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Bootstrapping for Propensity Score Analysis
November 26, 2013
By Jason Bryer
I am happy to announce that version 1.0 of the PSAboot package has been released to CRAN. This package implements bootstrapping for propensity score analysis. This deviates from typical
implementations such as boot in that it allows for separate sampling specifications for treatment and control units. For example, in the case where the ratio of treatment-to-control units is large,
one can bootstrap only the control units while always using all available treatment units. Additionally, this package will estimate treatment effects using multiple methods for each bootstrap sample.
In addition to adhering to Rosenbaum’s (2012) advise of “Testing One Hypothesis Twice in Observational Studies”, we can compare the performance of different methods across many samples. Lastly, a set
of functions to estimate and visualize balance across bootstrap samples and methods are provided.
You can get more details on the project page and the vignette. The project is hosted on Github project page. Download the latest version or submit bugs there.
This package supports stratification using ctree (from the party package), rpart, and quintiles (using fitted values from logistic regression) and well as matching using the MatchIt and Matching
packages. The project page outlines how to write custom methods.
The following example uses the tutoring dataset in the TriMatch package. This study examined the effects of tutoring on student grades in writing courses. The treatment group was defined a students
who used tutoring services during their course. The control group are students in a course section with at least one student who used the tutoring services. The PSAboot performs the bootstrap
analysis and returns an object of class PSAboot. The summary, plot, hist, boxplot, and matrixplot S3 methods are implemented.
# Loading required package: PSAboot
# Loading required package: PSAgraphics
# Loading required package: rpart
data(tutoring, package = "TriMatch")
tutoring$treatbool <- tutoring$treat != "Control"
covs <- tutoring[, c("Gender", "Ethnicity", "Military", "ESL", "EdMother", "EdFather",
"Age", "Employment", "Income", "Transfer", "GPA")]
# FALSE TRUE
# 918 224
tutoring.boot <- PSAboot(Tr = tutoring$treatbool, Y = tutoring$Grade, X = covs,
seed = 2112)
# 100 bootstrap samples using 5 methods.
# Bootstrap sample sizes:
# Treated=224 (100%) with replacement.
# Control=918 (100%) with replacement.
The summary function provides numeric results for each method including the overall estimate and confidence interval using the complete sample as well as the pooled estimates and confidence intervals
with percentages of the number of confidence intervals that do not span zero.
# Stratification Results:
# Complete estimate = 0.482
# Complete CI = [0.3, 0.665]
# Bootstrap pooled estimate = 0.476
# Bootstrap pooled CI = [0.332, 0.62]
# 100% of bootstrap samples have confidence intervals that do not span zero.
# 100% positive.
# 0% negative.
# ctree Results:
# Complete estimate = 0.458
# Complete CI = [0.177, 0.739]
# Bootstrap pooled estimate = 0.482
# Bootstrap pooled CI = [0.294, 0.67]
# 99% of bootstrap samples have confidence intervals that do not span zero.
# 99% positive.
# 0% negative.
# rpart Results:
# Complete estimate = 0.475
# Complete CI = [0.165, 0.784]
# Bootstrap pooled estimate = 0.45
# Bootstrap pooled CI = [0.212, 0.689]
# 84% of bootstrap samples have confidence intervals that do not span zero.
# 84% positive.
# 0% negative.
# Matching Results:
# Complete estimate = 0.479
# Complete CI = [0.388, 0.571]
# Bootstrap pooled estimate = 0.471
# Bootstrap pooled CI = [0.231, 0.711]
# 100% of bootstrap samples have confidence intervals that do not span zero.
# 100% positive.
# 0% negative.
# MatchIt Results:
# Complete estimate = 0.5
# Complete CI = [0.253, 0.747]
# Bootstrap pooled estimate = 0.513
# Bootstrap pooled CI = [0.293, 0.734]
# 100% of bootstrap samples have confidence intervals that do not span zero.
# 100% positive.
# 0% negative.
The plot function plots the estimate (mean difference) for each bootstrap sample. The default is to sort from largest to smallest estimate for each method separately. That is, rows do not correspond
across methods. The sort parameter can be set to none for no sorting or the name of any method to sort only based upon the results of that method. In these cases the rows then correspond to matching
bootstrap samples. The blue points correspond to the the estimate for each bootstrap sample and the horizontal line to the confidence interval. Confidence intervals that do not span zero are colored
red. The vertical blue line and green lines correspond to the overall pooled estimate and confidence for each method, respectively.
The hist function plots a histogram of the estimates across all bootstrap samples for each method.
# stat_bin: binwidth defaulted to range/30. Use 'binwidth = x' to adjust this.
# stat_bin: binwidth defaulted to range/30. Use 'binwidth = x' to adjust this.
# stat_bin: binwidth defaulted to range/30. Use 'binwidth = x' to adjust this.
# stat_bin: binwidth defaulted to range/30. Use 'binwidth = x' to adjust this.
# stat_bin: binwidth defaulted to range/30. Use 'binwidth = x' to adjust this.
The boxplot function depicts the distribution of estimates for each method along with confidence intervals in green. Additionally, the overall pooled estimate and confidence interval across all
bootstrap samples and methods are represented by the vertical blue and green lines, respectively.
The matrixplot summarizes the estimates across methods for each bootstrap sample. The lower half of the matrix are scatter plots where each point represents the one bootstrap sample. The red line is
a Loess regression line. The main diagonal depicts the distribution of effects and the upper half provides the correlation of estimates.
The balance function will provide balance statistics. The print, plot, and boxplot S3 methods are implemented.
tutoring.balance <- balance(tutoring.boot)
# Unadjusted balance: 0.117875835338968
# Complete Bootstrap
# Stratification 0.02923 0.03795
# ctree 0.04385 0.06913
# rpart 0.07846 0.08698
# Matching 0.04522 0.06668
# MatchIt 0.05078 0.05790
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Negative activation energies in molecular modeling: diagnosis and cures
This post is inspired by a recent discussion with
Paolo in the comments section of this post.
So, you've found that the energy of your transition state (TS) is lower than your reactants (i.e. you have a negative activation energy). Don't panic, instead pretend you're
Dr House
and go through it methodically:
Did you actually find the right TS?
A TS is a completely optimized geometry (no constraints, "zero" gradient) with one has one and only one imaginary frequency. Yes? OK, how big is the imaginary frequency and does the normal mode look
like what you would expect? Sometimes minima can have small (< ca 100 cm-1) imaginary frequencies due to numerical "noise", or the TS search algorithm will find a TS for, say, methyl rotation. No,
looks OK? (Consider computing an
do be really sure, just in case.) Let's move on.
How much lower?
Some processes have very low (< 1 kcal/mol) barriers. Numerical "noise" can cause these barriers to come out slightly negative (-0.1 to -1.0 kcal/mol) instead. If so, consider your reaction
barrier-less for all intends and purposes. No, much more negative? Let's move on.
What energy are we talking about here?
If the activation energy is calculated using
single point energies
(for example B3LYP/6-31G(d)//RHF/3-21G) then the B3LYP/6-31G(d) PES may not have a barrier or the B3LYP/6-31G(d) TS geometry looks very different from the RHF/3-21G TS geometry.
The easiest way to check for this particular case is to geometry optimize your reactant at the B3LYP/6-31G(d) level of theory. If the optimization results in the product geometry, then there is (very
likely) no barrier to the reaction. If the optimization gives you a reactant geometry, then one explanation is that 3-21G is not a reliable method for finding the TS and the solution is to use B3LYP/
6-31G(d) in your TS search. There are other explanations, so read on.
If the activation energy is calculated using
zero point energy
free energy corrections
and is negative (this is not uncommon for reactions whose PES barriers are less than 2-3 kcal/mol) then you can consider the reaction barrier-less. If you want to be really sure, you can add these
corrections along the entire
. However, there are also other explanations so read on.
Is your reaction unimolecular?
$R \rightarrow TS \rightarrow P$
OK, so you have found a reasonable-looking TS using some method and the electronic energy barrier computed using that method is quite negative. If your reaction is unimolecular then the most likely
explanation is that you have not found the lowest energy conformation of your reactant. If can't find a lower energy structure for your reaction, compute an
. An IRC follows the minimum energy path
to your reactant and product, so you
find structures with a
energy than your TS.
Is your reaction bimolecular?
$R_1 + R_2 \rightarrow R_1/R_2 \rightarrow TS \rightarrow P$
OK, so you have found a reasonable-looking TS using some method and the electronic energy barrier computed using that method is quite negative. The first question is how did you compute the barrier?
If you computed the barrier as the energy difference between the TS and the bound complex of your two reactants (R1/R2) then the most likely explanation is that you have not found the lowest energy
conformation of this complex (see the section on unimolecular reactions).
If you computed the barrier as the energy difference between the TS and the sum of the energies and your two isolated reactants [E(R1)+E(R2)] then there is not necessarily anything wrong with your
calculations. Read on.
How should I compute the activation energy of a bimolecular reaction? (This sub-section has been updated 2012.05.05)
The activation energy for a biomolecular reaction should be computed as
$E_a = E(TS) - [E(R_1) + E(R_2)]$
If this energy is negative then you would say that this is a barrier-less reaction and the reaction rate is proportional to the collision frequency.
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Portability non-portable (DeriveDataTypeable)
Stability unstable
Maintainer Marco Túlio Pimenta Gontijo <marcotmarcot@gmail.com>
Safe Haskell None
Chuchu is a system similar to Ruby's Cucumber for Behaviour Driven Development. It works with a language similar to Cucumber's Gherkin, which is parsed using package abacate.
This module provides the main function for a test file based on Behaviour Driven Development for Haskell.
Example for a Stack calculator:
Feature: Division
In order to avoid silly mistakes
Cashiers must be able to calculate a fraction
Scenario: Regular numbers
Given that I have entered 3 into the calculator
And that I have entered 2 into the calculator
When I press divide
Then the result should be 1.5 on the screen
import Control.Applicative
import Control.Monad.IO.Class
import Control.Monad.Trans.State
import Test.Chuchu
import Test.HUnit
type CalculatorT m = StateT [Double] m
enterNumber :: Monad m => Double -> CalculatorT m ()
enterNumber = modify . (:)
getDisplay :: Monad m => CalculatorT m Double
= do
ns <- get
return $ head $ ns ++ [0]
divide :: Monad m => CalculatorT m ()
divide = do
(n1:n2:ns) <- get
put $ (n2 / n1) : ns
defs :: Chuchu (CalculatorT IO)
= do
("that I have entered " *> number <* " into the calculator")
When "I press divide" $ const divide
Then ("the result should be " *> number <* " on the screen")
$ \n
-> do
d <- getDisplay
liftIO $ d @?= n
main :: IO ()
main = chuchuMain defs (`evalStateT` [])
chuchuMain :: (MonadIO m, Applicative m) => Chuchu m -> (m () -> IO ()) -> IO ()Source
The main function for the test file. It expects the .feature file as the first parameter on the command line. If you want to use it inside a library, consider using withArgs.
|
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Mega quiz
Mega quiz
How many of you guys have taken the megaquiz and have honestly i.e.
1. in 1st try
2. without using reference books,file, etc open
3. without testing the questions on another comp at the time
got 100%
Be honest. I did it a couple of days back and got 60%:eek:
I'm asking so that next time i'll post my doubts directly to you if you really got 100%
did it once when I first started here, got an 80 something cant' remeber, haven't looked back
hmmmm mabe I'll take it again
I got something like 90 with no help at all. With help I might have gotten 95. There's no way you can get 100 without knowing the questions in my opinion. Or you have an obsession with C. There
exists a healthy level, and that's not 100%.
I think I got 70% when I first took it (several months ago!). I might get a bit more now but i know the questions so that might be an unfair advantage!
Well question 1 is plain wrong I think. Is
it binary digits? decimal digits? Even then
it is not specified whether it is single-precision or double-precision.
I couldn't be arsed to bother taking it...
|
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Pgs 6 and 27 are on the same (double) sheet of newspaper. What are the pg no.s on the opposite side of the sheet? How... - Homework Help - eNotes.com
Pgs 6 and 27 are on the same (double) sheet of newspaper. What are the pg no.s on the opposite side of the sheet? How many pgs are there altogether?
The page numbers on the opposite side are (6-1) and (27+1) which is 5 and 28.
The number of pages is twice the (6+(27-1))/2
Therefore the number of pages = 2 xx (6+(27-1))/2
= 32.
The other way to get this is there are 5 more pages before 6, therefore there should be 5 more pages after 27 also. Therefore the number of pages = 27+5 = 32.
I don't know how to answer this but why there should be more 5 pages after 27??
Join to answer this question
Join a community of thousands of dedicated teachers and students.
Join eNotes
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|
Resolution: standard / high
Figure 5.
Directional accuracy in yeast model simulations and microfluidics experiments. (A) Simulations of yeast model at different gradient slopes. L[slp ]= 0.1, 0.01, and 0.001 nM μm^-1 (L[mid ]= 10 nM) and
σ = 3. Using the wild-type model, at least 20 Monte Carlo simulations were performed as described above. The mean value of active Cdc42 is plotted, and the mean ± SEM of cos(θ) is also shown for each
slope. (B) Directional projection accuracy at different gradient slopes. Experiments were performed for 1.5% gradient, 0.5%, 0.05%, and 0.005% μm^-1 gradients (L[slp]/L[mid]). Cells were counted in
the middle two sections of the gradient where L[mid ]~ 20 nM. Directional accuracy was measured in terms of cos(θ), and the mean ± SEM is shown for n = 3 trials. (C) Plotting yeast mating projection
directional accuracy as a function of gradient slope for both experiments and modeling. Simulation data (gray) is from (A) and experimental data (black) is from (B).
Chou et al. BMC Systems Biology 2011 5:196 doi:10.1186/1752-0509-5-196
Download authors' original image
|
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In the figure shown, the circle has center O and radius 50
Author Message
In the figure shown, the circle has center O and radius 50 [#permalink] 03 Jan 2011, 17:18
55% (medium)
Question Stats:
ajit257 32%
Senior Manager (02:09) correct
Joined: 28 Aug 2010 67% (01:03)
Posts: 265 wrong
Followers: 3 based on 105 sessions
Kudos [?]: 45 [0], Attachment:
given: 11
Circle.png [ 14.99 KiB | Viewed 5619 times ]
In the figure shown, the circle has center O and radius 50, and point P has coordinates (50,0). If point Q (not shown) is on the circle, what is the length of line segment PQ ?
