuit capacity must be included when adding a variable
frequency drive to the system. See Variable Frequency Drives: Source Impedance
and Line Reactors
[1] "Cutler-Hammer Consulting Application Catalog," published by Cutler-Hammer,
Division of Eaton Corporation, Pittsburgh, PA.
[2] "Short Circuit Capacity and Voltage Sag," IEEE (Institute of Electrical and Electronic
Engineers) Industry Application Society (IAS) Magazine, July/August 2000, page 38. You
might find this magazine in universities which have Electrical Enginneering programs.
Do not get it confused with IEEE Transactions on Industry Applications which is a
different publication.
[3] "Power Distribution Products" Catalog ATD-01, Acme Electric Corporation, 1995,
page 125.
Power Quality and Drives LLC
Need a transformer? Medium Voltage Cast Resin and Oil Immersed are available here | {"url":"http://www.powerqualityanddrives.com/short_circuit_transformer/","timestamp":"2014-04-21T01:59:14Z","content_type":null,"content_length":"27912","record_id":"<urn:uuid:afa4eed7-c4b4-4e4f-a8cd-016b5d6907bb>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00133-ip-10-147-4-33.ec2.internal.warc.gz"} |
Maintainer diagrams-discuss@googlegroups.com
Safe Haskell None
"Traces", aka embedded raytracers, for finding points on the edge of a diagram. See Diagrams.Core.Trace for internal implementation details.
data Trace v
Every diagram comes equipped with a trace. Intuitively, the trace for a diagram is like a raytracer: given a line (represented as a base point and a direction), the trace computes the distance from
the base point along the line to the first intersection with the diagram. The distance can be negative if the intersection is in the opposite direction from the base point, or infinite if the ray
never intersects the diagram. Note: to obtain the distance to the furthest intersection instead of the closest, just negate the direction vector and then negate the result.
Note that the output should actually be interpreted not as an absolute distance, but as a multiplier relative to the input vector. That is, if the input vector is v and the returned scalar is s, the
distance from the base point to the intersection is given by s * magnitude v.
Action Name (Trace v)
Show (Trace v)
Ord (Scalar v) => Semigroup (Trace v)
Ord (Scalar v) => Monoid (Trace v)
(Ord (Scalar v), VectorSpace v) => Traced (Trace v)
HasLinearMap v => Transformable (Trace v)
VectorSpace v => HasOrigin (Trace v)
Newtype (QDiagram b v m) (DUALTree (DownAnnots v) (UpAnnots b v m) () (Prim b v))
class (Ord (Scalar (V a)), VectorSpace (V a)) => Traced a
Traced abstracts over things which have a trace.
Traced b => Traced [b]
Traced b => Traced (Set b)
(Ord (Scalar v), VectorSpace v) =>
Traced (Trace v)
Traced t => Traced (TransInv t)
The trace of a single point is the empty trace, i.e. the one which returns positive infinity for every query. Arguably it should return a finite distance for
(Ord (Scalar v), VectorSpace v) => vectors aimed directly at the given point and infinity for everything else, but due to floating-point inaccuracy this is problematic. Note that the envelope for
Traced (Point v) a single point is not the empty envelope (see Diagrams.Core.Envelope).
Traced a => Traced (Located a) The trace of a Located a is the trace of the a, translated to the location.
Traced (FixedSegment R2)
Traced (Trail R2)
Traced (Path R2)
(Traced a, Traced b, ~ * (V a) (V b))
=> Traced (a, b)
Traced b => Traced (Map k b)
Traced (Segment Closed R2)
(HasLinearMap v, VectorSpace v, Ord (
Scalar v)) => Traced (QDiagram b v m)
(Ord (Scalar v), VectorSpace v,
HasLinearMap v) => Traced (Subdiagram
b v m)
Diagram traces
Querying traces
traceV :: Traced a => Point (V a) -> V a -> a -> Maybe (V a)
Compute the vector from the given point to the boundary of the given object in the given direction, or Nothing if there is no intersection.
traceP :: Traced a => Point (V a) -> V a -> a -> Maybe (Point (V a))
Given a base point and direction, compute the closest point on the boundary of the given object, or Nothing if there is no intersection in the given direction.
maxTraceV :: Traced a => Point (V a) -> V a -> a -> Maybe (V a)
Like traceV, but computes a vector to the *furthest* point on the boundary instead of the closest.
maxTraceP :: Traced a => Point (V a) -> V a -> a -> Maybe (Point (V a))
Like traceP, but computes the *furthest* point on the boundary instead of the closest.
Subdiagram traces
boundaryFrom :: (HasLinearMap v, Ord (Scalar v)) => Subdiagram b v m -> v -> Point vSource
Compute the furthest point on the boundary of a subdiagram, beginning from the location (local origin) of the subdiagram and moving in the direction of the given vector. If there is no such point,
the origin is returned; see also boundaryFromMay.
boundaryFromMay :: (HasLinearMap v, Ord (Scalar v)) => Subdiagram b v m -> v -> Maybe (Point v)Source
Compute the furthest point on the boundary of a subdiagram, beginning from the location (local origin) of the subdiagram and moving in the direction of the given vector, or Nothing if there is no
such point. | {"url":"http://hackage.haskell.org/package/diagrams-lib-0.7.1/docs/Diagrams-Trace.html","timestamp":"2014-04-20T11:21:45Z","content_type":null,"content_length":"28194","record_id":"<urn:uuid:01a0cae6-c0d8-41ee-983b-0b92d426bc8a>","cc-path":"CC-MAIN-2014-15/segments/1397609538423.10/warc/CC-MAIN-20140416005218-00054-ip-10-147-4-33.ec2.internal.warc.gz"} |
find power series
find the power series representation of the function. heres the equation: http://img260.imageshack.us/img260/9021/untitledph1.jpg need help. thanks.
We know that, $-1<x\leq 1$ $\tan^{-1}x=x-\frac{x^3}{3}+\frac{x^5}{5}-...$ Know if you multiply this term by term for $(1+x^2)$ we have, $x-\frac{x^3}{3}+\frac{x^5}{5}-....+x^3-\frac{x^5}{3}+\frac{x^
7}{5}-...$ $x+\frac{2x^3}{3\cdot 1}-\frac{2x^5}{5\cdot 3}+\frac{2x^7}{7\cdot 5}-\frac{2x^9}{9\cdot 7}+...$ I just need to make the restriction, $-1<x<1\,$ rather then the endpoint because the Cauchy
product applies only for the interval of absolute convergence. | {"url":"http://mathhelpforum.com/calculus/11358-find-power-series.html","timestamp":"2014-04-19T00:30:44Z","content_type":null,"content_length":"34375","record_id":"<urn:uuid:e9539e23-30e9-490f-9684-70fbb55f7805>","cc-path":"CC-MAIN-2014-15/segments/1397609535535.6/warc/CC-MAIN-20140416005215-00589-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Topic: Entropy of linear function?
Replies: 0
weschrist Entropy of linear function?
Posted: Dec 22, 2012 4:35 PM
Posts: 1
From: Big Hi all,
Registered: Full disclosure, I'm a hydrogeologist... I play in the dirt and water. I love math and use it often, but I'm by no means an expert.
My question is in regards to Shannon Entropy = H(x). As I understand it, it is basically a measure of information contained in a signal.
H(x) = - sum(pi*log(pi)), where pi is the probability of getting value i
A continuous series of the same value (i.e. all 1's) would have pi=1 for all i, which gives H(x)=0. A series of N values would have pi=ni/N where ni is the number of time value i occurs.
I hope that's right, otherwise I'm in worse shape than I thought. A random set of values where no two values are the same would have pi=1/N. As N increases, the entropy increases.
So my question is, if you have a linearly increasing set of values where every value is greater than the previous, do you really have high entropy (high information content)? Or is there
some assumption of stationarity or transformation to ensure stationarity (i.e. take out the linear trend and drop the entropy to zero)?
Any insight would be greatly appreciated. | {"url":"http://mathforum.org/kb/thread.jspa?threadID=2422171&messageID=7942692","timestamp":"2014-04-17T13:39:34Z","content_type":null,"content_length":"14647","record_id":"<urn:uuid:be9d195c-cbe4-4534-bf11-45d24ffdf9b3>","cc-path":"CC-MAIN-2014-15/segments/1397609530131.27/warc/CC-MAIN-20140416005210-00079-ip-10-147-4-33.ec2.internal.warc.gz"} |
Theoretical Mechanics big ideas
I’ve taught Theoretical Mechanics before using Standards-Based Grading, but I think I want to make some tweaks to the focus of the standards. Here’s the old list. In this post I want to get my
current thoughts down about where the big areas should be. Note that the work Danny Caballero has done at the University of Colorado has helped me with these thoughts a lot (though there the course
is joined with math methods and spread over two semesters).
Ok, here’s my list of big ideas, that I hope, once refined, will be the main categories of standards for the course (they’re numbered for ease in referral in the comments).
1. Differential equations
1. Equations of motion are very compact ways of describing physical laws
2. I think students should understand the form, solutions, approaches, visual representations, and numeric work surrounding these
3. Typical introductions to this include air resistance trajectories and oscillators
2. Calculus of variations
1. I did this very early last time, and I think it worked out. Having the Lagrangian approach as an early tool comes in handy for oscillations, gravitation, etc.
2. I usually only do Lagrangian, not Hamiltonian, as I’m still unconvinced that it provides a tool that can do a lot more than the Lagrangian approach, given the level of abstraction needed to
learn it.
3. Momentum is king
1. Last time I enjoyed the class-wide conversation we had about whether momentum and kinetic energy are both needed to describe motion. This is really a focus on Newton’s laws.
4. Oscillations
1. damped, driven, etc.
5. Central force potentials
1. Kepler’s laws
2. I don’t like that students don’t know an ellipse when they see the mathematical evidence for it, but it’s a tour-de-force of physics.
3. I love Hohmann transfer problems.
6. Systems of particles
1. usefulness of center of mass
2. conservation issues
7. Noninertial frames
8. Rigid bodies
1. Linear algebra is not a pre-requisite for this class, so I’ve got some awkward work arounds for this.
9. Normal modes
1. Linear algebra is not a pre-requisite for this class, so I’ve got some awkward work arounds for this.
The linear algebra thing is really a problem for some of the cool stuff that happens with rigid bodies and normal modes, but I’m pretty happy with my normal modes solution, and slightly happy with my
rigid body approach.
One notable absence is gravitation as a main topic. It’s a chapter in the main books I’ve looked at, but I like putting the focus on general central force stuff.
Any thoughts would be greatly appreciated.
16 Responses to Theoretical Mechanics big ideas
1. Here’s one big idea that I think encompasses some of these: advanced mechanics is the story of r(t), v(t) and a(t) (or you could replace v(t) and a(t) with p(t)). What I found last quarter is
that students were familiar with r, v, and a, but not as functions of time that could be manipulated or that formed the basis of other functions like the Lagrangian and Hamiltonian.
I disagree with your thoughts on H; I would never stop at L. In my experience, the Hamiltonian is much, much more useful than L for working with anything but the simplest problems. Moving away
from second order equations to sets of first order equations is a big win both conceptually (e.g. phase space) and practically (numerical solutions). It also ties in nicely with QM, of course.
□ I really like the r, v, and a approach you’re suggesting here. The idea that they now need to be considered functions is very important, and perhaps can be considered part of my “differential
equations” section.
As for L vs H, here are 2 issues: 1) 2nd vs 1st order doesn’t matter if most of what you’re dealing with needs computational approaches, and 2) the connection to QM is not something I have
time for in this course. Do you think that connection should be done in a single course, or have the students see the two ends of the connection in two different courses?
☆ For computational purpose, maybe Mma doesn’t care, but most methods I’m familiar with are for sets of first order equations. Is that not true? Maybe it’s just my bias because of my
background in dynamical systems, but I really think that getting to H buys you a lot.
I don’t spend much time on the connection to QM, but most of are students take QM and CM together (or in close succession), so I think seeing H in a second context (and seeing q and p on
equal footing) is very valuable.
☆ Yeah, I should have been more clear: Mathematica doesn’t care what the order is when numerically integrating differential equations.
I’m still not convinced that the H connection between classical and quantum mechanics is done well in either course. What book do you think takes a stab?
☆ For example, here’s how Griffith’s QM book introduces the Hamiltonian:
“They (solutions to the TI-SWE) are states of difinite total energy. In classical mechanics, the total energy (kinetic plus potential) is call the Hamiltonian.”
Eventually things like commutators are developed, but I don’t think there’s really a strong connection with the Hamiltonian approach in classical mechanics.
☆ “Chaos in Classical and Quantum Mechanics” by Martin Gutzwiller? :) (http://www.amazon.com/Chaos-Classical-Quantum-Mechanics-Gutzwiller/dp/3540971734)
You mean of undergraduate texts? None. But that’s not what I’m trying to say. Seeing H and q and p in two different contexts lends something of a spiral approach that would otherwise be
I would say something similar about finding principal axes or normal modes and the connection to eigenvalues and wavefunctions in QM. I don’t spend a lot of time on it, but I do point out
that the same sort of stuff goes on in QM.
2. I’m also interested that you don’t do the Hamiltonian. While it may not be crucial for the classical mechanics course as you teach it, I think tend to think it is valuable to expose students to
the idea of the Hamiltonian before they hit quantum mechanics. I like students to see different pieces of a concept in different courses as it provides an opportunity to make connections, and
revisit what they have learned before.
□ I really don’t like the way that Thornton, Marion deal with the Hamiltonian. It doesn’t really motivate it other than to point out when things are conserved, and it definitely doesn’t give
you much purchase on the slope to understanding the quantum application.
☆ Have you looked at how Taylor deals with the Hamiltonian? It’s much better IMO.
☆ I’m embarrassed to admit I haven’t because I don’t have that book. How far down the path to QM does it go?
3. One point in favor of the Hamiltonian formalism is that it takes you to conjugate variables, and those lead to the link between physical symmetries and conservation laws — Noether’s theorem. And,
philosophically, that’s pretty heady stuff. Even if you don’t “do that” in “detail”, throwing it in as a forward-reference lecture bit can help students appreciate that the physics they’re
learning is connected to bigger, deeper things that they (hopefully) look forward to learning some day.
My very wise mentor and former teacher, Bill Gerace, likes to say that every course he teaches is 20% review (of stuff they should have learned before but probably don’t really get as well as
they should), 60% new material they should get, and 20% forward-reference material that they won’t really get this time around but will get much better in a future course for having seen it now.
Honor, don’t hide, the dangling ends at the edges of your course!
□ I do like the connection with conjugate variables, especially when connecting to the Heisenberg Uncertainty Principle. I also like the 20-60-20 idea. As usual, I’m trying to bring focus and
efficiency to my class. I’m so happy I use my blog now instead of blank sheets of paper for these brainstorming sessions.
□ I also like the idea of introducing the Hamiltonian in mechanics, even if it’s not fully grasped. You can use Poisson brackets to write out the algebra of classical observables and to make
the connection to quantum mechanics. The Poisson bracket makes it easy to find how observables change in time: $\frac{\mathrm{d}\mathcal{O}}{\mathrm{d}t} = \{f, \mathcal{H}\} + \frac{\partial
\mathcal{O}}{\partial t}$ It’s interesting that quantization is like choosing a the “quadratic” part of the algebra. Hamiltonian mechanics also has so many beautiful connections to the
symplectic geometry of phase space. There’s the Hamilton-Jacobi equations.
Utility? Maybe not so much.
☆ Yeah, this is deeper than I would usually choose to go in this course, Brian. I’m really starting to wonder if I should do more of this in quantum, though I haven’t taught that in a
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Circles and Ellipses
6.2: Circles and Ellipses
Created by: CK-12
Learning Objectives
• Understand the difference between an “oval” and an ellipse.
• Recognize and work with equations for ellipses.
• Derive the focal property of ellipses.
• Understand the equivalence of different definitions of ellipses.
• Reconstruct Dandelin’s sphere construction.
• Know some of the different ways people have approached ellipses throughout history.
• Understand some of the important applications of ellipses.
Let’s begin with the first class of shapes discussed in the last section. When the plane makes a finite intersection with one side of the cone, we get either a circle or the “oval-shaped” object
illustrated in the previous section. It turns out that this is no ordinary oval, but something called an ellipse, a shape with special properties.
Like parallelograms, or any other shape with lots of interesting properties, ellipses can be defined by some of these properties, and then the other properties necessarily follow from the definition.
For example, a parallelogram is typically defined as a quadrilateral with each pair of opposite pairs of sides parallel. Once you define it this way, it follows that the opposite sides must also be
equal in length, and that the diagonals must bisect each other. Well, if you instead started by defining a parallelogram by one of these other properties, for instance opposite sides having equal
lengths, then you would end up with the same class of shapes. The same thing happens for ellipses. One way to define an ellipse is as a “stretched out circle”. It’s the shape you would get if you
sketched a circle on a deflated balloon and then stretched out the balloon evenly in two opposite directions:
It’s also the shape of the surface of water that results when you tilt a round glass:
Or an ellipse could be thought of as the shape of a circle drawn on a piece of paper when it is viewed at an angle.
Equations of Ellipses
This “stretching” can be represented algebraically. For simplicity, take the circle of radius 1 centered at the origin (0,0). The distance formula tells us that this is the set of points $(x, y)$
$D & = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2} \\1 & = \sqrt{(x - 0)^2 + (y - 0)^2} \\ 1 & = x^2 + y^2$$x^2 + y^2 = 1$$x-$$x$$a>1$
$\left (\frac{x}{a} \right )^2 + y^2 = 1$
Why does this stretch the circle horizontally? Well, the effect of dividing $x$$a$$y-$$(x, y)$$x$$a$$(x, y)$$(ax, y)$$a$$\left (\frac{x}{2} \right )^2 + y^2 = 1$
Generalizing the equation by allowing a stretch in the vertical direction, we get the following.
$\left (\frac{x}{a} \right )^2 + \left (\frac{y}{b} \right )^2 =1$
The factor $a$$b$$a=b$$a eq b$$b < a$$a < b$
$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$
This is called the standard form of the equation of an ellipse, assuming that the ellipse is centered at (0,0).
To sketch a graph of an ellipse with the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$x$$y$
Example 1 Sketch the graph of $\frac{x^2}{4} + \frac{y^2}{9} = 1$
Solution: This equation can be rewritten as $\frac{x^2}{2^2} + \frac{y^2}{3^2} = 1$$x = 0$$y = 0$$y-$$x-$
Example 2 Sketch the graph of $\frac{x^2}{16} + y^2 = 1$
Solution: This can be rewritten as $\frac{x^2}{4^2} + \frac{y^2}{1^2} = 1$
The segment spanning the long direction of the ellipse is called the major axis, and the segment spanning the short direction of the ellipse is called the minor axis. So in the last example the major
axis is the segment from (-4,0) to (4,0) and the minor axis is the segment from (0,-1) to (0,1).
The major and minor axes are examples of what are sometimes called reference lines. Apollonius, the Ancient Greek mathematician who wrote an early treatise on conics, used these and other reference
lines to orient conic sections. Though the Greeks did not use a coordinate plane to discuss geometry, these reference lines offer a framing perspective that is similar to the Cartesian plane that we
use today. Apollonius’ way of framing conics with reference lines was the closest mathematics came to the system of coordinate geometry that you know so well until Descartes’ and Fermat’s systematic
work in the seventeenth century.
Example 3 Not all equations for ellipses start off in the standard form above. For example, $25x^2 + 9y^2 = 225$is an ellipse. Put it in the proper form and graph it.
Solution: First, divide both sides by 225, to get: $\frac{x^2}{9} + \frac{y^2}{25} = 1$$\frac{x^2}{3^2} + \frac{y^2}{5^2} = 1$
Review Questions
1. It was mentioned above that when a round glass of water is tilted, the surface of the water is an ellipse. Using our working definition of an ellipse as “stretched out circle”, explain why you
think the water takes this shape.
2. Sketch the following ellipse: $36x^2 + 25y^2 = 900$
3. Now try sketching this ellipse where the numbers don’t turn out to be so neat: $3x^2 + 4y^2 = 12$
Review Answers
1. Answers may vary, but should explain why the shape that results stretches a circle in one direction because the width of the glass is constant.
The Focal Property
In every ellipse there are two special points called the foci (foci is plural, focus is singular), which lie inside the ellipse and which can be used to define the shape. For an ellipse centered at
(0,0) that is wider than it is tall, its major axis is horizontal and its foci are at $\left ( \sqrt{a^2 - b^2},0 \right )$$\left ( - \sqrt{a^2 - b^2},0 \right )$
What is the significance of these points? The ellipse has a geometric property relating to these points that is similar to a circle’s relationship with its center. Remember a circle can be thought of
as the set of points in a plane that are a certain distance from the center point. In fact, that is typically the definition of a circle. Well, the foci act like the center except that there are two
of them. An ellipse is the set of points where the sum of the distance between each point on the ellipse and each of the two foci is a constant number. In the diagram below, for any point $P$$F_1 P +
F_2P = d$$F_1$$F_2$$d$
This definition using the sum of the focal distance gives us a great way to draw ellipses. Sure, you can just graph them on your calculator. But why not utilize a simpler technology that does the job
just as well? As you know, a circle could be drawn by fixing a string to a piece of paper, tying the other end to a pencil, and then drawing the curve that keeps the string taut.
Similarly, an ellipse can be drawn by taking a string that is longer than the distance between two points, fixing the two ends of the string to the two points. Then drawing all points that can be
drawn when the string is taut and the pencil is touching it. In the diagram above, the dotted line represents the string of fixed length, which is attached at the foci $F_1$$F_2$$P$$F_1$$F_2$
Review Questions
4. Use string and tacks to draw ellipses that
1. are nearly circles
2. are very different than circles
For each of these, what can you say about how the distance between the foci and the length of the string compare?
Review Answers
4. Drawings may vary. For ellipses that are nearly circles, the distance between the foci is small compared to the length of string.
The foci can also be used to measure how far an ellipse is “stretched” from a circle. The symbol $\varepsilon$eccentricity of an ellipse, and it is defined by the distance between the foci divided by
the length of the major axis, or $\frac{\sqrt{a^2 - b^2}}{a}$$\frac{\sqrt{b^2 - a^2}}{b}$$a = b$
Review Questions
5. What is the full range of the eccentricity of an ellipse? What does it look like near the extremes of this range?
Review Answers
5. The interval of possible values is $\varepsilon \in [0,1)$$\varepsilon = 0$
Turning the Definition of Ellipses on its Head
Often, this focal property is not thought of as a property of ellipses, but rather a defining feature. To see that these are equivalent, we have some to work to do. Let’s start by proving that
stretched out circles actually have this focal property. We want to prove that for a “stretched out circle” defined by the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$a$$b$$\left ( \sqrt{a^2 - b
^2},0 \right )$$\left ( - \sqrt{a^2 - b^2},0 \right )$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$\left ( \sqrt{a^2 - b^2},0 \right )$$\left ( - \sqrt{a^2 - b^2},0 \right )$
Proof: Suppose a point $(x,y)$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$y$$x$
$\frac{y^2}{b^2} & = 1 - \frac{x^2}{a^2} \\y^2 & = \frac{a^2b^2 - b^2x^2}{a^2} \\y & = \sqrt{\frac{a^2b^2-b^2x^2}{a^2}}$
Now, using the distance formula to compute the sum of the distance between the pairs of points: $\left (x, \sqrt{\frac{a^2b^2-b^2x^2}{a^2}} \right )$$\left ( \sqrt{a^2 - b^2},0 \right )$$\left (x, \
sqrt{\frac{a^2b^2-b^2x^2}{a^2}} \right )$$\left(-\sqrt{a^2-b^2},0\right)$
$\sqrt{\left (x-\sqrt{a^2 - b^2} \right )^2 + \frac{a^2b^2-b^2x^2}{a^2}} + \sqrt{\left (x+\sqrt{a^2 - b^2} \right )^2 + \frac{a^2b^2-b^2x^2}{a^2}}$
This algebraic equation looks daunting, but simplifying the expressions inside these square roots results in a surprising result. This becomes
$& \sqrt{\frac{a^2 \left (x - \sqrt{a^2 - b^2} \right )^2 + a^2b^2 - b^2 x^2}{a^2}} + \sqrt{\frac{a^2 \left (x+ \sqrt{a^2 - b^2} \right )^2 + a^2 b^2 - b^2x^2}{a^2}} \\& = \frac{1}{a} \left (\sqrt{a^
2 \left (x- \sqrt{a^2 - b^2} \right )^2 + a^2b^2 - b^2x^2} + \sqrt{a^2 \left (x+ \sqrt{a^2 - b^2} \right )^2 + a^2b^2 - b^2x^2} \right ) \\& = \frac{1}{a} \left (\sqrt{a^2x^2 - 2a^2x \sqrt{a^2-b^2} +
a^4 - a^2b^2 + a^2b^2 - b^2x^2} + \sqrt{a^2x^2 - 2a^2x\sqrt{a^2 - b^2} + a^4 - a^2b^2 + a^2b^2 - b^2x^2} \right ) \\& = \frac{1}{a} \left ( \sqrt{a^2x^2 - 2a^2 x \sqrt{a^2 - b^2} + a^4 - b^2x^2} + \
sqrt{a^2x^2 - 2a^2x \sqrt{a^2 - b^2} + a^4 - b^2x^2} \right ) \\& = \frac{1}{a} \left (\sqrt{\left (a^2 - x \sqrt{a^2 - b^2} \right )^2} + \sqrt{\left (a^2 + x \sqrt{a^2 - b^2} \right )^2} \right ) \
\& = \frac{1}{a} \left (a^2 - x \sqrt{a^2 - b^2} + a^2 + x \sqrt{a^2 - b^2} \right ) \\& = \frac{1}{a} (2a^2) \\& = 2a$
What a miraculous collapse! One of the gnarliest algebraic expressions that I have ever encountered turned into the simple expression $2a$$x$$y$The sum of the distances between every point on the
ellipse and the two foci is always $2a$$2a$$x-$$(a,0)$
Review Questions
6. Compute the distance from the $x-$$(a,0)$$2a$
7. What is the sum of the distances to the foci of the points on a vertically-oriented ellipse?
Review Answers
6. The distance between the $x-$$(a,0)$$\left (\sqrt{a^2 - b^2}, 0 \right )$$a - \sqrt{a^2 - b^2}$$x-$$(a,0)$$\left (- \sqrt{a^2 - b^2}, 0 \right )$$a + \sqrt{a^2 - b^2}$$a - \sqrt{a^2 - b^2} + a +
\sqrt{a^2 - b^2} = 2a$
7. $2b$
Defining an Ellipse by Focal Distance
So all this computing simply means that “stretched-out circles”—what we’ve been calling ellipses—satisfy the focal property. What would be great is if we could define ellipses by the focal property.
This would be a nice generalization of the way we define circles.
Recall that circles are defined as the set of points in a plane that are a constant distance from a center point. Analogously, ellipses could be defined as a set of points in a plane for which the
sum of the distances to two focus points is a constant. In other words, this definition would yield the exact same set of shapes as the “stretched out circle” definition that we started with.
Before proceeding we need to make sure of one thing. We’ve already proved that stretched out circles satisfy the focal property, but how do we know that any shape satisfying the focal property is in
turn a stretched out circle? Well, the calculation above can simply be read backwards. In other words, suppose you have a set of points that satisfy the focal property, that each point whose sum of
the distances to the points $(f,0)$$(-f,0)$$d$$d$$f$$2f<d$$a$$b$$d = 2a$$f = \sqrt{a^2 - b^2}$$(x, y)$$\left (\sqrt{a^2 - b^2}, 0 \right )$$\left (- \sqrt{a^2 - b^2}, 0 \right )$$2a$
Review Questions
8. Explain why for any two positive numbers $d$$f$$2f<d$$a$$b$$d = 2a$$f = \sqrt{a^2 - b^2}$
9. We just told you that the above proof could be read backwards. But you need to be careful when following algebraic steps backwards, especially ones involving squares or square roots.
1. For example, what happens when you follow this argument backwards? $x & = -2 \\x^2 & = 4$
2. Write a convincing argument that it is okay to follow the steps backwards in the above proof that every stretched circle has the focal property.
Review Answers
8. Set $a = \frac{d}{2}$$f$$2f<d$$b$$f = \sqrt{a^2 - b^2}$$2f < d, 2f < 2a$$a$$d > 0$$f < a$$b$$f < a$$f$$a$$b$ Then the Pythagorean Theorem tells us that $f^2 + b^2 = a^2$$f = \sqrt{a^2 - b^2}$$a$
1. Taking the square root of both sides of $x^2 = 4$$x = \pm 2$$(x = -2)$$(-2)^2$$2^2$
2. The student’s reason should include the fact no information is lost (through squaring both sides or other operations) in any of these steps, so that each step is completely reversible.
Equation of an Ellipse Not Centered at the Origin
All the ellipses we’ve looked at so far are centered around the origin (0,0). To find an equation for ellipses centered around another point, say $(h, k)$$x$$x-h$$y$$y-k$$h$$h < 0$$k$$k < 0$
$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$
It is centered about the point $(h,k)$$b<a$$\left ( h + \sqrt{a^2 - b^2}, k \right )$$\left ( h - \sqrt{a^2 - b^2}, k \right )$$a<b$$\left ( h,k + \sqrt{a^2 - b^2} \right )$$\left ( h,k - \sqrt{a^2 -
b^2} \right )$
Example 4
Graph the equation $4x^2 + 8x + 9y^2 - 36y + 4 = 0$
Solution : We need to get the equation into the form of general equation above. The first step is to group all the $x$$y$$x^2$$y^2$
$4(x^2 + 2x) + 9(y^2 - 4y) = -4$
Now, we “complete the square” by adding the appropriate terms to the $x$$y$http://authors.ck12.org/wiki/index.php/Algebra_I-Chapter-10#Solving_Quadratic_Equations_by_Completing_the_Square for more on
completing the square.)
$4(x^2 + 2x + 1)+ 9(y^2 - 4y + 4) = -4 + 4 + 36$
Now we factor and divide by the coefficients to get:
$\frac{(x+1)^2}{9} + \frac{(y-2)^2}{4} = 1$
And there we have it. Once it’s in this form, we see this is an ellipse is centered around the point (-1,2), it has a horizontal major axis of length 3 and a vertical minor axis of length 2, and from
this we can make a sketch of the ellipse:
Review Questions
10. Explain why subtracting $h$$x-$$k$$y-$$h$$k$
11. Graph this ellipse. $x^2 - 6x + 5y^2 - 10y - 66 = 0$
12. Now try this one that doesn’t have such nice numbers! $16x^2 - 48x + 125y^2 + 150y + 61 = 0$
13. Now try this one. $3x^2 - 12x + 5y^2 + 10y - 3 = 0$
14. What about this one. $5x^2 - 15x - 2y^2 + 8y - 50 =0$
Review Answers
10. If $(x,y)$$\frac{(x-h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$$(x+h,y+k)$$\frac{(x-h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$$h$$k$
13. After completing the square, we have the sum of positive numbers equaling a negative number. This is an impossibility, so the equation has no solutions.
14. After completing the square, the $x$$y$
Difference Between an Ellipse and an Oval or Proving the Sliced Cone Definition of an Ellipse
There is still one critical step missing in our exploration of ellipses. We showed that “stretched-out circles” satisfy the focal property, and that any shape satisfying this property is in fact a
“stretched-out circle”. So these are actually the same class of shapes, and they are called ellipses. But not any oval-shaped curve is an ellipse. Draw a random oval and you’re not likely to be able
to find two points that satisfy the focal property. In particular, when we cut a cone with a tilted plane, how do we know that the oval-shaped curve that results is a “stretched out circle”
satisfying the focal property?