(1) The x-coordinate of point Q is – 30.
(2) The y-coordinate of point Q is – 40.
Spoiler: OA
Math: new-to-the-math-forum-please-read-this-first-77764.html
Gmat: everything-you-need-to-prepare-for-the-gmat-revised-77983.html
Re: figure shown, the circle has center O and radius 50 [#permalink] 03 Jan 2011, 20:19
Senior Manager
Joined: 13 Aug 2010
This post received
Posts: 316 KUDOS
Followers: 1 We need to have both the X and Y coordinate to find the distance.
Kudos [?]: 7 [1] ,
given: 1
Re: figure shown, the circle has center O and radius 50 [#permalink] 04 Jan 2011, 02:03
This post received
Expert's post
In the figure shown, the circle has center O and radius 50, and point P has coordinates (50,0). If point Q (not shown) is on the circle, what is the length of line segment PQ ?
Note that we are told that
point Q is on the circle
. Also as the radius of the circle is 50 then for any point (x,y) on the circle
(check for more here:
) Look at the diagram:
Circle.png [ 13.28 KiB | Viewed 5587 times ]
(1) The x-coordinate of point Q is – 30 --> point Q can be either on the position of Q1 or Q2 on the diagram, but in any case the distance between P and Q is the same --> for
point Q:
Bunuel (-30)^2+y^2=50^2
Math Expert -->
Joined: 02 Sep 2009 y^2=20*80=40^2
Posts: 17317 , so
Followers: 2874 CQ^2=y^2=40^2
Kudos [?]: 18377 [4 (no matter where Q actually is on Q1 or Q2) --> PQ which is the hypotenuse in PQC is equal to
] , given: 2348
. Sufficient.
(2) The y-coordinate of point Q is – 40 --> point Q can be either on the position of Q2 or Q3 on the diagram, and the distance between P and Q will be different for theses cases.
Not sufficient.
Answer: A.
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Intern Re: figure shown, the circle has center O and radius 50 [#permalink] 23 Feb 2012, 00:48
Joined: 01 Jan 2012 How doe we know that P lies on the diameter ? As per question, position of P relative to centre is not known. So how is it assumed that P is on the diameter? Wouldn't it change
answer ?
Posts: 1
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Math Expert Re: figure shown, the circle has center O and radius 50 [#permalink] 23 Feb 2012, 01:24
Joined: 02 Sep 2009 Expert's post
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Followers: 2874
Re: figure shown, the circle has center O and radius 50 [#permalink] 28 Apr 2012, 10:39
This post received
Joined: 14 Feb 2012 KUDOS
Posts: 228 Nice question I got it wrong because i just considered X co-ordinate ,forgot that things could be different for y
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Math Expert Re: In the figure shown, the circle has center O and radius 50 [#permalink] 22 Jul 2013, 21:17
Joined: 02 Sep 2009 Expert's post
Posts: 17317
Followers: 2874
Re: figure shown, the circle has center O and radius 50 [#permalink] 03 Aug 2013, 06:52
Bunuel wrote:
In the figure shown, the circle has center O and radius 50, and point P has coordinates (50,0). If point Q (not shown) is on the circle, what is the length of line segment PQ ?
Note that we are told that
point Q is on the circle
. Also as the radius of the circle is 50 then for any point (x,y) on the circle
(check for more here:
) Look at the diagram:
(1) The x-coordinate of point Q is – 30 --> point Q can be either on the position of Q1 or Q2 on the diagram, but in any case the distance between P and Q is the same --> for
stne point Q:
Manager (-30)^2+y^2=50^2
Joined: 27 May 2012 -->
Posts: 212 y^2=20*80=40^2
Followers: 0 , so
(no matter where Q actually is on Q1 or Q2) --> PQ which is the hypotenuse in PQC is equal to
. Sufficient.
(2) The y-coordinate of point Q is – 40 --> point Q can be either on the position of Q2 or Q3 on the diagram, and the distance between P and Q will be different for theses cases.
Not sufficient.
Answer: A.
why cannot the point Q have coordinates (-30,0) in this case point Q will lie on the Diameter of the circle and the distance from point P will be 80, so aren't we getting two
cases for A?
What am I missing here? Can anybody assist?
- Stne
Re: figure shown, the circle has center O and radius 50 [#permalink] 03 Aug 2013, 07:01
Expert's post
stne wrote:
Bunuel wrote:
In the figure shown, the circle has center O and radius 50, and point P has coordinates (50,0). If point Q (not shown) is on the circle, what is the length of line segment PQ ?
Note that we are told that
point Q is on the circle
. Also as the radius of the circle is 50 then for any point (x,y) on the circle
(check for more here:
) Look at the diagram:
(1) The x-coordinate of point Q is – 30 --> point Q can be either on the position of Q1 or Q2 on the diagram, but in any case the distance between P and Q is the same --> for
Verbal Forum point Q:
Joined: 10 Oct 2012
Posts: 626
Followers: 35
, so
Kudos [?]: 488 [0],
given: 135 CQ^2=y^2=40^2
(no matter where Q actually is on Q1 or Q2) --> PQ which is the hypotenuse in PQC is equal to
. Sufficient.
(2) The y-coordinate of point Q is – 40 --> point Q can be either on the position of Q2 or Q3 on the diagram, and the distance between P and Q will be different for theses cases.
Not sufficient.
Answer: A.
why cannot the point Q have coordinates (-30,0) in this case point Q will lie on the Diameter of the circle and the distance from point P will be 80, so aren't we getting two
cases for A?
What am I missing here? Can anybody assist?
You are forgetting that if Q is (-30,0), then the point will no longer be ON the circle. It will be IN the circle.
All that is equal and not-Deep Dive In-equality
Hit and Trial for Integral Solutions
Re: figure shown, the circle has center O and radius 50 [#permalink] 03 Aug 2013, 08:50
mau5 wrote:
stne wrote:
Bunuel wrote:
In the figure shown, the circle has center O and radius 50, and point P has coordinates (50,0). If point Q (not shown) is on the circle, what is the length of line segment PQ ?
Note that we are told that
point Q is on the circle
. Also as the radius of the circle is 50 then for any point (x,y) on the circle
(check for more here:
) Look at the diagram:
(1) The x-coordinate of point Q is – 30 --> point Q can be either on the position of Q1 or Q2 on the diagram, but in any case the distance between P and Q is the same --> for
point Q:
stne -->
Manager y^2=20*80=40^2
Joined: 27 May 2012 , so
Posts: 212 CQ^2=y^2=40^2
Followers: 0 (no matter where Q actually is on Q1 or Q2) --> PQ which is the hypotenuse in PQC is equal to
. Sufficient.
(2) The y-coordinate of point Q is – 40 --> point Q can be either on the position of Q2 or Q3 on the diagram, and the distance between P and Q will be different for theses cases.
Not sufficient.
Answer: A.
why cannot the point Q have coordinates (-30,0) in this case point Q will lie on the Diameter of the circle and the distance from point P will be 80, so aren't we getting two
cases for A?
What am I missing here? Can anybody assist?
You are forgetting that if Q is (-30,0), then the point will no longer be ON the circle. It will be IN the circle.
Correct! Realized this just after I posted it. Although couldn't it have been more clearer if it was mentioned that point Q lies on the
of the circle? Just
the circle can also mean anywhere in the circle,can it not?
- Stne
Re: figure shown, the circle has center O and radius 50 [#permalink] 03 Aug 2013, 08:54
Expert's post
Verbal Forum
Moderator stne wrote:
Joined: 10 Oct 2012
Correct! Realized this just after I posted it. Although couldn't it have been more clearer if it was mentioned that point Q lies on the circumference of the circle? Just ON the
Posts: 626 circle can also mean anywhere in the circle,can it not?
Followers: 35 No.A point lying on the circumference or a point lying on the circle is the same thing.IF it is on the circle, it can't be anywhere else.
Kudos [?]: 488 [0], _________________
given: 135
All that is equal and not-Deep Dive In-equality
Hit and Trial for Integral Solutions
Re: figure shown, the circle has center O and radius 50 [#permalink] 03 Aug 2013, 09:01
mau5 wrote:
stne stne wrote:
Correct! Realized this just after I posted it. Although couldn't it have been more clearer if it was mentioned that point Q lies on the circumference of the circle? Just ON the
Joined: 27 May 2012 circle can also mean anywhere in the circle,can it not?
Posts: 212 No.A point lying on the circumference or a point lying on the circle is the same thing.IF it is on the circle, it can't be anywhere else.
Followers: 0 Let me just cement my understanding,ON and circumference is the same thing, So if it says that a point lies on the circle, it always means that the point lies on the
circumference, is that correct.
- Stne
Re: figure shown, the circle has center O and radius 50 [#permalink] 03 Aug 2013, 09:05
This post received
Expert's post
stne wrote:
Verbal Forum
Moderator mau5 wrote:
Joined: 10 Oct 2012 stne wrote:
Posts: 626
Correct! Realized this just after I posted it. Although couldn't it have been more clearer if it was mentioned that point Q lies on the circumference of the circle? Just ON the
Followers: 35 circle can also mean anywhere in the circle,can it not?
Kudos [?]: 488 [1] No.A point lying on the circumference or a point lying on the circle is the same thing.IF it is on the circle, it can't be anywhere else.
, given: 135
Let me just cement my understanding,ON and circumference is the same thing, So if it says that a point lies on the circle, it always means that the point lies on the
circumference, is that correct.
Yes, that is correct.Any point on the circle is a point on the circumference.
All that is equal and not-Deep Dive In-equality
Hit and Trial for Integral Solutions
Re: figure shown, the circle has center O and radius 50 [#permalink] 03 Aug 2013, 09:19
mau5 wrote:
Yes, that is correct.Any point on the circle is a point on the circumference.
Joined: 27 May 2012
Thank you +1, it was an eyeopener, had been unsure about this.
Posts: 212
Followers: 0
- Stne
Re: In the figure shown, the circle has center O and radius 50 [#permalink] 03 Aug 2013, 23:16
Lets look at this in another way, Equation of Circle: x^2 + y^2 = 2500
Statement (1): The x-coordinate of point Q is – 30.
Substituting this in x^2 + y^2 = 2500,
we get y = 40 or -40
Now using distance formula,
Intern dist(P,Q) = \sqrt{(x - x')^2 + (y - y')^2}
= \sqrt{(50 - (-30))^2 + (0 - (40))^2} or = \sqrt{(50 - (-30))^2 + (0 - (-40))^2}
Joined: 20 Mar 2013 = \sqrt{(6400 + 1600)}
= \sqrt{8000}
Posts: 13
So in either case, dist(P,Q) = \sqrt{8000}. Hence, Sufficient.
Followers: 0
Statement (2): The y-coordinate of point Q is – 40.
Kudos [?]: 6 [0], Substituting this in x^2 + y^2 = 2500,
given: 1 we get x = 30 or -30
Now using distance formula,
dist(P,Q) = \sqrt{(x - x')^2 + (y - y')^2}
= \sqrt{(50 - (-30))^2 + (0 - (-40))^2} OR = \sqrt{(50 - (30))^2 + (0 - (-40))^2}
= \sqrt{(6400 + 1600)} OR = \sqrt{(400 + 1600)}
= \sqrt{8000} OR = \sqrt{2000}
So different answers depending on whether x = -30 or x = 30. Hence, Insufficient.
Correct Ans: A
Re: In the figure shown, the circle has center O and radius 50 [#permalink] 03 Aug 2013, 23:24
kumar23badgujar wrote:
Lets look at this in another way, Equation of Circle: x^2 + y^2 = 2500
Statement (1): The x-coordinate of point Q is – 30.
Substituting this in x^2 + y^2 = 2500,
we get y = 40 or -40
Now using distance formula,
Intern dist(P,Q) = \sqrt{(x - x')^2 + (y - y')^2}
= \sqrt{(50 - (-30))^2 + (0 - (40))^2} or = \sqrt{(50 - (-30))^2 + (0 - (-40))^2}
Joined: 29 Jul 2013 = \sqrt{(6400 + 1600)}
= \sqrt{8000}
Posts: 1
So in either case, dist(P,Q) = \sqrt{8000}. Hence, Sufficient.
Followers: 0
Statement (2): The y-coordinate of point Q is – 40.
Kudos [?]: 0 [0], Substituting this in x^2 + y^2 = 2500,
given: 0 we get x = 30 or -30
Now using distance formula,
dist(P,Q) = \sqrt{(x - x')^2 + (y - y')^2}
= \sqrt{(50 - (-30))^2 + (0 - (-40))^2} OR = \sqrt{(50 - (30))^2 + (0 - (-40))^2}
= \sqrt{(6400 + 1600)} OR = \sqrt{(400 + 1600)}
= \sqrt{8000} OR = \sqrt{2000}
So different answers depending on whether x = -30 or x = 30. Hence, Insufficient.
Correct Ans: A
+1 ,Simple explanation.
Re: figure shown, the circle has center O and radius 50 [#permalink] 14 Apr 2014, 23:46
HI Bunnel,
Joined: 10 Mar 2014
I am not clear about second statement.
Posts: 22
(2) The y-coordinate of point Q is – 40 --> point Q can be either on the position of Q2 or Q3 on the diagram, and the distance between P and Q will be different for theses cases.
Followers: 0 Not sufficient.
Kudos [?]: 0 [0], Please clarify.
given: 4
Math Expert Re: figure shown, the circle has center O and radius 50 [#permalink] 15 Apr 2014, 00:12
Joined: 02 Sep 2009 Expert's post
Posts: 17317
Followers: 2874
gmatclubot Re: figure shown, the circle has center O and radius 50 [#permalink] 15 Apr 2014, 00:12
Similar topics Author Replies Last post
A circle has a center O at point (0,0) and a radius of 50, joemama142000 5 05 Feb 2006, 12:51
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10 In the figure above, the radius of circle with center O is 1 over2u 15 27 Aug 2009, 11:03
5 A circle with center O and radius 5 is shown in the xy-plane violetsplash 4 26 Sep 2013, 06:25
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A bestiary of topologies on Sch
up vote 31 down vote favorite
The category of schemes has a large (and to me, slightly bewildering) number of what seem like different Grothendieck (pre)topologies. Zariski, ok, I get. Etale, that's alright, I think. Nisnevich?
pff, not a chance. There are various ideas about stacks I would like to test out, but the sites I am most familiar with have few application-rich topologies. (Smooth, finite-dimensional manifolds are
particularly boring in this respect, and topological spaces are not much better)
What I'm after is a table listing the well-known/common topologies on $Sch$ and their relative 'fineness'. Or, if you like, containment. We of course have the canonical topology - is there a
characterisation of that in terms of schematic properties, as opposed to the obvious categorical definition?
And furthermore, one expects that for nice schemes, various topologies will coalesce, say one sort of covers becoming cofinal in another, when restricted to a subcategory of $Sch$. Say those schemes
which are Noetherian, smooth or even just varieties.
Then there are things like when categories of sheaves, or 2-categories of stacks, are equivalent. But maybe this is asking too much.
Maybe I'm after something like 'Counterexamples in Grothendieck topologies'. Does such a thing exist, all in one place? I'm sure it is all there in SGA, or the stacks project, or in Vakil's
Foundations of Algebraic Geometry, but I'm after the distilled essence.
PS I am interested in things which are (pre)topologies even if they are not usually used as such for the purposes of sheaves.
EDIT: I'm not merely after examples of Grothendieck topologies on $Sch$, even though that is handy. I want a reference, if there is one, or just a straight-out answer, that compares the various
topologies on $Sch$, and under which circumstances (restricting $Sch$ to a subcategory) they coincide.
For example, does an fppf cover of a variety have local sections over an etale cover? Do the fppf and fpqc topologies give rise to the same sheaves over a nicely behaved scheme? Is the etale topology
strictly 'weaker' than some other topology no matter what schemes one looks at? Does one get the same Deligne-Mumford stacks for topology A and topology B?
(Grumble over)
ag.algebraic-geometry grothendieck-topology ct.category-theory
15 A common hierarchy is fpqc --> fppf --> syntomic --> etale --> Nisnevich --> Zariski. There are also the infinitesimal/crystalline sites, but these are in a somewhat orthogonal direction (and one
can superimpose e.g. the etale topology on the crystalline site). – Emerton Sep 5 '11 at 3:01
1 Ah, but surely the lattice of topologies is more interesting than some linear order... And I don't get when you say 'etale topology on crystalline site'. To me, saying the blah site means schemes
with the blah topology. I'm not an algebraic geometer, so please bear with me. :) – David Roberts Sep 5 '11 at 4:21
1 Emerton, I share David's lack of understanding of the grammar of "etale topology on the crystalline site". In the definition I know, a site is a category equipped with a (Grothendieck) topology.
So if S is a site then the phrase "topology on S" isn't one I can parse, unless it just means topology on the underlying category C of S; but in that case you might as well say "topology on C".
What am I missing? – Tom Leinster Sep 5 '11 at 4:42
3 What Emerton means by that phrase is (probably) some version the following: 'crystalline site' over $S$ means the category of divided power thickenings of $S$-schemes (over some other base,
usually something like $\mathbb{Z}_p$). This is of course not a site till we decree what the coverings are. There are various possibilities, among those being the choice of etale coverings of the
thickenings. – Keerthi Madapusi Pera Sep 5 '11 at 5:02
5 There are also the various flavors of the h-topology on Sch introduced by Voevodsky. See Friedlander's notes on them here: math.northwestern.edu/~eric/lectures/ihp/ihplec2.pdf – Keerthi Madapusi
Pera Sep 6 '11 at 1:40
show 6 more comments
4 Answers
active oldest votes
The basic answer is essentially as Emerton described in the comment. The most commonly used topologies on schemes are Zariski, Nisnevich, étale, smooth, syntomic, fppf, and fpqc, and this
list is totally ordered by increasing fineness. The canonical topology is finer than the fpqc topology, but I have never seen it explicitly used. You can see a discussion of these topologies
(other than Nisnevich) in the Stacks project chapter on Topologies on Schemes.
You ask about restricting to subcategories of schemes to get equivalent topologies, but I think you would have to take unusually small subcategories. For example, the étale and Nisnevich
topologies coincide on the spectra of fields only when the fields are separably closed. I think if the Nisnevich covers of a scheme are Zariski covers, then the scheme is zero dimensional.
Smooth and étale covers coincide if you restrict to say, varieties of a single fixed dimension. I think the same is true for syntomic versus smooth and fppf versus symtomic (but I am far
from sure). If you restrict your schemes to be locally finitely presented over a fixed base, then fppf and fpqc coincide. Even though the étale and smooth topologies are usually not
up vote equivalent, they give rise to equivalent categories of sheaves, because every smooth cover has an étale refinement.
14 down
vote The Stacks project has a list of properties that different topologies satisfy, in the Descent chapter. Bjorn Poonen also has a table of permanence properties in Appendix C of his notes on
Rational points on varieties.
If you're really hoping for a more interesting looking partially ordered set of topologies, you may consider more exotic examples like the cdh topology (finer than Nisnevich, incomparable
with étale), and the naïve fpqc topology, whose covers are faithfully flat quasi-compact maps (incomparable with most of the list). The latter is typically only used by people when they are
making mistakes.
3 "The latter is typically only used by people when they are making mistakes." :-)) – DamienC Oct 3 '11 at 9:55
add comment
``For example, does an fppf cover of a variety have local sections over an etale cover?'' (I realize the question is a "big picture" one so that to focus on one small aspect is to miss the
point, but I do so anyway.) No. Take a morphism $f:S\to C$ with $S$ a surface and $C$ a curve, both $S$ and $C$ smooth over an algebraically closed field $k$. Assume that the geometric
generic fiber is singular; there exist such in positive characteristic. Say $D\subset S$ is the locus of singularities; this is a curve, inseparable over $C$. Now localize: strictly henselize
up vote $C$ at a closed point $P$ and strictly henselize $S$ at a point on $D$ over $P$. We now have $S'\to C'$, an fppf cover of $C'$. If there were a section over an etale cover of $C'$ then there
5 down would be a section, say $E$. This is a curve in $S'$ having intersection multiplicity $1$ with the generic fiber, while it meets the fiber over $P$ with multiplicity at least $2$,
vote contradiction.
Or, for a simpler example, over a base ring k, consider A^1->A^1 by x->y^n for n>1. This morphism is finite free of rank n and thus fppf. Let p:S->A^1 be etale. Then \Omega^1_{S/k} is free
2 on p^*dx. If S->A^1 lifts through A^1->A^1 as q:S->A^1, then p^*dx = q^*(ny^{n-1}dy). Thus ny^{n-1} (and hence y, as n>1) pulls back to a unit on S. Since p^*x = q^*(y^n), we see that x
pulls back to a unit on S, and so p misses the subscheme (x=0) of A^1. – user2490 Sep 9 '11 at 15:17
add comment
Here is one small point in answer to this question:
The étale site and the fppf site have the same algebraic stacks.
Here I'm saying 'algebraic stack' for a stack of groupoids with a representable smooth surjection from a scheme.
If we further restrict to algebraic stacks with quasi-affine diagonal, then we have:
The étale site and the fpqc site have the same algebraic stacks of this sort.
I learned this from notes on stacks by Anatoly Preygel.
up vote 5 Some more data points from the Stacks Project:
down vote
• smooth covers can be refined by etale covers (tag 055V)
• tag 02LH could possibly also be useful, if we can identify what sort of map $\coprod_{i=1}^n T_i \to T \to S$ is:
Let $f : U \to S$ be a surjective etale morphism of affine schemes. There exists a surjective, finite locally free morphism $\pi : T \to S$ and a finite open covering $T = T_1 \cup
\ldots \cup T_n$ such that each $T_i \to S$ factors through $U \to S$. if
• Given an algebraic stack $X$ (defined as in Stacks Project), we can find a presentation by a smooth groupoid $R\rightrightarrows U$ in algebraic spaces (i.e. the source and target are
smooth, and we have an equivalence $[U/R] \to X$). Tag 04T5 tells us that $X$ is also equivalent to the stack $[U'/R']$ where $R' \rightrightarrows U'$ is a presentation with source and
target flat and locally of finite presentation (so there is a surjection which is flat and locally of finite presentation $U' \to X$).
add comment
Another partial answer.
To quote wikipedia's page on Nisnevich topology:
□ The cdh topology allows proper birational morphisms as coverings.
□ The qfh topology allows De Jong's alterations as coverings.
up vote 3 down vote □ The l′ topology allows morphisms as in the conclusion of Gabber's local uniformization theorem.
The cdh and l′ topologies are incomparable with the étale topology, and the qfh topology is finer than the étale topology.
Of course, can we compare the cdh and l' topologies with other topologies?
This is separate to the other answer, because it deals with a different aspect of my question. – David Roberts Oct 3 '11 at 5:02
add comment
Not the answer you're looking for? Browse other questions tagged ag.algebraic-geometry grothendieck-topology ct.category-theory or ask your own question.
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Citizendium - building a quality general knowledge free online encyclopedia. Click here to join and contribute—free
Many thanks March 2014 donors; special to Darren Duncan. April 2014 donations open; need minimum total $100. Let's exceed that.
Donate here. Donating gifts yourself and CZ.
Field (mathematics)
From Citizendium, the Citizens' Compendium
Fields are algebraic structures that generalize on the familiar concepts of real number arithmetic. The set of rational numbers, the set of real numbers and the set of complex numbers are all fields
under the usual addition and multiplication operations.
The term field carries other meanings in other areas of mathematics, notably in calculus and mathematical physics. This article deals exclusively with fields as used in abstract algebra.
Informal description and definition
When dealing with $\mathbb{Q}$, the set of rational numbers, we notice several things:
• The rational numbers form what mathematicians call an abelian group under addition.
• When we exclude the number 0, they form an abelian group under multiplication as well.
• For any triplet $a,b,c$, we have that $a ( b + c ) = a b + a c$ - both calculations yield the same answer.
Dealing with other sets, both finite and infinite, we often notice this behavior. Obvious examples are $\mathbb{R}$, the set of real numbers and $\mathbb{C}$, the set of complex numbers.
All these sets, and others where the three conditions listed above are fulfilled, are known as fields.
Less obvious examples are Z[2], the set {0,1} under addition and multiplication modulo 2. Other examples are Z[3] and Z[5], ... , Z[p]. In general, any set {0,1,...,p-1} under addition and
multiplication modulo p, for any prime number p, is a field.
It's important to notice that neither the field elements nor the binary operations necessarily have to be anything closely resembling numbers. It is sufficient that the actual system under study can
be conceptualized into a structure that satisfies the formal definition of a field.
Formal definition of a field
A field consists of a set F, along with a binary operation + on F such that F is a commutative group with an identity element 0; and another binary operation * on F such that F\{0} is a commutative
group with identity element 1. Distributivity of * over + holds: that is, for any $a,b,c \in F$, we have that $a * (b + c ) = a * b + a * c$.
The first binary operation is usually called addition and the second one multiplication.
Characteristic of a field
The characteristic of a field is the smallest natural number $k$ such that $0=1+1+\cdots+1$ (where there are $k$ ones on the right-hand side). If such a natural number does not exist, then the
characteristic of a field is taken to be 0. If the characteristic of a field is nonzero, it is a prime number because otherwise, the number $1+1+\cdots+1$, where the number of ones is a proper
divisor of the characteristic, would not have an inverse.
See also
Related topics
External links
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categories: nonpareil dinats
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categories: nonpareil dinats
Here is an amusing observation, doubtless of no importance. In the middle
of a talk at Boulder in, I think, 1987, Bob Pare' came up with a class of
exotic dinatural endotransformations on the homfunctor on Set. Namely,
for an endomorphism t: X --> X, let t |--> n where n depends only the
cardinality of the fixpoint set of t. Since, for f: X --> Y and g: Y -->
X, fg and gf have isomorphic fixpoint sets, this turns out to be
dinatural. The fact that Fix(fg) is isomoprhic to Fix(gf) is perfectly
general in any category that has the equalizer used to define them. In
particular, this is true in Set\op and so we can get non-Pare' dinats by
using the same formula, but making n depend instead on the cofixpoint set
of t, where that is the coequalizer of t and the identity. It is easy to
find examples of endomorphisms that have the same fixpoint set, but
different cofixpoint sets and vice versa, so tere are genuinely new. Are
there any others? I don't know.
I started thinking about this after a note from Vaughan Pratt who was
interested in Chu(Set,2) (Surprise!). He had noted that there was a full
subcategory of chusets of the form (A,0) and you could treat their
endomorphisms separately. That full subcategory is essentially Set and so
you on that subcategory you can use all the Pare' and non-Pare' dinats.
Leaving those aside, you can do dinatural endotransformations of the
internal hom functor in four ways: If (A,X) is an object, then an
internal endoarrow is a 4-tuple (f,s,a,x) where f: A --> A, s: X --> X, a
in A and x in X subject to <fa,x> = <a,sx> for all a in A and all x in X.
The nth power of such a 4-tuple is simply <f^n,s^n,a,x>. Then you can
define dinats by letting (f,s,a,x) |--> (f,s,a,x)^n where n depends on
Fix(f) and Fix(s) OR on Cofix(f) and Cofix(s) OR on Fix(f) and Cofix(s) OR
on Cofix(f) and Fix(s).
Qeustion: Are there any dinats on the internal homfunctor on vector
spaces? I almost have an argument for finite dimensional spaces, but it
depends on writing every endomorphism as a sum of rank one endomorphisms.
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from The American Heritage® Dictionary of the English Language, 4th Edition
• adj. Of, relating to, or characterized by conjecture; conjectural.
• adj. Statistics Involving or containing a random variable or variables: stochastic calculus.
• adj. Statistics Involving chance or probability: a stochastic stimulation.
from Wiktionary, Creative Commons Attribution/Share-Alike License
• adj. Random, randomly determined, relating to stochastics.
from the GNU version of the Collaborative International Dictionary of English
• adj. Conjectural; able to conjecture.
• adj. random; chance; involving probability; opposite of deterministic.
• adj. of or pertaining to a process in which a series of calculations, selections, or observations are made, each one being randomly determined as a sample from a probability distribution.
from The Century Dictionary and Cyclopedia
• Conjectural; given to or partaking of conjecture.
from WordNet 3.0 Copyright 2006 by Princeton University. All rights reserved.
• adj. being or having a random variable
Greek stokhastikos, from stokhastēs, diviner, from stokhazesthai, to guess at, from stokhos, aim, goal; see stegh- in Indo-European roots.