Amazingly, the Ancient Greeks had an argument for this fact over two millennia ago. While it is impressive that this problem was solved so long ago, the argument itself involves an intricate
construction that isn’t as illuminating as a more modern the one I’m going to show you instead. Most mathematicians prefer to use a more modern argument that is simply stunning in its simplicity.
This modern argument isn’t fancy—the Greeks had all the tools they need to understand it—they just didn’t happen to think of it. It wasn’t until 1822 that the French mathematician Germinal Dandelin
thought of this very clever construction. Dandelin found a way to find the foci and prove the focal property in one fell swoop. Here’s what he said.
Take the conic section in question. Then choose a sphere that is just the right size so that when it’s dropped into the conic, it touches the intersecting plane, as well as being snug against the
cone on all sides. If you prefer, you can think of the sphere as a perfectly round balloon that is blown up until it “just fits” inside the cone, still touching the plane. Then do the same on the
other side of the plane. After we’ve drawn both of these spheres we have this picture:
These spheres are often called “Dandelin spheres”, named after their discoverer. It turns out that not only is our shape an ellipse (which, like all ellipses satisfies the focal property), but these
spheres touch the ellipse exactly at the two foci. To see this, consider this geometric argument.
The first thing to notice is that the circles $C_1$$C_2$$l$$d$$C_1$$C_2$$C_1$$C_2$
The next thing to remember is a property of tangents to spheres that you may have learned in geometry. If two segments are drawn between a point and a sphere, and if the line containing each segment
is tangent to the sphere, then the two segments are equal. In the diagram below, $AB=AC$this description for more about this property.)
Now consider the point $P$$\overline{QR}$$d$$C_1$$C_2$$P$$d_1$$d_2$$d_1 =RP$$d_2 =PQ$$d_1 + d_2 = RP + PQ = QR = d$$d$$P$
Review Questions
15. What do the Dandelin spheres look like in the case of a circle?
16. What is the area of an ellipse with the equation $\frac{(x-h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$
17. What is the perimeter of an ellipse with the equation $\frac{(x-h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$
Review Answers
15. The Dandelin spheres for a circle lie directly above one another, and both touch the circle at the center point.
16. The area of an ellipse is $ab \pi$$\pi$$a$$x-$$b$$y-$$a$$b$$ab$$ab \pi$
17. This is actually a much more difficult question than the previous one. You’re on your own! Even the great Indian mathematician Ramanujan could only come up with an approximation: $p \approx \pi \
left[3 (a+b) - \sqrt{(3a+b)(a+3b)}\right]$
The number of places ellipses appear in the natural world is immense. Consider, for instance, how often the simplest kind of ellipse, the circle, appears in your life. You see circles emanating as
waves when you throw a rock into a pond. You see circular pupils when you look at a set of eyes. You see what is roughly a circle as the image of the sun or moon in the sky. The path of an object
swung around on a string is a circle. When any of these circles are viewed at an angle you see an elongated circle, or an ellipse. So it is important to keep in mind that the discussion that follows
covers only a few of the instances of ellipses in our daily lives.
Planetary Motion
When a planet orbits the sun (or when any object orbits any other), it takes an elliptical path and the sun lies at one of the two foci of the ellipse. Johannes Kepler first proposed this at the
beginning of the seventeenth century as one of three laws of planetary motions, after analyzing observational data of Tycho Brahe. His law is accurate enough to produce modern computations which are
still used to predict the motion of artificial satellites. A century later, Newton’s law of gravity offers an explanation of why this law might be true.
Review Questions
18. Though planets take an elliptical path around the sun, these ellipses often have a very low eccentricity, meaning they are close to being circles. The diagram above exaggerates the elliptical
shape of a planet’s orbit. The Earth’s orbit has an eccentricity of 0.0167. Its minimum distance from the sun is 146 million km. What is its maximum distance from the sun? If the sun’s diameter
is 1.4 million kilometers. Do both foci of the Earth’s orbit lie within the sun? Recall that the eccentricity of an ellipse is $\varepsilon = \frac{\sqrt{a^2 - b^2}}{a}$
19. While the elliptical paths of planets are ellipses that are closely approximated by circles, comets and asteroids often have orbits that are ellipses with very high eccentricity. Halley’s comet
has an eccentricity of 0.967, and comes within 54.6 million miles of the sun at its closest point, or “perihelion”. What is the furthest point it reaches from the sun?
Review Answers
18. Assume that the orbit of the sun is an ellipse centered at (0,0). Then we can use the distance from the origin to the focus $\sqrt{a^2 - b^2}$$146 + 146 + 2 \sqrt{a^2 - b^2} = 2a$$0.167 = \frac{\
sqrt{a^2 - b^2}}{a}$$a = 175.270, b = 175.245$$c = 2.927$$2(2.927)=5.854$
19. $\sim 3.25$
Echo Rooms
The National Statuary Hall in the United States Capital Building is an example of an ellipse-shaped room, sometimes called an “echo room”, which provide an interesting application to a property of
ellipses. If a person whispers very quietly at one of the foci, the sound echoes in a way such that a person at the other focus can often hear them very clearly. Rumor has it that John Quincy Adams
took advantage of this property to eavesdrop on conversations in this room.
The property of ellipses that makes echo rooms work is called the “optical property.” So why echoes, if this is an optical property? Well, light rays and sound waves bounce around in similar ways. In
particular, they both bounce off walls at equal angles. In the diagram below, $\alpha = \beta$
For a curved wall, they bounce at equal angles to the tangent line at that point:
So the “optical property” of ellipses is that lines between a point on the ellipse and the two foci form equal angles to the tangent at that point, or in other words, whispers coming from one foci
bounce directly to the other foci. In the diagram below, for each $Q$$\angle{\alpha} \cong \angle{\beta}$
This seems reasonable, given the symmetry of the ellipse, but how do we know it is true? First, let’s prove an important property that is a bit more general. Suppose you have two points, $P_1$$P_2$
A nice way to find this is to reflect $P_2$$L$$P_2'$
Since $l$$P_1$$P_2'$$P_1$$P_2'$$l$$P_1$$P_2'$$P_1$$P_2$$l$$P_2$$P_1$$P_2'$$P_1$$P_2$$P_1$$P_2'$$l$$P_2'$$l$$P_2$$P_1$$P_2$$l$$P_1 Q$$Q P_2$
The part of the above diagram that is going to help us prove the optical property is that since $\angle{1} \cong \angle{2}$$\angle{2} \cong \angle{3}$$\angle{1} \cong \angle{3}$$P_1$$P_2$$l$
Now to prove the optical property of the ellipse, apply the above situation to the ellipse. In the picture below, $P_1$$P_2$$Q$$l$
The optical property states that $\angle{\alpha} \cong \angle{\beta}$$Q$$l$$Q$$l$$P_1$$P_2$$\angle{\alpha} \cong \angle{\beta}$
Review Questions
20. Design the largest possible echo room with the following constraints: You would like to spy on someone who will be 3 m from the tip of the ellipse. The room cannot be more than 100 m wide in any
direction. How far from the person you’re spying on will you be standing.
Review Answers
20. The echo room has a major axis of 100 m and a minor axis of 34.12 m. Situating the room in the coordinate plane, the room can be represented by the equation: $\frac{x^2}{2500} + \frac{y^2}{291} =
Conic sections help us solve the problem of making a sundial. Depending on the season, the sun shines at a different angle. However, due to the elliptical nature of the Earth’s orbit about the sun, a
shadow-casting stick can be placed in such a way that the shadow always tells the correct time of day, no matter what the time of year, as long as the stick is lined up with the earth’s pole of
Review Questions
21. No matter what the orientation of a stick, if you trace out the path that the shadow of the tip makes on a flat surface, you will find it is an ellipse. Describe why this is true.(HINT: for
simplicity, you can assume that you are making the measurements throughout the course of one day and that with the exception of the earth rotating about a pole, the sun and the earth are fixed
with respect to one another.)
22. It was mentioned earlier in the chapter that when a round glass of water is tilted, the surface of the water is an ellipse. Or, in other words, this statement is claiming that the cross section
of a cylinder is an ellipse. Prove that this is true. (Hint: to prove that this is an ellipse all you need to do is show that for any cross section of a cylinder there exists a cone that has the
same cross section.)
23. From the exercise above, it appears that there is some overlap between “conic sections” and “cylindrical sections”. Are any of the classes of conic sections we found in the last section not
cylindrical sections? Are there any cylindrical sections that are not conic sections?
Review Answers
21. Answers may vary.
22. Answers may vary.
23. Answers may vary.
A conic section that can be equivalently defined as: 1) any finite conic section, 2) a circle which has been dilated (or “stretched”) in one direction, 3) the set of points in which the sum of
distances to two special points called the foci is constant.
Major axis
The segment spanning an ellipse in the longest direction.
Minor axis
The segment spanning an ellipse in the shortest direction.
One of two points that defines an ellipse in the above definition.
A measure of how “stretched out” an ellipse is. Formally, it is the distance between the two foci divided by the length of the major axis. The eccentricity ranges from 0 (a circle) to points
close to 1, which are very elongated ellipses.
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[Edu-sig] Basic dictionary question
Kirby Urner urnerk at qwest.net
Sun Oct 9 07:54:39 CEST 2005
Talk about fishing for expert help!
Thanks Guido.
It's a pruning algorithm where I strip way pieces that don't meet up at
(x,y,z) bridge points. Lots of symmetry about the origin so just pure
distance won't work (not unique enough).
I think might still get away with a tuple-indexed dict (tallying for pairs,
pruning the rest) if I boil my floating points down to fixed decimals -- a
chance to play with the new decimal type perhaps.
I appreciate the quick lecture on intransitive nature of fuzzy equality and
__hash__ vs. __eq__. Helps me think.
> You never said what you wanted to do with these points -- are you
> looking to find which points are in fact close enough to be considered
> the same point? I suppose you could sort them by the vector length
> (sqrt(x**2 + y**2 + z**2), that sort of thing) and then you only have
> to compare points that are close together in the sorted list (as long
> as the abs values are closer than your epsilon). Or you could divide
> 3-space up into cubes with sides equal epsilon, and then you'd only
> have to look for points that end up in the same or in adjacent cubes.
> --Guido
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Principles of hydrometer analysis
The physical principles of sedimentation underlying the hydrometer analysis are presented in a number of texts including Das (2002); we briefly review them here. The hydrometer analysis applies
Stokes’s law, which governs the terminal velocity at which spherical particles settle through a column of fluid (Craig, 1992). Stokes’s law assumes particles that (1) are rigid, spherical, and
smooth; (2) have similar density; (3) are separated from each other; (4) do not interact during sedimentation; and (5) are large enough so that settlement is not governed by Brownian motion. The law
is also strictly applicable to slow fluid movements that display laminar flow patterns (i.e., Reynolds number = <1) (Wen et al., 2002).
Hydrometer analysis begins after thoroughly mixing the sediment and water, after which particles settle out of the water column according to Stokes’s law. The density of a sediment-water suspension
depends on the concentration and specific gravity of the sediments present in the mixture. If the suspension is allowed to stand, particles will settle out of the suspension and the density of the
sediment-water suspension will decrease. A hydrometer measures the density of the suspension at a known depth below the surface.
The two basic calculations made during a hydrometer analysis are the particle diameter at a specific time and depth and the percentage of the original sample mass still left in suspension. We
calculate the particle diameter according to the following equation:
• D = equivalent sedimentation diameter of particle (millimeters),
• η = viscosity of water (grams seconds per square centimeter),
• G[s] = specific gravity of sediment,
• L = effective depth measured from water surface to center of gravity of hydrometer bulb (centimeters), and
• t = time measured from start of sedimentation (seconds).
The percentage of particles remaining in suspension finer than particle diameter, D, is
• G[s] = specific gravity of sediment,
• V = total water-sediment volume (1000 mL),
• M = dry sample mass (grams),
• R[h] = corrected hydrometer reading of slurry mixture (grams per liter), and
• B = hydrometer reading of reference mixture of dispersing agent and distilled water (grams per liter). | {"url":"http://publications.iodp.org/proceedings/308/205/205_4.htm","timestamp":"2014-04-17T12:36:03Z","content_type":null,"content_length":"10497","record_id":"<urn:uuid:0a1dacb6-acf1-4af0-bfc0-fdaea10027ed>","cc-path":"CC-MAIN-2014-15/segments/1397609530131.27/warc/CC-MAIN-20140416005210-00538-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Traveling Wavefronts of Competing Pioneer and Climax Model with Nonlocal Diffusion
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 725495, 12 pages
Research Article
Traveling Wavefronts of Competing Pioneer and Climax Model with Nonlocal Diffusion
School of Mathematics, South China Normal University, Guangzhou 510631, China
Received 17 November 2012; Revised 11 March 2013; Accepted 21 March 2013
Academic Editor: Dumitru Baleanu
Copyright © 2013 Xiaojing Yu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly cited.
We study a competing pioneer-climax species model with nonlocal diffusion. By constructing a pair of upper-lower solutions and using the iterative technique, we establish the existence of traveling
wavefronts connecting the pioneer-existence equilibrium and the coexistence equilibrium. We also discuss the asymptotic behavior of the wave tail for the traveling wavefronts as .
1. Introduction
As we know, the interactions among species are important in determining the process of evolution for the ecosystem, and the modeling accompanied with the mathematical analysis of the models can help
people to understand and control the propagation of species. In general, the per capital growth rate (i.e., fitness) for a species in the model is assumed to be a function of a weighted total density
of all interacting species. A well-known example is the standard Lotka-Volterra model; its fitness of a species is a linear function. It is natural to consider other kinds of fitness functions other
than the linear one, because of the various species and interaction rules. In this paper, we will analyze a reaction-diffusion model describing pioneer and climax species. This model describes
interaction among species with peculiar fitness functions.
A species is called a pioneer species if it thrives best at lower density but its fitness decreases monotonically with total population density for overcrowded. Thus, the fitness function of a
pioneer species is assumed to be a decreasing function. Pine and yellow poplar are the species of this type. A species is called a climax species if its fitness increases up to a maximum value and
then decreases of its total density. Hence, a climax population is assumed to have a nonmonotone, “one-humped” smooth fitness function. Oak and maple are the climax species.
A typical reaction-diffusion model for a pioneer-climax species is given by the following system: where and represent densities of the pioneer and climax species, respectively. and denote the pioneer
fitness function and climax fitness function, respectively, . By making changes of variables , , system (1) changes into the form (the tildes of and are dropped out) where we still use and as the new
coefficients without confusing.
From the previous introduction, we assume that the pioneer fitness function satisfies for some , and the climax fitness function satisfies Ricker [1] used the fitness function , Hassell and Comins [2
] used the fitness function , and Cushing [3] used the fitness function . It is obvious that these and have the curves in Figure 1.
There are some existing results about the stability and traveling wave solutions for (2) ([4–6]). About traveling wave solutions, Brown et al. [4] studied the traveling wave of (1) connecting two
boundary equilibria by singular perturbation technique, and Yuan and Zou [6] obtained the existence of traveling wave solutions connecting a monoculture state and a coexistence state by upper-lower
solution method combined with the Schauder fixed point theorem. Also see van Vuuren [7] for the existence of traveling plane waves in a general class of competition-diffusion systems and Murray [8]
for more biological description of traveling wave solutions.
For system without spatial diffusion, the model will be Selgrade and Roberds [9], Sumner [10] analyzed the Hopf bifurcation of (5), and Selgrade and Namkoong [11], Sumner [12] considered the stable
periodic behavior of (5). Because of the existence of rich equilibria and the various ranges of parameters, the dynamics of ordinary differential system (5) are complex, and a detailed review of all
equilibrium types can be found in Buchanan [13, 14].
Although the Laplacian operator is always used to model the diffusion of the species, it suggests that the population at the location can only be influenced by the variation of the population near .
As we know that the individuals can move freely, then the movement of individuals is bounded to affect the other individuals. So, the Laplacian operator may have some shortage to describe the
diffusion. One way to deal with this problem is to replace the Laplacian operator with a convolution diffusion term This implies that the probability distribution function for the population at
location moving to the location is . At time , the total individuals that move from the whole space into the location will be . Therefore, one may call it as a nonlocal diffusion, and,
correspondingly, call as a local diffusion. During the recent years, the models with the nonlocal diffusion have been attracted much more attentions (see [15–18]).
In this paper, instead of (2), we will concentrate on the following pioneer-climax species with nonlocal diffusion: where , are positive constants accounting for the diffusivity, is a kernel function
which is continuous on satisfying
We are interested in traveling wavefronts accounting for a mildinvasion of the two species (traveling wavefronts connecting a boundary equilibrium and the coexistence equilibrium). For system (2),
the sufficient condition for (2) to have a mildinvasion is (see [6]), but our condition in this paper for system (6) is , which reveals a fact that the nonlocal diffusion of either the pioneer
species or the climax species did affect the climax invasion and wave propagation. Please see Section 5 for the discussion.
The remaining of this paper is organized as follows. In Section 2, there are some preliminaries about the equilibria and the system is transformed into a cooperative one. In Section 3, we prove the
existence of traveling wavefronts by using an iteration scheme combined with a pair of admissible upper and lower solutions, which can be constructed obviously, and thus a criterion of the existence
for traveling wavefronts is obtained. We also give a discussion on asymptotic behavior for the traveling wavefront tail as in Section 4. At last, we give some concluding discussions in Section 5.
2. Preliminaries
It is evident that is a trivial equilibrium of (6). The system (6) has at least four equilibria and at most six equilibria. The existence of nonnegative steady states depends on the locations of the
three nullclines: The long-term behavior of solutions to (6) can be qualitatively different caused by the different number, distribution, and types of equilibria. The dynamics of the system (6) are
of course very rich and complex. However, in this paper, we will only consider the following case: The condition follows as a sequence. Under the previous assumption, (6) has four nontrivial
equilibria: , , , and except for , where It is obvious that . We further assume that for the technical reason. See Figure 2 for this situation.
As mentioned in the introduction, we are interested in the coexistence of the two species. That means that we will seek traveling wavefronts connecting equilibrium and equilibrium .
By making changes of variables , and dropping the tildes, system (6) becomes and the equilibria , are changed into , , respectively, where , .
3. Existence of Traveling Wavefronts
A traveling wavefront of (6) connecting equilibria and can be changed into a traveling wavefront of (11) connecting and . Therefore, we consider the system (11) hereby.
A traveling wave solution of (11) is a solution with the form and , where and is a wave speed. A traveling wavefront is a traveling wave solution which has finite limits . Denoting the traveling wave
coordinate still by , we derive the wave profile system from (11): Associated with (12), we consider its solutions subject to the following boundary value conditions:
For , , implies ; implies but ; implies . Furthermore, the norm in is the Euclidean norm. Define For some constants , , letting , , we define an operator by Then, (12) can be written as an equivalent
form: Denote by It is obvious that a traveling wave solution of the problem (12) and (13) is a fixed point of and vice verse.
The following lemma states the monotone property of .
Lemma 1. Assume that holds, for sufficiently large and , with and , ; one has(i),, (ii),for all .
Proof. In order to prove (i), let , . Then For , and sufficiently large , it follows that , . Thus, if and for , we have
For (ii), we know that and for . It follows that From , , we have Note that for . This leads to The proof is complete.
The conclusion of the following lemma is direct.
Lemma 2. Assume that , are sufficiently large. For with nondecreasing on , , are also nondecreasing on .
We can easily see that also enjoys the same properties as those for settled in Lemmas 1 and 2.
Let and It is clear that is a Banach space equipped with the norm defined by .
Definition 3. A pair of continuous functions , is called an upper solution and a lower solution of (12), respectively, if there exists a set such that and are differentiable in and the essential
bounded functions satisfy for .
In what follows, we assume that (12) has an upper solution and a lower solution , such that(P1) for ;(P2), ;(P3) and are nondecreasing.
Define the following profile set by It is obvious that is nonempty.
For and , define , and
Lemma 4. For , the functions , and , defined by (26) satisfy(i); (ii) for ;(iii) and is a pair of upper and lower solutions of (12);(iv) and are continuously differentiable on .
Proof. We only give the argument for , and the situation for can be obtained by mathematical induction. From Definition 3, we obtain Let , . For any , there exists some such that or . We then derive
where .
By similar arguments, we can get The previous arguments implies that the conclusion (ii) holds.
From the monotone property of , we can easily obtain that , are nondecreasing for , and therefore the conclusion (i) holds.
For , by Lemma 1, we have In a similar way, we can prove that for . This indicates that and are a pair of upper and lower solutions of (12).
The conclusion (iv) is obvious. The proof is complete.
Lemma 5. , and the convergence is uniform with respect to the decay norm .
Proof. We have from Lemma 4 that the following limit exists: It is easy to know that is a closed and convex set. By the nondecreasing property of , we have . In the following, we prove that the
convergence is uniform with respect to the decay norm.
Since for any , there exists a , such that for all , Now, we consider the sequences for . Note is nondecreasing on , and thus By , there exists a positive constant , such that for . Similarly, we can
prove that there exists a positive number , such that for .
From the previous estimates, we know that is equicontinuous on with respect to the supremum norm. On the other hand, we have from Lemma 4 (ii) that is uniformly bounded. By Arzéla-Ascoli theorem,
there exist subsequences of which are uniformly convergent in . Without loss of generality, we still express this subsequence as . Thus, there exists a positive integer , such that Furthermore, we
Summarizing the previous arguments, for we have The proof is complete.
Theorem 6. Assume that holds; if (12) has a pair of upper and lower solutions that satisfy (P1)–(P3), then the system (11) has a traveling wavefront satisfying (13).
Proof. By the Lebesgue's dominated convergence theorem and the iteration scheme (26), we have Therefore, is a fixed point of , which also satisfies (12). Furthermore, (P2) indicates that satisfy . On
the other hand, we have from the monotonicity of that exists. Furthermore, since , we know that . By using L’Hôspital’s rule, we obtain Similarly, one can obtain . That is, is an equilibrium of (12).
Note that the assumption (10) implies that there is only one positive equilibrium of (12) satisfying . Therefore, and is a traveling wavefront satisfying (13). The proof is complete.
In order to construct a pair of admissible upper-lower solutions for (12), we linearize (12) at and obtain Thus, we consider the following characteristic equation: Note that The convex property of
leads to for any . In view of the previous observation, we have the following lemma directly.
Lemma 7. The following conclusions are true. (i)There exists a , , such that (ii)For , for .(iii)For , the equation has two zeros such that
Now we are ready to construct the upper solution of (12).
Lemma 8. Define Then for , is an upper solution of (12).
Proof. Let be such that Notice that , we have .
If , , , we have from the fact for and that Similarly, we have from the fact for , and for that
If , , , we have The proof is complete.
Let satisfying , we can obtain from (45) that
Lemma 9. Define where is a constant to be chosen later. Then is a lower solution of (12). Furthermore, is a lower solution of (12) satisfying
Proof. Let be such that , it follows that
If , , , and for , we have
Let By the fact that we have Note that (58) leads to for ; it follows that and thus Therefore, Let sufficiently large; we can have hence,
If , , , and for , we have
From the previous arguments, we obtain that is a lower solution of (12). By Lemma 4, we can get that is also a lower solution. Furthermore, for , , by direct calculation, we have for . Therefore, for
. Similarly, we have for . The proof is complete.
Theorem 10. Assume that and hold. Then for any , the system (11) has a traveling wavefront with speed , which connects and .
Proof. The conclusion for can be obtained from the previous discussions. We only need to establish the existence of wave fronts when .
Let with . For , (12) with admits a nondecreasing solution such that Without loss of generality, we assume that . Obviously, , , and satisfy
As the same argument in Lemma 5, we can obtain that is uniformly bounded and equicontinuous on ; using Arzéla-Ascoli theorem and the standard diagonal method, we can obtain a subsequence of , still
denoted by , such that uniformly for in any bounded subset of , as . Clearly, is nondecreasing and .
By the dominated convergence theorem and (67), it follows that Since and exist, using L’Hôspital rule leads to Thus, is a traveling wavefront of the system (11) connecting and .
Remark 11. We say that the is the minimal wave speed in the sense that (11) has no traveling wavefront with . We could briefly explain this in the following. In fact, the linearization of (12) at
zero solution is (41), and the function is obtained by substituting in the second equation of (41). For , we know from (ii) of Lemma 7 that for any . We have from the second equation of (12) and the
second equation of (41) that (12) cannot have a solution that satisfies .
Theorem 12. Assume that and hold. Then for any , the system (6) has a traveling wavefront with speed c, which connects and .
4. Asymptotic Behavior for Traveling Wavefronts
In this section, we discuss the asymptotic behavior for the traveling wavefronts obtained in the previous section as .
Theorem 13. Let be a traveling wavefront of (11) decided by Theorem 10; then where is the smallest root of the characteristic equation (42).
Proof. Note that Then we have which implies that
Note that we have from and that which is uniformly on . By the second equation of (12), we have from (73), and convergence theorem that | {"url":"http://www.hindawi.com/journals/aaa/2013/725495/","timestamp":"2014-04-17T10:25:10Z","content_type":null,"content_length":"1048595","record_id":"<urn:uuid:b0ee6ff9-3574-4158-bbc8-8c78d56092c6>","cc-path":"CC-MAIN-2014-15/segments/1398223206672.15/warc/CC-MAIN-20140423032006-00643-ip-10-147-4-33.ec2.internal.warc.gz"} |
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2. Students will be able to read, understand and trace the execution of programs written in C++ language.
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3. Students will be able to provide code for a Java class given objects’ attributes and behaviors.
4. Students will be able to use existing Java classes to perform required tasks.
CSCI 150
1. Students will be able to manipulate data at the bit and byte levels.
2. Students will be able to identify the components of a computer and the organization of those components.
3. Students will be able to describe disk storage systems and file systems.
4. Students will be able to use assembly language instructions to write small programs.
CSCI 170
1. Students will be able do basic UNIX OS administration tasks, including account management.
2. Students will be able to use the Unix file system
3. Students will be able to perform basic UNIX networking tasks including setting up a LAN using NIS
4. Students will be able to use Unix programming tools: compilers, Make utility, debugger, profiler, version control.
5. Students will be able to read-understand-write short scripts in a Unix shell.
CSCI 190
1. Students will be able to use truth table for propositional calculus.
2. Students will be able to use math induction and recursive definitions and algorithms.
3. Students will be able to understand the terminology of finite graphs and trees and use the basic algorithms for traversal, shortest path, graph coloring.
4. Students will be able to use basic counting techniques, combinatorics concepts and binomial coefficients.
CSCI 210
1. Students will be able to use Boolean algebra for algebraic simplification.
2. Students will be able to use truth tables, maps, and tabular reduction methods in combinational network design.
3. Students will be able to use state tables and diagrams in sequential network design.
4. Students will be able to differentiate between combinational and sequential logic networks.
CSCI 220
1. Students will be able to analyze problems and select the appropriate data structure.
2. Students will be able to estimate running time given an algorithm.
3. Students will be able to implement and use linear data structures including sets, stacks, queues, and lists.
4. Students will be able to implement and use trees including binary tree, binary search trees, and heaps.
CSCI 230
1. Students will be able to implement efficient searching techniques including hash tables and skip lists.
2. Students will be able to implement and analyze running time for various sorting algorithms.
3. Students will be able to represent graphs and implement well-known graph algorithms.
4. Students will be able to differentiate the costs between memory access and disk access. | {"url":"http://math.mtsac.edu/slo_cs.html","timestamp":"2014-04-18T05:42:40Z","content_type":null,"content_length":"15352","record_id":"<urn:uuid:1958b5af-76f6-4fd1-a8cf-e443f06c6573>","cc-path":"CC-MAIN-2014-15/segments/1398223205137.4/warc/CC-MAIN-20140423032005-00178-ip-10-147-4-33.ec2.internal.warc.gz"} |
A measure of the likelihood of a given process occurring in an accelerator, i.e., of particles colliding or otherwise interacting. Cross-section is expressed as an effective target area, which can be
related to the quantum-mechanical probability of interacting by multiplying by factors such as the flux of particles entering the interaction region. The basic unit of cross-section for particle
physics is the barn (b), equal to 10^-24 cm^2. Typical hadron collisions are measured in millibarns. Typical neutrino collision cross-sections are about 10^-11 cm^2.
Related categories | {"url":"http://www.daviddarling.info/encyclopedia/C/cross-section.html","timestamp":"2014-04-18T08:48:27Z","content_type":null,"content_length":"5988","record_id":"<urn:uuid:badd45fa-b283-47a9-b8e0-4fc4d99ab1dd>","cc-path":"CC-MAIN-2014-15/segments/1397609533121.28/warc/CC-MAIN-20140416005213-00482-ip-10-147-4-33.ec2.internal.warc.gz"} |
Patent US5068788 - Quantitative computed tomography system
1. Field of the Invention
This invention relates to the field of quantitative computed tomography and, in particular, to a system for reproducibly measuring the CT number of in vivo tissue samples.
2. Description of the Related Art
Computed Tomography (CT) was developed as a method for qualitatively analyzing soft tissues by non-invasively obtaining three-dimensional pictures of affected areas in a patient. Diagnosis of many
pathological conditions, however, requires quantitative analysis of the affected organs or tissues. This has historically necessitated painful and often dangerous invasive methods of obtaining tissue
samples for in vitro analysis, for example to determine the relative density or composition of the affected organs.
Because of the non-invasiveness of CT analysis and the ability to scan a precise three-dimensional cross section of the patient, it has been proposed to use the data resulting from a CT scan
quantitatively as well as qualitatively.
One type of condition recognized as suitable for quantitative CT analysis is osteoporosis. Osteoporosis evaluation involves the measurement of bone mineral density in affected spinal tissues.
However, quantitative CT analysis of osteoporosis has suffered from practical disadvantages which have severely limited its accuracy, despite the potential advantages obtained by such non-invasive
quantitative CT analysis, and the widespread availability of CT scanners.
The primary problem in utilizing quantitative CT analysis has been calibration of the CT numbers obtained in a CT scan to account for patient variability and positioning of the patient. The
attenuation of the X-ray spectrum used in a CT scan is affected not only by tissues in the region of interest, for example a particular vertebra, but also by the thickness and density of tissues
around the region of interest which also attenuate the X-ray beam. Since the density of the affected bone tissues is itself variable, it is not possible to calibrate the CT scan without an additional
Recently, attempts have been made to correct for the problem of CT number variability of a tissue situated in a heterogeneous surrounding of other tissues by providing a reference, known as a
"phantom" , located in the path of the CT scan but external to the patient. Such reference "phantoms" have been partially successful in accounting for effects of X-ray beam "hardening" and scatter
that are correlated with patient variability, but are nevertheless subject to such problems as local differences in X-ray spectra and in scatter distribution between the phantom and the target
tissue, variations resulting from patient movement with respect to the reference phantom, and volume averaging difficulties resulting from the current practice of taking the mean of the CT
In order to overcome the problems associated with the use of external reference phantoms, it has been suggested that internal reference tissues located near the affected tissues could be utilized.
Potential in vivo reference tissues include fat and muscle, whose relatively invariable densities and close proximity to vertebral bone tissues make them suitable candidates for use in calibrating CT
scans without the need for external reference phantoms.
However, it has heretofore been impossible to reproducibly obtain meaningful CT numbers for in vivo fat and muscle tissues because no method has been available for eliminating the inevitable effects
of fat and muscle tissue mixing, and to account for the nonuniform distribution of such tissues throughout the CT slice.