(American Heritage® Dictionary of the English Language, Fourth Edition)
From Ancient Greek στοχαστικός (stokhastikos), from στοχάζομαι (stokhazomai, "aim at a target, guess"), from στόχος (stokhos, "an aim, a guess"). (Wiktionary)
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• work encompassed by the "stochastic arts"--those that "diagnose and fix things that are variable, complex, and not of our own making." He includes in this category mechanics and medical
practitioners and those with other occupations which, because of the constant risk of failure, at least potentially prevent self-absorption, and instead "cultivate not creativity, but the less
glamorous virtue of attentiveness"
• Funny, this came up when I was 'randoming.'
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= Preview Document = Member Document = Pin to Pinterest
Practice reading graphs and interpreting data with this rainforest-themed set. Four pages.
Set of color posters explaining bar graphs and tally charts for multiple-choice testing format.
Answer the questions by interpreting the data presented in the bar graph. Created using abctools.
Use a picture graph to determine how many tacos were sold at Mr. Diaz's shop.
• Make your own graphs with this graph form with 12 columns going to 20 by 2s.
Patterns for eye color picture graph. Each student colors their own to add to the classroom bar graph.
• [member-created with abctools] Count each shape, fill each graph to the right height, and circle the shape that appears most often.
An enlarged version of abcteach's Pets Bar Graph & Tally Chart, for use on a board. Graph measures the number of children who have various types of pets.
Chart for students to record the weather. Includes cute graphics to use with the chart.
Use the line graph to find out how many baseball cards Brandon has collected over a five-month period.
• Use a line graph to determine the number of trees planted on Arbor Day.
Use these picture patterns to make a classroom picture graph. Each student colors a picture that represents them and adds it to the classroom graph.
Set of four color bar graphs, each with a set of four tally cards in multiple-choice testing format.
A line graph that illustrates average temperatures for spring and summer months with a set of ten questions on the same page, plus an answer sheet.
A line graph that illustrates the number of birthdays occuring in a class over a five month period of the school year, with a set of eight questions on the same page, and a separate answer sheet.
Students practice reading basic graphs with this scholastic-themed worksheet.
Students practice reading basic graphs with this sunflower-themed worksheet.
A line graph illustrating the number of students in a class who claim one of each of five flavors of pizza as their favorite, with a set of eight questions on the same page, plus an answer sheet.
• Practice number recognition (to 12), graphing and cooperation with this fun "fishing" activity.
• Use the bar graph to find out which careers these middle school students prefer. Common Core: Grade 6 Statistics and Probability: 6.SP.A.1, 6.SPA.2 Common Core: Grade 6 Statistics and
Probability: 6.SP.A.1, 6.SPA.2
• Students color the crayon boxes their favorite color, and use them for a classroom picture graph.
Interpret the bar graph in order to answer the questions. Document has questions regarding jump roping.
Use the data table to create the line graph and interpret the questions that follow. Created using abctools.
Read the bar graph to find out which summer activities are the most popular.
An enlarged version of abcteach's Lunch Bar Graph and Tally Chart file. Graph measures the number of students who choose chicken, spaghetti or sack lunch.
An enlarged version of abcteach's Jelly Bean Color Bar Graph file, for use on a board.
Students use these worksheets to graph the weather conditions during each month of the year. Graphs up to 15 days for each of 7 weather conditions.
Students label a bar graph and then use the graph to determine the numbers of Easter bonnets sold by color.
Students use these worksheets to graph the weather conditions during each month of the year. Graphs up to 15 days for each of 7 weather conditions. Some weather graphics are cute (snowman, daisy
with umbrella).
|
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Looking for a particular family of C.Y quintics
up vote 6 down vote favorite
It is possible to construct (in many ways) a family of Calabi-Yau quintics $\mathcal{X}\rightarrow \Delta$, over disk, such that the fiber over $0$ has a singularities locally given by the equation
$x_1^2+x_2^2+x_3^2+x_4^2=0$ and the others are smooth quintics. In such families there is vanishing cycle in generic fiber which is a Lagrangian isomorphic to $S^3$ and collapses to the node at
central fiber. This Lagrangian $S^3$ is obtained by the deformation of the local equation: $$x_1^2+x_2^2+x_3^2+x_4^2=0 \rightarrow x_1^2+x_2^2+x_3^2+x_4^2=\epsilon$$
Here is my questions: Consider the Fermat quintic $z_1^5+\cdots+z_5^5=0$ and its real locus which is a Lagrangian isomorphic to $S^3 /\mathbb{Z}_2$. Is there any one parameter family of C.Y 3-folds
with one fiber isomorphic to Fermat quintic, such that real locus of quintic apears as vanishing cycle for this family ?
Comment: I think if there is such family then the singular fiber has to have orbifold singularities given by local equation: ${x_1^2+x_2^2+x_3^2+x_4^2=0}/\mathbb{Z}_2$ and I think its impossible to
write a family of degree 5 equations with this type local singularities!
ag.algebraic-geometry sg.symplectic-geometry mirror-symmetry deformation-theory
aren't those $A_1$ singularities? – Vivek Shende Jan 9 '11 at 19:18
Which one do you mean? The one with Z_2 quotient or without ? There might be a mistake with naming but I think my question is clear. Is not it? :) – Mohammad F. Tehrani Jan 9 '11 at 21:36
2 In the first part of the question it seems that you just ask for a Lefschetz pencil? I guess if you take a generic pencil of hyperplane sections of $P^4$ in the 5-fold Veronese embedding it is
Lefschetz (all singularities of fibers are ODP's). Then you can choose one singular fiber and a small neighborhood of the point of the base of the pencil. – Sasha Jan 10 '11 at 6:09
This is a great question, I don't understand why no one apart from me seems to like it :)... To Sasha and Sandor -- it seems to me that you misread the question. The real question start after:
"Here is my questions..." , prior to this Mohammad just explains basic things about vanishing cycles, is not he? – Dmitri Jan 10 '11 at 9:23
@ Sasha: Please look at Dmitri's comment. – Mohammad F. Tehrani Jan 10 '11 at 15:10
show 1 more comment
1 Answer
active oldest votes
Let me say first, that I really like this question. Very unusual question about such well known things (in fact I did not know even that the real quintic $\sum_i x_i^5=0$ is $\mathbb RP^3$).
This is not an answer, but more like my interpretation of this question (hopefully correct one). The first comment is about vanishing cycles. Usually, when we speak about vanishing cycles in
symplectic geometry, we speak about lagrangan spheres in a fiber of a Lefshetz fibraton constructed using parallel transport given by the symplectic connection on the fibers. But Lefshetz
pencil is just the most simple object of algebraic geometry. We can consider instead other one-parameter families of algebraic varieties, that can have singularities more complicated than
double points $A_1$. In this case again we can ask what will be the shape of the vanishing cycle constructed via symplectic parallel transport? How does it look like? The point that I want to
make is that in a large case of situations, this vanishing cycle will not look at all like a manifold.
Namely, we will consider the example coming from a function on $\mathbb C^n$ with isolated singularity. I.e., we have an analytic function $F:\mathbb C^n\to \mathbb C$ with isolated
singularity at $0$ and consider its level sets: hyper-surfaces $F_t:=F^{-1}(t)$ intersected with $B_\varepsilon(0)$ (the ball of radius $\varepsilon$). Then we know that the homotopy type of
$F_t\cap B_{\varepsilon}$ is a bucket of $k$ spheres, where $k$ is the Milnor number of the singularity. Now, I think (and here I can not provide the proof), that the vanishing cycle is a
up vote deformation retract of $F_t\cap B_{\varepsilon}$. Hence it can not be diffeomorphic to a manifold unless $k=1$. In this case it has to be homotopy equivalent to a sphere, i.e., homeomoerphic
5 down to it (by Poincare).
In the case when the singularity of $F$ is non-degenerate (double point), the vanishing cycle is of course diffeomorphic to $S^n$, but maybe if we consider Brieskorn singularities we will be
able to get vanishing cycles that are exotic spheres as well?
The above situation concerns the case when the total space of the fibration is smooth. In this case we does not seem to be able to get $\mathbb RP^{n-1}$ as a vanishing cycle. But if we allow
singularities in the total space of the fibration, we can get it. Indeed if we consider the function $\sum_i x_i^2=0$ on the total space $\mathbb C^n/\mathbb Z_2$, where $\mathbb Z_2$ is
acting by $z\to -z$, the vanishing cycle will be diffomorphic to $\mathbb RP^{n-1}$. In sum, it seems to me that the following question is interesting: can we describe the class of manifolds
what appear as vanishing cycles of algebraic one-parameter families?
As for the question itself, I don't know what is the answer...
All you said is correct, and my question is some how exactly what you asked at the end: Can a family of Quintics develop a singulary given by the local equation $x_1^2+\cdots+x_4^2$ on $\
mathbb{C}/\mathbb{Z}_2$ ? – Mohammad F. Tehrani Jan 10 '11 at 15:16
add comment
Not the answer you're looking for? Browse other questions tagged ag.algebraic-geometry sg.symplectic-geometry mirror-symmetry deformation-theory or ask your own question.
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Convert cm to kg - Conversion of Measurement Units
›› Convert centimetre to kilogram
Error: We couldn't find a conversion between cm and kg [incompatible types]
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›› Definition: Centimeter
A centimetre (American spelling centimeter, symbol cm) is a unit of length that is equal to one hundreth of a metre, the current SI base unit of length. A centimetre is part of a metric system. It is
the base unit in the centimetre-gram-second system of units. A corresponding unit of area is the square centimetre. A corresponding unit of volume is the cubic centimetre.
The centimetre is a now a non-standard factor, in that factors of 10^3 are often preferred. However, it is practical unit of length for many everyday measurements. A centimetre is approximately the
width of the fingernail of an adult person.
›› Definition: Kilogram
The kilogram or kilogramme, (symbol: kg) is the SI base unit of mass. A gram is defined as one thousandth of a kilogram. Conversion of units describes equivalent units of mass in other systems.
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ConvertUnits.com provides an online conversion calculator for all types of measurement units. You can find metric conversion tables for SI units, as well as English units, currency, and other data.
Type in unit symbols, abbreviations, or full names for units of length, area, mass, pressure, and other types. Examples include mm, inch, 100 kg, US fluid ounce, 6'3", 10 stone 4, cubic cm, metres
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A strange example of typing in Ocaml
up vote 5 down vote favorite
it is quite strange that this ocaml snippet is well typed by the toplevel. Look at the structure, if g is of type int->int as shown in the toplevel, the h x = g x part of the structure would not be
able to get type-unified. So can any one clarify a bit?
module Mymodule : sig
val h:int ->int
val g: string-> string
end = struct
let g x = x
let h x = g x
This is the topelevel's response:
module Mymodule : sig val h : int -> int val g : string -> string end
ocaml typing
add comment
2 Answers
active oldest votes
The important thing to understand here is that OCaml performs type inference in a compositional manner, i.e., it will first infer type of struct ... end and only then it will match
the inferred types against sig ... end to verify that the structure really does implement the signature.
For example, if you write
module Monkey : sig val f : int -> int end =
let f x = x
then OCaml will be happy, as it will see that f has a polymorphic type 'a -> 'a which can be specialized to the required type int -> int. Because the sig ... end makes Monkey opaque,
i.e., the signature hides the implementation, it will tell you that f has type int -> int, even though the actual implementation has a polymorphic type.
In your particular case OCaml first infers that g has type 'a -> 'a, and then that the type of h is 'a -> 'a as well. So it concludes that the structure has the type
sig val g : 'a -> 'a val h : 'a -> 'a end
Next, the signature is matched against the given one. Because a function of type 'a -> 'a can be specialized to int -> int as well as string -> string OCaml concludes that all is
well. Of course, the whole point of using sig ... end is to make the structure opaque (the implementation is hidden), which is why the toplevel does not expose the polymorphic type
of g and h.
Here is another example which shows how OCaml works:
module Cow =
let f x = x
up vote 7 down let g x = f [x]
vote accepted let a = f "hi"
module Bull : sig
val f : int -> int
val g : 'b * 'c -> ('b * 'c) list
val a : string
end = Cow
The response is
module Cow :
val f : 'a -> 'a
val g : 'a -> 'a list
val a : string
module Bull :
val f : int -> int
val g : 'a * 'b -> ('a * 'b) list
val a : string end
add comment
I'd say that the string -> string typing isn't applied to g until it's exported from the module. Inside the module (since you don't give it a type) it has the type 'a -> 'a.
up vote 8 down (Disclaimer: I'm not a module expert, trying to learn though.)
Your answer is perfectly correct. – gasche Jun 19 '12 at 20:57
Excellent, thanks! :-) – Jeffrey Scofield Jun 19 '12 at 21:10
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= Preview Document = Member Document = Pin to Pinterest
Practice reading graphs and interpreting data with this rainforest-themed set. Four pages.
Set of color posters explaining bar graphs and tally charts for multiple-choice testing format.
Answer the questions by interpreting the data presented in the bar graph. Created using abctools.
Use a picture graph to determine how many tacos were sold at Mr. Diaz's shop.
• Make your own graphs with this graph form with 12 columns going to 20 by 2s.
Patterns for eye color picture graph. Each student colors their own to add to the classroom bar graph.
• [member-created with abctools] Count each shape, fill each graph to the right height, and circle the shape that appears most often.
An enlarged version of abcteach's Pets Bar Graph & Tally Chart, for use on a board. Graph measures the number of children who have various types of pets.
Chart for students to record the weather. Includes cute graphics to use with the chart.
Use the line graph to find out how many baseball cards Brandon has collected over a five-month period.
• Use a line graph to determine the number of trees planted on Arbor Day.
Use these picture patterns to make a classroom picture graph. Each student colors a picture that represents them and adds it to the classroom graph.
Set of four color bar graphs, each with a set of four tally cards in multiple-choice testing format.
A line graph that illustrates average temperatures for spring and summer months with a set of ten questions on the same page, plus an answer sheet.
A line graph that illustrates the number of birthdays occuring in a class over a five month period of the school year, with a set of eight questions on the same page, and a separate answer sheet.
Students practice reading basic graphs with this scholastic-themed worksheet.
Students practice reading basic graphs with this sunflower-themed worksheet.
A line graph illustrating the number of students in a class who claim one of each of five flavors of pizza as their favorite, with a set of eight questions on the same page, plus an answer sheet.
• Practice number recognition (to 12), graphing and cooperation with this fun "fishing" activity.
• Use the bar graph to find out which careers these middle school students prefer. Common Core: Grade 6 Statistics and Probability: 6.SP.A.1, 6.SPA.2 Common Core: Grade 6 Statistics and
Probability: 6.SP.A.1, 6.SPA.2
• Students color the crayon boxes their favorite color, and use them for a classroom picture graph.