Any quantitative distribution obtained from a CT scan for areas of fat and muscle will invariably include areas representing fat and muscle mixtures, as well as background from other tissues in the
path of the scan. Because the mixing is essentially random, CT numbers representing such mixtures are of no use in deriving a reference standard for correlating CT numbers with tissue density.
The invention provides a method and means for solving the problems of the prior art by providing a way of analyzing a suitably chosen CT number histogram to delete the effects of background scatter
and intermixing of tissues.
CT numbers of individual tissues are obtained by locating leading edges of histogram distribution curves in regions of the histogram representing the individual tissues. The leading edge values are
used as a starting point for construction of model curves representative of pure tissue samples against which the actual histogram distribution can be measured, for example by calculating and
adjusting moments of the curve, following subtraction of assumed values for background and intermixing derived from the leading edge values. The adjusted CT numbers are used to create a reference
plot by which other CT numbers can be converted to a physical quantity such as density for use in analyzing other tissues.
FIGS. 1(a) through 1(e) illustrate the process by which a meaningful curve is obtained from an initial histogram distribution.
FIGS. 2(a) and 2(b) illustrate the process of converting CT numbers to known quantities for use as a reference standard.
FIGS. 3(a) and 3(b) are flowcharts detailing the method of the preferred embodiment.
FIGS. 1(a) to 1(e) and FIGS. 2(a) and 2(b) illustrate the effects of implementing the preferred method, shown in FIGS. 3(a) and 3(b). Each of the method steps is implemented by means of appropriate
software and hardware associated with a CT scanner.
The first step (100) is to take a CT scan which includes an affected vertebra and proximate regions of fat and muscle tissue. The region of interest is selected to include as high a percentage of fat
and muscle in the area of the affected bone tissue as possible. A physician or qualified lab technician will be able to select a likely region for analysis and enter the spatial parameters of the
three-dimensional region into a computer. The higher the percentage of fat and muscle in the defined area, the more accurate the calibration.
The computer then calculates a histogram of the pixel values resulting from the CT scan of the selected region, as indicated by step 101. The pixel values correspond to "CT numbers" which represent
x-ray beam attenuation by the various tissues in the path of the beam.
At this point, histogram values which are clearly the result of system errors and background effects can be filtered out by averaging the histogram over groups of adjacent CT numbers to eliminate
sharp discontinuities in the histogram, as indicated by step 102.
FIG. 1(a) shows an exemplary filtered histogram of CT number distribution in the area of a vertebra. The distribution of CT number frequencies results not only from individual fat and muscle tissues,
but also from the inevitable mixing of fat and muscle, as well as the effects of attenuation and scatter of the X-ray by surrounding body tissues. Actual histograms will, of course, vary from patient
to patient and according to the type of scanner employed.
In a preferred embodiment of the invention, separated muscle and fat CT numbers are derived from this histogram for use in analyzing the vertebral tissues. The CT numbers obtained are then plotted
against known density values for fat and muscle to obtain a CT number versus density reference curve 16, shown in FIG. 2(b). The manner in which the reference curve is obtained will be described in
detail below, but first a description of the method by which the adjusted curves are obtained will be given.
After obtaining a filtered histogram, the search parameters for locating fat and muscle CT numbers for the histogram must be defined (step 104). This is accomplished by recognizing certain general
characteristics of such histograms. As shown in FIG. 1(a), the CT number for fat clearly lies in the area of the histogram curve designated by reference numeral 1 and having CT numbers of between
approximately -140 and -20, while the CT numbers for muscle lie between approximately 20 and 100. It has been found that, depending on the scanner, the above-mentioned ranges will account for
virtually all spinal muscle and fat peaks. Thus, the search is defined to include those areas of the histogram, as will be explained in more detail below.
Furthermore, the search routine makes use of the fact that, in the absence of background effects and intermixing, the fat and muscle distributions would be separate "gaussian" curves.
The effects of background attenuation and scatter can be seen in the non-zero CT number frequencies in areas along the CT number axis outside of the "fat" and "muscle" areas, and the influence of
mixing is found in the deviation of the distributions from a "delta" or gaussian distribution. This is especially true on the right side of the fat "peak" and on the left side of the muscle "peak",
as shown in FIG. 1(a), because the CT number, range for muscle is located to the right of the range for fat.
The basic approach of the invention is to calculate the deviation of the actual CT number distribution curve from the theoretical gaussian curves representing pure fat and muscle by locating the
leading edges of the respective fat and muscle curves. This is important because these are the areas least likely to be affected by intermixing. As shown in FIG. 1(a), the left side 3 of the fat
distribution is defined to be a "leading" edge, while the right side 4 of the muscle distribution is defined to also be a "leading" edge.
The base of a leading edge is determined by searching for areas at which the slope of the curve exceeds a threshold slope for a sufficient number of samples. This is because spikes in the background
will create false slopes, as indicated by reference numeral 5 in FIG. 1(a).
In order to locate the leading edges, the search parameters must be defined, as noted above. The most critical search parameters are, in no particular order, threshold, search start, search
direction, search end, and base width.
The first parameter mentioned, "threshold" is a minimum slope expected to be indicative of the beginning of a peak. The threshold is defined according to the size of the region of interest. The
larger the region of interest, the smaller the slope deemed necessary to define a peak.
The search start is defined to be outside of the CT number region of the histogram in which potential fat and muscle peaks have been observed to lie. In the histogram illustrated in FIG. 1, a search
start for fat is chosen as a number less than -140 and a search start for muscle is chosen as a number greater than +100, illustrated in FIG. 1.
It should be noted that by starting the search outside of the area of the histogram which includes the target tissue, an important advantage is obtained over possible alternative methods of analysis.
This is because the search does not require a precise starting point, as would be the case if one attempted to immediately inscribe the highest probability fat and muscle areas in order to obtain a
mean CT number. The search at this point is merely for "leading" edges, and not for exact muscle or fat locations.
The search direction is then selected, for example in terms of a search "increment" . In the preferred embodiment, a search increment for fat of +1 means that the search will retrieve each CT number
frequency for analysis, moving to the right along the histogram as indicated by arrow 7 in FIG. 1(b). Similarly, the search increment for muscle would be negative as indicated by arrow 8.
An end search point is also selected as part of step 103. If the search proceeds beyond the point at which a leading edge is likely to be found, the search is terminated. In the illustrated
histogram, +20 would be a reasonable endpoint for a muscle search, while 60 would be a reasonable endpoint for a fat search.
Finally, a "basewidth" is selected. The "basewidth" essentially is the width at twenty percent of the height of an ideal pure fat or muscle histogram distribution. For both fat and muscle, the base
width is selected to be 80 for the histogram shown in FIG. 1. This "basewidth" will be used to locate a presumed trailing edge for further analysis.
Once the search parameters have been selected and defined as variables in the computer program, the search begins (step 104). As the histogram values are retrieved from a memory buffer, the
derivative of the distribution curve is calculated and compared with the "threshold" (step 105). If the derivative exceeds the threshold for a sufficient number of increments, a leading edge is
deemed to be detected (step 106).
Once the location of the leading edge is established, the CT number frequency to the left of the curve is used to establish a baseline indicative of the background contribution at the leading edge.
As noted above, if the fat and muscle CT numbers were not effected by background effects, the baseline would be zero.
Furthermore, if there were no background or fat and muscle mixing, the frequency on the trailing side of the curve would theoretically be zero. The method of the invention assumes that the effects of
tissue mixing linearly decrease across the peak, and everything below the theoretical base of the base of the peak is in effect deleted.
The baseline contribution of background effects is determined, in effect, by taking the frequency value or height 11 (See FIG. 1(c)) of the curve at the leading edge (step 107). The height of the
leading edge would be zero if no background and no mixing were present. To ensure a more accurate value for the background baseline, an average of values immediately preceding the leading edge may be
The value of the "basewidth", described above, is then added to the CT number of the leading edge to locate an assumed trailing edge (step 108). Again, it is presumed that the height of the trailing
edge would be zero in the case of a pure tissue sample with no background or tissue intermixing.
At this time, the presumption that the effects of intermixing of the CT numbers decrease linearly from the trailing edge to the leading edge is utilized . The height of the trailing edge minus the
height of background baseline is taken to reflect the amount of tissue intermixing at the trailing edge of the tissue "peak".
Essentially, in steps 109 and 110, a line is drawn from the leading to the trailing edge, and everything below the line is deleted from the histogram. This is done by calculating the slope which such
a line would follow, incrementing along the X-axis and subtracting the base line value plus the Y-value from the Y-value of the histogram curve (for example, if the histogram curve is defined by the
function f(x), the slope of the line is m, and the base line value is b, then the subtracted curve f'(x) would be given by f'(x)=f(x) -(mx +b)).
The result is a uniquely adjusted histogram curve which substantially eliminates the effects of both background and intermixing. The same process of background subtraction, using appropriate search
parameters, is performed for both fat and muscle regions of the histogram. By drawing a line as described for the fat region, the effect of intermixing is extracted, making accessible as much of the
intermixed fat CT numbers as possible for subsequent analysis, together with the CT numbers representative of fat tissue in the pure fat region of the histogram, indicated by the portion of the fat
curve which is above the line. Similarly, by drawing the line in the muscle region of the histogram, a corresponding extraction of the effect of intermixing occurs, resulting in a separation of the
intermixed muscle CT numbers for subsequent analysis.
The background-subtracted curve is then subjected to a curve fitting algorithm (step 112) to determine its deviation from a true gaussian curve. As explained above, the parameters or "moments" of the
gaussian curve are indicative of the parameters of a curve uninfluenced by surrounding tissue and tissue intermixing. In step 116, the mode of the curve is selected on the basis of this analysis, and
taken to be the measured CT number for use in calibrating the scanner.
A variety of statistical analysis techniques are known which may be used to analyze the curve for fit and to find the mode. It is also possible to analyze the curve in terms of other gaussian
moments, although no other moment is believed to be as meaningful as the mode, which is the CT number of highest frequency.
The specific preferred method of analysis is the "chi-squared" test, details of which are known to those skilled in the art and form no part of this invention. Once the curve has been transferred to
"chi-squared space," its width, defined to be the width of the curve at 20% of its height, is compared with a theoretically acceptable width, for example the above-described basewidth, and the curve
is rejected if it is not deemed to be representative of pure fat or muscle.
The width obtained during the curve fitting analysis can then be substituted for the originally presumed base width, and the process of subtraction can be repeated in an iterative process for greater
accuracy (step 115). FIG. 1(e) shows the results of the iterative process and also includes a histogram of trabecular tissue for comparison.
It is also possible to use parameters other than the base width for analyzing whether the curve is acceptable. For example, the actual slope of the leading edge may be a more reliable indicator of
the acceptability of the curve than the width.
Once a CT number for fat and muscle has been obtained, the numbers may be used to obtain a calibrated plot of CT number versus density (step 118). Referring to FIGS. 2(a) and 2(b), the CT numbers are
plotted along the Y-axis of a graph of CT number versus "equivalent bone density." The equivalent bone densities of both muscle and fat are known and are plotted along the X-axis of the graph. A line
16 is then drawn which shows the relationship between measured CT number and equivalent bone density.
The equivalent bone densities will of course be different depending on the energy of the scanning beam. Also, it is noted that while the preferred equivalent bone densities are expressed in units of
mg/cc of cortical bone, numerous other units of equivalent bone density are also known.
As shown in FIG. 2(b), line 16 can then be used to obtain the equivalent bone density of the subject trabecular tissue.
The resulting density values are highly reproducible, and are not dependent on precise positioning of the patient, unlike values obtained by prior art methods. Reproducibility studies on 33 images
with intentional region of interest mispositioning of 1.5 to 2.5 mm yielded a sample standard deviation of 1-2 CT numbers for trabecular bone. This translates to a precision of better than 1% for
normal and 1-2% for osteoporotic patients. Prior art methods yielded a reproducibility error slightly less than 2% for normal patients and as high as 3-4% for osteoporotic patients.
In conclusion, the invention provides a useful tool for convenient, reliable derivation of quantitative CT values for an in vivo tissue. The tool does not require external calibration phantoms, and
is therefore practical for clinical use. The invention permits a significant improvement in reproducibility due to the decreased dependence on patient positioning.
It is contemplated that the method and means of the invention will also be applicable to the evaluation of conditions involving such tissues as lung, kidney, or liver tissues, and to tumor
evaluation. Numerous other modifications will also occur to those skilled in the art. Therefore, the invention is not to be limited to the described embodiment but is to be defined solely by the
appended claims. | {"url":"http://www.google.com/patents/US5068788?ie=ISO-8859-1&dq=6188988","timestamp":"2014-04-18T17:18:50Z","content_type":null,"content_length":"99600","record_id":"<urn:uuid:34cf1452-be52-4b7c-a237-db42ca11c12f>","cc-path":"CC-MAIN-2014-15/segments/1397609533957.14/warc/CC-MAIN-20140416005213-00237-ip-10-147-4-33.ec2.internal.warc.gz"} |
Area of Circles - Concept
The area of circles is derived by dividing a circle into an infinite number of wedges formed by radii drawn from the center. When these wedges are rearranged, they form a rectangle whose height is
the radius of the circle and whose base length is half of the circumference of the circle. The area of circles are also used in sectors, segments and annuluses.
If we want to find the area bound by a circle, well we're going to have to chop this up into pieces of something that we know. And we know the area of a parallelogram is equal to its base times its
corresponding height.
So what I'm going to do is I'm going to imagine that I can cut this circle up into an infinite number of pieces. But since I can't draw an infinite number of pieces, I'm just going to start by
drawing in let's say about 16 different pieces here. So if I could cut out all these pieces and rearrange them, what I would be able to do is I could, let's say we just looked at one of these pieces.
So you're going to have the little curvature on the outside form the circle and that both of these are going to be a radius. So what I'm going to do is I'm going to set all of these up in opposing
order and if I just continue doing this, I would be able to from a parallelogram where each of these is going to be one of the wedges that I cut out.
So I said that my height here is going to be r and I need to find out what is my corresponding base. Well, if I look at this, this outside is part of my circle and this outside right here is also the
other part of the circle. So if the whole circumference is equal to 2 pi r, then one of these bases is going to be half of this. And, if I divide this by 2, then half of my circumference is just pi
times r. So this distance right here is pi times my radius. So if I know my height is r, and I know my base is pi times r, then I can say the area is just r times r which is r squared times pi.
So if I could cut this up into a infinite testable amount of small wedges, I could create a rectangle where I know that my height is r and one of my bases is pi times r and these areas would be
equal. So I can say that the area of my circle, where I know the radius is pi times the radius squared.
circle radius circumference area | {"url":"https://www.brightstorm.com/math/geometry/area/area-of-circles/","timestamp":"2014-04-19T19:37:19Z","content_type":null,"content_length":"66764","record_id":"<urn:uuid:32bc1a6d-f6ca-4a23-8c46-9e5a93f7e1e7>","cc-path":"CC-MAIN-2014-15/segments/1398223206118.10/warc/CC-MAIN-20140423032006-00522-ip-10-147-4-33.ec2.internal.warc.gz"} |
Summary: UNIVERSITY OF REGINA
Department of Mathematics and Statistics
Speaker: D. Farenick
Date: 17 February 2006
Time: 3.30 o'clock
Location: College West 307.20 (Math & Stats Lounge)
Title: Local multiplier algebras
Abstract: The multiplier algebra M(A) of a (nonunital) C
-algebra A is the maximal
unitisation of A as a C
-algebra. In the late 1970s, a number of very important results
were obtained by G.A. Elliott that characterised those separable C
-algebras A for which
every derivation of A is determined by a multiplier of A--that is, is the restriction to
A of an inner derivation of M(A).
The notion of local multiplier applies to any C
algebra A; in this case, one considers
the multiplier algebras of essential ideals of A. Via direct limits on the partially ordered
set of essential ideals of A, the local multiplier algebra of A is obtained. This lecture will | {"url":"http://www.osti.gov/eprints/topicpages/documents/record/038/2500754.html","timestamp":"2014-04-16T23:06:43Z","content_type":null,"content_length":"7964","record_id":"<urn:uuid:6b287ce1-4534-4cff-98f8-49e825b4b2b5>","cc-path":"CC-MAIN-2014-15/segments/1397609525991.2/warc/CC-MAIN-20140416005205-00287-ip-10-147-4-33.ec2.internal.warc.gz"} |
Archives of the Caml mailing list > Message from Jacques Garrigue
[Caml-list] Comments on type variables
Date: -- (:)
From: Jacques Garrigue <garrigue@k...>
Subject: Re: [Caml-list] Comments on type variables
From: Alain Frisch <frisch@clipper.ens.fr>
> I'd like to make a few comments on the manipulation of type variables
> in OCaml. They range from some points that could be worth mentioning
> somewhere (FAQ, manual) to "features" that may be considered as bugs,
> IMHO.
At least I'll try to define what is a feature and what is a bug.
In Caml named type variables are handled in the following way:
1) Annotations and coercions
Type annotations (expr : t), let ... : t = expr, etc are constraints
applied to inferred types. As such type variables in them have just
an existential meaning, and they only affect sharing. In an alias
(t as 'a), 'a is handled as an instance variable.
Since sharing can happen with any other annotation in the same
expression, we have to be careful when generalizing types.
Caml chooses to only allow generalizing them at the level of the
whole expression/definition. That is, just like for recursive
functions, named typed variables inside an expression are
monomorphic during the typing of this expression.
If you don't need sharing, you can use "_" as variable name, which
can be generalized as needed.
2) Type definitions
All type variables are universally bound in the head of the
They may be constrained, but only explicitly. That is, they are
quantified universally,
type 'a t = texp1 constraint 'a = texp2
type t = /\'a < texp2. texp1
where 'a is bound to be an instance of texp2.
3) Interfaces
The semantics is the same as for type definitions, but the
quantification is implicit. Constraint may appear from the use of
val f : (#c as 'a) -> 'a
f : /\'a < #c. 'a -> 'a
4) Classes
They mix features from expressions and types, but the semantics of
type variables is still the same: they are all bound at the level at
the level of the class. Inside a definition group the quantification
is existential, even in interfaces (some constraints may be
inherited, or appear due to mutual recursion).
A small discrepancy appears in constraints on the class type:
class [tvars] c : ctype = cexpr
Here tvars are shared in both ctype and cexpr, but variables
appearing in ctype are not shared with does appearing in cexpr.
I'm not sure of why it is so, but Jerome Vouilon may be able to
5) Polymorphic methods
They form an exception to above rules.
While their types appear inside class definitions or descriptions,
they are quantified explicitly at the level of the method:
method fold : 'a. (t -> 'a -> 'a) -> 'a -> 'a = ...
In interfaces, the quantification is implicit for newly introduced
method fold : (t -> 'a -> 'a) -> 'a -> 'a
As you can see the exception is in the location of the
quantification, otherwise they behave like type of value descriptions.
6) Local modules
They do not have any specific behaviour, but definitions inside them
have their type variables quantified independently, as you would
expect if the module were defined at toplevel.
So there are lots of cases to consider, but I would not describe it as
messy: the basic idea is the scope is the definition, and you cannot
generalize before getting out of the scope.
> Generalization of explicit variables
> ------------------------------------
> Explicit variables are generalized only at toplevel let-declarations
> (see bug report 1156:
> http://caml.inria.fr/bin/caml-bugs/open?id=1156;page=3;user=guest):
> # let f (x : 'a) : 'a * 'a = (x,x) in (f 1, f true);;
> This expression has type bool but is here used with type int
Yes. This is a side-effect of using named variables.
> Variables do not add polymorphism
> -----------------------------------
> Contrary to the intuition one could have, variable never allow
> more polymorphism (exception for polymorphic methods); indeed, they
> _restrict_ the polymorphism.
Yes. This is a property of ocaml that dropping parts of a term can
only improve its polymorphism.
> One could expect polymorphic recursion to be allowed when
> function interfaces are explicitely given; this is not the case:
Yes. Polymorphic recursion is not a magical feature. The existential
semantics of type annotations do not allow to infer anything about
polymorphism from a type annotation.
> A paragraph in the FAQ is somewhat misleading:
> << Polymorphism appears when we introduce type schemes, that is types
> with universally quantified type variables.
> ...
> The polymorphism is only introduced by the definition of names during
> a let construct (either local, or global).
> >>
> The last sentence seems to contradict the first; the programmer doesn't
> introduce type schemes.
The first instance is about types, the second about terms.
In fact, to recover symmetry, the first sentence should probably be
<< Polymorphism appears when we introduce values whose types are
schemes, ... >>
> Variables do not guaranty polymorphism
> --------------------------------------
Fully yes for expressions. Partially yes for type definitions and
interfaces: a variable can only be constrained explicitly. If their is
no constraint on a variable, then it is guaranteed polymorphic.
> We have to rely on the module system:
> # module M : sig val f : 'a -> 'a end = struct let f x = x + 1 end;;
> Would it be possible to declare variables that have to remain polymorphic
> (I don't know how to formalize this; just an idea ...), so as to reject
> the following ?
> let 'a. f (x : 'a) : 'a = x + 1;;
Perfectly possible. But I'm not sure this is sufficient, as we are
also interested by constrained variables.
Maybe allow
let 'a = texp. f .....
Another way is to integrate it with explicit polymorphism:
let f : 'a. 'a -> 'a = ...
The real question is what we are going to do with this information.
If we don't use it, then interfaces are probably sufficient.
But we could use it for polymorphism recursion, or first class
> Variable scoping rules
> ----------------------
> The scoping rules for type variables do not seem uniform.
There are only two exceptions, and they concern classes.
Otherwise the scoping is the single phrase, or a group of phrase
connected by and for term definitions.
Maybe restricting to a single phrase even for simultaneous term
definitions would be more uniform.
> In a structure, the scope of a variable in an expression is the toplevel
> element:
> # module M = struct let f (x : 'a) = x let g (x : 'a) = x + 1 end;;
> module M : sig val f : 'a -> 'a val g : int -> int end
> ... whereas for objects, the scope is the class definition:
> # class o = object method f (x : 'a) = x method g (x : 'a) = x + 1 end;;
> class o : object method f : int -> int method g : int -> int end
Yes, a class is not a structure. A class is both an expression and a
type definition, and as such rules common to expressions and type
definitions apply.
> Interaction with local modules
> ------------------------------
> Inside a local module, type variables introduced outside the module are
> invisible:
> # let f (x : 'a) =
> let module M = struct type t = 'a list end in ();;
> Unbound type parameter 'a
To be expected by the rules above. Functions with local modules are
not functors. 'a is not a type constructor.
> # let f (x : 'a) =
> let module M = Set.Make(struct type t = 'a end) in ();;
> Unbound type parameter 'a
> # let f (x : 'a) =
> let module M = struct let g (y : 'a) = y end in M.g 2;;
> val f : 'a -> int = <fun>
Side-effect of the independence of members in a structure.
Is there a better semantics?
> Type declaration flushes the introduced variables:
> # let f (x : 'a) = let module M = struct type t end in (2 : 'a);;
> val f : 'a -> int = <fun>
That one is a bug. Bad one. And not immediate to correct, as type
variables are reset all over the place.
Thanks for reporting.
> Too bad, when one need such a variable to restrict polymorphism:
Indeed. If class typing were principal, there would be no such
> Objects
> -------
> At least it is clear for polymorphic methods where type variables
> are introduced.
> For class types, the explicit quantification is optional:
> # class type o = object method f : 'a -> 'a method g : 'b -> 'b end;;
> class type o = object method f : 'a. 'a -> 'a method g : 'b. 'b -> 'b end
> # class type ['a] o = object method f : 'a -> 'a method g : 'b -> 'b end;;
> class type ['a] o = object method f : 'a -> 'a method g : 'b. 'b -> 'b end
> # type 'a t = { x : 'a; y : 'b };;
> type 'a t = { x : 'a; y : 'b. 'b; }
That one is not valid any more. In records, one might expect
quantification to be existential (at the level of the whole record),
so it seemed better to make the quantification explicit.
> BTW, I don't understand the following error message:
> # class type o = object method f : 'a -> 'a method g : 'a -> 'a end;;
> The method g has type 'a -> 'b but is expected to have type 'c
Such error message occurs when 'a or 'b is a universal variable, and
you try to unify it with a type variable, against quantification
rules. I must find a way to get better error messages.
This is the case here because 'a is universal after the typing of f,
but is reused (and shared) in the typing of g.
I'm undecided about whether reuse on non-explicitly quantified
universal variables should be allowed. This might be confusing if the
intent was really sharing.
> Also, why are class parameters enclosed in brackets ?
> type 'a t = ...
> but:
> class ['a] t = ...
> and not:
> class 'a t = ...
That's a purely syntactic problem
class c = [int] t
must be distinguished from
class c = int t -> ...
at parsing time.
> '_a variables
> -------------
> It is not possible to specify "impure" type variables:
> # let z = let g y = y in g [];;
> val z : '_a list = []
> # let z : '_a list = [];;
> val z : 'a list = []
Indeed, there is no such concept as "impure" variable in input type
expressions. What it should mean is not so clear.
> Then, how do we specify an interface for the module struct let x = ref [] end ?
> # module M = (struct let x = ref [] end : sig val x : '_a list ref end);;
You must choose a type. Anyway, you will have to choose one at some
time, so why wait?
Note that it is illegal to export a non-generic type variable from a
compilation unit.
> Variable name in error message
> ------------------------------
> Error messages would be more readable if they retained (when possible)
> variable names:
> # let f (a : 'a) (b : 'b list) = b + 1;;
> This expression has type 'a list but is here used with type int
I see. As Francois Pottier pointed out, it's not clear what it would
mean when type variables are only quantified existentially, and can be
merged all over the place.
Note also that the unification algorithm may introduce new type
variables, or create type which were not in the input. Hard to follow
Hope this helps.
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Optimization Problem: Find a point on the line y = 2x+5 that is closest to the origin
• one year ago
• one year ago
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distance of origin from a pt. on the line is sqrt.[(x)^2 + (2x+5)^2].................square it since the distance and its square maximise simoultaneously and then differentiate with respect to x
and find the minima....the corresponding value of x and y will give you the required point....
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Thanks, that is correct sir. Except I'm not sure why you differentiate D^2 instead of D. Will differentiating D give you the same answer? My textbook mentions the same thing, but it doesn't
really explain why
Best Response
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yeah differentiating D would also yield the same answer but differentiating a square root is a tedious job so instead we diff. D^2......if D is min. for some value of x, it's square will also be
min. at same value of x as compared to the other values....got it..???
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oh okay, that's pretty intuitive i guess
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Recall The Ac Circuit Of Problem H8. Copy It To ... | Chegg.com
Here is HW8, it is needed to solve the following problem
Here is the problem:
Image text transcribed for accessibility: Recall the ac circuit of Problem H8. Copy it to the diagram below (onto the dotted lines a-n and a'-n') and name it "phase a" Make an exact copy of the phase
a circuit (replace nodes a and a' with b and b'), and copy it onto the diagram above, changing the voltage to Make an exact copy of the phase a circuit (replace nodes a and a' with c and c'), and
copy it onto the diagram above, changing the voltage to: Solve for the currents (a,b,c). Solve for the line voltages (ab,bc,ca). Complete the phasor diagram below to scale (9 phasors: 6 voltages, 3
currents). Use three colors Compute the neutral current (n). Now disconnect the neutral (n-n'). Did the phase currents change? Based on (h), is the neutral really necessary? Define the circuit of (a)
to be a 4-wire (abcn) balance 3-phase system. Define the modified circuit with the neutral removed to be a 3-wire (abc) balanced 3-phase system. Complete the 3-wire circuit diagram. In the 3-wire
(abc) balanced 3-phase system, the loads and sources are said to be "wye- connected", where Re-draw the 3-wire circuit, keeping the wye-connected sources, but replacing the wye-connected load
elements with delta-connected load elements. Note that node n' disappears. Compute the load currents: Using the values in (k), compute the phase currents (a,b,c). Do these values check with the
results in (d)? Should they? Compute the 3-phase load in the wye. Complete the 3-phase power triangle.
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Theory Seminar
Friday Dec. 13 Adam Marcus (Yale)
WWH 1314 A solution to Weaver's $KS_2$
Friday Dec. 6 CCI meeting
Friday Nov. 22 New York Area Theory Day
9:30AM -- 4:10PM
Auditorium 109 Program
Friday Nov. 15 Leonid Gurvits (CUNY)
WWH 1314 Breaking e^n barrier for deterministic poly-time approximation of the permanent and settling Friedland's conjecture on the Monomer-Dimer Entropy
Friday Nov. 8 CCI meeting
Thursday Nov. 7 Ali Kemal Sinop (IAS)
WWH 1314 Towards a better approximation for sparsest cut?
Friday Nov. 1 Thomas Vidick (Newton Institute)
WWH 1314 Three-player entangled XOR games are NP-hard to approximate
Friday Oct. 25 Ravishankar Krishnaswamy (Princeton)
WWH 1314 Capacitated Network Design: Algorithms and Hardness
Friday Oct. 18 CCI meeting
Thursday Oct. 17 Brendan Juba (Harvard)
WWH 1314 Efficient reasoning in PAC Semantics
Friday Oct. 11 Zeev Dvir (Princeton)
WWH 1314 Incidence theorems and their applications
Friday Oct. 4 Aleksandar Nikolov (Rutgers)
WWH 1314 Approximating Hereditary Discrepancy (without privacy)
Friday Sept. 27 Clément Canonne (Columbia)
WWH 1314 Testing probability distributions using conditional samples
Thursday Sept. 19 Ben Lee Volk (Technion)
WWH 1314 Boolean functions with small spectral norm, sparse boolean functions and decision tree complexity
Thursday Sept. 5 Ronald de Wolf (CWI)
WWH 1302 Approximate degree bounds for all and almost all Boolean functions | {"url":"http://cs.nyu.edu/web/Calendar/colloquium/cs_theory_seminar/Fall-13.html","timestamp":"2014-04-16T05:13:10Z","content_type":null,"content_length":"40567","record_id":"<urn:uuid:f3a34ced-53a7-44a8-ab94-70f26654d10a>","cc-path":"CC-MAIN-2014-15/segments/1397609521512.15/warc/CC-MAIN-20140416005201-00543-ip-10-147-4-33.ec2.internal.warc.gz"} |
Homework Help
Posted by Jon on Friday, March 7, 2008 at 1:38pm.
Should the triangle be solved beginning with Law of Sines of Law of Cosines. Then solve the triangle. Round to the nearest tenth. A=56 degrees, B=38 degrees, a=13.
Sines. I get confused on the formula. I know C=86 degrees
• Trig - Reiny, Friday, March 7, 2008 at 2:18pm
To solve a triangle you must be given 3 independent bits of information.
2 sides and 1 angle
3 sides
1 side and 2 angles
(note 3 angles is not "3 independent pieces of information, since if you know 2 angles, you automatically know the third)
general simple rule:
If you are given a side and its opposite angle, use the sine law
if not, use cosine law.
so for yours, clearly sine law.
• Trig(I know all of that I just need to know how to use the formula) - Jon, Friday, March 7, 2008 at 2:47pm
I know I need to use the law of sines but I don't really know how and which one to use like sinA/a= sin B/b and sinB/b =sinC/c and sinA/a = sinC/c. Thats where I get confused.