Interpret the bar graph in order to answer the questions. Document has questions regarding jump roping.
Use the data table to create the line graph and interpret the questions that follow. Created using abctools.
Read the bar graph to find out which summer activities are the most popular.
An enlarged version of abcteach's Lunch Bar Graph and Tally Chart file. Graph measures the number of students who choose chicken, spaghetti or sack lunch.
An enlarged version of abcteach's Jelly Bean Color Bar Graph file, for use on a board.
Students use these worksheets to graph the weather conditions during each month of the year. Graphs up to 15 days for each of 7 weather conditions.
Students label a bar graph and then use the graph to determine the numbers of Easter bonnets sold by color.
Students use these worksheets to graph the weather conditions during each month of the year. Graphs up to 15 days for each of 7 weather conditions. Some weather graphics are cute (snowman, daisy
with umbrella).
|
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|
FOM: basic concepts; structuralism
Stephen G Simpson simpson at math.psu.edu
Mon Oct 20 23:47:11 EDT 1997
Dear FOM people,
I'm delighted that our discussion of foundations of mathematics
(f.o.m.) is proceeding so vigorously.
Recent comments by Hilary Putnam, Vaughan Pratt, and John Baldwin
touch on some issues in f.o.m. which I regard as crucial.
Hilary Putnam writes:
> What makes a concept of mathematics "basic"? I am inclined to think
> that very few concepts are *essential* to mathematics: perhaps only
> the (better: a certain) concept of *proof*.
Hilary does not provide any context for his remark about proof.
Therefore, I find the remark very difficult to understand. In the
context provided by my definition of f.o.m. at
my reaction to Hilary's remark would be: How can you talk about
mathematical proof in the absence of basic mathematical concepts? In
the final analysis, doesn't *proof* entail *reduction to basic
concepts*? Perhaps Hilary is distinguishing between "basic concepts"
and "essential concepts"? If so, what's the distinction?
On the positive side, although I find Hilary's remark maddeningly
cryptic, nevertheless I will be happy if this evolves into a
discussion of certain issues in f.o.m. which I regard as important.
Among those issues are: what are the most basic mathematical concepts,
why are they basic, etc.
Vaughan Pratt replied to Hilary's remark by saying that sets are
probably essential to mathematics, but membership is not; at least I
think this was the gist of it. If I'm not mistaken, Vaughan's remarks
are informed by a category-theoretic perspective, wherein "set" is
interpreted as "object in a topos", but "elements of sets" are murky.
I know a little about topos theory, but I have never really understood
the way category theorists seem to view the rest of mathematics, not
to mention the rest of human knowledge. I know that some topos
theorists such as Peter Freyd have sometimes asserted that category
theory or topos theory is or can be turned into a good foundational
theory for all of mathematics or at least a good portion of it. I
have never understood how this can be. It seems wildly exaggerated.
Could somebody please try to explain it clearly? (I.e., no
For example, we know how standard set-theoretic foundations explains
real analysis in set-theoretic terms. This explanation is successful
in a certain sense. It may not be very illuminating or satisfying,
but right now that's beside the point. The question right now is, how
can category theory be used to explain real analysis? And if it
can't, then what does category theory have to do with f.o.m.?
John Baldwin writes:
> Vaughn Pratt's note reminded me of a fancy word I learned at
> the meeting for Bill Tait's retirement that seems relevant to
> this discussion. The word was structuralism and I while I'll
> yield to the philosophers for the proper technical usage my
> interpretation was this. One should understand mathematical
> structures in terms of the relationships amongst the objects
> of the structure with our regard to their internal properties.
> Thus we study the reals as a complete ordered field, not
> Cauchy sequences of equivalence classes of integers.
> So I suppose the question to Steve is whether any analysis
> beginning at this level could be viewed as foundational?
John has asked me a question, so I'll try to answer it.
Yes, this kind of analysis might be viewed as foundational, because
the concept of number is probably basic in some sense. But to make
this analysis succeed qua f.o.m., you have to show that this concept
of real number (i.e. a real number as an element of a complete ordered
field) is truly basic and does not call for any further explanation in
terms of more basic mathematical concepts. And that is questionable,
because this particular explanation of real number appears to depend
on several other concepts: field, order, complete. In particular,
it's natural to ask how completeness of the reals would be defined in
the absence of the concept of an arbitrary bounded set of reals. I
think it was considerations such as this that led to set-theoretic
foundations, in the late 19th and early 20th century (Dedekind et al).
This probably doesn't answer what John intended, but I think it does
answer what he asked.
John also didn't ask me what I think of structuralism, but I'll
comment anyway! The main point that I would like to make is that
structuralism as a general intellectual trend is ultimately
anti-foundational and anti-scientific. The structuralist creed is
that each subject exists in its own internal world or framework and is
understandable only from that narrow viewpoint. Thus structuralism
tends to isolate subjects and cut off all of their connections with
the rest of human knowledge. It's the opposite of the foundational
approach, which regards the unity of human knowledge as paramount.
This critique of structuralism applies to John's example, as follows:
If you understand the real numbers "structurally" as a complete
ordered field and nothing else, then how can you connect real numbers
to the rest of human knowledge? It seems you are left with no way to
make such connections.
If anyone else would care to comment on structuralism vis a vis
f.o.m., I'd be most interested.
John Baldwin continues:
> Another point that I made in my essay and think bears repeating is
> that it seems fruitful to look not only a single global FOM
> but a foundationS of mathematics where different techniques and
> viewpoints are appropriate for various areas of mathematics.
I'm not sure, but I think John is harking back to his earlier point
about "local foundations". The point was stated obscurely and
elliptically. Was this deliberate, in order to blur f.o.m.? In any
case, my guess is that John is referring to what I would call
"foundations of particular branches of mathematics".
E.g. "foundations of topology", "foundations of algebraic geometry"
(Zariski etc), "foundations of analysis", "foundations of model
theory", etc. To me it seems obvious that these specialized
"foundationS" (as John might call them) are interesting subjects, but
it's also obvious that their character is very different from that of
"foundations of mathematics as a whole", i.e. f.o.m. And it is also
clear (to me at least) that these specialized foundational subjects
are interesting only to pure mathematicians, whereas f.o.m. is of
general intellectual interest to the educated public. This distincion
is elaborated in my essay referred to above.
The pure mathematicians and their little brothers may want to
obliterate genuine f.o.m., and may therefore attempt to disagree with
the point I have just made, i.e. the distinction between f.o.m. and
Baldwin-style "local foundations". But they will have a very hard
time explaining how results in sheaf theory, general topology, model
theory, etc should be or could be of interest to the general
scientific/philosophical public. By contrast, a number of results in
genuine f.o.m. (e.g. G"odel's completeness and incompleteness
theorems) do have this quality of general scientific interest.
-- Steve
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CAPM - Capital Asset Pricing Model
From Bogleheads
In finance, the capital asset pricing model (CAPM) ^[1] is used to determine a theoretically appropriate required rate of return of an asset, if that asset is to be added to an already
well-diversified portfolio, given that asset's non-diversifiable risk. The model takes into account the asset's sensitivity to non-diversifiable risk (also known as systemic risk or market risk),
often represented by the symbol beta (β), as well as the expected return of the market and the expected return of a theoretical risk-free asset.
Risk and return
In general, investors expect that it is necessary to take a high risk to receive a high return. Conversely, investors are willing to sacrifice return (accept less than their current return) to reduce
An asset exhibits two types of risk: systematic and unsystematic
Unsystematic risk only affects an individual security or portfolio and does not affect the market as a whole. Unsystematic risk is treated as "random noise" in a portfolio. Consider, for example,
volatility of returns. Through the use of diversification (adding many securities), the random noise component will eventually have a mean of zero (the definition of random noise). Standard deviation
of returns is also reduced as the number of securities in the portfolio increases.
If there are enough assets in a portfolio such that diversification cannot affect the performance, the volatility of the portfolio's returns matches that of the overall market's returns. Risk that
can not be mitigated through diversification is known as systematic risk. The portion of its volatility of returns which is considered systematic is measured by the degree to which its returns vary
relative to those of the overall market.
The parameter beta (β) is used to describe the relationship between the returns of a security or portfolio (an asset) and the returns of the market as a whole; it combines the correlation of the
asset's returns and the market's returns with the relative volatility of those returns:
β = cov(r[A], r[M]) / σ[M]^2 = ρ(r[A], r[M]) × (σ[A] / σ[M])
□ r[A] is the set of returns of the asset
□ r[M] is the set of returns of the market
□ σ[M]^2 is the variance of the returns of the market
□ cov(r[A], r[M]) is the covariance between the returns of the asset and the returns of the market
□ ρ(r[A], r[M]) is the correlation between the returns of the asset and the returns of the market
□ σ[A] is the standard deviation of the returns of the asset
□ σ[M] is the standard deviation of the returns of the market
Market risk (β) is calculated using historical returns for both the asset and the market, with the market portfolio being represented by a broad index such as the S&P 500 or the Russell 2000.
The Capital Asset Pricing Model (CAPM) attempts to quantify the relationship between the beta of an asset and its corresponding expected return. Several assumptions are made:
1. Investors care only about expected returns and volatility of returns. Therefore, expected returns are maximized for any given level of expected volatility of returns.
2. All investors have homogeneous beliefs about the risk/reward trade-offs in the market.
3. There is only one risk factor common to a broad-based market portfolio, called systematic market risk. Investors are assumed to hold diversified portfolios. As a result, the CAPM model states
that if an asset's beta is known, the corresponding expected return can be predicted.
Model description
There are three areas of interest:
1. β = 0: An asset that has no volatility of returns (no risk) does not have returns that vary with the market and therefore has a beta of zero and an expected return equal to the risk-free rate.
2. β = 1: An asset that moves with a volatility of returns exactly equal to the market's has a beta of one. In other words, the returns are perfectly positively correlated. By definition, its
expected return is equal to the market's expected return:
E(r[A]) = E(r[M])
3. β > 1: An asset that experiences greater swings in periodic returns than the market, which, by definition, has a beta greater than one. This asset is expected to earn returns superior to those of
the market as compensation for this extra risk.
Making a lot of generalizations leads to the CAPM model:
E(r[A]) = r[f] + β[A](E(r[M]) - r[f])
□ E(r[A]) is the expected return of the asset
□ r[f] is the risk-free rate
□ E(r[M]) is the expected return of the market portfolio
(Note: the quantity E(r[M]) - r[f], which is the expected excess return of the market above the risk-free rate, is called the market risk premium, often abbreviated MRP.)
The general idea of CAPM is that investors should be compensated in two ways: time value of money and risk.
The time value of money is represented by the risk-free (r[f]) rate in the formula and compensates the investors for placing money in any investment over a period of time.
The other part of the formula represents risk and calculates the amount of compensation the investor needs for taking on additional risk. This is done by taking an estimate of risk, (β[A]), and
multiplying by the MRP, (E(r[M]) - r[f]).
An asset is expected to earn the risk-free rate plus a reward for bearing risk as measured by that asset’s beta. The chart below demonstrates this predicted relationship between beta and expected
return – this line is called the Security Market Line.
For example, a stock with a beta of 1.5 would be expected to have an excess return of 15% in a time period where the overall market beat the risk-free asset by 10%.
The CAPM model is used for pricing an individual security or a portfolio. For individual securities, the security market line (SML) and its relation to expected return and systematic risk (beta)
shows how the market must price individual securities in relation to their security risk class.
As the CAPM predicts expected returns of assets or portfolios relative to risk and market return, the CAPM can also be used to evaluate the performance of active fund managers. The difference is
“excess return”, which is often referred to as alpha (α). If α is greater than zero, the portfolio lies above the Security Market Line.
Shortcomings of the CAPM model
Several shortcomings of the CAPM model exist. Incorrectly predicting results compared to realized returns and the affect of other risk factors have put this model under criticism. The assumption of a
single risk factor limits the usefulness of this model.
Eugene F. Fama and Kenneth R. French found that on average, a portfolio’s beta explains about 70% of its actual returns. For example, if a portfolio were up 10%, about 7% of the return can be
explained by the advance of all stocks and the other 3% is the result of other factors not related to beta. This observation led to the development of the Fama and French three-factor model.
Other uses of CAPM
CAPM is also used to cost equities in applications other than investing. For example, in the Weighted Average Cost of Capital (the rate that a company is expected to pay to finance its assets), CAPM
is used to calculate the cost of equity:
Weighted Average Cost of Capital (for a firm) = (% of the firm in debt, at market value) × K[d][cost of debt] × (1 - marginal tax rate for the firm) +
(% of the firm in equity, at market value) × K[e][cost of equity]
(Note: this formula assumes the firm has only debt and common equity; i.e., no preferred equity.)
K[e] is estimated using the CAPM:
K[e] = r[f] + β[e](E(r[M]) - r[f])
□ K[e] is the cost of equity
□ r[f] is the risk-free rate of return
□ E(r[M]) is the expected return of the market
□ E(r[M]) - r[f] is the market risk premium and is held to be, normally, between 4% - 5% (real) per year (the last 110 years of UK data, similar to the US)
(Another method of calculating K[e] uses the Dividend Discount Model.)
See also
1. ↑ Press Release, 16 October, 1990. William F. Sharpe won the The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 1990 for his pioneering work on the CAPM.
External links
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solving a system of transcendental equations
April 16th 2009, 04:32 AM #1
Apr 2009
solving a system of transcendental equations
I want to solve the system of three transcendental equations for A, B & C.
0.97*cos(B)*cos(C)-0.18*cos(B)*sin(C)+ 0.01*sin(B)=0.6304
0.97*(sin(A)*sin(C)+cos(A)*sin(B)*cos(C)-0.18*-sin(A)*cos(C)+cos(A)*sin(B)*sin(A))-0.01*(cos(A)*cos(B))= -0.2981
Kindly suggest any analytical or numerical way of solving it. I have already tried the symbolic toolbox in matlab and it gives incomprehensible solutions.
I also tried eliminating A from equations ----2 and ----3 and solving the resultant two equations. This approach results in 0=0 kind of problem.
Many thanks for your inputs,
Yours Truely
I want to solve the system of three transcendental equations for A, B & C.