• Trig - Reiny, Friday, March 7, 2008 at 2:56pm
clearly you are given a side and its opposite angle, a and A, so that is obviously the ratio you are going to use
SinA/a = SinB/b
sin56/13 = sin38/b crossmultiply and solve for b.
b = 13sin38/sin56
= 9.65
do the same to find c
• Trig - Jon, Friday, March 7, 2008 at 3:23pm
b=9.7(i had to round)
im not sure if im doing this right.
SinB/b = SinC/c
Sin38/9.7 = Sin86/c
c = 9.7sin38/sin86
c= 5.98 or 6 since I have to round to nearest tenth.
□ Trig - Reiny, Friday, March 7, 2008 at 3:30pm
why would you not stick with the exact ration sin56/13, which uses the original numbers given
You used an answer that you obtained after you rounded off that answer, so you are compounding your error.
• Trig - Jon, Friday, March 7, 2008 at 3:36pm
so to find c all I have to do is
Sin56/13 = Sin86/c
c = 13Sin56/Sin86
c = 10.8
□ Trig - Reiny, Friday, March 7, 2008 at 3:50pm
Sin56/13 = Sin86/c now cross-multiply
csin56 = 13sin86 divide by sin56
c = 13sin86/sin56
c = 15.6
check: in any triangle, your smallest side should be across from the smallest angle, the largest side should be across the largest angle.
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Types of Angles Study Guide
LearningExpress Editors
Updated on Oct 5, 2011
Introduction to Types of Angles
Lesson Summary
This lesson will teach you how to classify and name several types of angles. You will also learn about opposite rays.
Many of us use the term angle in everyday conversations. For example, we talk about camera angles, angles for pool shots and golf shots, and angles for furniture placement. In geometry, an angle is
formed by two rays with a common endpoint. The symbol used to indicate an angle is
Naming Angles
People call you different names at different times. Sometimes, you are referred to by your full name, and other times, only by your first name or maybe even a nickname. These different names don't
change who you are—just the way in which others refer to you. You can be named differently according to the setting you're in. For example, you may be called your given name at work, but your friends
might call you by a nickname. Confusion can sometimes arise when these names are different.
Just like you, an angle can be named in different ways. The different ways an angle can be named may be confusing if you do not understand the logic behind the different methods of naming.
If three letters are used to name an angle, then the middle letter always names the vertex. If the angle does not share the vertex point with another angle, then you can name the angle with only the
letter of the vertex. If a number is written inside the angle that is not the angle measurement, then you can name the angle by that number. You can name the following angle any one of these names: ,
Right Angles
Angles that make a square corner are called right angles (see p. 26 for more details about what makes an angle a right angle). In drawings, the following symbol is used to indicate a right angle:
Straight Angles
Opposite rays are two rays with the same endpoint that form a line. They form a straight angle. In the following figure,
Classifying Angles
Angles are often classified by their measures. The degree is the most commonly used unit for measuring angles. One full turn, or a circle, equals 360°.
Acute Angles
An acute angle has a measure between 0° and 90°. Here are two examples of acute angles:
Right Angles
A right angle has a 90° measure. The corner of a piece of paper will fit exactly into a right angle. Here are two examples of right angles:
Obtuse Angles
An obtuse angle has a measure between 90° and 180°. Here are two examples of obtuse angles:
Straight Angles
A straight angle has a 180° measure. ABC is an example of a straight angle:
Practice problems for these concepts can be found at: Types of Angles Practice Questions.
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Array-Based Algorithm for
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An Array-Based Algorithm for Simultaneous Multidimensional Aggregates
Yihong Zhao, Prasad M. Deshpande, and Jeffrey F. Naughton
Summary by: Steve Gribble and Armando Fox
One-line summary: Efficent relational on-line analytical processing (ROLAP) algorithms for computing the data cube have been developed; this paper presents a detailed analysis of an efficient
multidimensional on-line analytical processing (MOLAP) algorithm for gleaming the cube.
Data cubes are relevant, ergo efficient data cube computation is relevant, and this algorithm seems to work quite well.
• this paper is a bear to read - they could do much better with more diagrams and less text
• I'd like to see an analysis which results in the number of I/Os as a function of memory size, data size, dimension size, and number of dimensions. This seems like it should be possible (although
not easy), and it would give me a better idea of the tradeoffs in this algorithm.
Overview/Main Points
• Check out the data cube summary to read all about data cubes.
• ROLAP: use relational tables as data structure. Thus, a cell in a multidimensional space is a tuple, with some attributes representing location in the space, and one or more representing the
value of that cell. Efficient ways to compute the cube given this structure:
□ use grouping operation on the dimension attributes to cluster related tuples (e.g. sorting or hashing)
□ use grouping performed on behalf of sub-aggregate as partial grouping to speed computation of another sub-aggregate
□ compute an aggregate from another aggregate, rather than from the much larger base table. (e.g. compute 1-d aggregate from one of the 2-d aggregates rather than the N-d data.)
• MOLAP: stores data as sparse arrays. The position of the value within the sparse array encodes the position of the value in the multidimensional space.
• All about arrays:
□ array is most likely far too big to fit in memory. Must therefore split array into chunks. Row-major or column-major is not efficient; each chunk should be a small N-d piece of the N-d space,
and chunk size should correspond to block size of disk.
□ even with chunking, many cells of chunks will be empty - use compression for these sparse arrays
□ array may need to be built from non-array (e.g. table) data. Use a 2-pass algorithm: first pass assigns chunk numbers to tuples, second pass clusters together tuples by their chunk number.
• Simple data cube computing algorithm
□ Let's say we have a 3-D cube (dimensions labelled A, B, C) and we want to compute the aggregation across dimension C, which corresponds to a plane of size |A| * |B| of aggregate values.
□ One possibility is to sweep the entire plane down the C dimension. Better to use chunking: sweep a chunk of size |Ac|*|Bc| (where Ac is the chunk size in the A dimension and Bc is the chunk
size in the Bc dimension) down, then that chunks neighbor, etc., until have slowly built up the aggregation plane in patchwork fashion. Less memory is required for this.
□ For a data cube, we need multiple aggregates (AB plane, BC plane, AC plane, A edge, B edge, C edge, and the total aggregation point). Far better to compute A from AB than A from entire data
set ABC, etc. Think of aggregates in cube computation as a lattice, with ABC as the root, and ABC having children AB, BC, and AC; AC has children A and C, etc. Goal is to embed a tree in this
lattice, and compute each aggregate from its parent in the tree. Trick is to find the most efficient tree.
□ array algorithm: be sure to compute any lower-dimensional group-by from the higher-dimensional parent. For example, read in each chunk of ABC and use to produce an aggregate chunk of AB. Once
this chunk of AB is finished, output to disk, and repeat for next chunk of AB.
□ This algorithm is careful to use parent aggregates to save computation for child subaggregates, but it computes subaggregats independently (e.g. scan entire ABC to make AB, then rescan ABC
entirely to produce BC.) Can do better.
• multi-way algorithm
□ n-d data cube has 2^n group bys, one for each subset in the power set of the cube.
□ ideally, have large enough memory to store all group-bys, and finish cube in one scan. Unfortunately this is usually not possible; goal is then to minimize memory needed for each computation
to achieve maxmimum overlap in aggregation computations.
□ single-pass algorithm: define an order over which you will scan the entire cube (row major order, scanned across each dimension in some order, e.g for 2-d table AB, scan
a0b0,a1b0,a2b0,a0b1,a1b1,a2b1,a0b2,a1b2,a2b2). Can compute how much memory is required to compute aggregates across each dimension. If memory order is ABC, only need 1 chunk for the BC
aggregate (as we always scan down the A dimension), but need 4 for the B dimension, and 16 for the C dimension. Still need structure to coordinate overlapping computation - use a lattice with
a minimum spanning tree. Nasty formulae are given for computing memory required for a given dimension order and resulting minimum spanning tree, from which one can deduce the optimal
dimension order.
□ If don't have enough memory for optimal dimension order, need to split up aggregation into multiple passes using the multi-way algorithm. End up storing incomplete trees, then filling them in
on subsequent passes.
• Performance studies show that array based algorithm does better than the ROLAP algorithms, even! Factors that affect performance are # of valid data entries, the dimension size (how many elements
in each dimension), and number of dimensions.
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February 2013
Math Digest
Summaries of Media Coverage of Math
Edited by Mike Breen and Annette Emerson, AMS Public Awareness Officers
Contributors: Mike Breen (AMS), Claudia Clark (freelance science writer), Lisa DeKeukelaere (2004 AMS Media Fellow), Annette Emerson (AMS), Brie Finegold (University of Arizona), Baldur Hedinsson
(2009 AMS Media Fellow), Allyn Jackson (Deputy Editor, Notices of the AMS), and Ben Polletta (Harvard Medical School)
February 2013
Brie Finegold summarizes blogs on girls and science and on green mathematics:
"Girls and science: why the gender gap exists and what to do about it," by Emma G. Keller. The Guardian UK--US News Blog, 5 February 2013.
This ambitiously titled blog post opens by highlighting a recent international study that found that on average, teenage girls outperform their male peers in math and science, but not in the United
States. The study centers around a test that has been given every three years since 2000 to 15-year-olds in developed nations, most recently in 2009 when teens in over 65 countries took the exam. The
test, the Programme for International Student Assessment, is administered by the Organisation for Economic Cooperation and Development and covers mathematics, science, and reading. The test also
includes a questionnaire concerning students' attitudes and backgrounds. You can view sample test questions.
The blog post quickly departs from a discussion of the study to give advice on how to help readers' daughters improve their beliefs concerning their abilities. Much of the advice seems overly general
or lacking in supporting evidence. For instance, the author encourages parents to bring up mathematics when their girls want to "buy something too expensive." Also, it conforms to stereotypes of
girls being more interested in cooking and shopping than in other, more academic, pursuits. A tongue-in-cheek response entitled "Boys and Science: The Gender Gap and How to Maintain It" also appeared
in a different Guardian blog. In addition, there was a response entitled "Pseudoscience and Gender-Stereotyping Won't Solve Gender Inequality in Science."
While Emma Keller may be aiming to offer parents some practical advice on ways to encourage their daughters to pursue science and math, the measures she suggests do not seem likely to close the
gender gap. Even now that we see ample evidence of strong female role models in science and math, the media often struggle to portray females in science without resorting to tired stereotypes like
the ones we saw just a few months ago in the video Science: It's a Girl Thing. Maybe we should focus less on marketing the sciences to women as if they were a product and more on educating the
general public concerning the achievements of female scientists who have enriched their fields.
"Prospects for a Green Mathematics," by John Baez and David Tanzer. Azimuth, 15 February 2013.
John Baez, mathematical physicist, and David Tanzer, a computer scientist, invite you to participate in the Azimuth Project, an international collaboration to create a focal point for scientists and
engineers interested in saving the planet.
While you may not see your mathematical expertise as being pertinent to solving the world's most pressing problems, the authors provide some interesting examples that may alter your views. For
example, leaves may grow according to network optimization, and chemical reactions may be modeled by stochastic Petri nets. Even those who study abstract ideas like quantum field theory may be able
to apply their knowledge to networks. There is plenty of new, "green" mathematics to be found in the study of networks and how they apply to biology. Baez and Tanzer seem to truly believe that
mathematicians and theorists have something to offer in identifying those tipping points at which our biosphere might drastically change. As 2013 is being called the year of "Mathematics of Planet
Earth," this post is apropos. The official start of this special year is on March 5, and you can find the description of this broad project on the website.
--- Brie Finegold
Return to Top
"Newsmakers: Three Q's," Science, 22 February 2013, page 891.
Science's news section asked MIT physicist Max Tegmark three questions about his symposium 'Is Beauty Truth?'. In the symposium professor Tegmark made the case for the universe and even reality
itself being mathematical structures. "There is no evidence there is anything at all in our universe that is not mathematical," explained Tegmark. He described the mathematical universe hypothesis as
a fundamentally optimistic way of looking at reality. If it is right, he said, "the road ahead is open and our future understanding is really only limited by our imagination."
--- Baldur Hedinsson
Return to Top
"A mathematician puts Fermat's Last Theorem on an axiomatic diet," by Julie Rehmeyer. Science News, 20 February 2013.
Nearly two decades after Andrew Wiles showed the world his complicated proof for Fermat's Last Theorem, mathematician Colin McLarty presented an alternative, pared-down proof at the 2013 Joint
Mathematics Meetings requiring fewer axioms and mathematical leaps of faith. Wiles's proof relied heavily on principles of algebraic geometry, in particular an assumption that allows for building
larger and larger sets while avoiding a well-known paradox. Given the simplicity of the theorem, and the fact that modern algebraic geometry was in its infancy when Fermat first penned the theorem
for which he claimed he had a "marvelous demonstration" that unfortunately would not fit in the margin of the page, mathematicians have speculated about the existence of a simpler alternative to
Wiles's proof, based instead on basic set theory. Enter McLarty, who has now shown that the theorem can be proven using only the basic axioms of mathematics.
--- Lisa DeKeukeleare
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"Don’t Let Economists and Politicians Hack Your Math," by Edward Frenkel. Slate, 8 February 2013.
Edward Frenkel writes about a cunning, behind-the-scenes political plan to raise taxes and slash social benefits, and how the public's mathematical knowledge is the only way to stop sneaky
politicians from fixing the numbers. Taxes, Social Security and Medicare are all pegged to the Consumer Price Index (CPI) as a measure of inflation. A proposal to change how this cost-of-living index
is calculated was included in the fiscal cliff deal reached by the White House and Congress late last year until the very last moment. This change would have raised taxes and cut social benefits with
just a little tinkering with a mathematical formula. He then brings forward the importance of the general public possessing the mathematical knowledge needed to protect the citizen from arbitrary
decisions by the powerful few in the increasingly math-driven world.
--- Baldur Hedinsson
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"Honoring the Best and Brightest" and "Lubchenco Earns Spanish Research Prize." News. Science, 8 February 2013, page 633.
These two news items higlight recent award winners, including four mathematicians.
The first is about the National Medal of Science winners who were honored at a White House ceremony in early February. Among the winners were Barry Mazur (Harvard University) and Solomon Golomb
(USC). As Notices Senior Writer and Math Digest contributor Allyn Jackson wrote in an AMS news item, Mazur "made important contributions early on to topology and has been an influential figure more
recently in number theory and arithmetic algebraic geometry. Golomb has made significant contributions in mathematics and engineering, especially in coding theory."Photos: Barry Mazur (left) by Jim
Harrison, Solomon Golomb (right) by Noe Montes/USC.
The second article is about the 400,000-Euro Frontiers of Knowledge Award, which is made by the BBVA Foundation in Madrid. Two of the six awardees were Ingrid Daubechies (Duke University) and David
Mumford (Brown University), pictured above, who shared the award for their work in data compression and pattern recognition. Read more about Daubechies and Mumford and link to more information about
the award. Photos: Ingrid Daubechies, Duke Photography; David Mumford, Brown University.
--- Mike Breen
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"Mathematician discovers largest prime (so far)," by Tia Ghose. The Christian Science Monitor, 5 February 2013.
On January 25, the largest known prime number--and the 48th known Mersenne prime--was discovered on the computer of University of Central Missouri mathematician Curtis Cooper. (Mersenne primes have
the form 2^p – 1 where p is a prime number.) The 17,425,170-digit number is 2 raised to the 57,885,161 power, minus 1. After the discovery, other researchers independently verified the prime using
different software programs. Cooper is a volunteer with the Great Internet Mersenne Prime Search (GIMPS), a giant network of volunteers who run the GIMPS software on their computers in the hopes of
finding the largest known of these numbers. The GIMPS project, formed in 1996 by computer scientist George Woltman, has been responsible for identifying all 14 of the largest known Mersenne primes.
The article reports that Cooper, who has found two other of the largest known Mersenne primes since 2005, is eligible for a $3,000 GIMPS award for this discovery.
--- Claudia Clark
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"Hard to Put Red-Light Violations Under a Lens," by Carl Bialik. The Wall Street Journal, 1 February 2013.
The Wall Street Journal's Numbers Guy looks into the statistics of whether cameras at red lights improve traffic safety. These cameras record motorists who run red lights and who are consequently
fined. The cameras can be found in over 500 U.S. urban areas and are intended to deter drivers from rushing through intersections after the light has just turned red and thus preventing dangerous
crashes. But depending on how you do the numbers, the effect of these cameras on traffic safety varies. Some cities have found that crashes increased at intersections where cameras were installed.
For example, including rear-end hits in the statistics, can mean that setting up cameras at an intersection increases the total number of accidents since drivers are more likely to slam on the brakes
when they notice the cameras or see warning sings. Determining whether the pros outweigh the cons can be tricky for city administrators, but many researchers find that if the number of severe crashes
declines overall, the increase in minor collisions is worthwhile.
--- Baldur Hedinsson
Return to Top | {"url":"http://ams.org/news/math-in-the-media/mathdigest-md-201302-toc","timestamp":"2014-04-21T07:36:03Z","content_type":null,"content_length":"26242","record_id":"<urn:uuid:98cb9593-2c2a-4b04-bf57-15744a8ee0de>","cc-path":"CC-MAIN-2014-15/segments/1397609539665.16/warc/CC-MAIN-20140416005219-00261-ip-10-147-4-33.ec2.internal.warc.gz"} |
Physics Forums - View Single Post - Entropy: Heat addition to surrounding.
If 12007 kJ of heat is lost to the surroundings with an ambient temperature of 25 degrees centigrade during a cooling process, and the ambient temperature of the surroundings is unaffected by the
heat addition, what is the entropy change of the surroundings?
If Δs=∫δQ/T, then Δs=ΔQ/T=12007 kJ/(25+273.15)K= 40.272 kJ/K.
Is my thinking process here correct? | {"url":"http://www.physicsforums.com/showpost.php?p=4151663&postcount=1","timestamp":"2014-04-16T13:40:32Z","content_type":null,"content_length":"8765","record_id":"<urn:uuid:b9ce2459-52a6-4408-9bee-1512b8453e76>","cc-path":"CC-MAIN-2014-15/segments/1397609523429.20/warc/CC-MAIN-20140416005203-00653-ip-10-147-4-33.ec2.internal.warc.gz"} |
Summary: A VECTORIAL VARIATIONAL MODE SOLVER AND
O. (Alyona) Ivanova, Remco Stoffer, Manfred Hammer, E. (Brenny) van Groesen
University of Twente, MESA+
Institute for Nanotechnology, P.O. Box 217, 7500 AE Enschede, The Netherlands
Abstract A Variational Vectorial Mode Solver for 3-D dielectric waveguides with arbitrary 2-D cross-
sections is proposed. It is based on expansion of each component of a mode profile as a superposition of
some a priori defined functions defined on one coordinate axis times some unknown continuous coefficient
functions, defined on the other axis. By applying a variational restriction procedure the unknown coefficient
functions are determined as an optimum approximation of the true vectorial mode profile. A couple of
examples illustrate the performance of the method.
Optical waveguides are main ingredients in many integrated optical circuits, and design tools for these waveguides
are of great importance. Detailed overviews of such techniques are presented in [1, 2]. In this paper we extend
the scalar Variational Mode Expansion Method [3] to fully vectorial simulations of lossless isotropic dielectric
waveguides with in principle arbitrary refractive index distribution. The validity of the method is checked for
a waveguide with non-rectangular piecewise-constant refractive index distribution and a diffused waveguide.
A. Problem Formulation | {"url":"http://www.osti.gov/eprints/topicpages/documents/record/516/3112878.html","timestamp":"2014-04-16T08:14:04Z","content_type":null,"content_length":"8923","record_id":"<urn:uuid:d2d336e2-1ebd-4e28-b538-c74c331f1c3d>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00347-ip-10-147-4-33.ec2.internal.warc.gz"} |
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...I am familiar with importing and exporting many different forms of data, and also merging data sets. Some of the PROC functions that I've used include the following: PROC REG PROC CORR PROC
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34 Subjects: including geometry, chemistry, writing, English | {"url":"http://www.purplemath.com/pico_rivera_ca_geometry_tutors.php","timestamp":"2014-04-21T15:21:05Z","content_type":null,"content_length":"23988","record_id":"<urn:uuid:04acf480-246f-4519-b9f4-cb89a2f96a0f>","cc-path":"CC-MAIN-2014-15/segments/1397609540626.47/warc/CC-MAIN-20140416005220-00387-ip-10-147-4-33.ec2.internal.warc.gz"} |
A comparative study of survival models for breast cancer prognostication based on microarray data: does a single gene beat them all?
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Bioinformatics. Oct 1, 2008; 24(19): 2200–2208.
A comparative study of survival models for breast cancer prognostication based on microarray data: does a single gene beat them all?
Motivation: Survival prediction of breast cancer (BC) patients independently of treatment, also known as prognostication, is a complex task since clinically similar breast tumors, in addition to be
molecularly heterogeneous, may exhibit different clinical outcomes. In recent years, the analysis of gene expression profiles by means of sophisticated data mining tools emerged as a promising
technology to bring additional insights into BC biology and to improve the quality of prognostication. The aim of this work is to assess quantitatively the accuracy of prediction obtained with
state-of-the-art data analysis techniques for BC microarray data through an independent and thorough framework.
Results: Due to the large number of variables, the reduced amount of samples and the high degree of noise, complex prediction methods are highly exposed to performance degradation despite the use of
cross-validation techniques. Our analysis shows that the most complex methods are not significantly better than the simplest one, a univariate model relying on a single proliferation gene. This
result suggests that proliferation might be the most relevant biological process for BC prognostication and that the loss of interpretability deriving from the use of overcomplex methods may be not
sufficiently counterbalanced by an improvement of the quality of prediction.
Availability: The comparison study is implemented in an R package called survcomp and is available from http://www.ulb.ac.be/di/map/bhaibeka/software/survcomp/.
Contact: bhaibeka/at/ulb.ac.be
Supplementary information: Supplementary data are available at Bioinformatics online.
During the last two decades, several clinical and pathological indicators such as histological grade, tumor size and lymph node involvement have been used for the survival prediction of breast cancer
(BC) patients independently of treatment, also known as prognostication. Examples of clinical guidelines to the selection of patients who should receive adjuvant therapy are the St Gallen consensus
criteria (Goldhirsh et al., 2003), the NIH guidelines (Eifel et al., 2001), the Nottingham prognostic index (NPI, Galea et al., 1992) and Adjuvant! Online (AOL, Olivotto et al., 2005). Although BC
prognostication has been the object of intense research, a still open challenge is how to detect patients who needs adjuvant systemic therapy.
The advent of array-based technology and the sequencing of the human genome brought new insights into breast cancer biology and prognosis. Interestingly, several research teams conducted
comprehensive genome-wide assessments of gene expression profiling and identified prognostic gene expression signatures. Examples of gene signatures which were obtained by studying the relationship
between gene expression profiles and clinical outcome, are the 70-gene (van't Veer et al., 2002) and 76-gene (Wang et al., 2005) signatures. With respect to clinical guidelines, these signatures were
shown to correctly identify a larger group oflow-risk patients not requiring treatment. This is particularly relevant for clinicians, since reducing treatments means also reducing potential side
effects and cutting costs. Another example of gene signature is reported in Sotiriou et al. 2006b). This study is focused on histological grade, a well-established pathological indicator rooted in
the cell biology of breast cancer. In fact, clinicians encounter problems when confronted with patients with intermediate-grade tumors (Grade 2). These tumors, which represent 30–60% of cases, are a
major source of inter-observer discrepancy and may display intermediate phenotype and survival, making treatment decisions for these patients a great challenge, with subsequent under- or
over-treatment. By means of a supervised analysis, the authors developed a gene expression grade index based on 128 probes. The associated genes were mainly involved in cell-cycle regulation and
proliferation and were consistently differentially expressed between low- and high-grade breast carcinomas. This signature, which essentially quantifies the degree of similarity between the tumor
expression pattern of these genes and the tumor grade, was able to separate patients labeled with histological Grade 2 tumors into two groups having distinct clinical outcomes similar to those of
histological Grades 1 and 3, respectively.
Other research groups have proposed gene expression signatures that are predictive of the clinical outcome in breast cancer [see Sotiriou and Piccart (2007) for a review]. However, since different
risk prediction methods, different accuracy measures and different validation sets were used, it is not easy to compare their performance in terms of BC prognostication.
The purpose of this work is 2-fold: first set up a common and independent assessment framework to compare the performance of existing gene signatures and several state-of-the-art risk prediction
methods; second, compare the prediction accuracy of these methods in several BC microarray prognostication tasks to elucidate the key characteristics of a successful risk prediction method and to
bring additional insights into BC biology.
Every risk prediction model aims to assign risk values or survival probabilities to patients on the basis of the information that is available at the time of diagnosis. This is known to be a
difficult task because of several issues specific to survival microarray data. First of all, censored information cannot be exploited by traditional supervised classification and regression methods,
but demands the adoption of specific survival analysis techniques, like the semi-parametric Cox's proportional hazards model (Cox, 1972). A second issue is the high dimensionality of microarray data.
When the number of explanatory variables exceeds by far the number of patients in the sample cohort (high feature-to-sample ratio), overfitting of naively applied data mining methods and
overoptimistic performance assessment lie in wait. At the same time, it is very difficult to select the most relevant variables for prediction, because of their interdependency and the reduced power
of the statistical inference procedure for high feature-to-sample ratio datasets (Bontempi, 2007). As a consequence, it is common to select variables that fit nicely the training set and fail
dramatically on independent validation sets, thus leading to unstable gene signatures (Ein-Dor et al., 2005; Michiels et al., 2005) and poor prediction models. A third issue is the lack of standards
in performance assessment for risk prediction models. Indeed, there exist few accuracy measures for risk prediction, and, to the best of our knowledge, no articles studied their agreement on the same
set of methods and datasets. Lastly, the validation and the comparison of BC microarray prognostication methods are made difficult due to the lack of independent data.
In this work, we compare the performance of 13 risk prediction methods on more than 1000 patients. This is made possible thanks to the recent publications of several large microarray datasets in gene
expression databases, such as the Gene Expression Omnibus (GEO, Barrett et al., 2005) An important outcome of the analysis is that, in spite of the large number of samples, there is no statistical
evidence that complex methods outperform the simplest BC prognostication techniques. This result suggests that the loss of interpretability deriving from the use of overcomplex data analysis
strategies may not be sufficiently counterbalanced by an improvement in the quality of prediction.
Finally, it is worth to mention that the present article complies with the research reproducibility guidelines proposed in Gentleman (2005) in terms of availability of the code and reproducibility of
results and figures.^1 A list of acronyms used throughout the article is given in Supplementary Table 1.
2 METHODS
2.1 Notations for survival analysis
Throughout the article we will adopt the following notation: upper case and lower case letters represent random variables and their realization, respectively, while bold letters denote vectors or
matrices. Let us denote the time as t. We suppose that a sample cohort of n patients is available and that for each patient we observed a p-dimensional vector of covariates x[i] with 1≤i≤n at the
time of diagnosis t=0, as well as the evolution of the survival status. Since we limit our study to microarray data, the covariate x[i] denotes the expression of the whole genome of the i-th patient.
Survival data for the i-th patient are denoted as follows: t[i] stands for the event time, c[i] for the censoring time and δ[i] for the censoring indicator (δ[i]=1 if t[i]≤c[i] and δ[i]=0 if t[i]>c[i
]). We introduce the counting process d[i](t)=1 if t[i]≤t and d[i](t)=0 if t[i]>t to denote survival status at any time t where d[i](t)=1 indicates that patient i experienced an event prior to time t
2.2 Risk prediction methods
The aim of a risk prediction model is to predict future survival status for all patients in the cohort. All the risk prediction models considered in this study return a risk score denoted by R, that
is a continuous value which quantifies the risk of a patient to experience an event. Clinicians often use the risk score to derive risk groups, denoted by G, on the basis of quantiles of the risk
score distribution. Although the discretization of individual risk scores into a finite (and often small) set of risk groups may introduce bias (Gerds and Schumacher, 2001), this approach is very
intuitive and conforms to the daily doctors' decision making process, e.g. the attribution of either low or high risk to patients. In the following, the quantity r[i] and g[i] will denote the risk
score and the risk group for patient i, respectively. G is either 0 or 1 for a low- or high-risk patient, respectively.
In this experimental study, we decided to focus on a set of 13 state-of-the-art methods (summarized in Table 1) with the ambition of being representative of a large number of risk prediction
strategies. The first risk prediction method is also the simplest one and defines the risk score as the expression of a single proliferation gene (AURKA) well studied in literature (Hanahan and
Weinberg, 2000). The following 10 methods (from 2 to 11) are characterized by the type of observed genotype (input data), the dimension reduction strategy, the structure of the model, the learning
algorithm and the predicted phenotype (outcome variable).
Characteristics of the risk prediction methods studied in this work
Genotype: it can be the expression of a single proliferation gene (AURKA), the expression of a biologically driven selection of genes of interest (BD) or the expression of the whole genome (GW).
AURKA and the small set of genes in BD were selected to represent several biological processes in BC (Hanahan and Weinberg, 2000). The selected genes were AURKA (also known as STK6, 7 or 15), PLAU
(also known as uPA), STAT1, VEGF, CASP3, ESR1 and ERBB2, representing the proliferation, tumor invasion/metastasis, immune response, angiogenesis, apoptosis phenotypes and the ER and HER2 signaling,
Dimension reduction strategy: we use either a simple univariate ranking (RANK) of the k most relevant features or a selection of the first k principal components (PCA). Univariate ranking uses
Wilcoxon rank sum test (Wilcoxon, 1945) in the case of binary outcome or Cox's proportional hazards model (Cox, 1972) in the case of survival outcome. The signature size k is either fixed or tuned by
cross-validation (CV) as described in Section 2.2.1. It is worth to note that no dimension reduction was performed for BD input data due to the low dimensionality of the input space.
Structure of the model: we adopt either a multivariate (MULTIV) model or a linear combination of univariate models (COMBUNIV), particularly interesting in a high-dimensional setting (Haibe-Kains et
al., 2008; Kittler et al., 1998).
Learning algorithm: we consider four types of learning algorithms: (i) the linear combination of gene expressions weighted by the significance computed from the Wilcoxon rank sum test (WILCOXON) that
allowed for identifying the most relevant genes to discriminate the patients with histological Grades 1 and 3 tumors, (ii) the multivariate linear regression model (LM), (iii) the linear combination
of gene expressions weighted by the significance computed from the univariate Cox's proportional hazards model (COX) and (iv) the multivariate Cox's model with L1 regularization (RCOX) as implemented
in Park and Hastie (2007).
Phenotype: we use three different phenotypical informations to fit the prediction models: (i) the binary class defined by histological grades 1 and 3 (HG), (ii) the censored survival data (SURV) and
(iii) the time of events (TOE), i.e. the times from diagnosis until the patients experienced an event. In the following, we will denote each of the 10 models with a unique label obtained by
concatenating the acronyms referring to its characteristics (Table 1). For instance, BD.COMBUNIV.COX.SURV refers to a combination of univariate Cox's proportional hazards models fitted from a
biologically driven selection of genes.
The last two models taken into consideration are the published GENE76 (Wang et al., 2005) and GGI (Sotiriou et al., 2006b) models. The GENE76 model is defined as a hierarchical model using two linear
combinations of the top gene expressions with respect to a ranking based on Cox's proportional hazards model. The choice of the linear combination to compute the risk score depends on the estrogen
receptor status of the patient. The GGI model consists of a linear combination of the expressions of the top probes ranked according to their standardized mean difference (Hedges and Olkin, 1987)
between patients with histological grades 1 and 3 tumors. The weights of the linear combination are simply the signs of the ranking statistics.