0.97*cos(B)*cos(C)-0.18*cos(B)*sin(C)+ 0.01*sin(B)=0.6304
0.97*(sin(A)*sin(C)+cos(A)*sin(B)*cos(C)-0.18*-sin(A)*cos(C)+cos(A)*sin(B)*sin(A))-0.01*(cos(A)*cos(B))= -0.2981
Kindly suggest any analytical or numerical way of solving it. I have already tried the symbolic toolbox in matlab and it gives incomprehensible solutions.
I also tried eliminating A from equations ----2 and ----3 and solving the resultant two equations. This approach results in 0=0 kind of problem.
Many thanks for your inputs,
Yours Truely
This is not the most elegant method of doing this, but is certainly the fastest.
Rewrite your equations in the form:
Form the objective function:
$<br /> O(A,B,C) =f_1(A,B,C)^2+f_2(A,B,C)^2+f_3(A,B,C)^2<br />$
Then use the Excel solver to minimise $O$. One result of this are:
$A\approx3.46,\ B\approx -3.07,\ C\approx 2.08$
However I doubt that if this is an approximation to an exact solution that the exact solution is unique. (to the machine prescission this is a solution and to the same prescission it is not
An approximate solution with all variables in the range $\pm \pi$ is:
$A\approx 0.092,\ B \approx -0.370,\ C\approx -0.994$
Last edited by CaptainBlack; April 16th 2009 at 11:36 PM.
April 16th 2009, 10:44 PM #2
Grand Panjandrum
Nov 2005
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Mathematics of the 19th-century - mathematical logic, algebra, number-theory, probability-theory - Kolmogorov,AN, Yushkevich,AP
Marsden, B. (1994) Mathematics of the 19th-century - mathematical logic, algebra, number-theory, probability-theory - Kolmogorov,AN, Yushkevich,AP. British Journal for the History of Science, 27
(93). pp. 236-237. ISSN 0007-0874. (The full text of this publication is not available from this repository)
• Depositors only (login required):
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Algebra of automata versus Process algebra
Algebra of automata versus Process algebra
The work I have been pursuing for some years with Nicoletta Sabadini and collaborators (Piergiulio Katis, Luisa de Francesco Albasini) has been the study of distributed and concurrent systems using
algebras of automata
, with the main operations being sequential and communicating-parallel composition.
This is in contrast with the mainstream of concurrency which studies
process algebras
Why our contrary attitude? I would like to explain the point of view.
In a nutshell our view is that
semantics should come before syntax
. Process algebra takes the opposite point of view, syntax before sematics.
Let's take an analogous development in mathematics, the study of numbers.
Numbers come first. Then, after a long time, operations on numbers. Then a language for talking about numbers and their operations - polynomials. Finally, equations between polynomials.
We believe the same sequence should occur in the theory of systems. Discrete systems have states and transitions, that is, are graphs. They have interfaces and sequential and (communicating) parallel
operations (including feedbacks). The languages for describing systems are the free algebras. Finally equations in the algebra (recursion) may be used to specify systems.
Process algebras take the opposite point of view. Beginning with a vague intuition about systems, processes are defined to be solutions of equations of a free algebra. The danger of this sudden jump
to the language before a careful mathematical analysis of systems and their operations is that the wrong algebra with the wrong operations may be taken. Milner's CCS does not have a sequential
composite of systems (only that a process may be preceded by a transition). Union is used instead of disjoint union in the other sequential operation. The communicating parallel operation is based on
a vague broadcast idea.
Of course, in the end of a development as proposed by us, free algebras and recursion arise, but only when the correct operations have been identified. At this stage also
other semantics
may be identified (in the example of numbers, real numbers, complex numbers, rings of functions, ...). If the algebra of systems has been developed correctly, at least a part should have continuous
systems as models, thus allowing the confrontation between discrete and continuous.
Some years ago we produced
a process algebra
based on our idea of parallel. Recently Pawel Sobocinski is developing a
process algebra
based on our idea of communicating parallel (but not our view of sequential).
Labels: category theory, computing, Optimistic
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Cryptology ePrint Archive: Report 2007/169
On the Security of Protocols with Logarithmic Communication ComplexityMichael Backes and Dominique UnruhAbstract: We investigate the security of protocols with logarithmic communication complexity.
We show that for the security definitions with environment, i.e., Reactive Simulatability and Universal Composability, computational security of logarithmic protocols implies statistical security.
The same holds for advantage-based security definitions as commonly used for individual primitives. While this matches the folklore that logarithmic protocols cannot be computationally secure unless
they are already statistically secure, we show that under realistic complexity assumptions, this folklore does surprisingly not hold for the stand-alone model without auxiliary input, i.e., there are
logarithmic protocols that are statistically insecure but computationally secure in this model. The proof is conducted by showing how to transform an instance of an NP-complete problem into a
protocol with two properties: There exists an adversary such that the protocol is statistically insecure in the stand-alone model, and given such an adversary we can find a witness for the problem
instance, hence yielding a computationally secure protocol assuming the hardness of finding a witness. The proof relies on a novel technique that establishes a link between cryptographic definitions
and foundations of computational geometry, which we consider of independent interest. Category / Keywords: foundations / complexity theoryDate: received 7 May 2007Contact author: unruh at cs uni-sb
deAvailable format(s): PDF | BibTeX Citation Version: 20070507:211846 (All versions of this report) Discussion forum: Show discussion | Start new discussion[ Cryptology ePrint archive ]
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all 11 comments
[–]picado1 point2 points3 points ago
sorry, this has been archived and can no longer be voted on
It's not solvable algebraically (what they teach in algebra). Most equations aren't. You can't solve cos(x)=x or x^x = 2 either.
And those aren't exceptions but are the norm; the ones you are used to are the exceptions. Pretty much any equation where you just pick a random expression and set it equal to something can't be
You can solve it numerically, it's computable in the sense that you can write a program to eventually come up with as many digits as you like.
You can express it in terms of product log and its analytic continuations. But that's kind of a dead end.
So it comes down to where you're trying to go with this.
[–]agentflare[S] 0 points1 point2 points ago
sorry, this has been archived and can no longer be voted on
[–]picado0 points1 point2 points ago
sorry, this has been archived and can no longer be voted on
[–]agentflare[S] 0 points1 point2 points ago
sorry, this has been archived and can no longer be voted on
[–]picado0 points1 point2 points ago
sorry, this has been archived and can no longer be voted on
Newton's method and power series are each a bit too much to type into a text box. I wish I could, but they're long chapters in the calculus textbooks that are written by people smarter than I am, no
joke here.
Why you can't do it without something like that, I know the result but I couldn't type it in if I had all the space in the world. It's differential Galois theory which is way above my head.
[–]Thomas_Henry_Rowaway0 points1 point2 points ago
sorry, this has been archived and can no longer be voted on
What level of maths are you at? It is possible that someone gave you this as a joke. I don't think it has a solution in terms of elementary functions (things like sin(x), e^x , and log(x)). I threw
it into Wolfram Alpha which confirms this (to the extent that it can't find an elementary solution) although obviously this doesn't prove anything.
I will be happy to explain anything you can't get on that page in the morning).
Edit: Actually I don't think I was answering what you were answering.
There are no values of A for which the equation you showed is valid for every single x (The unhelpful link I gave above gives equations which give you the x value which each A (in a certain range)
solves the equation for)
[–]agentflare[S] 0 points1 point2 points ago
sorry, this has been archived and can no longer be voted on
[–]marpocky0 points1 point2 points ago
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Clearly 0 < A < 1 is part of the solution. After doing some numerical investigation, it seemed that the rest was approximately 1 < A <= 1.45. In fact, I think it's 1 < A <= e^1/e [~1.444667861], but
I can't quite explain why.
Note that log_(e^1/e)(e) = e, so A = e^1/e itself does work, and that for A < [some value which I think is e^1/e] there are two solutions.
[–]wiggyword0 points1 point2 points ago
sorry, this has been archived and can no longer be voted on
By using wolfram alpha to plot ((A^x)/x)^100 for different values of A, I was able to see when (A^x)/x took on a minimum value of 1. If you plug the following in:
plot y = ((1.444667861)^x/x)^100, y = 1 from 2.718 to 2.719
you can see that that value is pretty darn close. I have no idea how you're "supposed" to come up with an answer, but if 10 significant figures is good enough...
[–][deleted] ago
sorry, this has been archived and can no longer be voted on
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Intermediate Algebra with Applications and Visualization, Fourth Edition
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Generalization of traces
up vote 1 down vote favorite
Hello all,
I already asked this question here, I hope it is ok to repeat it.
A trace can be defined for endomorphisms of dualizable objects in a closed symmetric monoidal category. More concretely, in the category of $R$-modules for any associative ring $R$, a trace is
defined for endomorphisms of finitely generated projective $R$-modules.
The question can be stated in the more general setting from the beginning, but for simplicity: Is there a useful notion of a trace for (all!) endomorphisms of a more general class of modules than
f.g.p. modules?
I was thinking about something like: If $M$ is an $R$-module and $0\to P_n\to\dots\to P_1\to M\to0$ is a projective resolution of $M$ where every $P_k$ is not only projective but finitely generated,
then $tr(f)$ can be defined by the formula $\sum(-1)^ktr(f_k)=0$, where the $f_k$ are an extension of $f$ and $f_0=f$. If I am not mistaken, this definition is independent of the resolution.
Are there references for such a trace, or are their reasons that this does not make real sense? In particular at the moment I am unable to show/disprove that this trace is additive on short exact
sequences, which certainly it should be.
Thanks, D.
homological-algebra ra.rings-and-algebras
Totally unrelated to your question, but you've got one too many http's in your linked url. It should read: math.stackexchange.com/questions/233602/generalized-traces – Yuichiro Fujiwara Nov 10 '12
at 20:04
The category of $R$-modules for a noncommutative ring $R$ isn't automatically equipped with a monoidal structure, so let's pretend that you meant "commutative." Then, as you say, a trace can be
defined for endomorphisms of dualizable objects in a closed symmetric monoidal category. $R$-modules are such a category. So are chain complexes of $R$-modules... On the other hand, it seems to me
that there is no reasonable notion of trace of an arbitrary endomorphism on an infinite-dimensional vector space, so you should probably be more specific about your intended application. – Qiaochu
Yuan Nov 10 '12 at 21:00
Of course you are right about the monoidal structure, nevertheless there is a trace for endomorphisms of $R$-modules with $R$ non-commutative (living in the abelianization of $R$). I was intending
to get a trace for reasonably behaved finitely generated modules, so no infinite dimensions in sight. As I have sketched, modules with a finitely generated projective resolution seem promising
candidates. As for applications, I would like to be able to pass to the homology of a fgp chain complex and obtain a trace in homology. This is usually not projective, so I need some weaker
condition. – DaniW Nov 10 '12 at 21:48
I think you want something like dualizable objects in the derived category for that. I don't know the details though. – Qiaochu Yuan Nov 10 '12 at 22:57
No idea whether it is related to your question, but here is a reference: H. Rohrl, A categorical setting for determinants and traces, Nagoya Math. J. 34 (1969), 35-76 projecteuclid.org/euclid.nmj/
1118797662 – Pasha Zusmanovich Nov 14 '12 at 14:50
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IntroductionResults and DiscussionCosmic Self-SimilarityDocumented ExamplesNucleon PackingGeneral Covalence in MoleculesFibonacci PhyllotaxisThe Solar SystemAtomic StructureMolecular StructureThe Structure HypothesisMolecular ShapeAlkanesPolypeptidesSilica NanostructuresDiscussionSpace-time Geometry4D CrystallographyThe Quantum PotentialThe Quantum MoleculeSpace-Time TopologyConclusionsKolbe and van’t HoffReferencesFigures
The widespread perception that all structures in the cosmos share a common design principle was given some theoretical content by Oldershaw’s statement of cosmic self-similarity [7]. The idea of
self-similarity is closely related to fractal geometry and the golden ratio as expressed in the properties of the golden logarithmic spiral. Oldershaw distinguishes between three hierarchical
structures at the atomic or subatomic, stellar or substellar and galactic scales. The basic premise is that the fractal nature of the universe leads to self-similarity, or invariance with respect to
scale transformation, in which small parts of a structure have geometrical properties that resemble the whole structure or large parts thereof. The most familiar example of self-similarity is the
Mandelbrot set and the logarithmic spiral is the only smooth curve that is self-similar at all scales.
Self-similarity between objects on different scales implies the existence of a mathematical operation which is equally effective on different cosmic scales and which operates on symmetry elements in
any number of dimensions from one to four. A quaternion is such an object, whose operation is best understood in terms of rotation in the complex plane, described by the complex number ( a , b ) = a
+ i b ≡ r ( cos θ + i sin θ ) = r e i θwhich is isomorphic with the matrix ( a − b b a ), e.g., ( cos θ − sin θ sin θ cos θ ), under addition and multiplication. For example, the complex number i is
represented by the matrix ( 0 − 1 1 0 ) that corresponds to a counter-clockwise rotation of π/2 about the origin.
The locus of a continuously rotating point is a circle. By also allowing continuous dilatation of the radius, the locus becomes an equiangular, or logarithmic spiral, best known in the classical form
r = A e θ cot φwhere A and φ are constants. It is common practice to set A = 1 to obtain a unit spiral. For φ = 72.83° the spiral r = Ae^θτ/2 is virtually identical to a golden spiral, which develops
on removing a square (gnomon) from a golden rectangle and continuing the process indefinitely on the newly created golden rectangles of diminishing size, as shown in Figure 1. The inscribed golden
spiral converges to the intersection of diagonals.
More generally, in terms of the complex number a + ib: r = A e ( a / b ) θ , b ≠ 0It is instructive to note that rotation in the complex plane has no counterpart in three dimensions, as
first discovered by William Hamilton. The direction of a rotation axis in three dimensions is perpendicular to the complex plane and provides no extra information about the rotation. An equivalent
conclusion is that three-dimensional complex numbers are undefined. The simplest hypercomplex number, defined in four dimensions, is known as a quaternion: q = a + i b + j c + k dwhere i, j, k are
generalizations of − 1, such that i 2 = j 2 = k 2 i j = k , j k = 1 , k i = j ; j i = − k , k j = − 1 i k = − jand q^2 = a^2 + b^2 + c^2 + d^2. A four-dimensional rotation is now
defined by the quaternion Q e θ ( i α + j β + k γ ) = Q { cos θ + sin θ ( i α + j β + k γ ) }Whereas a complex number is defined by a vector and a phase, a quaternion is specified by a tensor and a
The demonstration that both Lorentz transformation and quantum spin are the direct result of quaternion rotation implies that all relativistic and quantum structures must have the same symmetry. This
is the basis of cosmic self-similarity. The observation that the golden mean features in many self-similarities is interpreted to show that τ represents a fundamental characteristic of space-time
curvature. The existence of antimatter and the implied CPT symmetry of space-time favours closed metric-free projective geometry with involution; the only topology that automatically generates the
gauge invariance that links quantum mechanics to the electromagnetic field [8]. This topology is consistent with constant space-time curvature, locally distorted by large gravitating masses. It seems
reasonable to assume that the logarithmic spiral (1) follows the general curvature in two-dimensional projection, characteristic of stable structures and growth patterns in tangent Euclidean space.