2.2.1 Tuning of hyperparameters
The GGI and GENE76 models did not require any tuning of hyperparameters since they were fully defined in previous publications. Only the models based on dimension reduction and regularization
required the tuning of an hyperparameter. For GW models, a simple ranking or a principal components analysis was used to select the k most relevant features. The hyperparameter k was either set to
30, this signature size being reported as a good trade-off between relevance and model complexity in the comparison study of Dudoit et al. (2002), or tuned using a 5-fold CV procedure (see Section 1
in Supplementary Material). The dimension reduction strategies using the latter procedure are referred as RANKCV and PCACV for the univariate ranking and the principal component analysis,
respectively. For methods using RCOX as learning algorithm, the hyperparameter for the penalty term was tuned by using a 5-fold CV as in Park and Hastie (2007).
2.3 Performance assessment
In order to assess the performance of the risk prediction methods, we used five accuracy measures: the time-dependent receiver operating characteristic (ROC) curve (Heagerty et al., 2000), the
sensitivity and specificity, the concordance index (Harrell et al., 1996), the Brier score (Brier, 1950; Graf et al., 1999) and the traditional hazard ratio (HR) from Cox's proportional hazards model
(Cox, 1972).
2.3.1 Time-dependent ROC Curve
The ROC curve is a standard technique for assessing the performance of a continuous variable for binary classification (Sweets, 1988). A ROC curve is a plot of sensitivity versus 1−specificity for
all the possible cutoff values of the continuous variable, denoted by c. In survival analysis, the continuous variable is the risk score, denoted by R, and the binary class to predict is the event
occurrence, denoted by D(t). As the event occurrence is time-dependent, time-dependent ROC curves are more appropriate than conventional ones. In Heagerty et al. (2000), the authors proposed to
summarize the discrimination potential of a risk score R, estimated at the diagnosis time t=0, by calculating ROC curves for cumulative event occurrence by time t. Once we define the sensitivity SE
and the specificity SP as follows
the ROC curve ROC(t) at time t is the plot of SE(c, t, r) versus 1−SP(c, t, r), where the cutoff point c is the parameter. In order to estimate the conditional probabilities in (1) and (2),
accounting for possible censoring, we used the nearest neighbor estimator for the bivariate distribution function proposed by Akritas (1994).
From the ROC curve ROC(t) we can derive the area under the curve (AUC) quantity, denoted by AUC(t). Since AUC depends on time t, we define the integrated area under the curve (IAUC) as the area under
AUC(t), ∀ tT. Note that, in this study, the larger the AUC at time t, the better is the predictability of time to event (TTE) at time t. Similarly, the larger IAUC, the better is the average
predictability of TTE.
2.3.2 Sensitivity and specificity
A widely used performance criterion for a clinical test is the pair {sensitivity, specificity} (Simon, 2005). However, the calculation of these values from survival data requires estimators
accounting for TTE and possible censoring. We used the estimators defined in (1) and (2) for sensitivity and specificity, respectively. For risk score prediction, we estimated the specificity for a
sensitivity of 90% in accordance with the St Gallen (Goldhirsh et al., 2003) and National Institutes of Health (Eifel et al., 2001) treatment guidelines. For risk group prediction, we estimated both
the sensitivity and the specificity of the binary classification returned by all the methods. Note that the larger the sensitivity and the specificity, the better is the predictability of TTE.
2.3.3 Concordance index
The concordance index (C-index) computes the probability that, for a pair of randomly chosen comparable patients, the patient with the higher risk prediction will experience an event before the lower
risk patient. The C-index takes the form
where r[i] and r[j] stand for the risk predictions of the i-th and the j-th patient, respectively, and Ω is the set of all the pairs of patients {i, j} who meet one of the following conditions: (i)
both patients i and j experienced an event and time t[i]t[j] or (ii) only patient i experienced an event and t[i]c[j]. In the case of risk group prediction, an additional condition must be met, that
is the risk predictions are different for patients i and j (no ties in r).
Note that the C-index is a generalization of the AUC(t), though it is unable to represent the evolution of performance with respect to time (Harrell et al., 1996).
Standard errors, confidence intervals and P-values for the C-index are computed by assuming asymptotic normality (Pencina and D'Agostinno, 2004). Note that, in this study, the larger C-index, the
better is the predictability of TTE.
2.3.4 Brier score
The Brier score, denoted by BSC, is defined as the squared difference between an event occurrence and its predicted probabilities at time t. Probabilities of event, denoted by Q, can be derived from
Cox's proportional hazards model fitted with the risk score R or risk group G predictions. Intuitively, if a patient experiences no event at time t, the event predicted probability should be close to
zero. Symmetrically, if the patient experiences an event the probability should be close to one. The BSC formalizes this intuition by computing the time dependent quantity
where the weights W are used to remove a large sample censoring bias (Gerds and Schumacher, 2006; Graf et al., 1999).
A summary of the predictability error over times is returned by the integrated Brier score, denoted by IBSC. Note that the lower the BSC, the better is the predictability of TTE at time t. Similarly,
the lower the IBSC, the better is the average predictability of TTE.
For judging the (I)BSC, we will rely on the score of a benchmark risk prediction model which is obtained with the overall Kaplan–Meier estimator (Kaplan and Meier, 1958) for the survival function
(this model is called KM in further sections). This simple risk prediction model corresponds to a model which assigns the same risk prediction to all patients. It ignores the information contained in
explanatory variables completely and thus provides a suitable benchmark value similar as the one obtained with the null model in linear regression.
2.3.5 Hazard ratio
In this work we used the HR as an accuracy measure for the risk group prediction in order to keep it interpretable and comparable between different risk prediction methods as the scale of predictions
is well defined (see Section 2.1). HR is a summary of the risk difference between several survival curves estimated by Cox's proportional hazards model (Therneau and Grambsch, 2000). Cox's model
assumes that the relative risk of event between groups is constant at each interval of time. The hazard function for a patient i as defined by Cox's proportional hazards model, can be written as
Given the nature of the variable G, λ[0](t) is the hazard function for a patient in the low-risk group. Moreover, the hazard function for any patient in the high-risk group is ψλ[0](t) (proportional
hazards), so ψ is the HR with ψ=exp(β). Note that, in this study, the larger the HR, the larger is the difference in survival probabilities between the groups of patients, and consequently the better
is the discrimination between low- and high-risk groups.
2.4 Performance comparison
To test whether a method performs significantly better than another one, we used two types of statistical tests: (i) a paired Student t-test based on the assumption of normality for the natural
logarithm of the hazard ratio (i.e. the coefficient β in Cox's proportional hazards model) and the concordance index; (ii) a paired Wilcoxon rank sum test of event occurrence with respect to the time
t for the AUC(t) and BSC(t). We considered that a method performs significantly better than another one if its performance is significantly better with a P < 0.05 and the difference between the
performance estimates of the two methods is larger than 1% of the lowest value. Note that we did not statistically compare the estimations of sensitivity and specificity due to the lack of standard
statistical test.
The concordance indices and the hazard ratios for all the risk prediction methods are represented using a forest plot (Lewis and Clarke, 2001). The accuracy measures are shown as squares centered on
the point estimate of the performance of each method. A horizontal line runs through the square to show its 95% confidence interval.
3 RESULTS
3.1 Breast cancer datasets
In order to compare different risk prediction methods with published gene signatures, we used four large microarray BC datasets collected with Affymetrix microarray platform (22 283 common probes),
called VDX (Wang et al., 2005), TBG (Desmedt et al., 2007), TAM (Haibe-Kains et al., 2008; Loi et al., 2007) and UPP (Miller et al., 2005). These datasets are publicly available from the GEO database
^2 through accession numbers {"type":"entrez-geo","attrs":{"text":"GSE2034","term_id":"2034"}}GSE2034, {"type":"entrez-geo","attrs":{"text":"GSE7390","term_id":"7390"}}GSE7390,
{"type":"entrez-geo","attrs":{"text":"GSE6532","term_id":"6532"}}GSE6532/{"type":"entrez-geo","attrs":{"text":"GSE9195","term_id":"9195"}}GSE9195 and {"type":"entrez-geo","attrs":
{"text":"GSE3494","term_id":"3494"}}GSE3494, respectively. VDX includes the gene expressions of 286 untreated node-negative BC patients and was used to build GENE76 and to validate GGI (see end of
Section 2.2). Only TBG exhibited the same criteria for the selection of patients (198) than VDX, i.e. untreated node-negative BC patients, and was used as an official validation of GENE76 and GGI.
TAM was composed of 354 ER-positive BC patients (the largest molecular group of BC) being homogeneously treated by tamoxifen therapy. UPP was composed of 251 patients being treated with heterogeneous
therapies. Although the selection of patients was different for VDX and TBG, TAM and UPP datasets might contain important prognostic information as well and could, therefore, be used as additional
validation sets. Due to their homogeneity in selection of patients, we should consider VDX and TBG as the most important datasets in this comparative study. The results obtained with TAM and UPP made
possible a more thorough assessment of the performance.
We considered the distant metastasis free survival of BC patients as the survival endpoint for VDX, TBG and TAM. This endpoint refers to the appearance of distant metastasis only. We considered the
relapse free survival of BC patients, i.e. appearance of local, regional or distant relapses, in UPP as the information on distant metastasis was not available. All the survival data were censored at
10 years as in Desmedt et al. (2007).
We used VDX as training set and TBG, TAM and UPP as validation sets. Although this choice was guided by the original publications in BC prognostication, we also performed all the analyses using TBG
as training set to ensure that our results were not driven by the choice of the training set (Michiels et al., 2005). We obtained similar results that let the conclusions of this study unchanged (see
Section 10 in Supplementary Material).
We assessed the performance in the training set and in the three validation sets using all the risk prediction methods summarized in Table 1. The AURKA model was used as reference because of its low
complexity. Both risk score and risk group predictions were compared.
3.2 Risk score prediction
This section presents the results for the performance assessment of the risk score predictions using the five performance criteria presented in Section 2.3.
The specificity for a sensitivity of 90%, is reported for all the risk prediction methods in Table 2. We observed values consistent with the literature (Buyse et al.; Desmedt et al., 2007; Foekens et
al., 2006; van de Vijver et al., 2002). GGI was the best method in two validation sets, yielding larger specificity values than the simplest model AURKA. The increase was estimated to 17.4, 2.8 and
1.5% for TBG, TAM and UPP datasets, respectively.
Specificity for a sensitivity of 90% for risk score prediction in the training set (VDX) and the three validation sets (TBG, TAM and UPP)
The performance assessment using the concordance index, the time-dependent ROC curve and the Brier score are given in Supplementary Figures 1, 2–5 and 6–9, respectively.
Looking at the most complex models, i.e. multivariate (MULTIV) survival (SURV) models using genome-wide data (GW) and GENE76, we observed overoptimistic performance estimates in training set compared
to the validation sets. Although we used advanced machine learning techniques to control overfitting, i.e. linear combination of univariate models or L1 regularization in Cox's proportional hazards
model, these complex models failed to outperform simpler ones in a validation setting.
The simple AURKA model was competitive in all the datasets. As mentioned earlier, we statistically compared the performance of all the models with AURKA (Table 3). Only GGI was significantly better
in at least two validation sets whatever the accuracy measure. It is worth to note that AURKA outperformed KM, the benchmark model for the Brier score (see Section 2.3.4), in all the datasets except
for TBG.
Performance for risk score prediction in the training set (VDX) and the three validation sets (TBG, TAM and UPP)
As we did not observe a significant improvement using cross-validated dimension reduction strategies (see Section 2.2.1), we reported the performance of the methods using either RANKCV or PCACV in
Supplementary Tables 3 and 4.
We computed all the pairwise performance comparisons (see Section 9.1 in Supplementary Material) and observed again that complex models performed poorly in validation sets compared to simpler ones.
According to the IAUC performance criterion in TBG dataset, the GW methods using SURV phenotype are significantly better than the other risk prediction methods. However, these significant results are
not confirmed in the other validation sets. We noticed also that, unlike GGI, GENE76 was consistently worse than most of the other considered methods.
3.3 Risk group prediction
This section presents the results for the performance assessment of the risk group predictions using the five performance criteria presented in Section 2.3. We used the tertile to define the risk
groups (see Section 2.2), leaving 33% of the patients in the low-risk group and the remaining 66% in the high-risk group. This proportion is usually observed in BC prognostication (Buyse et al., 2006
; Desmedt et al., 2007; van't Veer et al., 2002; Wang et al., 2005).
The sensitivity and the specificity for all the risk prediction methods are reported in Table 4. Again, we observed values for sensitivity and specificity that are consistent with the literature. GGI
was the best method in two validation sets, yielding larger sensitivity and specificity values than the simplest model AURKA. However, the difference is small except for TBG dataset.
Sensitivity and specificity for risk group prediction in the training set (VDX) and the three validation sets (TBG, TAM and UPP)
The performance assessment of the risk group predictions using the concordance index, the HR and the Brier score are given in Supplementary Figures 10, 11 and 12–15, respectively.
Similarly to the risk score prediction, the simple AURKA model was competitive in all the datasets. We statistically compared the performance of all the models with AURKA (see Table 5). In order to
illustrate the gain of using GGI over AURKA for risk groups prediction, we compared the survival curves for each dataset separately (see Supplementary Figs 16–19). Because neither AURKA nor GGI were
fitted on VDX, this dataset is not considered as a training set. We observed a substantial improvement in risk group prediction in terms of survival probabilities in the low-risk group in using the
risk groups predicted by GGI compared to AURKA. At 5 years, the increase in survival probabilities in the low-risk group was estimated to 2, 6, 0 and 1% for VDX, TBG, TAM and UPP, respectively.
Performance for risk group prediction in the training set (VDX) and the three validation sets (TBG, TAM and UPP)
As we did not observe a significant improvement using cross-validated dimension reduction strategies, we reported the results of the methods using either RANKCV or PCACV in Supplementary Tables 5 and
We computed all the pairwise comparisons (see Section 9.2 in Supplementary Material) and made similar observations than for the risk score prediction. This holds true if we restrict the analysis to
TBG dataset. However, less significant differences between methods were detected. This can be explained by the loss of prognostic information due to the risk group creation.
We assisted recently to intense research in BC prognostication due to a growing availability of high-dimensional genomic information that could potentially be used for risk prediction (Simon, 2005).
The situation is often characterized by a relatively small number of patients and a large number of explanatory variables. This high-dimensional setting can be seen as an opportunity to create better
risk prediction models compared to those solely based on clinical data and/or single markers. At the same time this prevents from a straightforward use of classical approaches of statistical modeling
and data analysis.
To the best of our knowledge, the present study is the first that has systematically compared the performance of state-of-the-art survival methods for BC prognostication from gene expression data by
using a training/validation framework and several accuracy measures for survival prediction. The public availability of large microarray BC datasets and the recently introduced measures for
performance assessment in survival analysis allow to perform an in-depth comparative study in order to elucidate the key characteristics of a successful risk prediction method and to bring new
insights into BC prognostication.
We used four large microarray BC datasets (one for training and three for validation) in order to compute unbiased estimates of five accuracy measures (see Section 2.3) for 13 risk prediction methods
(see Section 2.2). As expected, we observed that complex methods, e.g. multivariate survival models fitted using dimension reduction from genome-wide data or GENE76, performed very well in the
training set. However, the performances in the validation sets were poorer and they failed to outperform consistently the simplest model, i.e. AURKA, in spite of the use of machine learning
strategies (namely combination of univariate models or regularization) to reduce the risk of overfitting. These results highlighted the fact that the loss of interpretability deriving from the use of
overcomplex methods in survival analysis of BC microarray data might be not sufficiently counterbalanced by an improvement in the quality of prediction.
Interestingly, AURKA, the simplest model defining the risk score as the expression of a single proliferation gene, performed well in all the survival prediction tasks. From Tables 3 and and5,5, we
noticed that AURKA was significantly better than KM, the benchmark model ignoring all genetic information, except only for the risk score prediction in TBG. Several other methods outperformed
consistently KM as shown in Supplementary Figures 22 and 25. These results are very encouraging for BC prognostication as it was shown that, in diffuse large B-cell lymphoma prognostication for
instance, most prediction models were not better than KM in a validation setting [see discussion of Schumacher et al. (2007)]. Moreover, GGI was the only model that outperformed AURKA in at least two
validation sets whatever the accuracy measure for risk score and risk group predictions. As GGI is a linear combination of proliferation gene expressions (see Section 2.2), these results highlight
the importance of proliferation measured by gene expression profiling in BC prognostication, and confirm the results of Sotiriou et al., (2006a, b).
In order to go further in the comparison of the different risk prediction methods, we computed all the pairwise performance comparisons for all the methods (see Section 9 in Supplementary Material)
and we observed that models using only the biologically driven selection of genes of interest (BD) led to similar performance than models using genome-wide data (GW) with ranking (RANK) or principal
components analysis (PCA). This suggests that finding a combination of relevant variables in a high-dimensional setting is a difficult task since simple dimension reduction methods did not succeed to
significantly improve the models from genome-wide data compared to simpler models using a very small set of genes selected from literature. Our results are consistent with earlier studies focused on
stability of feature selection for GW methods (Ein-Dor et al., 2005; Michiels et al., 2005). Moreover, we observed that models fitting the histological grade (HG) as phenotype performed globally
better in validation sets than models fitting survival data (SURV or TOE) suggesting that we did not succeed to catch additional information about prognostication in using survival models. The fact
that the performance of GGI was the best in validation sets reinforced this observation as the GGI was built using a method similar to our weighted combination of relevant genes for the histological
grade (see Section 2.2).
It is worth to mention that all the accuracy measures were in nice agreement in our comparative study. Smaller significant differences in performance estimates were detected by the IAUC and the IBSC
criteria, probably due to the type of statistical test [paired Wilcoxon rank sum test for AUC(t) or BSC(t) compared to paired Student t-test for C-index or HR, see Section 2.3].
A final analysis concerns the performance comparison with the classical indicators, such as the histological grade (Scarff and Torloni, 1968), AOL (Olivotto et al., 2005) and NPI (Galea et al., 1992
). In our comparative study, GGI was consistently better than the histological grade (data not shown). We were not able to compute the risk scores for AOL and NPI on TAM and UPP datasets due to lack
of information. As shown in Supplementary Table 10, AURKA and GGI outperformed consistently AOL in the VDX and TBG datasets, except for the IAUC in TBG. However, this is not the case for NPI. These
results suggest that the superiority of microarray-based risk prediction methods is not obvious and need further investigations.
In conclusion, our results challenge the use of microarray technology to screen the whole genome for BC prognostication of global populations of patients. Indeed, we found that models using a single
gene or a small set of biologically driven selected genes yielded similar or even better performance than models fitted from genome-wide data. Although GGI, the model yielding the best performances
in validation sets, uses a set of 128 probes, it can be considered as a simple extension of AURKA, i.e. a quantification of proliferation using more genes. Moreover the authors recently showed that
we could yield similar performance in using only a small subset of these 128 probes (Durbecq et al., 2007). The use of high-sensitivity gene expression profiling technologies such as the reverse
transcription polymerase chain reaction, in addition to be cheaper and more user friendly, might improve the performance of these risk prediction models.
The relevance of proliferation for BC prognostication was previously reported by several other research groups. Indeed, Thomassen et al. (2007) found that cell cycle and cell proliferation
represented the predominant overlaps in gene ontology categories of the nine prognostic signatures they compared. Yu et al. (2007) also conducted pathway analyses of five published prognostic gene
signatures and found that the signatures had many pathways in common such as cell cycle, regulation of cell cycle, mitosis, apoptosis, etc. Our group also investigated in a large meta-analysis of
publicly available gene expression data, how different gene lists may give rise to signatures with equivalent prognostic performance and found by dissecting these signatures according to the main
molecular processes involved in breast cancer, that proliferation may be the common driving force of several prognostic signatures (Sotiriou et al., 2006a).
Until now, the generation of the prognostic signatures has been done on global populations of BC patients. However, since it is clear that breast cancer is a molecular heterogeneous disease, with
subgroups defined primarily by the estrogen (ER) and HER2 receptors (Perou et al., 2000; Sotiriou et al., 2003), prognosis could be refined to these molecularly homogeneous subgroups of patients. We
showed, for example, in a meta-analysis recently published by our group that proliferation is the strongest parameter predicting clinical outcome in the ER+/HER2 − subgroup of patients only (group of
patients representing more than 66% of the global population), whereas immune response and tumor invasion appear to be the main biological processes associated with prognosis in the ER-/HER2- and
HER2+subgroups, respectively (Desmedt et al., 2008; Sotiriou et al., 2007). These recent results suggest that we could improve BC prognostication by restricting the genome-wide analysis to specific
molecular subtypes. This will be the subject of further investigations.
Supplementary Material
[Supplementary Data]
The authors would like to thank Martin Schumacher and Thomas Gerds for providing the R package, and Yann-Aël Le Borgne for his constructive comments.
Funding: This work was supported by the Belgian National Foundation for Scientific Research FNRS (B.H.-K., C.D., C.S.), and by the MEDIC Foundation (C.S.).
Conflict of Interest: C. Sotiriou, M. Delorenzi and M. Piccart are named inventors on a patent application for the Gene expression Grade Index used in this study. There are no other conflicts of
^1Raw gene expression and clinical data are publicly available in the GEO Gene Expression Omnibus!GEO public database and the Sweave version of the article including the standalone R code (R
Development Core Team, 2007) is available from http://www.ulb.ac.be/di/map/bhaibeka/survcompaper/.
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Bounding the modular discriminant of an elliptic curve in the j-invariant
up vote 6 down vote favorite
Consider an elliptic curve $X=\mathbf{C}/ (\mathbf{Z}+\tau \mathbf{Z})$, where $\tau$ is an element in the complex upper half plane. We define $$\Vert \Delta\Vert(X) = (\Im \tau)^6 \vert q\prod_{k=1}
^\infty (1-q^k)^{24}\vert,$$ where we write $q=\exp(2\pi i \tau)$ as usual. This is called the modular discriminant of $X$.
Assume $j$ is algebraic, i.e., $X$ can be defined over a number field.
Question. Can the function $\log\Vert \Delta \Vert(-)$ (on the moduli space of elliptic curves over $\mathbf{C}$) be bounded (from above or below) in terms of the (height of the) $j$-invariant?
Firstly, one should be able to answer this question ineffectively. That is, to give a yes or no answer to the question. An effective bound (if it exists) might be a bit harder to obtain.
I heavily edited this old question. Therefore, the first four comments below might not make sense anymore.
elliptic-curves ag.algebraic-geometry riemann-surfaces
1 It doesn't seem that the morphism $\pi$ play a role in your question, or am I missing something? – François Brunault Dec 7 '10 at 12:09
1 Actually it probably does. I first thought I had solved the question simply by taking the Weierstrass function and using that the modular discriminant is s^2(s-1)^2 but now I doubt this being
correct. That's why I removed it again. The point is that the degree of \pi should come into play at some point, right? – Ari Dec 8 '10 at 8:20
1 Ow maybe I removed it too quickly and it was actually correct.... – Ari Dec 8 '10 at 8:24
1 Why do you take the absolute value of the imaginary part of $\tau$? It looks like it's already a positive real (which is more than I can say for that product in $q$). – S. Carnahan♦ Dec 8 '10 at
3 Height only makes sense if $j$ is algebraic. Your formulation implies $X$ is any elliptic curve over $\mathbb{C}$. – Felipe Voloch Jul 25 '11 at 17:38
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2 Answers
active oldest votes
The new $\| \Delta \|$, defined as $\mathop{\rm Im}(\tau)^6$ times the absolute value of the usual modular form $\Delta$, is invariant under the full modular group $\Gamma = {\rm PSL}_2({\
bf Z})$ acting on the upper half-plane $H$. This $\| \Delta \|$ is nonzero and continuous on the quotient $H / \Gamma$, and approaches zero exponentially as $\tau$ approaches the one cusp
of $H / \Gamma$. Hence $\|\Delta\|$ is uniformly bounded above, without any hypothesis on $j$; and $\|\Delta\|$ is bounded below if we have an upper bound on $|j|$. The latter bound is
completely effective, namely $$ \| \Delta \| \gg (\log|j|)^6 / |j| {\rm\ \ \ \ as\ \ \ \ } |j| \rightarrow \infty, $$ and indeed $\| \Delta \| \sim C (\log|j|)^6 / |j|$ for some universal
constant $C$, which is $(2\pi)^{-6}$ if I did this right. Now if you bound the height of $j$ from above then you impose an upper bound on the absolute value of any conjugate of $j$, and
up vote 9 thus on $\| \Delta \|$.
down vote
accepted Whether and how this lower bound depends on the height of $j$ then hinges on which flavor of height you're using, i.e. whether you normalize according to the degree $[{\bf Q}(j) : {\bf Q}]
$, and whether you take logarithms. There is no such bound in the other direction: large height of $j$ does not force small $\| \Delta \|$ because it does not force $j$ to have a large
conjugate (e.g. $j$ could be $1 / 10^{100}$).
1 How does one obtain the effective lower bound for $\Vert \Delta\Vert$ in terms of the absolute value of $j$? – Ari Jul 26 '11 at 9:10
1 @Ariyan: As $j(\tau) \rightarrow \infty$ with $\tau$ in the usual fundamental domain, $q \rightarrow 0$ with $q \sim 1/j$. Thus $|q| \sim 1/|j|$. Also Im$(\tau) = \log(1/|q|)/(2\pi) \
sim \log|j|/(2\pi)$. This accounts for two of the factors of $\|\Delta\|$, and the remaining factor $\prod_{n=1}^\infty (1-q^n)^{24}$ approaches $1$ as $q \rightarrow 0$. This gives the
asymptotic formula for $\|\Delta\|$ in terms of $|j|$, and all the error estimates are readily seen to be effective. – Noam D. Elkies Jul 26 '11 at 10:42
Thank you very much. This is a great answer. Do you think there's any hope in making the uniform upper bound on $\Vert \Delta \Vert$ explicit? – Ari Jul 26 '11 at 13:00
You're welcome!$$ $$Getting an explicit upper bound on $\|\Delta\|$ is a calculus exercise. We may assume $\tau = x+iy$ with $y^2 \geq 3/4$ (fundamental domain). Then the factor $\prod_
1 {n=1}^\infty (1-q^n)^{24}$ of $\|\Delta\|$ is at most $\prod_{n=1}^\infty (1+e^{-\sqrt{3}\pi n})^{24}$, while the rest is $y^6 e^{-2\pi y}$ which is maximized at $y = 3/\pi$, etc.$$
$$It seems (and is probably known) that in fact the max occurs at the sharper corner $\tau = (\pm 1 + \sqrt{3}i)/2$ of the fundamental domain, where $\|\Delta\| = .002+ = (2\pi^2/9)^6 /
\Gamma(2/3)^{36}$. – Noam D. Elkies Jul 27 '11 at 1:55
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As it stands, I think this question is still too vague to be answerable in generality. What kind of expression are you permitting for the bound? Certainly one can construct an artificial
bound which doesn't even involve $j$ at all, which would be completely silly and certainly not what you have in mind.
up vote 3 As a first step, it's easy to see that no rational function of $j$ bounds $\Delta$. Indeed, if $R$ is a rational function such that $|\Delta| \leq |R(j)|$ everywhere on $X=\mathbb{H}/PSL
down vote (2, \mathbb{Z})$, then the bounded meromorphic function $\Delta/R(j)$ must be a constant. But this implies that $\Delta$ has weight $0$, which is false.
Thank you for this very important remark. It made me realize that I'm actually looking for a bound on the logarithm of the modular discriminant. Sorry for the vagueness. I'm just
interested in this particular discriminant $\log \Vert \Delta \Vert$, because from an Arakelov-theoretic point of view it coincides up to a constant with the Faltings delta invariant. –
Ari Jul 25 '11 at 16:52
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whole wheat percentage
February 22, 2009 - 10:36am
I was thinking about how we often describe mixtures of flour in ratios, such as 50/50 whole wheat/white, 20/80, etc. This is referring to the ratio of a measure of whole wheat flour to a measure of
In reality, though, whole wheat flour is white flour -- 100%, in fact -- plus a little extra. Or, perhaps more accurately, white flour is 70% of the whole wheat grain.
So, I'm curious as the most accurate way to describe a "50/50" mix. What about a 20/80 mix (white/ww)? 80/20? Does anyone have the math power to figure these out?
This is purely an academic exercise, and probably not helpful for most home bakers. However, I have learned that when my left brain cries out for sustenance, it is best to feed him before he gets out
of hand and does things like making lists and organizing. | {"url":"http://www.thefreshloaf.com/print/10838","timestamp":"2014-04-24T07:46:50Z","content_type":null,"content_length":"6844","record_id":"<urn:uuid:2bb100ed-453f-44fb-abb5-761be79a0876>","cc-path":"CC-MAIN-2014-15/segments/1398223205375.6/warc/CC-MAIN-20140423032005-00005-ip-10-147-4-33.ec2.internal.warc.gz"} |
[SciPy-Dev] chi-square test for a contingency (R x C) table
Neil Martinsen-Burrell nmb@wartburg....
Thu Jun 17 19:43:18 CDT 2010
On 2010-06-17 11:59, Bruce Southey wrote:
> On 06/17/2010 10:45 AM, josef.pktd@gmail.com wrote:
>> On Thu, Jun 17, 2010 at 11:31 AM, Bruce Southey<bsouthey@gmail.com> wrote:
>>> On 06/17/2010 09:50 AM,josef.pktd@gmail.com wrote:
>>>> On Thu, Jun 17, 2010 at 10:41 AM, Warren Weckesser
>>>> <warren.weckesser@enthought.com> wrote:
>>>>> Bruce Southey wrote:
>>>>>> On 06/16/2010 11:58 PM, Warren Weckesser wrote:
>>>>>> The handling for a one way table is wrong:
>>>>>> >>>print 'One way', chisquare_nway([6, 2])
>>>>>> (0.0, 1.0, 0, array([ 6., 2.]))
>>>>>> It should also do the marginal independence tests.
>>>>> As I explained in the description of the ticket and in the docstring,
>>>>> this function is not intended for doing the 'one-way' goodness of fit.
>>>>> stats.chisquare should be used for that. Calling chisquare_nway with a
>>>>> 1D array amounts to doing a test of independence between groupings but
>>>>> only giving a single grouping, hence the trivial result. This is
>>>>> intentional.
>>> In expected-nway, you say that "While this function can handle a 1D
>>> array," but clearly it does not handle it correctly.
>>> If it was your intention not to do one way tables, then you *must* check
>>> the input and reject one way tables!
>>>>> I guess the question is: should there be a "clever" chi-square function
>>>>> that figures out what the user probably wants to do?
>>> My issue is that the chi-squared test statistic is still calculated in
>>> exactly the same way for n-way tables where n>0. So it is pure
>>> unnecessary duplication of functionality if you require a second
>>> function for the one way table. I also prefer the one-stop shopping approach
>> just because it's chisquare doesn't mean it's the same kind of tests.
>> This is a test for independence or association that only makes sense
>> if there are at least two random variables.
> Wrong!
> See for example:
> http://en.wikipedia.org/wiki/Pearson's_chi-square_test
> "Pearson's chi-square is used to assess two types of comparison: tests
> of goodness of fit and tests of independence."
> The exact same test statistic is being calculated just that the
> hypothesis is different (which is the user's problem not the function's
> problem). So please separate the hypothesis from the test statistic.