In four-dimensional space-time the curvature is more appropriately described by a formula such as ρ ( x μ ) = A e ( a / b 2 + c 2 + d 2 ) θwhich describes spherical rotation in quaternion notation.
For our purpose all structures which can be related unequivocally in terms of the golden ratio and/or a golden spiral are assumed universally self-similar. This includes the packing of nucleons,
nuclide periodicity, general covalency, nanostructures, botanical phyllotaxis, biological growth, tropical hurricanes, gaps in the asteroid belt, the distribution of moons, rings and planets in the
solar system, eddies in globular clusters and the structure of spiral galaxies. An obvious gap in this hierarchy occurs at the level of chemical molecules, although scale symmetries [9] and code
regularities [10] have been reported for DNA structures. Self-similarity within series of dendrimers has been observed. However, no mechanism to explain these phenomena at the molecular level has
been proposed.
The lowest level at which golden-ratio symmetry has been mooted is in the packing of neutrons and protons in atomic nuclei. The direct evidence is that the fractional ratio Z/(A − Z) of protons to
neutrons, for stable (non-radioactive) nuclides, which decreases with mass number, converges to τ = 1 2 ( 5 − 1 ) = 0.61803 … . . . from an initial value of unity for low mass number. This ratio
occurs naturally as the limiting quotient, n/(n + 1), of successive Fibonacci numbers in the series: F n = 1 1 2 3 5 8 13 … n n + 1
…The space-filling two-dimensional spiral arrangement of growing seedbuds can always be traced along secondary spiral arms with n and n + 1 units in opposite directions. A similar three-dimensional
distribution of protons and neutrons would explain [11] the convergence from 1 to τ. It leaves a surface excess of protons, x[e] = Z − Nτ, which converges to zero at Z = 102, N = 165, A = 267. These
numbers represent limiting values for 100 natural elements and 300 nuclides.
Scatter plots of x[e] vs A, N or Z define a periodic function that arranges the 264 stable nuclides in an 11 × 24 matrix which contains the magic numbers of nuclear physics and the periodic table of
the elements as subsets. The corollary is that the periodic table appears as a nuclear property, independent of the extranuclear distribution of electrons as traditionally assumed. All of chemistry
is then seen to emerge as a topological property of space-time or the vacuum.
Molecules are the product of atomic associations. Their formation is best understood for the simplest type of molecule, consisting of two identical atoms. The obvious interpretation is to ascribe the
association to a redistribution of extranuclear electrons in the field of two nuclei and stabilization of the diatomic unit by electromagnetic interaction. In order to interact, two atoms need to be
activated. If, in the process, an electron reaches the ionization energy level of the atom it is decoupled from its nucleus and free to associate with another, similarly activated, atom. Such an
activated atom is said to be in the valence state.
The redistribution of charge that stabilizes the diatomic molecule is readily simulated by the quantum-mechanical method, first developed by Heitler and London, to model the formation of a hydrogen
(H[2]) molecule, or alternatively by classical electrostatics, using point charges to model electron density and monopositive atomic cores in the calculation of dissociation energy as a function of
interatomic distance. This relationship, in dimensionless units, valid for all homonuclear diatomic molecules, has the simple form shown in Figure 2 [12].
It fits precisely into a golden rectangle and reaches maximum binding energy of 2τ at an interatomic distance of τ. This result is interpreted to mean that the minimum space needed for a pair of
electrons to co-exist is limited by the golden ratio as a universal property of space-time. This exclusion principle was first postulated by Pauli on empirical grounds to account for the observed
fine structure of atomic spectra.
In chemical terminology the interaction is known as a covalent bond, with maximum strength, like the maximum charge density, conditioned by the golden ratio. If this means that covalent interaction
depends on the golden ratio and on the topology of space-time, it is more than likely that the shape of covalent molecules should be self-similar with other structures based on the golden ratio. We
propose to investigate this likelihood.
The best known, but by no means unique, example of Fibonacci phyllotaxis is the distribution of growing seedbuds in a sunflower head. The seedbuds that increase in size from the central growth point
are self-similar in shape and arranged along a logarithmic spiral. Although the primary spiral is not all that obvious, the secondary spiral arms invariably extend over n and n + 1 seeds and count in
opposite sense, where n and n + 1 are successive Fibonacci numbers. Exactly the same symmetry is observed on a nautilus shell where the separate chambers of increasing size are self-similar, with the
logarithmic spiral now the most conspicuous feature.
Variation on this central theme is repeated endlessly in botanical and primitive zoological structures. On closer inspection the golden-ratio curvature that underlies this symmetry is also evident in
the bone structures, such as horns and tusks of higher zoological species. Interspecies variation as observed in the fossil record can also be generalized in terms of this same symmetry [13] and we
now postulate that biological growth develops on a template that reflects both space-time topology and golden-ratio symmetry. Again, we cannot ignore the likelihood that self-similar features of
chemical molecules underpin this biological regularity.
All numerical regularities that had been observed in the arrangement of heavenly bodies in the solar system were recently shown [14] to arise from the same symmetries that occur in biological
In botany the arrangement of leaves on a stalk is optimized by the requirement of equal exposure to sunlight, which is achieved by avoiding direct overlap of leaves at different levels on the stalk.
The theoretically most efficient and botanically preferred arrangement corresponds to the emergence of a new leaf at an angle of 2πτ^2 ≡ 137.5° with respect to the previous one. This is the angle
that divides a circle in golden ratio: 2πτ^2/2πτ = 2πτ/2π = τ; τ^2 + τ = 1.
The arrangement of successive points, generated with this divergence angle, starting counterclockwise from 0, and labelling three points per cycle, is shown in Figure 3.
This procedure may be continued indefinitely without causing exact overlap, irrespective of the number of cycles. This is a feature of the irrational nature of τ.
In the case of satellite accretion along a spiral, a different criterion determines the optimal divergence angle. Compared to the botanical divergence angle of 2πτ^2, which is an area effect,
competition for intermediate material along a spiral arm is linear. The preferred divergence angles for planets and moons are multiples of π/5 ≃ πτ/3.
We now show that the distribution of extranuclear electrons that surround an atomic nucleus, like leaves on a stalk, moons around a planet, and planets around the sun, is subject to optimization
governed by a logarithmic spiral.
The most likely structure of an atom is that of a positively charged massive nucleus surrounded by discreet spherical shells of electrons, in sufficient number to balance the nuclear charge, at
characteristic distances from the nucleus. Whereas atomic size is wave-mechanically poorly defined, self-similarity suggests that the spacing between electron shells could relate to some divergence
angle measured along a golden spiral. What wave mechanics does show, is that a degeneracy of 2n − 1 regulates the number of electrons with the highest angular momentum in a shell of principal quantum
number n [15]. On this basis a divergence angle for optimal spacing of electron shells may be assumed to vary as 4π/(2n − 1).
Orbital radii, derived graphically, using this prescription, are shown in Figure 4, predicting r/a[0] = 1, 4, 9, 16, etc., in agreement with the Bohr radii of r[n] = n^2a[0]. Spectroscopic
measurement [16] shows the radii of Rydberg atoms to obey the same formula.
It is of interest to note how the botanical divergence angle solves leaf placement directly; in astronomy it identifies a planar orbit and in the atom, a spherical shell – one, two and
three-dimensional problems respectively. Molecular shape will be shown to be a four-dimensional problem. There is no anomaly here: quaternions describe rotations in all sub-spaces of four-dimensional
The periodic table of the elements is correctly predicted in detail as a function of the ratio Z/(A − Z) as it converges from unity to the golden mean. Wave mechanics cannot account for more than a
few general features of the observed periodicity and fails to explain the appearance of eight, rather than ten, elements in each transition series.
Electron-pair covalency as a function of the ratio d/r[0] of interatomic distance over ionization radius accounts for the observed dissociation energies, D, and identifies the exclusion principle at
the convergence of d/r[0] → τ and D → 2τ, in dimensionless units. Wave-mechanical treatment of the problem is limited to a few simple molecules, such as H[2]^+ and H[2].
As demonstrated in the previous paragraph, even atomic size, only obliquely related to quantum principles, is directly predicted by a simple model based on self-similarity. We infer with confidence
that the shape of isolated molecules is more likely to be revealed by reference to other structures deemed to be a function of space-time curvature.
The molecular-structure hypothesis is arguably the most controversial issue in theoretical chemistry, but perhaps, the most readily accepted by practising chemists. Not only are three-dimensional
structures routinely observed experimentally, by techniques such as X-ray crystallography, but also rationalized, even at junior high-school level, in terms of elementary orbital-hybridization
models. Both of these norms are seriously flawed.
In crystallographic analysis an observed electronic charge distribution is assumed to map the inherent geometrical structure of individual chemical species. However, in reality it maps an array of
molecules, condensed into periodic alignment and suitably distorted into a space-filling assembly by thermodynamic environment. Occurrence of the exact same conformation, inferred from
crystallographic analysis, is hard to substantiate in the liquid state, or in solution. Molecular structure in the gas phase is equally elusive.
Theoretical study of molecular shape is even more uncertain. Quantum observables, as traditionally defined, are associated with suitable operators which generate the eigenvalues that satisfy the wave
equation. Such an operator has never been defined for molecular shape, which remains quantum-mechanically undefined. An obvious fall-back position is to use an observed molecular structure as
boundary condition in solving for a molecular electronic wave function in the study of a whole range of molecular properties. Except for some success with the smallest of molecules, no results of
practical importance have ever been obtained by this method. At a further level of approximation, molecular wave functions, constructed from modified hydrogen-like atomic functions, are used as trial
functions to minimize molecular energies within the assumed molecular framework.
With computerized procedures this technique has now been extended into the so-called ab-initio LCAO-MO (Linear Combination of Atomic Orbitals – Molecular-Orbital) method, in the belief that the
results, including an optimized molecular structure, are equivalent to solution of the many-body molecular wave equation. The reason for this strange conviction is that the overwhelming majority of
users have no understanding of the commercialized software that they use and trust. The argument is confounded by a number of suspect assumptions:
The LCAO is based on real functions, whereas quantum-mechanical wave functions are complex;
Real hydrogenic functions are disallowed by the exclusion principle;
Minimization of electronic energy, which is a scalar can never produce a three-dimensional shape, which is a vectorial construct;
The many-body problem cannot be solved, neither classically nor non-classically;
A rigid molecular structure is at variance with the uncertainty principle; is a classical concept and cannot be optimized quantum-mechanically.
LCAO-MO computations, widely referred to as Quantum Chemistry, has a qualitative counterpart, known as orbital-hybridization theory, which is the de facto working model of modern chemists and the
form in which wave mechanics is handled pedagogically. The sad fact is that this model, which pretends to use rigorous wave-mechanical concepts, vulgarizes quantum theory. It claims Schrödinger’s
eigenfunction solution for the hydrogen electron as basis. In order to avoid the use of complex functions, linear combinations, claimed to eliminate imaginary parts, without loss of information, are
used instead. These constructs are known as atomic orbitals. An immediate problem is that a degenerate set of such orbitals, considered as eigenfunctions, would have the same quantum numbers n, l, m
[l], in violation of the exclusion principle. To avoid this embarrassment quantum numbers are said (see [12], page 64) not to be required any more. This is tantamount to admitting that orbitals have
no quantum-mechanical meaning.
The simple explanation of this dilemma is that a linear combination of eigenfunctions does not produce new eigenfunctions, but only rotates the polar axis and hence, also the complex plane. As an
example, consider the three-fold degenerate set of, so-called, p-functions: p l m l : p 0 = z r , p 1 = x + i y r , p − 1 = x − i y rBy linear combination 1 2 ( p 1 + p − 1 ) = x r ;
1 2 ( p 1 − p − 1 ) = i y rBy choosing a new polar direction along x rather than z, the set becomes p 0 = x r , p 1 = z + i y r , p − 1 = z − i y ra simple rotation of the coordinate
axes. Note that x/r and z/r cannot co-exist.
Like MO quantum chemistry, the theory of orbital hybridization is a classical model, dressed up in wave-mechanical garb and does not offer a non-classical description of molecular shape.
The disadvantage of a LCAO is that it eliminates orbital angular momentum as a wave-mechanical variable with eigenvalues proportional to m[l]. In so far as m[l] = ±1 describes orbital
angular-momentum vectors it has the latent ability to decide the relative orientation of interacting chemical fragments. By invoking the principle of optimal quenching of angular momentum during
molecule formation, it has been used [12] to predict the structure of small molecules with a single central atom and of torsionally rigid molecules, such as ethylene. The same principle provides a
successful physical interpretation of optical activity and the Faraday effect.
This is as far as wave mechanics can account for molecular shape. The question of torsional freedom remains unresolved. Treated classically, by the methods of molecular mechanics, torsional
parameters, based on empirical models of non-bonded interaction, are used routinely to reproduce experimentally observed structures. The shape and conformation of an isolated molecule in the vacuum
cannot be modelled.
We now make the conjecture that first-neighbour interaction is a function of valence forces and orbital angular momentum, whereas torsional interaction, in the absence of environmental factors,
depends on long-range intramolecular effects and the curvature of space-time. We consider a number of relevant examples.
An alkane molecule with regular torsion angles of π about all C–C bonds may be assumed to have cylindrical symmetry in the sense that each chain of symmetry-related atoms describes a helix around the
central (cylinder) axis. This symmetry is more readily visualized for a molecule such as C[n]H[n][+2]Br[n], as shown in the Newman projection down the C^n^+1– C^n link.
Such structures are known for long-chain hydrocarbons in the crystalline state. However, because of the low level of torsional rigidity, the free molecules are unlikely to assume the same stretched
configuration in the gas phase. Because of chaotic entanglement the polymeric structure of polyethylene cannot be used as a guideline. However, to make sense of biological growth structures it seems
reasonable to assume a uniform increase of C–C torsion angle, in the absence of distorting influences, such as crystal packing forces. Should the C–C torsion angle deviate slightly from π, e.g. as 6/
(5τ^2) or 4 τ, the cylindrical axis, at the molecular scale, will curve away, almost imperceptibly from linearity, becoming noticeable only on the macro scale.