It is only the exact same test statistic if we know the expected cell
counts. How these expected cell counts are determined depends
completely on the type of test that is being carried out. In a
goodness-of-fit test (chisquare_oneway) the proportions of each cell
must be specified in the null hypothesis. For an independence test
(chisquare_nway), the expected cell counts are computed from the given
data and the null hypothesis of independence. The fact that the formula
involving observed and expected numbers is the same should not obscure
the fact that the expected numbers come from two completely different
assumptions in the n=1 and n>1 cases. Can you explain how the expected
cell counts should be determined in the 1D case without the function
making assumptions about the user's null hypothesis?
I believe that we CANNOT separate the test statistic from the user's
null hypothesis and that is the reason that chisquare_oneway and
chisquare_nway should be separate functions. The information required
to properly do a goodness-of-fit test is qualitatively different than
that required to do an independence test. I support your suggestion to
reject 1D arrays as input for chisquare_nway. (With appropriate checking
for arrays such as np.array([[[1, 2, 3, 4]]].)
>> I don't like mixing shoes and apples.
> Then please don't.
Great. I'm glad to see that we all agree that chisquare_oneway and
chisquare_nway should remain separate functions. :)
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[Scipy-tickets] [SciPy] #1466: More accurate roots/weights for Gauss-Hermite quadrature
SciPy Trac scipy-tickets@scipy....
Mon Jun 27 23:28:03 CDT 2011
#1466: More accurate roots/weights for Gauss-Hermite quadrature
Reporter: Bogdan | Owner: pv
Type: enhancement | Status: new
Priority: normal | Milestone: Unscheduled
Component: scipy.special | Version: devel
Keywords: |
System: OSX 10.6.7, Python 2.6.1 (the one provided with system), Numpy
2.0.0.dev-3071eab and SciPy 0.10.0dev (installed via SciPy Superpack)
I am using scipy.special.orthogonal.hermite and .h_roots to perform
decomposition into basis of harmonic oscillator eigenfunctions (which is
essentially G-H quadrature). I noticed that decomposition is not behaving
well when the number of modes is big (>40 in my case). After some testing
I narrowed it down to inaccurate polynomial roots provided by h_roots().
As far as I understand from the code, scipy implementation uses Golub-
Welsch algorithm, which is general and fast because it employs linear
algebra. But in our imperfect world with finite precision numbers it
starts to lose stability with large matrix sizes. So I took recursive root
finding algorithm from Numerical Recipes (3rd ed., section 4.6.1) and it
showed much better accuracy (see attached script with comparison).
I understand that this algorithm is less general (because it uses good
guesses for first roots of Hermite polynomials) and definitely slower than
G-W, but there are applications when its accuracy is necessary. I think it
should be present in SciPy, at least as an alternative to the current
Ticket URL: <http://projects.scipy.org/scipy/ticket/1466>
SciPy <http://www.scipy.org>
SciPy is open-source software for mathematics, science, and engineering.
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Random walk on the hypercube
up vote 1 down vote favorite
Let $H_N=\{0,1\}^N$ the N-dimensional hypercube. We define the following random walk $X_n$ on $H_N$:
• start from a point $x \in H_N$
• pick at random an integer $k$ in $[1,N-1]$ and exchange $x(k)$ and $x(k+1)$
• go on like that...
For instance, with n=4, the random walk looks like: $X_0=0011$, $X_1=0101$, $x_2=0110$, ...
Do you know any article/textbook where this random walk is studied in details ? I have been looking in many classical textbook without success...
Beyond this general request, I am interested in the following question: take a given point $x^* \in H_N$ and define $d_n$ the Hamming distance between $X_n$ and $x^*$. Is is possible to estimate the
mean of the first time when $d_n$ hits $0$ (as a function of $x*$ and the starting point of the chain $X_0$) ? Of course we suppose that $X_0$ and $x^*$ have the same number of ones... I hope you
will enjoy this problem.
Thank you !
This looks like the sort of thing that R. Graham et al would have studied at Bell Labs. Does a paper such as www-stat.stanford.edu/~cgates/PERSI/papers/… provide any guidance? – Benjamin Dickman
Mar 15 '13 at 21:27
"Go on like that" is not entirely clear to me. Do we pick a new random $k$ at each step? Or do we proceed $k$, $k+1$, ..., which is consistent with the example you gave? Also, I suggest that it
will probably help to visualize the problem to think of these 0,1 strings as lattice paths from $(0,0)$ to $(t,n-t)$, where we read 1's as horizontal steps and 0's as vertical steps. The basic swap
move which you describe looks at two adjacent steps. If they are both horizontal or both vertical, nothing happens; otherwise, a single box is added or subtracted from the region under the path. –
Hugh Thomas Mar 16 '13 at 0:42
One upper bound would be the time to reach a particular permutation by adjacent transpositions in the symmetric group. However, this value should be much lower, since you only need to hit one of
$k! (n-k)!$ permutations. – Douglas Zare May 18 '13 at 22:56
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Autonomous perturbations of LISA orbits
by Pucacco, Giuseppe and Bassan, Massimo and Visco, Massimo
27pages, 20 figures
We investigate autonomous perturbations on the orbits of LISA, namely the effects produced by fields that can be expressed only in terms of the position, but not of time in the Hill frame. This first
step in the study of the LISA orbits has been the subject of recent papers which implement analytical techniques based on a “post-epicyclic” approximation in the Hill frame to find optimal
unperturbed orbits. The natural step forward is to analyze the perturbations to purely Keplerian orbits. In the present work a particular emphasis is put on the tidal field of the Earth assumed to be
stationary in the Hill frame. An accurate interpretation of the global structure of the perturbed solution sheds light on possible implications on injection in orbit when the time base-line of the
mission is longer than that assumed in previous papers. Other relevant classes of autonomous perturbations are those given by the corrections to the Solar field responsible for a slow precession and
a global stationary field, associated to sources like the interplanetary dust or a local dark matter component. The inclusion of simple linear contributions in the expansion of these fields produces
secular solutions that can be compared with the measurements and possibly used to evaluate some morphological property of the perturbing components. | {"url":"http://brownbag.lisascience.org/arxiv1005-2976/","timestamp":"2014-04-16T07:31:53Z","content_type":null,"content_length":"30540","record_id":"<urn:uuid:3eebb4a5-b476-4fc2-b35e-abcec07382fb>","cc-path":"CC-MAIN-2014-15/segments/1397609521558.37/warc/CC-MAIN-20140416005201-00489-ip-10-147-4-33.ec2.internal.warc.gz"} |
FOM: Hersh's incoherent attack on formalism and logicism
Stephen G Simpson simpson at math.psu.edu
Thu Oct 1 12:16:38 EDT 1998
Reply to Hersh 30 Sep 1998 16:19:42.
1. Frege and modern logic
Hersh writes:
> You say "you must dismiss Frege's work as a failure." Not at all.
> I wrote, page 141 of W.I.M.R.: "Frege's introduction of quantifiers is
> consdered the birth of modern logic."
Do *you* consider Frege's work to be the birth of modern logic?
Do *you* think modern logic is of value for philosophy of mathematics?
You don't seem to think so.
2. The axiom of infinity: Hilbert's program and formalism
> You research program doesn't respond to my remark.
Which remark, and which research program?
One of your remarks (12 Sep 1998 18:06:45) was as follows:
> One famous difficulty is [the] axiom of infinity. You can't do
> modern math without it.
One of "my" research programs (actually it involves a number of
people) responds directly to your remark. It does so by examining
the extent to which modern mathematics is reducible to finitism.
This is in the context of Hilbert's program, which you have
dismissed as a failure.
Are you willing to rescind your dismissal? Are you willing to
examine evidence against your remark?
> You say, "If you are unwilling to study the role of infinity in
> mathematics, then how can you expect anyone to take your comments
> on it seriously?"
> My comment, that the axiom of infinity is not inituitively
> plausible as an axiom of logic, is not mine. It has been made by
> others ....
Even if the comment has been made by others, you repeated it, so you
must take some responsibility for it. You can't hide behind others.
However, I wasn't referring to that particular comment. (More on
that comment below, in connection with logicism.) Rather, I was
referring to another of your comments concerning the axiom of
> One famous difficulty is [the] axiom of infinity. You can't do
> modern math without it.
I say again: How can you expect anyone to take this comment
seriously, if you are not willing to examine evidence for and
against it?
> You say, "You present a a caricature of Hilbert's work, then
> attack the caricature." No. I used the word "formalism in the
> common, colloquial sense, not in Hilbert's sense. There is no
> caricature and no attack.
Here you seem to be evading the fact that Hilbert is generally
regarded as the originator of formalism. Do you dispute this
conventional view of the history of formalism?
But, all right, let's take you at your word and assume that you
never attacked Hilbert's formalism. Let's assume that you were
attacking somebody else's formalism.
Who are these hitherto unnamed formalists? Do you recognize a
difference between their views and those of Hilbert? Or are you
merely attacking coffee-room chatter, as Martin Davis suggested?
> You say, "You were arguing that it's OK to dismiss Hilbert's
> views without a hearing." As I keep trying to explain, I never
> referred to Hilbert's views at all. The word formalism has more
> than one meaning. I can't believe you're unaware of that.
I'm *not* aware of that. I accept the conventional view that
Hilbert is the originator of formalism. If you have some other kind
of formalism in mind, please tell me who originated it and how it
differs from Hilbert's formalism.
Here are the real questions:
Do *you* think Hilbert's program is of any actual or potential value
for philosophy of mathematics?
Do *you* think the research of your other formalists (who are they?)
has any actual or potential value for philosophy of mathematics?
3. The axiom of infinity: set theory and logicism
Hersh writes:
> You seem incapable of dealing with this well known fact.
Here you are referring to the well known fact that the axiom of
infinity is not generally regarded as a logical axiom. I accept
that fact, and I understand the reasons for it, at least in the
context of Russell's type theory and ZF set theory. In this sense,
one could say that these theories do not represent a *total*
vindication of the logicist program. But it's going too far to say,
as you do, that the logicist program as a whole is a mistake or a
By the way, there is an alternative set theory known as New
Foundations (= NF), going back to Quine. I don't know too much
about it, but my impression is that it attempts to carry out the
logicist program by deriving the axiom of infinity and others from
some logical principles. Naturally there are costs to this. As I
say, I am not an expert on this. The FOM subscriber list includes
some experts on NF: Thomas Forster, Randall Holmes.
Also, there is some recent work of Harvey Friedman about motivating
the axioms of set theory in a more logical way, as an outcome of a
theory of mathematical predication.
Do you regard this kind of f.o.m. research as legitimate? Do you
regard it as having potential interest for philosophy of
4. Demonization
Hersh writes:
> what do you mean, "demonize"? When you attribute such motives to
> me, it's I who am being demonized. To criticize or even reject
> foundationalism isn't demonizing anything. It's what people do
> in the course of finding their philosophical beliefs.
You have gone beyond what I regard as legitimate philosophical
criticism. You have attacked foundationalism as anti-"humanistic",
anti-life in a sense, and you have tried to artificially link
foundationalism with religion and with authoritarian or totalitarian
politics. I don't think "demonize" is too strong a term to describe
your behavior.
> It's weird to tell me I regard the pursuit of certainty as "evil
> incarnate."
It's not at all weird to tell you this, in light of your attempt to
demonize foundationalism, on the explicit grounds that the
foundationalists (Frege, Brouwer, Hilbert, ...) were motivated by a
quest for certainty.
> to be fair, you'll have to accuse Sol of demonizing, attacking,
> and being "so hostile" to fom!!
Not at all. Sol Feferman has never attempted to demonize f.o.m. by
saying that it is anti-humanistic and linking it to totalitarian
politics, as you routinely do.
5. Misinterpretation
Reuben Hersh writes:
> I asked why you consistently persist in misinterpreting me.
> You didn't answer, of course.
OK, I'll answer. The answer is that I don't accept the premise of
your question. The premise of your question is that I am
misinterpreting you. I don't accept that premise. I don't think I am
misinterpreting you. I think my interpretation of you is correct. To
put it colloquially, I think I've "got your number".
-- Steve
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Next: Parameter setup Up: Cross-correlation Previous: Definitions
Figure 4 shows an example of a cross-correlation function for a sum of squared differences. The reference image is a bright galaxy, and the compared image is a shifted version of the reference with a
translation vector of (+5,-5). This plot clearly shows that the cross-correlation criterion is minimal at the expected point. A search for a local minimum will yield the right translation vector.
At this point, a subpixel offset detection is applied by fitting a parabola to the cross-correlation signal in x and y, and looking for the subpixel minimum. This method is theoretically precise up
to a 1/100 pixel precision in the ideal case of noiseless images. The precision obtained with SOFI data is about 1/10 pixel.
Nicolas Devillard | {"url":"http://www.eso.org/projects/dfs/papers/jitter99/node11.html","timestamp":"2014-04-18T19:52:35Z","content_type":null,"content_length":"3209","record_id":"<urn:uuid:041cb2f0-a3af-4a40-805c-9c33520fc080>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.7/warc/CC-MAIN-20140416005215-00097-ip-10-147-4-33.ec2.internal.warc.gz"} |
HUP in QFT and QM:virtual particles
and I was thinking that feynman rules are derived first,then the gauge is fixed (which is supposed to be the key advantage of path integral formalism)
You have to be careful; there are the so-called [itex]R_\xi[/itex] gauges which are a generalization of the Lorentz gauge [itex]\partial_\mu A^\mu = 0[/itex]; in the [itex]R_\xi[/itex] gauges one
adds a gauge
term to the Lagrangian in the path integral
[tex]\delta\mathcal{L}_\xi = -\frac{(\partial_\mu A^\mu)^2}{2\xi}[/tex]
which is a Gaussian located at [itex]\partial_\mu A^\mu = 0[/itex] with width [itex]\xi[/itex] in the "gauge field space". Via this mechanism one has a "family of gauge fixings" labelled by the
continuous parameter [itex]\xi[/itex]; for [itex]\xi\to 0[/itex] the gauge breaking term in the action reduces to a delta functional in the PI fixing the theory to ordinary Lorentz gauge.
Another possibility is to introduce the axial gauge condition [itex]n_\mu A^\mu = 0[/itex] where the global direction [itex]n_\mu[/itex] remains a s a free parameter in the theory on the level of
Feynman diagrams.
It is true that via this mechanism one introduces a free parameter into the Feynman rules and that chosing a specific gauge (i.e. a specific value for [itex]\xi[/itex], [itex]n^\mu[/itex], ...) is
done afer deriving the Feynman rules.
But this is not what I refer to. What I mean is that one first fixes a
family auf gauges
, which may depend on a free parameter and then derives the Feynman rules for this family. | {"url":"http://www.physicsforums.com/showthread.php?p=4157362","timestamp":"2014-04-16T07:45:24Z","content_type":null,"content_length":"44584","record_id":"<urn:uuid:9ee8052e-d757-43a1-9ea8-3731cc1990fe>","cc-path":"CC-MAIN-2014-15/segments/1398223202457.0/warc/CC-MAIN-20140423032002-00040-ip-10-147-4-33.ec2.internal.warc.gz"} |
Fundamentals of College algebra (10th Ed)
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ACT Section 5: Writing
Test Name: Section Five: Writing
# of Questions: 65 Study Guide Pages
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Variable name Definition and construction Source
GDP growth rate Percentage growth rate of real GDP per capita in constant prices. Reference year: 1996, Laspeyres index. Penn World Tables 6.3
GDP volatility Standard deviation of real GDP growth rate. Yearly series. Penn World Tables 6.3
Consumption volatility Standard deviation of real consumption growth rate (real GDP times consumption share of GDP). Yearly series. Penn World Tables 6.3
Investment volatility Standard deviation of real Investment growth rate (real GDP times investment share of GDP). Yearly series. Penn World Tables 6.3
Gov. consumption volatility Standard deviation of real public consumption growth rate (real GDP times government cons.share of GDP). Yearly series. Penn World Tables 6.3
Investment share of GDP Log level of the investment share of real GDP. Yearly series. Penn World Tables 6.3
Initial GDP Log level of GDP of the 1st year of the window of which the corresponding volatility is computed. Yearly series. Penn World Tables 6.3
Population growth rate Percentage growth rate of population. Yearly series. Penn World Tables 6.3
Education Logarithm of the percentage of secondary schooling attained in population aged 25 years and over. Barro-Lee dataset (2010) | {"url":"http://www.hindawi.com/journals/isrn.economics/2013/381368/tab3/","timestamp":"2014-04-20T21:54:40Z","content_type":null,"content_length":"3254","record_id":"<urn:uuid:aceff731-d3a4-4b1e-b27c-5c7b06ad9336>","cc-path":"CC-MAIN-2014-15/segments/1397609539230.18/warc/CC-MAIN-20140416005219-00236-ip-10-147-4-33.ec2.internal.warc.gz"} |
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The following text was written a few years ago, but much of it never got published. So, I thought that this might be a good opportunity to make it available, since what it says is still true today.
Since a phylogenetic tree is interpreted in terms of the monophyletic groups that it hypothesizes, it is important to quantitatively assess the robustness of all of these groups (i.e. the degree of
support for each branch in the tree) — is the support for a particular group any better than would be expected from a random data set? This issue of clade robustness is the same as assessing branch
support on the tree, since each branch represents a clade. Many different techniques have been developed, including:
1. analytical procedures, such as interior-branch tests (Nei et al. 1985; Sneath 1986), likelihood-ratio tests (Felsenstein 1988; Huelsenbeck et al. 1996b), and clade significance (Lee 2000);
2. resampling procedures, such as the bootstrap (Felsenstein 1985), the jackknife (Lanyon 1985), topology-dependent permutation (Faith 1991), and clade credibility or posterior probability (Larget
and Simon 1999); and
3. non-statistical procedures, such as the decay index (Bremer 1988), clade stability (Davis 1993), and spectral signals (Hendy and Penny 1993).
Of these, far and away the most popular and widely used method has been the bootstrap technique (Holmes 2003; Soltis and Soltis 2003).
The bootstrap
This method was first introduced by Efron (1979) as an alternative method to jackknifing for producing standard errors on estimates of central location other than the mean (e.g. the median), but it
has since been expanded to cover probabilistic confidence intervals as well (Efron and Tibshirani 1993; Davison and Hinkley 1997). It was introduced into phylogenetic studies by Penny et al. (1982)
and then formalized by Felsenstein (1985), who suggested that it could be implemented by holding the taxa constant and resampling the characters randomly with replacement, the tree-building analysis
then being applied to each of the bootstrap resamples.
Bootstrapping is a monte carlo procedure that it generates "pseudo" data sets from the original data, and uses these new data sets for its inferences. That is, it tries to derive the population
inferences (i.e the "true" answer) from repeated generation of new samples, each sample being constrained by the characteristics of the original data sample. It thus relies on an explicit analogy
between the sample and the appropriate population: that sampling from the sample is the same as sampling from the population. Clearly, the strongest requirement for bootstrapping to work is that the
sample be a reasonable representation of the population.
Bootstrap confidence intervals are only ever approximate, especially for complex data structures, as they are a fundamentally more ambitious measure of accuracy than is a simple standard error (SE).
For example, the usual formula for calculating a confidence interval (CI) when the population frequency distribution is assumed to be normal is: CI = t * SE, where t is the Student t-value associated
with the particular sample size and confidence percentage required. However, the main use of bootstrapping is in situations where the population frequency distribution is either indeterminate or is
difficult to obtain empirically, and so this simple formula cannot be applied. Getting from the standard error to a confidence interval is then not straightforward. As a result, there are actually
several quite distinct procedures for performing bootstrapping (Carpenter and Bithell 2000), with varying degrees of expected success.
Types of bootstrap
The original technique is called the
percentile bootstrap
. It is based on the principle of using the minimum number of ad hoc assumptions, and so it merely counts the percentage of bootstrap resamples that meet the specified criteria. F§or example, to
estimate the standard error of a median, the median can be calculated for each bootstrap resample and then the standard deviation of the resulting frequency distribution will be the estimated
standard error of the original median. The method is thus rather simplistic, and is often referred to as the
naïve bootstrap
, because it assumes no knowledge of how to calculate population estimates. It is a widespread method, as it can be applied even when the other methods cannot. However, it is known to have certain
problems associated with the estimates produced, particularly for confidence intervals, such as bias and skewness (especially when the parent frequency distribution is not symmetrical). These were
pointed out right from the start (Efron 1979), and efforts have subsequently been made to deal with them. Nevertheless, this is the form of bootstrap introduced by Felsenstein (1985), and it is the
one used by most phylogeny computer programs. It is therefore the one that will be discussed in more detail below.
These known problems with the naïve bootstrap can be overcome by using bias-corrected (BC) bootstrap estimates — that is, the bias is estimated and removed from the calculation of the confidence
interval. Possible dependence of the standard error on the parameter being estimated, which creates skewness, can be dealt with by using
bias-corrected and accelerated (BCa) bootstrap
estimates, so that the bias and skewness are both estimated and removed from the calculation of the confidence interval. The BCa method is the one usually recommended for use (Carpenter and Bithell
2000), because it corrects for both bias and skewness. This method is much slower to calculate than the simple percentile bootstrap, because it requires an extra parameter to be estimated for each of
the bias and skewness corrections, and the latter correction is actually estimated by performing a separate jackknife analysis on each bootstrap resample (which means that the analysis can take 100
times as long as a naïve analysis). There have been several attempts to apply this form of correction methodology to bootstrapping in a phylogenetic context (Rodrigo 1993; Zharkikh and Li 1995; Efron
et al. 1996; Shimodaira 2002), but while these can be successful at correcting bias and skewness (Sanderson and Wojciechowski 2000) these have not caught on, possibly because of the time factor
Alternatively, we can decide not to be naïve when calculating confidence intervals, and to therefore calculate them in the traditional manner, using the standard error and the t-distribution.
However, we then need to overcome any non-normal distribution problems of these two estimates by estimating both of them using bootstrapping. That is,
bootstrapped-t confidence intervals
are derived by calculating both the standard error and the t-value using bootstrapping, and then calculating the confidence interval as ±t * SE. To many people, this is the most natural way to
calculate confidence intervals, since it matches the usual parametric procedure, and thus it is frequently recommended (Carpenter and Bithell 2000). Once again, this method is much slower to
calculate than the percentile bootstrap, because the t-value is actually estimated by performing a separate bootstrap analysis on each bootstrap resample (which means that the analysis can take 100
times as long as a naïve analysis). This methodology seems not to have yet been suggested in a phylogenetic context, and in any case the time factor may be restrictive.
It is also possible to calculate
test-inversion confidence intervals
. This idea is based on the reciprocal relationship of statistical tests and confidence intervals, where (for example) non-overlapping 95% confidence intervals indicate statistically significant
patterns at p<0.05 and vice versa. Thus, if we work out the situation where the pattern has a probability of p=0.05 of occurring by chance, then this defines the 95% confidence limit of the pattern.
Clearly, this can be a complex process, especially for two-sided tests (which double the required number of calculations), as it can only be done by iteratively modifying the pattern and
re-calculating the probability until the solution is reached. Once again, no-one yet seems to have suggested this methodology in a phylogenetic context, which is not unexpected given the general
problems in deciding how to test branches statistically.
The above methods all count as non-parametric bootstrap methods. More recently,
parametric bootstrapping
methods have also been developed, which make the more restrictive assumption that a parametric model can be applied to the data (e.g. that the standard deviation of the parameter can be reliably
estimated). In parametric bootstrapping we generate simulated datasets based on the assumed frequency distribution of the data, rather than by resampling from the data set itself. That is, instead of
sampling from the sample, we sample from the assumed theoretical distribution to generate the set of bootstrap samples. We can then apply the percentile, BCa or bootstrap-t methods, described above,
in the usual way. Clearly this method assumes that we know the appropriate frequency distribution; and the method will only be appropriate if this assumption is true, but not otherwise. However, if
the assumption is correct, then this can be the most powerful method (Huelsenbeck et al. 1996a; Newton 1996) because it is not dependent on the representativeness of the data sample. The method has
been introduced into phylogenetics in several contexts (Goldman 1993; Adell and Dopazo 1994; Huelsenbeck et al. 1996a), but the appropriate frequency distribution for branch support is not obvious
(i.e. a phylogeny is a complex structure and cannot be represented by a single number but rather requires a model of sequence evolution and a model tree) and so it is not used for this purpose.
Issues with the bootstrap
Thus, for several reasons, all of the best bootstrapping methods are not likely to be available when assessing the robustness of a phylogenetic tree, and we are left with the naïve percentile
bootstrap, which can be expected a priori to provide biased and skewed estimates of confidence intervals (because the frequency distribution associated with tree branches will not be symmetrical).
Sadly, these problems have been repeatedly confirmed for the assessment of branch support in phylogenetic tree-building, both theoretically (Zharkikh and Li 1992a, 1992b; Felsenstein and Kishino
1993; Li and Zharkikh 1994; Sitnikova et al. 1995; Berry and Gascuel 1996; Efron et al. 1996; Huelsenbeck et al. 1996a; Newton 1996; Sanderson and Wojciechowski 2000; Suzuki et al. 2002; Alfaro et
al. 2003; Erixon et al. 2003; Galtier 2004; Huelsenbeck and Rannala 2004) and empirically (Sanderson 1989; Hillis and Bull 1993; Buckley et al. 2001; Buckley and Cunningham 2002; Wilcox et al. 2002;
Taylor and Piel 2004).
An example of the relationship between naïve bootstrap probabilities and the true probability of a false positive result, showing that percentile bootstrap indices >75% tend to be underestimates of
the amount of support while they are overestimates below this level. The graph is based upon 1000 bootstrap resamples of 100 simulated characters for a clade of three taxa plus outgroup (based on
data presented by Zharkikh and Li 1992a). The true probability represents the amount of character support for the clade in the simulated data, while the bootstrap probability is the proportion of
resamples that included the clade.
These studies have demonstrated that the probability of bootstrap resampling supporting the true tree may be either under- or overestimated, depending on the particular situation. For example,
bootstrap values >75% tend to be underestimates of the amount of support, while they may be overestimates below this level, as shown in the first graph (above). That is, when the branch support is
strong (i.e. the clade is part of the true tree) there will be an underestimation and when the support is weak (i.e. the clade is not part of the true tree) there will be an overestimation. This
situation has been reported time and time again, with various theoretical explanations (e.g. Felsenstein and Kishino 1993; Efron et al. 1996; Newton 1996), although there are dissenting voices (e.g.
Taylor and Piel 2004) as would be expected for a complex situation. Unfortunately, practitioners seem to ignore this fact, and to assume incorrectly that bootstrap values are
Just as importantly, the theoretical studies show that the pattern of over- and underestimation depends on (i) the shape of the tree and the branch lengths, (ii) the number of taxa, (iii) the number
of characters, (iv) the evolutionary model used, and (v) the number of bootstrap resamples. This was first reported by Zharkikh and Li (1992a), and has been reconfirmed since then. For example, with
few characters the bootstrap index tends to overestimate the support for a clade and to underestimate it for more characters. This is particularly true if the number of phylogenetically informative
characters is increased or the number of non-independent characters is increased; and the index becomes progressively more conservative (i.e. lower values) as the number of taxa is increased.
Moreover, these patterns of under- and overestimation are increased with an increasing number of bootstrap replications, as shown in the next graph — this called "being wrong, with confidence".
An example of the relationship between the true clade probability and the observed non-parametric bootstrap proportion for two simulated data sets with different numbers of characters (as shown). The
lines are based on data presented by Zharkikh & Li, (1995) for 1000 bootstrap resamples of a clade of three taxa plus outgroup.
The following graph pair of graphs show the effect of varying the evolutionary model used to generate the data, where under-specification of the analysis model leads to a general over-estimate of the
true probability (cross-over at p=0.8, as shown in the first graph of the pair), while matching the generating and analysis models leads to a general under-estimation (cross-over at p=0.3, as shown
in the second graph of the pair).
An example of the relationship between the true tree probability and the difference between the observed percentile bootstrap proportion and the true probability for two simulated data sets. The
label in the bottom corner shows the substitution model used to simulate the data, then the model assumed in the bootstrap analysis (the sequence length is 100 nucleotides); JC69 = Jukes-Cantor, GTRG
= general time- reversible + gamma-distributed among-site rate variation. The points are based on data presented by Huelsenbeck & Rannala (2004).
These are serious issues, which seem to be often ignored by practitioners. We can't just assume that the "true" support value is larger than our observed bootstrap value. In particular, this means
that bootstrap values are not directly comparable between trees, even for the same taxa, and thus there can be no "agreed" level of bootstrap support that can be considered to be "statistically
significant". A bootstrap value of 90% on a branch on one tree may actually represent
support than a bootstrap value of 85% on another tree, depending on the characteristics of the dataset concerned and the bootstrapping procedure used (although within a single tree the values should
be comparable).
This complex situation means that we have to consider carefully how best to interpret bootstrap values in a phylogenetic context (Sanderson 1995). The bootstrap proportion (i.e. the proportion of
resampled trees containing the branch/clade of interest) has variously been interpreted as (Berry and Gascuel 1996):
1. a measure of reliability, telling us what would be expected to happen if we repeated our experiment;
2. a measure of accuracy, telling us about the probability of our experimental result being true; and
3. a measure of confidence, interpreted as a conditional probability similar to those in standard statistical hypothesis tests (i.e. measuring Type I errors or false positives).
The bootstrap was originally designed for purpose (1), and all of the problems identified above relate to trying to use it for purposes (2) and (3). The values derived from the naïve bootstrap need
correcting for purposes (2) and (3), and the degree of correction depends on the particular data set being examined (Efron et al. 1996; Goloboff et al. 2003).
The issue of support values depending on the number of bootstrap replicates is also of interest. It is usually recommended that at least 1,000–2,000 bootstrap resamples are taken for estimating
confidence intervals, and this generality has been applied to phylogenetic trees (Hedges 1992). However, it is important to recognize that these suggestions relate to the precision of the confidence
estimates not to their accuracy. Accuracy refers to how close the estimates are to the true value (i.e. correctness) while precision refers to how variable are the estimates (i.e. repeatability).
Accuracy depends on a complex set of characteristics many of which have nothing to do with bootstrap replication. Precision, on the other hand, is entirely to do with the number of bootstrap
replicates and the expected accuracy of the estimates. As shown in the next graph, 100 replicates at a conventional level of accuracy produces estimates that are expected to be within ±4% of the
"true" values while 2,000 replicates produces estimates ±1%. This needs to be borne in mind when deciding whether to call a particular value "significant support" or not.
The number of bootstrap replicates needed to achieve a specified amount of precision, given statistical testing at two different levels of probability. For example (as shown by the dotted line), 100
bootstrap replicates means that, if the bootstrap value is accurate at the 95% confidence level, then the estimated bootstrap percentage will be precise to ±4.3%. In order to get ±1% precision then
nearly 2,000 bootstrap replicates are needed.
There have also been attempts to overcome some of the practical limitations of bootstrapping for large data sets by adopting heuristic procedures, including resampling estimated likelihoods for
maximum-likelihood analyses (Waddell et al. 2002) and reduced tree-search effort for the bootstrap replicates. However, approaches using reduced tree-search effort produce even more conservative
estimates of branch support, and the magnitude of the effect increases with decreasing bootstrap values (DeBry and Olmstead 2000; Mort et al. 2000; Sanderson and Wojciechowski 2000).
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In Australia at the time I was born, the most popular first name for boys was "David" and the second most popular was "Andrew". Not unexpectedly, the most popular middle name was "Andrew" and number
two was "David". It then comes as no surprise to you that I ended up with this pair of given names.
Names come and go in popularity (these are called fads), and if your parents have no imagination then you will grow up knowing that you are not unique, because half the people in your classroom will
have the same name as yourself. You may even end up being numbered (David #1, David #2, etc). What's worse, if you are not careful then you may end up doing the same thing to your own children.
Indeed, having a common name has only one known advantage — no matter where you go in the world everyone can recognize it, although they may not always spell it and pronounce it the way you expect
(David, Davide, Dawit ...). Therefore, you will have no problems making restaurant bookings where ever you happen to be (see Leonard S. Bernstein. 1981. Never Make a Reservation in Your Own Name.
Rand McNally).
These days in Australia, "David" struggles to be in the top 100 in popularity for boys. However, currently it appears to be in the top 10 in places like Armenia, Austria, Hungary, Italy, Spain and
Israel (in 2012 or 2013), as well as the top 20 in Poland and Portugal. This information comes from The Baby Name Wizard. This site has current lists for many countries (Popular Names From Around the
World), but has historical data only for the USA.
So, let's look at the U.S. data in more detail. As for Australia, the peak popularity in the USA was from 1955-1965, as shown in the first graph.
Note that the peak is truncated from 1950-1960.
The site's Name Mapper web page has annual data for each state from 1960-2009, which is precisely 50 years. These data show the ranking of names by popularity within each state. The average rank for
the name "David" across the 50 states is shown in the next graph. "David" was one of the top 10 names for boys born from 1936-1992, the #1 name in 1960, and it remains inside the top 20 to this day.
We can also look at the data for each state individually, as shown in the next graph, where darker shading represents greater popularity of the name. From the peak in the 1960s there was a steady
decrease in almost every state until 1995, after which the popularity has been more erratic. For example, in 1960 "David" was the #1 boys name in 28 of the 50 states (and in the top 5 in every
state), but by 1968 it was not #1 anywhere. The last time it was ranked #1 was in Utah in 1970, which was also the last year in which it was in the top 6 in every state.
Note that the states are grouped and colored geographically / culturallu.
Only in California and Texas has the name stayed in the top 10 over the past 50 years. In the other states it has stayed in the top 50 or so, except for North Dakota, where it is currently struggling
to stay in the top 100. In Nevada and Alaska it has even made a bit of a comeback in the past 10 years.
We can look at the relationship between the states using a phylogenetic network. The next graph is a NeighborNet (based on the manhattan distance) of the 1960-2009 data for the popularity ranking of
"David" as a boy's name. States near each other in the network have a similar naming popularity, while states further apart are progressively more different from each other. The network shows a
simple trend of increasing average popularity of "David" from the top-left to the bottom-right.
I have also colored the states using the same color scheme as for the previous graph (ie. geographically / culturally). Note that the orange, red, yellow and blue states are fairly neatly grouped,
indicating that their alleged geographical / cultural similarity extends to the popularity of given names ("David" has continued to be popular in all of these states). The purple, brown and green
states are not grouped very much, indicating much more diversity in the popularity of "David". For example, "David" has continued to be popular in New York and New Jersey but not in Maine, New
Hampshire or Vermont. The extreme disinterest of North Dakotans in the name is very clear.
The fall of "David" is not as bad as that of "James" and "John", which were in the top 3 most popular names in the USA all the way from the 1880s to the 1950s, but which are now in 17th and 27th
place, respectively (see the timeline graph in the Name Voyager).
I am not sure what has led to the eclipse of these names, other than the whims of faddishness. For example, in Britain and Ireland the name "Harry" has shot to the top in recent years (guess why!),
while it still languishes near #700 in the USA. Otherwise, "Noah" and "Liam" seem to have the most widespread popularity for boys in the western world at the moment.
Footnote: I actually got the name Andrew because it is my father's middle name, and his father's before him, and his father's first name. | {"url":"http://phylonetworks.blogspot.se/","timestamp":"2014-04-20T13:31:11Z","content_type":null,"content_length":"137622","record_id":"<urn:uuid:fd2cf84f-e833-40ff-b95f-8bc958ad79d5>","cc-path":"CC-MAIN-2014-15/segments/1397609538787.31/warc/CC-MAIN-20140416005218-00473-ip-10-147-4-33.ec2.internal.warc.gz"} |
Sorting Algorithm Breaks Giga-Sort Barrier, With GPUs
15690032 story
Posted by
from the quick-like-double-time dept.
An anonymous reader writes
"Researchers at the University of Virginia have recently open sourced an algorithm capable of sorting at a rate of one billion (integer) keys per second using a GPU. Although GPUs are often assumed
to be poorly suited for algorithms like sorting, their results are several times faster than the best known CPU-based sorting implementations."
This discussion has been archived. No new comments can be posted.
Sorting Algorithm Breaks Giga-Sort Barrier, With GPUs
Comments Filter:
• The video card in question.. (Score:5, Informative)
by black3d (1648913) on Sunday August 29, 2010 @10:33PM (#33412038)
Specifically, a GTX480 (just over 1 B keys/sec), followed up by a Tesla 2050 at around 75% of the speed of the GTX480. (745 M keys/sec)
• Link to Technical Paper (Score:5, Informative)
by PatPending (953482) on Sunday August 29, 2010 @10:59PM (#33412126)
Revisiting Sorting for GPGPU Stream Architectures [virginia.edu] (PDF)
• Re:x86 (Score:5, Informative)
by emmons (94632) on Monday August 30, 2010 @12:14AM (#33412374) Homepage
GPUs are highly parallel processors, but most of our computing algorithms were developed for fast single core processors. As we figure out how to implement new solutions to old problems to take
advantage of these highly parallel processors, you'll continue to see stories like this one. But, there's a limit to how good they can be at certain types or problems. Read up on Amdahl's law.
Basically, traditional x86 processors are good at lots of stuff. Modern GPUs are great at a few things.
• No (Score:5, Informative)
by Sycraft-fu (314770) on Monday August 30, 2010 @12:29AM (#33412408)
GPUs are special kinds of processors, often called stream processors. They are very efficient at some kinds of operations, and not efficient at others. Some things, they run literally a thousand
times faster than the CPU. Graphics rasterization would be one of these (no surprise, that's their prime job). However other things they run much slower. For something to run fast on a GPU it has
to meet the following requirement, the more it matches them, the faster it is:
1) It needs to be parallel to a more or less infinite extent. GPUs are highly parallel devices. The GTX 480 in question has 448 shaders, meaning for max performance it needs to be working on 448
things in parallel. Things that are only somewhat parallel don't work well.
2) It needs to not have a whole lot of branching. Modern GPUs can branch, but they incur a larger penalty than CPUs do. So branching in the code needs to be minimal. It needs to mostly be working
down a known path.
3) When a branch happens, things need to branch the same way. The shaders work in groups with regards to data and instructions. So if you have half a group branching one way, half the other,
that'll slow things down as it'll have to be split out and done separately. So branches need to be uniform for the most part.
4) The problem set needs to fit in to the RAM of the GPU. This varies, 1GB is normal for high end GPUs and 4-6GB is possible for special processing versions of those. The memory on board is
exceedingly fast, over a hundred gigabytes per second in most cases. However the penalty for hitting the main system RAM is heavy, the PCIe bus is but a fraction of that. So you need to be able
to load data in to video RAM and work on it there, only occasionally going back and forth with the main system.
5) For very best performance, your problem needs to be single precision floating point (32-bit). That is what GPUs like the best. Very modern ones can do double precision as well, but at half the
speed. I don't know how their integer performance fares over all, they can do it, but again not the same speed as single precision FP.
Now this is very useful. There are a lot of problems that fall in that domain. As I said, graphics would be one of the biggest, hence why they exist. However there are many problems that don't.
When you get ones that are way outside of that, like, say, a relational database, they fall over flat. A normal CPU creams them performance wise.
That's why we have the separate components. CPUs can't do what GPUs do as well, but they are good at everything. GPUs do particular things well, but other things not so much.
In fact this is taken to the extreme in some electronics with ASICs. They do one and only one thing, but are fast as hell. Gigabit switches are an example. You find that tiny, low power, chips
can switch amazing amounts of traffic. Try it on a computer with gigabit NICs and it'll fall over. Why? Because those ASICs do nothing but switch packets. They are designed just for that, with no
extra hardware. Efficient, but inflexible.
• Re:Ugh. (Score:3, Informative)
by metacell (523607) on Monday August 30, 2010 @02:26AM (#33412784)
Dude, an algorith which is O(n*log(n)) is not faster than O(n) just because n*log(n) < n.
When an algorithm is O(n* log(n)), it means the actual time requirement is p*n*log(n)+q, where p and q are constants specific to the algorithm.
The O(n*log(n)) algorith is faster than the O(n) one when
p1*n*log(n)+q1 < p2*n+q2
... and for any n, it is possible to choose p1, p2, q1 and q2 so that the O(n) algorith becomes faster.
This means, for example, that an algorithm which is O(n*log(n)) isn't automatically faster than an algorithm which is O(n) on lists with three elements or more. The O(n*log(n)) algorithm may take
a hundred times longer to sort a list of two elements than the O(n) one (due each step being more complex), and in that case the lists will need to grow some before the O(n*log(n)) algorithm
becomes faster.
• Re:The video card in question.. (Score:2, Informative)
by FilipeMaia (550301) on Monday August 30, 2010 @02:36AM (#33412818) Homepage
The reason for the GTX480 being faster is that it has 15 SM compared to 14 from the Tesla 2050. Also the GTX 480 runs at a higher clock speed (700 compared to 575). Put together this is 575/
700*14/15 = 76.7% which comes pretty close to the 75%.
• Re:Um... (Score:3, Informative)
by Anonymous Coward on Monday August 30, 2010 @02:43AM (#33412832)
Typically, I hear researchers describe the parallelism of an algorithm separately from its computational complexity (big oh notation) using the idea of "speedup."
The perfect scaling in your first example has linear speedup, and the second example has logarithmic speedup (that is, the speedup is log(p)).
Here is the relevant Wikipedia article [wikipedia.org].
• Re:Also (Score:1, Informative)
by Anonymous Coward on Monday August 30, 2010 @03:32AM (#33412986)
Doesn't matter what the theory says, if the hardware can do X faster than Y, then X is better according to users.
Normally big-O notation is applied on a purely theoretical level where all operations are assumed to have the same base cost in terms of time to execute. This does not make the notation invalid
for real world applications and implementations, however. But when doing so you have to adjust your formulas by adding the proper weighting to execution time. In practice this is usually a waste
of effort as it's generally faster to just write an implementation and time it on various pieces of hardware.
In the article we're talking about, they are comparing a single implementation of a single algorithm on several pieces of hardware. So first of all the summary shouldn't be shouting about
breaking any kind of record- they weren't trying to hit any particular benchmark it's a relative test of the hardware, not the algorithm or implementation.
The reason why using big-O and making comparisons is useful, is because if all we use is this type of test the answer to any speed problem is simply "Get faster hardware, or buy a piece of
hardware which runs this code faster". In all likliehood, there are other methods which, when implemented on the same hardware, may yield much faster results. Heck, it's possible someone else's
implementation of the same algorithm may yield faster results as well.
In regards to your comments about raytracing vs. polygon rendering, all I'm going to say is that you don't have a very good concept of what raytracing really is if you think a sphere created from
a million triangles will raytrace faster than one modeled as a mathematical sphere. It won't- those demonstrations are a pure head-to-head comparison operating on a scene which has actually been
optimized for a non-raycasting technique.
• Re:No (Score:2, Informative)
by FlawedLogic (1062848) on Monday August 30, 2010 @03:33AM (#33412992)
The GTX480 can actually do a double precision op per clock cycle. Fermi was designed with DP supercomputing in mind which is why it's so bloody expensive. To get the price down for consumer cards
they removed that ability since graphics doesn't generally need it. Consumer cards need four ticks to do the equivalent DP op.
• Re:Excel Charts (Score:4, Informative)
by w0mprat (1317953) on Monday August 30, 2010 @05:51AM (#33413324)
Amen. Some tools like that would be a godsend. It could be coming. http://en.wikipedia.org/wiki/Linked_Data [wikipedia.org] http://linkeddata.org/ [linkeddata.org] - Not what you are talking
about, but what you describe may result from it.
• Re:The video card in question.. (Score:2, Informative)
by ericcj (1574601) on Monday August 30, 2010 @09:52AM (#33414430)
Chips on the GTX 480, C2050, and C2070 come from the exact same die and wafer. The C20XX GPUs actually run at a lower clock speed for 32-bit floating-point and integer operations than a GTX 480.
C20XX series hardware is intended for physics/science/engineering calculations, where double-precision is preferred. The C20XX series is 4 times faster at double-precision calculations than the
GTX 480. This is the sweet spot it is tuned for.
• by Anonymous Coward on Monday August 30, 2010 @11:05AM (#33415228)
His naivety hit the real world and exploded.
Related Links Top of the: day, week, month. | {"url":"http://developers.slashdot.org/story/10/08/30/0133203/Sorting-Algorithm-Breaks-Giga-Sort-Barrier-With-GPUs/informative-comments","timestamp":"2014-04-24T13:32:16Z","content_type":null,"content_length":"109972","record_id":"<urn:uuid:bdc960f2-53e5-41ca-851e-0ca4072ad07a>","cc-path":"CC-MAIN-2014-15/segments/1398223206147.1/warc/CC-MAIN-20140423032006-00585-ip-10-147-4-33.ec2.internal.warc.gz"} |
Deptford Township, NJ Calculus Tutor
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Nano and Giga Abstract
Large Scale Dynamics with Quantum Mechanical Forces: The Transfer Hamiltonian
The question of 'predictability' in multi-scale materials simulations is a very important one, since we want the results to be reliable, qualitatively or quantitatively. The frist step in achieveing
this largely depends upon having a source of accurate quantum chemical forces that underlie the classical dynamics. However, unlike small molecules, where highly sophisticated and predictive methods
like coupled-cluster theory can be applied, for any multi-scale simulation hundreds to thousands of atoms need to be described by the quantum chemical methods with an efficientcy that permits tying
to dynamics. To solve this problem we have embarked upon the concept of a 'transfer Hamiltonian' which is formally defined by coupled-cluster theory whose eigenfunction is a single determinant, yet
whose energy and forces are exact. Furthermore, it also permits the treatment of different electronic or ionized states permitting optical properties and state specificity to be described as part of
the simulation. This transfer Hamiltonian can be evaluated purely from first principles, or from determining parameters that define the Hamiltonian for the particular phenomena of interest. We will
illustrate this approach in comparison with others like density functional theory in problems involving the fracture of silica including the presence of water. | {"url":"http://www.asdn.net/moscow/software-abstracts/bartlett/","timestamp":"2014-04-16T21:52:00Z","content_type":null,"content_length":"3789","record_id":"<urn:uuid:1b3ebe8e-7f5f-4272-b5a3-222ecf809f52>","cc-path":"CC-MAIN-2014-15/segments/1397609525991.2/warc/CC-MAIN-20140416005205-00602-ip-10-147-4-33.ec2.internal.warc.gz"} |
Bessel Function Zeros - File Exchange - MATLAB Central
Please login to add a comment or rating.
Comments and Ratings (21)
14 Jul Utilizei a rotina para calcular a solução analítica da equação da difusão 2D. Ficou muito boa.
01 Mar what are the advantages of using this over fzero? thanks
02 Feb Mostly seems to work. However, the zeros for the Bessel functions of the second kind are missing the first zero (0.893576966279). That is, the first Y_n zero it gives is actually the
2010 second, the second is actually the third, and so on.
Hi all
I am using the function in Matlab R2008a
I entered the command besselzero(1/2,1,2)
24 Jul and the result is different than the value from
2009 Mathematica s BesselYZero[1/2,1]
I also tried some other combinations which seemed fine.
I just wonder if this is a special case or I should check
the numbers given by this Matlab function with some other online sources. Thanks
08 Oct i dont have comments jeje sorry but this informaion is good ! ! thank you!!!
02 Mar Thank you, very useful
29 Jan The algorithm is fascinating and the speed is satisfactory. But there is a small defect: For Y function, the initial guess is not correctly chosen and the routine misses the first root in
2008 some cases.
02 Oct
05 May
01 May It works just fine
23 Apr j'ai male a trouver la solution de l'equation dde la gaine fini et infini de la fibre vouler vous m'aider svp.
20 Mar Nice work. You've saved me a lot of time.
26 Nov works great
15 Nov Good -- more commenting / discussion would be nice.
13 Oct I am using it. But not sure about its accuracy.
18 Aug
Very good. Works perfectly.
11 Jul Thanks
2006 T.S
BGU University Israel
20 Apr
05 Dec French student thank you.
2005 We hope you'll have a good life.
I used this m-file to generate the zeros for a bessel function of the first kind and order zero and it worked just fine.
03 Dec
2005 When you run the function, you should use besselzero(n,k,kind) I interpreted the documentation to suggest that the function needs only 2 inputs, but it really requires 3 to work.
04 Aug It seems the besselzero(n,k,1) and besselzero(n,k,2) works fine. Checked with multiple plot(besselzero(n,100,kind)).
2005 Good work ... | {"url":"http://www.mathworks.com/matlabcentral/fileexchange/6794-bessel-function-zeros","timestamp":"2014-04-21T04:49:26Z","content_type":null,"content_length":"36033","record_id":"<urn:uuid:4d6af810-3c91-4afb-9301-c9a2db62643e>","cc-path":"CC-MAIN-2014-15/segments/1397609539493.17/warc/CC-MAIN-20140416005219-00575-ip-10-147-4-33.ec2.internal.warc.gz"} |
How TeleConverters Affect Magnification
Prepared 2007-02-28 (169/12917) by Bill Claff
TeleConverters (TCs) operate by the same lens combination principles as close-up lenses.
The basic formulas are:
P = P[1] + P[2] - d[m] * P[1] * P[2]
P, P[1], and P[1] are lens powers in diopters
d[m] is the distance between lens nodes of P[1] and P[2] in meters
H[11]H[1] = d[m] * P[2] / P
H[22]H[2] = -d[m] * P[1] / P
For the distance the resulting lens nodes shift as a result of a non-zero d[m]
For close-up lenses we often disregard d[m] which simplies calculations considerably.
(See How Close-up Lenses Affect Magnification)
However, the design of a TC is more complex since in addition to changing focal length the rear node must also move to maintain infinity focus.
Because of the placement and movement of the rear node, the term d[m] in the power equations cannot be disregarded.
So without complete lens and TC information, only the effect on focal length at infinity focus and the effect on magnification at any focus position can be easily calculated.
In a suprising twist, the effect of the TC multiplier decreases as the magnification of the primary lens increases and can go below 1 resulting in a decrease of focal length as you focus closer.
For example, the 105mm f/2.8G ED-IF AF-S VR Micro-Nikkor has infinity and closest focus focal lengths of 105.2mm and 75.1mm.
With the addition of a Kenko 1.4xTC the infinity focal length increases a factor of 1.4 from 105.2mm to 148.8mm but the closest focus focal length drops from 75.1mm to 52.9mm.
All of the above values are actual measurements with the exception of the 148.8mm value which has been computed.
Although focal length may not be as expected at closest focus, magnification is increased as expected; and a TC can be a suprisingly good way to get more magnification for close-ups. | {"url":"http://home.comcast.net/~NikonD70/GeneralTopics/Close-up/How_TeleConverters_Affect_Magnification.htm","timestamp":"2014-04-20T09:20:46Z","content_type":null,"content_length":"6749","record_id":"<urn:uuid:7d17d038-1eea-4c83-9bda-ccb659c010e6>","cc-path":"CC-MAIN-2014-15/segments/1397609538110.1/warc/CC-MAIN-20140416005218-00372-ip-10-147-4-33.ec2.internal.warc.gz"} |
Parkside, PA Math Tutor
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Chapman, S
Franz, B. and Flegg, M. B. and Chapman, S. J. and Erban, R. (2013) Multiscale reaction-diffusiion algorithms: PDE-assisted Brownian dynamics. SIAM Journal of Applied Mathematics, 73 (3). pp.
Franz, B. and Flegg, M. B. and Chapman, S. J. and Erban, R. (2012) Multiscale reaction-diffusion algorithms: pde-assisted
Brownian dynamics. SIAM Journal on Applied Mathematics . (Submitted)
Erban, R
Franz, B. and Flegg, M. B. and Chapman, S. J. and Erban, R. (2013) Multiscale reaction-diffusiion algorithms: PDE-assisted Brownian dynamics. SIAM Journal of Applied Mathematics, 73 (3). pp.
Franz, B. and Flegg, M. B. and Chapman, S. J. and Erban, R. (2012) Multiscale reaction-diffusion algorithms: pde-assisted
Brownian dynamics. SIAM Journal on Applied Mathematics . (Submitted)
Franz, B. and Erban, R. (2011) Hybrid modelling of individual movement and collective behaviour. In: Dispersal, individual movement and spatial ecology: A mathematical perspective. Springer.
Flegg, M
Franz, B. and Flegg, M. B. and Chapman, S. J. and Erban, R. (2013) Multiscale reaction-diffusiion algorithms: PDE-assisted Brownian dynamics. SIAM Journal of Applied Mathematics, 73 (3). pp.
Franz, B. and Flegg, M. B. and Chapman, S. J. and Erban, R. (2012) Multiscale reaction-diffusion algorithms: pde-assisted
Brownian dynamics. SIAM Journal on Applied Mathematics . (Submitted)
Franz, B
Franz, B. and Flegg, M. B. and Chapman, S. J. and Erban, R. (2013) Multiscale reaction-diffusiion algorithms: PDE-assisted Brownian dynamics. SIAM Journal of Applied Mathematics, 73 (3). pp.
Franz, B. and Flegg, M. B. and Chapman, S. J. and Erban, R. (2012) Multiscale reaction-diffusion algorithms: pde-assisted
Brownian dynamics. SIAM Journal on Applied Mathematics . (Submitted)
Franz, B. and Erban, R. (2011) Hybrid modelling of individual movement and collective behaviour. In: Dispersal, individual movement and spatial ecology: A mathematical perspective. Springer.
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Case of the week # 367 - Moshe Bronshtein, Segal Yeoushua
Case of the week # 366 - Fabrice Cuillier
Case of the week # 365 - Lilit Hovsepyan
Case of the week # 364 - F Grochal
Case of the week # 363 - F Grochal
Case of the week # 362 - Alberto Sosa Olavarría_Eduardo Caleiras
Case of the week # 361 - Frantisek Grochal
Case of the week # 360 - Philippe Deblieck
Case of the week # 359 - Philippe Deblieck
Case of the week # 358 - Juan Pablo Gallo
Case of the week # 357 - Emmanuel Julien
Case of the week # 356 - A Krasnov, A Averyanov, I Glazkova, A Kurkevich, L Golovakha, A Malov
Case of the week # 355 - M Bronshtein, I Naroditsky, F Grochal
Case of the week # 354 - Moshe Bronshtein
Case of the week # 353 - AS Olavarria, JLD Corral, A Cortez, F Boscaja
Case of the week # 352 - Martin Juhas; Jozef Janovsky
Case of the week # 351 - Othman Al-Asali; F Yassen
Case of the week # 350 - Frantisek Grochal
Case of the week # 349 - Alberto Sosa
Case of the week # 348 - Mayank Chowdhury
Case of the week # 347 - Moshe Bronshtein
Case of the week # 346 - Moshe Bronshtein
Case of the week # 345 - F Cuillier, Mardamootoo D, Balu M, Alessandri JL
Case of the week # 344 - Elena Andreeva
Case of the week # 343 - Shona Whitmell, MRT(N), RDMS, CRGS
Case of the week # 342 - Murad Esetov
Case of the week # 341 - Tudor Iacovache
Case of the week # 340 - Murad Esetov, MD, Gylnabat Bekeladze, MD, Elina Gyseinova, MD
Case of the week # 339 - Murad Esetov
Case of the week # 338 - Fabrice Cuillier, MD.
Case of the week # 337 - Nguyen Ha, MD
case of the week # 336 - Francesco Contarin
Case of the weeks # 335 - Moshe Bronshtein, MD
Case of the weeks # 334 - Moshe Bronshtein, MD
Case of the weeks # 333 - F Ventriglia; V Martucci; A Cerekja; A Caiaro; I Orsola; L Manganaro
Case of the week # 332 - Fabrice Cuillier
Case of the week # 331 - Elena Andreeva
Case of the week # 330 - Elena Andreeva, MD;
case of the week # 329 - Vu D. Nguyen
Case of the week # 328 - Nguyen Ha
Case of the week # 327 - Marcos Antonio Velasco Sanchez
Case of the weeks # 326 - Moshe Bronshtein
Case of the weeks # 325 - Claudio Gomes, MD
Case of the week # 324 - Hanane Saadi
Case of the week # 323 - Fabrice Cuillier, Mardamootoo .
Case of the week # 322 - Emmanuel Julien, MD.
Case of the week # 321 - J. Van Keirsbilck, MD., Ph. Moerman, MD. PhD, K. Devriendt, MD.PhD, L. De Catte, MD. PhD
Case of the week # 320 - Fabrice Cuillier, MD
Case of the week # 319 - Fabrice Cuillier, MD.1, Michel J.L.,MD.2, Harper,MD., Alessandri J.L,MD
Case of the week # 318 - Fabrice Cuillier ,MD., Arrazola G.,MD., Cartault F.,MD., Alessandri J.L.,MD.
Case of the week #317 - F Grochal
Case of the week #316 - M Bronshtein
Case of the week #315 - F Grochal, J Krsiakova, E Kaufmann, D Mardamootoo, F Cuillier, F Cartault
Case of the week # 314 - Fabrice Cuillier ,MD.*,Mardamootoo D. , Midwife *,Alessandri J.L.,MD.**,Balu M.,MD.***.
Case of the week # 313 - Fabrice Cuillier
Case of the week #312 - M Bronshtein; I Naroditsky
Case of the week #311 - A Krasnov, E Buryachenko, A Averyanov, I Glazkova, R Abdullin
Case of the week #310 - F Grochal, A Cunderlik, P Klesken, P Szwarc, P Calda, R Vlk, P Babjak, D Murgasova, V Frisova
Case of the week #309 - Philippe Jeanty
Case of the week #308 - E Andreeva; L Zhuchenko; N Odegova; F Lagkueva
Case of the week #307 - Philippe Deblieck
Case of the week #306 - Fabrice Cuillier; JL Alessandri
Case of the week #305 - F. Cuillier MD, Y Rio MD, J.L. Alessandri MD.
Case of the week #304 - Moshe Bronshtein, MD.
Case of the week #303 - Amr Hamza, MD.
Case of the week #302 - Cuillier F. MD, Alessandri J.L.MD, Dr Cartault F.MD.
Case of the week # 301 - Kozeta Mustafaraj, MD, Shuaip Beqiri, MD, Arben Haxhihyseni, MD.
Case of the week #300 - Philippe Jeanty, MD PhD., Eva Leinart, MD, PhD.
Case of the week #299 - E. Andreeva, MD, L. Zhuchenko, MD, I. Geilis, MD.
Case of the week #298 - Philippe Jeanty, MD PhD., Eva Leinart, MD, PhD.
Case of the week #297 - WL Lau, MD, LL Chan, MD, KS Chan, MD, WC Leung MD
Case of the week #296 - Sivasamy Manohar, MD.
Case of the week #295 - Moshe Bronshtein, MD
Case of the week #294 - Elena Andreeva, MD.
Case of the week #293 - A. Mazzocco MD, R. Manara MD, A.de Nardi MD, A.Cerekja MD PhD
Case of the week #292 - Vu Dinh Nguyen, MD.
Case of the week #291 - Fabrice Cuillier, MD, A.Bertha
Case of the week #290 - Elena Andreeva, MD., Ludmila Juchenko, MD.
Case of the week #289 - A. Sosa Olavarria
Case of the week #288 - Elena Andreeva, MD.
Case of the week #287 - D. Markov MD PhD, E. Pavlova MD, D. Atanassova MD, P. Markov, MD
Case of the week #286 - O. Al-Asali MD, I. Abo-Almaged MD, A. Al Taher MD, M Al-Hakeem MD
Case of the week #285 - Fabrice Cuillier, MD, J.L. Alessandri, MD, F.Cartault, MD, L.Vinatier MD.
Case of the week #284 - Binodini M. Chauhan, MD.
Case of the week #283 - G. Perez-Canto Ch, A. Sosa Olavarria, H. Parra Venero
Case of the week #282 - Nguyen Ha, M.D.
Case of the week #281 - Héctor Gonzalo Quiroga Pacheco, MD.
Case of the week #280 - Moshe Bronshtein, MD
Case of the week #279 - Ajit Gandhi MD, Anish Dekhane MD
Case of the week #278 - Nguyen Ha, M.D.
Case of the week #277 - Elena Andreeva, MD., Ludmila Juchenko, MD.
Case of the week #276 - Tihonenko I.MD, Venchikova N.MD, Nerovnya A. MD, Yushchenko O.
Case of the week #275 - Ilyina H.G. MD, Novikova I.V. MD, Lishtvan L.M. MD, Savenko L.A. MD, Kasetskaya T.V. MD, Miheeva N. MD.
Case of the week #274 - Elena Andreeva, MD.
Case of the week #273 - Cuillier F.MD,Elad T.MD, Prost F.MD, Possati P.MD
Case of the week #272 - Nattinee Srisantiroj, M.D.
Case of the week #271 - Sheryl Rodts-Palenik, MD.
Case of the week #270 - A Krasnov, MD, , I Glazkova MD, A.Averyanov MD, I Mokryk MD.
Case of the week #269 - N. Srisantiroj MD, P.Padungkiatwattana MD,S.Sirijareonthai MD,P.Jaruyawong MD.
Case of the week #268 - S. Manohar, MD, M.Mohan Karthikeyan, MD
Case of the week #267 - Christa Faschingbauer, M.D.
Case of the week #266 - S.Manohar, M.D, M.Mohan Karthikeyan, DMRD
Case of the week #265 - Natalya Miheeva, MD.
Case of the week #264 - Tibisay G. Chin Aleong, MD., Montserrat Alegre, MD., Maria V.Huerta Anaya, MD.
Case of the week #263 - Emmanuel Julien, MD., Philippe Juhel, MD.
Case of the week #262 - C Gar El, MD, AB Gelot, MD PhD, M Moutard, MD, R Grigorescu, MD, L Gourand, MD.
Case of the week #261 - Binodini M. Chauhan, MD.
Case of the week #260 - Trinh Nguyen, M.D.
Case of the week #259 - JZ. Peralta, MD., FL.Garcia, MD., OA. López, MD.
Case of the week #258 - P. Budziszewska, MD., D.Kuka, MD., K. Sodowski, MD., P.Jeanty, MD., M. Hernanz-Schulman, MD.
Case of the week #257 - Ha To Nguyen, MD.
Case of the week #256 - Elena Andreeva, MD.
Case of the week #255 - A Krasnov, MD, G Gorokhova, L Golovakha, I Glazkova
Case of the week #254 - F Cuillier, JL Alessandri, KC Dillon
Case of the week #253 - Andrés Arencibia Molina, MD.
Case of the week #252 - Elena Andreeva, MD.
Case of the week #251 - Irina Tihonenko, MD, Tatiyana Koipish, MD.
Case of the week #250 - Emmanuel Julien
Case of the week #249 - Fabric Cuillier
Case of the week #248 - S. Manohar, DMRD, MD.
Case of the week #247 - Adam Paer, MD, Philippe Jeanty, MD.
Case of the week #245 - Margarita Alvarez de la Rosa Rodriguez, MD
Case of the week #246 - Irina Tihonenko
Case of the week #244 - Héctor Gonzalo Quiroga Pacheco, MD.
Case of the week #243 - Andrea L. Gonzales ,Luis A. Izquierdo-Encarnacion ,Samantha Long
Case of the week #242 - S. Manohar, DMRD, MD.
Case of the week #241 - S. Manohar, DMRD, MD.
Case of the week #240 - Nattinee Srisantiroj ,Prapat Wanitpongpan ,Vitaya Titapant
Case of the week #239 - Binodini M. Chauhan, MD.
Case of the week #238 - Nguyen Ha ,Francois Manson
Case of the week #237 - Nickolay Petrovich Veropotvelyan, MD.
Case of the week #236 - F Cuillier ,H M’Lamali ,X Baron ,A Berta
Case of the week #235 - Elena Andreeva
Case of the week #234 - E Racanska ,R Gerychova ,P Janku.
Case of the week #233 - Calvo A, MD ,Azumendi Guillermo, MD ,J. R. Herrero, MD ,Marisa Borenstein, MD.
Case of the week #232 - Fabrice Cuillier ,J. Bideault ,D. Daguindeau.
Case of the week #231 - F Cuillier ,D Daguindeau ,J Bideault ,JL Travers
Case of the week #230 - WL Lau ,HSW Lam ,WC Leung ,RKH Chin.
Case of the week #229 - Francois Manson, MD.
Case of the week #228 - Alexander Krasnov ,Irina Glazkova ,Andrey Averyanov.
Case of the week #227 - Jayprakash Shah
Case of the week #226 - A Volkov ,A Rymashevsky ,A Lukach
Case of the week #225 - F Cuillier ,D Daguindeau ,J Bideault ,A Bertha
Case of the week #224 - Nguyen Ha ,Philippe Jeanty
Case of the week #223 - Elena Andreeva ,Natalia Odegova ,Natalia Bortnovskaya ,Alexandr Michin
Case of the week #222 - Trinh Nguyen, MD.
Case of the week #221 - Fabrice Cuillier ,J. Chouchani ,A. Hamas.
Case of the week #220 - Fabrice Cuillier ,A. S. Charpentier ,L. Lagarde.
Case of the week #219 - F Cuillier ,P Dussautoir ,G Kah ,JP Riviere
Case of the week #218 - Trinh Nguyen, MD.
Case of the week #217 - Luis Guillermo Diaz Guerrero, MD.
Case of the week #216 - Meyer Serrano Riano, MD.
Case of the week #215 - Andrés Arencibia Molina, MD ,Ludmila Ocón, MD.
Case of the week #214 - Irina Glazkova, MD.
Case of the week #213 - Maha Tulbah, MD.
Case of the week #212 - Ha To Nguyen, MD
Case of the week #211 - F Cuillier ,T Gervais ,J M Scemama ,JM Laville ,F Salmeron ,JP Riviere ,A Bertha
Case of the week #210 - A Volkov ,E Andreeva ,A Rymashevsky
Case of the week #209 - F Cuillier ,J Bideault ,P Lemaire ,M Deshayes ,JM Travers
Case of the week #208 - Christa Faschingbauer
Case of the week #207 - Elena Andreeva, MD ,Svetlana Tchuchvaga, MD.
Case of the week #206 - Elena Andreeva, MD ,Svetlana Tchuchvaga, MD.
Case of the week #205 - BH Syla ,S Fetiu ,S Tafarshiku
Case of the week #204 - Trinh Nguyen, MD
Case of the week #203 - Fabrice Cuillier, MD ,J. Narboni, MD ,J. L. Alessandri, MD.
Case of the week #202 - C Dalmon ,G Brodaty ,B Bessières ,V Mirlesse ,L Gourand
Case of the week #201 - Francois Manson, MD.
Case of the week #200 - Sheryl Rodts-Palenik, MD.
Case of the week #199 - Breck Collins, MD ,Philippe Jeanty, MD, PhD ,Ian and Kristi Hay,
Case of the week #198 - Robert Dankovcik, MD, PhD.
Case of the week #197 - F Contarin, MD ,J Suárez, MD ,J Rangel, MD ,J Visconti, MD ,B Rodriguez, MD.
Case of the week #196 - Fabrice Cuillier, MD ,K. Comalli Dillon, BA, RDMS ,Edouard Kaufman, MD ,H. Randrianaivo, MD
Case of the week #195 - Carlos Canetti, MD ,Alejandra Medina, MD
Case of the week #194 - Heron Werner, MD ,Iugiro Kuroki, MD ,Erick Malheiro L. Martins, MD
Case of the week #193 - Vavilala Suseela, MD.
Case of the week #192 - Héctor Gonzalo Quiroga Pacheco, MD.
Case of the week #191 - Joan Acosta Diez, MD.
Case of the week #190 - Kathleen Comalli Dillon, BA, RDMS ,Fabrice Cuillier, MD ,C, P. Lemaire, MD.
Case of the week #189 - S. Manohar, DMRD, MD
Case of the week #188 - Miguel Merino, MD
Case of the week #187 - S. Manohar, MD, DMRS
Case of the week #186 - F. Cuillier, MD ,J. L. Alessandri ,J. M. Laville ,D. Durandeu ,J. P. Bourdil ,K. C. Dillon
Case of the week #185 - Giuseppe Cali, MD ,Francesco Labate, MD
Case of the week #184 - Montse Alegre, MD
Case of the week #183 - Alvaro Teran, MD
Case of the week #182 - Cuillier F*, MD ,J. Chouchani**, MD ,Abdou M**, MD ,Bideault J***, MD ,Fossati P****, MD
Case of the week #181 - Mariliza Volpe, MD
Case of the week #180 - Luc Gourand, MD
Case of the week #179 - Fabrice Cuillier, MD
Case of the week #177 - Cuillier F, MD ,Carasset G, MD ,Lemaire P ,Deshayes M ,DeNapoli S, MD
Case of the week #176 - Debabrata Das, MD, Vaijayanthi Raja, MD
Case of the week #175 - Luc Gourand, MD, Veronique Mirlesse, MD
Case of the week #174 - Lachlan De Crespigny, MD
Case of the week #173 - Debabrata Das, MD, Sulochana Neelakandam, MBBS
Case of the week #172 - Philippe Jeanty, MD, PhD ,Cheryl Turner, BS, RDMS, Juliana Leite, MD
Case of the week #171 - Ron Auslander, MD, Goldberg Yael, MD
Case of the week #170 - David Fox, MD
Case of the week #169 - Luis Diaz Guerrero, MD ,Thuy Van Nguyen, MD
Case of the week #168 - Cuillier F, MD, Avignon MS, MD
Case of the week #167 - Novakov Mikic A, MD ,Ivanoviæ Lj, MD ,Luèiæ M, MD ,Kiralj A, MD ,Dobriæ Lj, MD ,Koprivšek , MD
Case of the week #166 - Cuillier, MD ,Alessandri JL, Bideault J, MD ,Rabenja A, MD
Case of the week #165 - Cuillier F, MD ,Baroche G, MD ,Alessandri JL, MD
Case of the week #164 - Fabrice Cuillier, MD
Case of the week #163 - Heron Werner, MD
Case of the week #162 - Fabrice Cuillier, MD
Case of the week #161 - Heron Werner, MD
Case of the week #160 - Carlos Elorza, MD
Case of the week #159 - Graham Parry, MD
Case of the week #158 - Giuseppe Calì, MD ,Francesco Labate, MD ,Adriano Cipriani, MD ,Sergio Di Liberto, MD
Case of the week #157 - Montse Alegre, MD
Case of the week #156 - Yagel Simcha, MD
Case of the week #155 - João Mendes, MD
Case of the week #154 - Ilan Timor, MD
Case of the week #153 - Fabrice Cuillier, MD ,Elad T., MD, Fossati P.,MD
Case of the week #152 - Fabrice Cuillier, MD ,Deshayes M.,MD, Michel J.L., MD
Case of the week #151 - S. Manohar, DMRD, MD
Case of the week #150 - Fabrice Cuillier, MD
Case of the week #149 - Philippe Jeanty ,Pam Ross ,Dorris Baier
Case of the week #148 - Juan Carlos Mejia Quintero, MD
Case of the week #147 - Drs. Gui T.Mazzoni Jr ,Marcos M.L.Faria ,César C.Sabbaga ,Linei A.B.D.Urban, et al
Case of the week #146 - Hector Quiroga, MD
Case of the week #145 - Emmanuel Julien, MD
Case of the week #144 - Dr.Cuillier F ,Dr Bideault J. ,Dr Alessandri J.L. ,Dr Cartault J.F.
Case of the week #143 - Philippe Jeanty, MD, PhD
Case of the week #142 - Fabrice Cuillier, MD
Case of the week #141 - Philippe Jeanty, MD, PhD
Case of the week #140 - Rodolpho Lambruschini, MD
Case of the week #139 - Philippe Jeanty, MD, PhD
Case of the week #138 - Claudia Elena Teodorescu, MD
Case of the week #137 - Sheryl Rodts-Palenik, MD
Case of the week #136 - Ghada Mansour, MD ,Sobhi Abou-Louz, MD
Case of the week #135 - S.Manohar, MD, DMRD
Case of the week #134 - Philippe Jeanty, MD, PhD ,Chaitali Shah, MD ,Cerine Jeanty
Case of the week #133 - Philippe Jeanty, MD, PhD
Case of the week #132 - José Luis Volpacchio, MD ,Pablo Giuliani, MD, et al.
Case of the week #131 - Patrick Bailleul, MD
Case of the week #130 - Helio Camargo, MD ,Marcia Camargo, MD
Case of the week #129 - Ivan Lee, MD
Case of the week #128 - Fabrice Cuillier, MD
Case of the week #127 - Wayne Persutte, MS, RDMS ,John Hobbins, MD
Case of the week #126 - Montse Alegre, MD
Case of the week #125 - Fabrice Cuillier, MD
Case of the week #124 - François Jacquemard*, MD ,Danièle Pariente MD ,Hélène Martelli, MD ,Luc Gourand*, MD
Case of the week #123 - Lívia Rios,MD
Case of the week #122 - Rajesh Garge, MD
Case of the week #121 - Miguel Octavio Sosa
Case of the week #120 - Luc Druart MD ,Helene Dessuant MD ,Luc Gourand MD ,Bettina Bessieres MD ,Fernand Daffos MD
Case of the week #119 - Fabrice Cuillier, MD
Case of the week #118 - Luc Gourand, MD
Case of the week #117 - Raul Martinez, MD
Case of the week #116 - Fernando Velazquez, MD
Case of the week #115 - Juan Carlos Quintero MD ,Eduardo Romero, MD
Case of the week #114 - Adrian Clavelli, MD ,Horacio Ahielo, MD
Case of the week #113 - Alberto Sosa Olavarria, MD
Case of the week #112 - Maher Sarraf, MD
Case of the week #111 - Fabrice Cuillier, MD ,Philippe Lemaire, MD
Case of the week #110 - Cheryl Turner, RDMS, BS, Sefa Kelekci, MD ,Jos Offermans, MD, PhD ,Philippe Jeanty, MD, PhD.
Case of the week #109 - Ana Bircher, Philippe Jeanty
Case of the week #108 - Adrian Clavelli MD
Case of the week #107 - Giuseppe Calì, MD , Francesco Labate, MD
Case of the week #106 - François Duchatel
Case of the week #105 - Claudia Scarpetta, MD, Gustavo Vasquez, MD and Juan Carlos Quintero MD
Case of the week #104 - A. Sosa Olavarría ,L. Díaz Guerrero ,G. Giugni Chalbaud
Case of the week #103 - Hector G Quiroga, MD
Case of the week #102 - Alberto Sosa Olavarría
Case of the week #101 - Miguel Octavio Sosa
Case of the week #100 - Raul Martinez, MD
Case of the week # 99 - Fabrice Cuillier
Case of the week # 98 - Eleni Tzachrista
Case of the week # 97 - Boopathy Vijayaraghavan, MD, DMRD ,R.Lalitha, MBBS, DGO ,Geethanjali, MBBS, DMRD
Case of the week # 96 - Luc Gourand
Case of the week # 95 - Nayana Parange, MD
Case of the week # 94 - Waldo Sepulveda, MD and Sebastian Illanes, MD
Case of the week # 93 - Ronaldo Levy MD ,Luc Gourand MD ,Jean Simon Arfi MD ,Bettina Bessières MD,
Case of the week # 92 - Luis Izquierdo
Case of the week # 91 - Alberto Sosa, MD
Case of the week # 90 - Moshe Bronshtein MD and Etan Z. Zimmer MD
Case of the week # 89 - Boopathy Vijayaraghavan.S, MD., DMRD, ,Suma Natarajan, MD, DGO, ,Ravikumar.V.R.MS, MCh.
Case of the week # 88 - Thomas ,Goolaertsl ,Watkins ,Autin ,Barlow
Case of the week # 87 - Duchatel
Case of the week # 86 - Johns ,Bircher ,Jeanty ,Heredia ,Dev
Case of the week # 85 - Mirlesse ,Levy ,Brodaty ,Sonigo ,Teillac ,Gourand ,Daffos ,Heredia
Case of the week # 84 - Luis Izquierdo ,Amanda Cotter
Case of the week # 83 - Fernando Heredia ,Hector Figueroa
Case of the week # 82 - Gourand ,Jacquemard ,Bessières ,Fallet ,Daffos
Case of the week # 81 - Ana Bircher ,Fernando Heredia ,Philippe Jeanty
Case of the week # 80 - Raúl Martínez ,Fernando Heredia
Case of the week # 79 - Ana Maria BIrcher, MD, Philippe Jeanty, MD, PhD
Case of the week # 78 - Guillermina M. Ochua ,Ricardo De Loredo ,Roque A. Carpio ,Javier B. Gardey ,Silvina Fernàndez
Case of the week # 77 - Sebastian Illanes, MD ,Cristian Kottmann, MD ,Waldo Sepulveda, MD
Case of the week # 76 - Jose Sierra, MD ,Philippe Jeanty, MD, PhD
Case of the week # 75 - Alberto Sosa Olavarría, MD, PhD ,Gelsy Giugni Chalbaud ,Luis Díaz Guerrero, MD ,Miguel A. Granda MD
Case of the week # 74 - Philippe Jeanty, MD, PhD ,Jose Sierra, MD ,Eleni Tzachrista, MD
Case of the week # 73 - Pavel Vlašin, MD, Pavel Eliáš, MD*
Case of the week # 72 - Renato Ximenes, MD
Case of the week # 71 - Moshe Bronshtein MD ,Etan Z. Zimmer MD
Case of the week # 70 - Alberto Sosa Olavarría, MD, PhD ,Luis Díaz Guerrero, MD ,Maria Miraz MD ,Aldo Reigosa MD ,Emilio Mac
Case of the week # 69 - Philippe Jeanty, MD, PhD
Case of the week # 68 - Geneviève Brodaty, MD ,Bettina Bessières, MD ,Luc Gourand, MD
Case of the week # 67 - Alberto Sosa Olavarría, MD ,PhD. Luis Díaz Guerrero, MD ,Reigosa Yanis, MD
Case of the week # 66 - Raul Martinez MD
Case of the week # 65 - Luís Flávio Gonçalves, MD
Case of the week # 64 - Philippe Jeanty, MD PhD ,Ana Bircher, MD
Case of the week # 63 - Fabrice Cuillier, MD ,Jacques Bidault, MD
Case of the week # 62 - Alberto Sosa Olavarría, MD, PhD ,Gonzalo Pérez Canto, MD ,Luis Díaz Guerrero, MD
Case of the week # 61 - Philippe Jeanty, MD, PhD
Case of the week # 60 - Moshe Bronshtein, MD ,Etan Zimmer, MD
Case of the week # 59 - Rama Murthy, MD
Case of the week # 58 - Marcos Tawil, MD ,Norma Tello, MD
Case of the week # 57 - Luiz Eduardo Machado, MD
Case of the week # 56 - Philippe Jeanty, MD, PhD ,Cheryl Turner, RDMS
Case of the week # 55 - Luis A. Izquierdo, MD, RDMS ,Victor H. Gonzalez-Quintero, MD ,Mahsomeh Haghayegh, RDMS
Case of the week # 54 - Angela Regina Capelanes, MD ,Gloria Valero, MD ,Philippe Jeanty, MD, PhD
Case of the week # 53 - Alessandro Giuffrida, MD ,Saveria Cantone, MD ,Fabiola Galvani, MD ,Claudio Giorlandino, MD.
Case of the week # 52 - Philippe Jeanty, MD, PhD
Case of the week # 51 - Laurie Briare, RDMS ,Dianne Glassford, RDMS ,Dianna Heidinger, RDMS ,Van R. Bohman, MD
Case of the week # 50 - Connie Jörgensen, MD, PhD
Case of the week # 49 - Gustavo Malinger, MD ,Tally Lerman-Sagie, Dorit Lev ,Mordechai Tamarkin ,Debora Kidron.
Case of the week # 48 - Juan Carlos Quintero Mejia MD ,Philippe Jeanty, MD, PhD
Case of the week # 47 - Christine Comstock, MD
Case of the week # 46 - Bettina Bessieres, MD ,François Jacquemard, MD ,Luc Gourand, MD ,Fernand Daffos, MD
Case of the week # 45 - Elke Sleurs, MD ,Luc De Catte, MD
Case of the week # 44 - Dominique Thomas, MD ,Isabelle Mahillon, MD, ,France Hayez-Delatte, MD, ,Françoise Rypens, MD, Joelle
Case of the week # 43 - Elke Sleurs, MD ,Luc De Catte, MD
Case of the week # 42 - Elke Sleurs, MD ,Luc De Catte, MD
Case of the week # 41 - Maria Verônica Muñoz Rojas, MD ,Luís Flávio Gonçalves, MD ,Rodrigo Dias Nunes, MD ,Jorge Abi Saab N
Case of the week # 40 - Anupama Patil, MD ,Philippe Jeanty, MD, PhD
Case of the week # 39 - Daniel Margulies, MD ,Nestor Matarasso, MD
Case of the week # 38 - Fernando M. Heredia, MD ,Víctor G. Quiroz, MD ,Eliana C. Selman, MD ,Carlos B. Henríquez, MD, Ph
Case of the week # 37 - Elke Sleurs, MD ,Luc de Catte, MD
Case of the week # 36 - Alberto Sosa Olavarria, MD, PhD ,Luis Díaz Guerrero, MD
Case of the week # 35 - L. Gourand MD ,V. Mirlesse MD ,B. Bessieres MD ,F. Jacquemard MD ,R. Levy MD ,F. Daffos MD
Case of the week # 34 - Phillip Ramm
Case of the week # 33 - Cheryl Turner, RDMS Philippe Jeanty, MD, PhD
Case of the week # 32 - Luiz Eduardo.Machado, MD ,L. Chamusca, MD ,K. Chagas, MD ,C. Cadete, MD ,AF.Santos, MD
Case of the week # 31 - Philippe Jeanty, MD, PhD ,Ghada Mahmoud Mansour, MD
Case of the week # 30 - Carlos Alberto Mejia Escobar, MD ,Jorge Ramirez, MD ,Jaime Gomez, MD ,Oscar Medina, MD
Case of the week # 29 - Fernando Heinen MD ,Diego Elias, MD ,Marcelo Pietrani MD ,P. Verdaguer MD
Case of the week # 28 - Luis Flavio Goncalves
Case of the week # 27 - Luc de Catte, MD
Case of the week # 26 - Elke Sleurs, MD, and Luc de Catte, MD
Case of the week # 25 - Peter Twining
Case of the week # 24 - Alberto Sosa Olavarría, MD, PhD ,Luis Díaz Guerrero, MD ,A Reigosa Yanis, MD ,Perinatology Unit, Car
Case of the week # 23 - Marcos Tawil, MD, Mexico, ,Philippe Jeanty
Case of the week # 22 - Sosa Olavarría, A. MD, PhD ,Luis Díaz Guerrero, MD ,García M.MD ,Pereira D. MD ,Martines M.MD, Perin
Case of the week # 21 - Philippe Jeanty, MD, PhD
Case of the week # 20 - Sandra Silva, MD, Sao Paolo Brazil , Philippe Jeanty, MD, PhD
Case of the week # 19 - Guillermo Diaz Guerrero
Case of the week # 18 - Gerald Mulligan, MD
Case of the week # 17 - Claudio Giorlandino, MD
Case of the week # 16 - Luc de Catte
Case of the week # 15 - Adrian Clavelli
Case of the week # 14 - Philippe Jeanty, MD, PhD ,Gianluigi Pilu, MD
Case of the week # 13 - Philippe Jeanty, MD, PhD
Case of the week # 12 - Alberto Sosa Olavarria, MD ,Luis Guillermo Diaz Guerrero, MD, Valencia, Venezuela
Case of the week # 11 - Submitted by Luis Izquierdo, MD, Pensacola, Florida
Case of the week # 10 - Peter Twining, Nottingham, MD
Case of the week # 9 - Luc de Catte, Brussels, Belgium
Case of the week # 8 - Luis A. Izquierdo, Pensacola, Florida
Case of the week # 7 - Luc De Catte, MD,Brussel, Belgium
Case of the week # 6 - Cheryl Norris ,Philippe Jeanty
Case of the week # 5 - Adrian Clavelli ,Buenos Aires , Argentina
Case of the week # 4 - Hytham M. Imseis, MD, MAHEC
Case of the week # 3 - Peter Twining
Case of the week # 2 - Glynis Sacks, MD
Case of the week # 1 - Jeanty | {"url":"http://www.sonoworld.com/TheFetus/Listing.aspx?Id=2","timestamp":"2014-04-18T11:01:42Z","content_type":null,"content_length":"391793","record_id":"<urn:uuid:f98ee882-e4f9-42fa-822a-718184a546fc>","cc-path":"CC-MAIN-2014-15/segments/1397609533308.11/warc/CC-MAIN-20140416005213-00085-ip-10-147-4-33.ec2.internal.warc.gz"} |
Ernie's 3D Pancakes
Congratulations and thank you for your contribution to Clinical Psychology. Now that the book is published, we need your help to get some 5 star reviews posted to both Amazon and Barnes & Noble
to help support and promote it. As you know, these online reviews are extremely persuasive when customers are considering a purchase. For your time, we would like to compensate you with a copy of
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it was all a mistake
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Lunar Perigee and Apogee Calculator
To display the date, time, and distance of lunar perigees and apogees for a given year, enter the year in the box below and press “Calculate”. Depending on the speed of your computer, it may take a
while for the results to appear in the text boxes. This page requires your browser to support JavaScript, and that JavaScript be enabled; all computation is done on your own computer so you can, if
you wish, save this page in a file and use it even when not connected to the Internet.
All dates and times are Universal time (UTC); to convert to local time add or subtract the difference between your time zone and UTC, remembering to include any additional offset due to summer time
for dates when it is in effect. For each perigee and apogee the distance in kilometres between the centres of the Earth and Moon is given. Perigee and apogee distances are usually accurate to within
a few kilometres compared to values calculated with the definitive ELP 2000-82 theory of the lunar orbit; the maximum error over the years 1977 through 2022 is 12 km in perigee distance and 6 km at
The closest perigee and most distant apogee of the year are marked with “++” if closer in time to full Moon or “--” if closer to new Moon. Other close-to-maximum apogees and perigees are flagged with
a single character, again indicating the nearer phase. Following the flags is the interval between the moment of perigee or apogee and the closest new or full phase; extrema cluster on the shorter
intervals, with a smaller bias toward months surrounding the Earth's perihelion in early January. “F” indicates the perigee or apogee is closer to full Moon, and “N” that new Moon is closer. The sign
indicates whether the perigee or apogee is before (“−”) or after (“+”) the indicated phase, followed by the interval in days and hours. Scan for plus signs to find “photo opportunities” where the
Moon is full close to apogee and perigee.
This table gives the time of all new and full Moons in the indicated year, as well as the last phase of the preceding year and the first phase of the next year.
Click on titles to order books on-line from
Meeus, Jean. Astronomical Algorithms . Richmond: Willmann-Bell, 1998. ISBN 0-943396-63-8.
The essential reference for computational positional astronomy. The calculation of perigee and apogee time and distance is performed using the algorithm given in Chapter 48.
Meeus, Jean. Astronomical Formulæ for Calculators, Fourth Edition . Richmond: Willmann-Bell, 1988. ISBN 0-943396-22-0.
This book, largely superseded by the more precise algorithms given in Astronomical Algorithms, remains valuable when program size and speed are more important than extreme precision. The date and
time of the phases of the Moon are calculated using the method given in Chapter 32, and are accurate within 2 minutes, more than adequate for our purposes here. The more elaborate method in
Chapter 47 of Astronomical Algorithms reduces the maximum error to 17.4 seconds (and mean error to less than 4 seconds), but would substantially increase the size and download time for this page,
and the calculation time for each update.
Chapront-Touzé, Michelle and Jean Chapront. Lunar Tables and Programs from 4000 B.C. to A.D. 8000 . Richmond: Willmann-Bell, 1991. ISBN 0-943396-33-6.
If you need more precise calculation of the Moon's position than given in the references above, you're probably going to end up here. This book presents the ELP 2000-85 theory which, while less
accurate than ELP 2000-82, has been tested for stability over a much longer time span. ELP 2000-85 generates predictions of lunar longitude accurate to 0.0004 degrees for the years 1900 through
2100, and 0.0054 degrees for the period 1500 through 2500.
Chapront-Touzé, Michelle and Jean Chapront. Lunar solution ELP 2000-82B.
This is the most precise semi-analytical theory of the Moon's motion for observations near the present epoch. Machine-readable files for all of the tables and a sample FORTRAN program which uses
them to compute lunar ephemerides may be obtained from the Astronomical Data Center at the NASA Goddard Space Flight Center by FTP across the Internet, or on CD-ROM, along with a wide variety of
other astronomical catalogues and tables. This material is intended for experts in positional astronomy and computation. If you can't figure it out, don't ask me for help.
by John Walker
May 5, 1997
This document is in the public domain. | {"url":"http://fourmilab.ch/earthview/pacalc.html","timestamp":"2014-04-18T23:15:05Z","content_type":null,"content_length":"9417","record_id":"<urn:uuid:80b0f412-7ee1-4289-95be-7a79f1b83ffb>","cc-path":"CC-MAIN-2014-15/segments/1397609535535.6/warc/CC-MAIN-20140416005215-00557-ip-10-147-4-33.ec2.internal.warc.gz"} |
FOM: Recursion theory question
Harvey Friedman friedman at math.ohio-state.edu
Sat Feb 16 16:58:14 EST 2002
The following is well known.
THEOREM. "The set of all n such that there exists e < n with phi_e(0) = n"
is of Turing degree 0'.
Normally, one easily proves that this is effectively simple, and quotes
Donald Martin's theorem that every effectively simple set is of Turing
degree 0', whose usual proof is to use Donald Martin's theorem about DNR.
However, there is a simple direct proof of this Theorem that does not
involve this interesting machinery. This is also probably well known.
But now what about
CONJECTURE. "The set of all n such that there exists e < n such that n is
the largest element of W_e" is of Turing degree 0'.
Is this well known, or at least known? If not, then can you prove this? I
am confident that an expert in recursion theory can do this.
More information about the FOM mailing list | {"url":"http://www.cs.nyu.edu/pipermail/fom/2002-February/005289.html","timestamp":"2014-04-17T04:22:39Z","content_type":null,"content_length":"3048","record_id":"<urn:uuid:9ab22c0c-2d85-446a-88ca-b35bf8a98adc>","cc-path":"CC-MAIN-2014-15/segments/1397609526252.40/warc/CC-MAIN-20140416005206-00025-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Topic: [ap-calculus] Re: From the Moderator + AP versus non-AP calculus
+ An AP myth
Replies: 1 Last Post: Nov 27, 2004 4:28 PM
[ap-calculus] Re: From the Moderator + AP versus non-AP calculus
+ An AP myth
Posted: Nov 27, 2004 4:28 PM
Hello all,
There have been a variety of comments on the above three subjects over
the past few days. Here are mine:
1: The article quoted in the Moderator's message should be viewed in
light of the data at
<a href="http://measuringup.highereducation.org/2000/">http://measuringup.highereducation.org/2000/</a>
For example, in 2002 the five-year completion rate for college
students in Texas was 41%; this should be compared with the rate on
the chart in the article. Go to the web page to see the various
qualifiers listed, qualifiers which are not included in the article.
I have not read the book on which the article is based, so I can not in
fairness judge the quality of its statistics. However, the numbers in
the Post article in and of themselves have no statistical value; they
are just a table of numbers. I would be hesitant to draw any
conclusions from them.
2: If calculus is going to be offered in high school, it should be AP
calculus. A course in STATISTICS (Freedman, Aliaga, ...) or FINITE
MATHEMATICS" (COMAP, Hathaway, ...) would be a much better choice for
both the academic and civic futures of the students. In particular, a
course of either type would require students to set up models, make
judgments, and draw conclusions. In other words, they would have to
think about what they are doing.
In my experience, students who take AP calculus, but do not get a 4 or a
5 on the AP test, are very similar to students who do take a non-AP
calculus class in high school; too many of think they know calculus.
Their performance on their first college calculus examination is usually
a shock and, hopefully, a wake-up call. As for the 4's and 5's, their
performance in more advanced courses, again in my experience, ranges from
very good to very bad. I realize that these comments run contrary to many
of the submissions to this list but, again, why are schools reexamining
the way they award AP calculus credit?
3: Students who get college credit for two or three AP classes
typically are not going to save any money. Most institutions of higher
learning charge "full-time" students the same tuition whenever
they take a full course load, which may range from 12 to 18 hours per
term. On the other hand such a student can shave a semester off of their
undergraduate work through the appropriate use of summer classes, which
usually cost less, per course, than full time tuition during the regular
academic year. Of course, the structure of their major courses has to be
amenable to this approach. Similar comments apply to students who get
credit for six or more AP courses except, again their major requirements
willing, they can get out a year earlier with the help of summer courses.
Richard J. Maher
Mathematics and Statistics
Loyola University Chicago
6525 N. Sheridan Rd.
Chicago, Illinois 60626
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series - converge or diverge
November 10th 2009, 02:56 AM #1
Junior Member
Mar 2008
series - converge or diverge
Hi all,
I am trying to determine if the following series converges or diverges:
$<br /> \sum_{n=1}^{\infty}\frac{sin^2(n)}{n^2}<br />$
Not sure if what i did is correct:
-1 <= sin(n) <= 1
1 <= sin^2(n) <= 1
1/n^2 <= [sin^2(n)]/ n^2 <= 1/n^2
Note: sin^2(n) = [sin(n)]^2
I then compared the series to 1/n^2 and determined that the series 1/n^2 is a harmonic p series, where p > 1 and so it converges. Thus, the series $<br /> \sum_{n=1}^{\infty}\frac{sin^2(n)}{n^2}
<br />$ converges.
Could someone tell me if i am doing this correctly
Thanks in advance,
You mean -1<= sin^2(n)<= 1 but the comparison test uses absolute values that doesn't matter.
1/n^2 <= [sin^2(n)]/ n^2 <= 1/n^2
Note: sin^2(n) = [sin(n)]^2
I then compared the series to 1/n^2 and determined that the series 1/n^2 is a harmonic p series, where p > 1 and so it converges. Thus, the series $<br /> \sum_{n=1}^{\infty}\frac{sin^2(n)}{n^2}
<br />$ converges.
Could someone tell me if i am doing this correctly
Thanks in advance,
Except for the unimportant sign error, yes.
November 10th 2009, 02:59 AM #2
MHF Contributor
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