The dependence of molecular shape on torsion angle within a chain is a guiding principle in the characterization of structure type in polypeptides and proteins.
Different conformations, such as an α-helix, depend on characteristic values of the torsion angles φ and ψ. Hydrogen bonds and the nature of R-groups have a decisive influence on the allowed values
of these angles. The flexibility of long fibrous structures in collagen is probably responsible for the characteristic curvature of freely-growing horns and tusks of higher vertebrates.
An interesting example of a curved molecular surface that spontaneously adapts to a geometrical arrangement related to the golden ratio has been observed [17]. Patterns, self-similar to botanical
phyllotaxis were observed in nanometre-thick silica shells grown on spherical silver cores of 10 micrometre diameter. As the silver core shrinks on cooling, the stress pattern that develops in the
silica surface consists of spherules that line up along 5,8 and 13,21 Fibonacci spirals, exactly as in botanical structures.
There is a good reason why wave mechanics fails to model molecular systems. By the theory of special relativity we live in a four-dimensional space-time continuum with an interval between events
given by s = x 0 2 + x 1 2 + x 2 2 + x 3 3in which x[0] = ict represents the time coordinate, together with the three usual cartesian coordinates. A transformation between point events is defined by
a Lorentz rotation through a complex angle. Such a rotation relates the four coordinates on an equivalent basis, which means that space and time coordinates are not separable in the usual way.
The four-dimensional Laplacian equation □ 2 Ψ = 0which describes potential balance in four-space is also written in the form of a wave equation ∇ 2 Ψ = 1 c 2 ∂ 2 Ψ ∂ t 2which is the basis of
Schrödinger’s equation, interpreted to describe matter waves in three-dimensional space. This equation is routinely solved by separating space and time variables. The eigenfunctions, obtained in this
way, are formulated as complex functions, whereas solutions to equation (2) are strictly defined in hypercomplex four-dimensional space. A vital aspect of physical reality is lost in the
As a consequence we find that the nexus between molecular structure and space-time curvature has disappeared. The classical three-dimensional structure, commonly assigned to a molecule, is no more
than a crude caricature of the actual quantum structure. That is why so many properties of molecules, such as intramolecular rearrangement, are impossible to reconcile with an assumed classical
structure. Even phase transitions, observed to occur in single crystals, often appear to be sterically impossible. That is why the structure of ammonia cannot be correctly formulated in three
dimensions [18] and many other molecules cannot be described in terms of Lewis structures.
An extreme case of how molecular structure adapts to environmental factors is observed in the way that the perchlorate ion facilitates crystallization by occupying voids in an otherwise close-packed
arrangement [20]. When fitting tightly into a void the ion adopts its familiar tetrahedral structure. In larger cavities it shows up crystallographically as electron density, uniformly spread
throughout the entire cavity around the central atom. The effect of temperature [21] confirms that this is not due to static disorder. This result is interpreted to show that the free perchlorate ion
is essentially formless and only develops structure under environmental pressure. This phenomenon cannot be addressed classically.
It is beginning to make sense why molecular phenomena should correlate better with space-time geometry than with wave mechanics. Number systems in closed form, are only defined in space of 1, 2, 4 or
8 dimensions, known as linear, complex, quaternion and octonion space respectively [22]. All physical evidence suggests that the universe exists in four-dimensional curved space, locally perceived as
three-dimensional tangent space, in line with uncritical casual observation. This is akin to the apparent flat-earth geometry on the curved surface of the planet. Any physical model that fails to
take all dimensions into account must produce a distorted picture of reality. Molecular matter exists in four-dimensional space and therefore cannot be described correctly in terms of a
three-dimensional theory. Self-similar symmetry is a feature of four-dimensional space and connects all structures in the cosmos, without describing any of their intrinsic details. We conjecture that
only by solving equation (2) without separation of the variables can a quantum-mechanical model of molecular shape be formulated. However, only a three-dimensional projection is observable in tangent
space. This awareness stems from the observation that atoms, molecules, covalence, botanical phyllotaxis, zoological skeletons, solar systems, globular clusters and spiral galaxies are related in
self-similar fashion by the symmetry of space time as perceived in tangent space.
The geometry of space-time is often simplified and routinely presented in two-dimensional Minkowski space, M 2. Compared to Euclidean geometry that only has one type of non-zero vector, in M 2 there
are three, depending on the value of the dot product: A[1] · A[2] = −c^2t[1]t[2] + x[1]x[2], called spacelike if A[1] · A[2] > 0, lightlike if A[1] · A[2] = 0 and time-like if A[1] · A[2] < 0. It is
of special interest to note that the distance between any pair of lightlike points, | A | = A 1 ⋅ A 2 = 0, is zero.
In quantum systems defined by (2), although more complicated, a similar counterintuitive situation prevails and intramolecular distances can therefore not be specified in Euclidean measure. This
means that the very concept of bond length is quantum-mechanically undefined, which confirms the earlier conclusion that molecular structure is a strictly classical concept.
The behaviour of quantum systems that emerges from this analysis is much more non-classical than traditionally assumed. Direct interaction is not limited to occur between first-neighbour pairs, but
operates among all atoms, as could be inferred from non-local interaction mediated by the quantum potential of Bohmian mechanics.
The familiar three-dimensional structure is a projection of the four-dimensional shape which is generated as a function of the molecular environment. Structures obtained by crystallographic analysis
are the most familiar. As pointed out before [23] crystallographic analysis is also conditioned by the assumption of three-dimensional translational symmetry. To quote:
“. . . the electron-density transform, formulated in terms of space coordinates only, would always project a static ensemble average, even from a situation that fluctuates periodically at the unit
cell level. Such fluctuation would add a fourth element of translational symmetry in the time coordinate, but which remains undetected unless a coherent radiation source is used to record a
diffraction pattern as a function of time”.
This statement largely anticipates the conclusions reached here on more fundamental grounds.
The molecular structure that emerges is radically different from crystallographic structures. The idea that atoms in a crystal vibrate around fixed positions falls away, to be replaced by an
arrangement that fluctuates periodically with time. This non-classical arrangement is conveniently pictured in M 2.
Label the complex coordinate axis ict as v. The effect of time is modelled by a shift of the unit cell origin. An atom at position a(x, v) appears to wander through the unit cell and returns to the
same position in the cell after h cycles of x and k cycles (in another context this is known as a Poincaré recurrence: Every closed dynamic system reverts in time towards its previous state) of v. As
the isotropic Minkowski diagonal of the cell moves with respect to atomic positions in the cell, the latter appear to fluctuate from time-like to space-like with respect to the origin. At the same
time the separation between any two atoms fluctuates continuously from being space-like, through light-like to time-like and this fluctuation is never in phase for any two pairs. This is typical
quantum behaviour.
The situation is the same in four-dimensional hypercomplex space ℍ^4, which we recognize as the essential quantum system, with four, instead of two indices of periodicity. In projection along v the
geometry changes to the classical ℝ^3 and all quantum effects disappear.
It is instructive to note that by substituting a wave function in complex polar form Ψ = R e i S / ħinto Schrödinger’s equation ∂ Ψ ∂ t = ( − i ħ 2 m ∇ 2 + V ) Ψthe real and imaginary parts
separate into a quantum Hamilton–Jacobi equation ∂ S ∂ t + ( ∇ S ) 2 2 m − ħ 2 ∇ 2 R 2 m R + V = 0and an equation of continuity ∂ R 2 ∂ t + ∇ ( R 2 ∇ S m ) = 0Equation (3) differs from the classical
H–J equation only in the term V q = − ħ 2 ∇ 2 R 2 m R ,which Bohm [24] identified as a quantum potential energy to distinguish between classical and non-classical systems. It has the unusual property
of showing the amplitude R in both numerator and denominator, which is interpreted to mean that quantum interactions operate non-locally, i.e., instantaneously over large distances. Exactly this type
of interaction is inferred to occur within a four-dimensional quantum molecule between sites in space-like relationship.
The unit charge associated with an electron is defined as e ∫ − ∞ ∞ R 2 ( x ) d xwhere the charge density at a point, ρ(x) = |R^2(x)|. The quantum potential depends on the wave function over all
space, V q = − ħ 2 2 m ∫ − ∞ ∞ ∇ 2 R ( x ) R ( x ) d xwhich is a continuous function. The quantum potential energy associated with a pair of hypothetical sub-electronic charge elements V[q](x[1], x
[2]) is independent of the coordinates of these elements and depends holistically on the quantum state of the whole system. There is no three-dimensional distance-related interaction between charge
elements and hence nothing that corresponds to the self-energy of quantum electrodynamics.
A quantum molecule shares many attributes with an electron, with the difference that the charge distribution is discreet, rather than continuous, and of two types—negative and positive. The total
molecular wave function is a product function ψ = Π i n ϕ i ( x i , t ) and the quantum potential is the sum over n terms: V q = − ħ 2 2 ∑ i = 1 n ∇ i 2 R i ( x i , t ) m i R i ( x i , t )In contrast
to the holistic electron, a molecule is said to constitute a partially holistic system.
A quantum molecule has four-dimensional structure which disappears, together with the phase factor, on projection into three-dimensional classical space. There are no chemical bonds in a quantum
molecule. It responds holistically to any local distortion and has the ability to rearrange spontaneously via intermediates that may appear to be sterically impossible in three dimensions. The
internal wave structure of a molecule is periodic and the time-average projection is observed as connectivity in Euclidean space. It is particularly sensitive to photochemical distortion that affects
the angular momentum of electron waves. A final complication which has been ignored is the effect of space-time curvature on molecular structure.
An additional factor that affects the growth of self-similar structures in Nature is the topology of space-time. Although not known in detail there is sufficient evidence from general relativity to
confirm that space-time is topologically closed and most probably of projective geometry [25]. An immediate implication is that the structure of molecules in free space is defined in four-dimensional
non-Euclidean space-time. Any classical structural parameter that varies with molecular flexibility should stabilize in free space at a value commensurate with space-time topology. We propose that
this equilibrium argument is responsible for cosmic self-similarity, which appears to be related to the golden logarithmic spiral, r = ae(θτ/2), with homothetic points separated by θ = 2τ, such that
Δr = ae(nπτ). This provides powerful evidence that the curvature of hypercomplex space-time is a function of the three fundamental irrational numbers e, π and τ, and − 1: e for growth, π for rotation
and τ for dilatation.
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Design of the CWP Optimization Library
Next: COOOL Classes Up: The CWP Object-Oriented Optimization Previous: A Quick Look
The three criteria behind the development of COOOLare:
1. It should provide a consistent application programming interface for solving optimization problems. By ``consistent'', we mean that optimization algorithms as well as objective-function formats
should be relatively transparent to application users. This feature provides the flexibility for users to choose optimization methods easily and concentrate on the specific problem rather than
struggling to fit the requirements of the various optimization algorithms.
2. It should allow incremental development of the library. There should be easy access (either adding or modifying) to each level of objects, with minimum influence on others. On the other hand,
reusability of existing objects should be maximized.
3. It should allow for application packages (objects) to be built from the existing library. This is equivalent to saying that new, higher, levels can be easily added to the library. For example, we
should be able to build an object for a certain travel-time inversion problem with the flexibility of choosing any of the mathematical optimization methods.
Sun Feb 25 12:08:00 MST 1996
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function in excel 2007
I have a next function that stopped working in Excel 2007, what should be changed?
Function GetFolderPath() As String
Dim oShell As Object
Set oShell = CreateObject("Shell.Application"). _
BrowseForFolder(0, "Select folder to retrieve files:", 0, "c:")
If Not oShell Is Nothing Then
GetFolderPath = oShell.Items.item.Path
GetFolderPath = vbNullString
End If
Set oShell = Nothing
End Function
The function is showing GetFolderPath message box.
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Particle Model for RBC
A particle-based model for the transport of erythrocytes in capillaries
S. Majid Hosseini & James J. Feng
Chem. Eng. Sci. 64, 4488-4497 (2009)
Abstract - This paper presents a particle-based model for the red blood cell, and uses it to compute cell deformation in simple shear and pressure-driven flows. The cell membrane is replaced by a set
of discrete particles connected by nonlinear springs; the spring law enforces conservation of the membrane area to a high accuracy. In addition, a linear bending elasticity is implemented using the
deviation of the local curvature from the innate curvature of the biconcave shape of a resting red blood cell. The cytoplasm and the external liquid are modeled as homogeneous Newtonian fluids, and
discretized by particles as in standard smoothed-particle-hydrodynamics (SPH) solution of the Navier-Stokes equations. Thus, the discrete particles serve not only as a numerical device for solving
the partial differential equations, but also as a means for incorporating microscopic physics into the model. Numerically, the fluid flow and membrane deformation are computed, via the particle
motion, by a two-step explicit scheme. The model parameters are determined from experimental measurements of cell viscosity and elastic moduli for shear, areal dilatation and bending. In a simple
shear flow, the cell typically deforms to an elongated shape, with the membrane and cytoplasm undergoing tank-treading motion. In a Poiseuille flow, the cell develops the characteristic parachute
shape. These are consistent with experimental observations. Comparison with prior computations using continuum models shows quantitative agreement without any fitting parameters, which is taken to be
a validation of the particle-based model and the numerical algorithm.
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Essential Details of One Representative Gödel Machine
Next: Proof Techniques Up: Gödel Machines: Self-Referential Universal Previous: Limitations of Gödel Machines
Essential Details of One Representative Gödel Machine
Notation. Unless stated otherwise or obvious, throughout the paper newly introduced variables and functions are assumed to cover the range implicit in the context.
Theorem proving requires an axiom scheme yielding an enumerable set of axioms of a formal logic system modus ponens combined with unification, e.g., [10].
The remainder of this paper will omit standard knowledge to be found in any proof theory textbook. Instead of listing all axioms of a particular
Next: Proof Techniques Up: Gödel Machines: Self-Referential Universal Previous: Limitations of Gödel Machines Juergen Schmidhuber 2005-01-03
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The attempt to load metrics for this article has failed.
The attempt to plot a graph for these metrics has failed.
(a) Fifth-order, 3D spectroscopy has five strong pump pulses which generate a fifth-order polarization, which interferes with a local oscillator, LO. (b) The four double-sided Feynman diagrams for
the phase matching direction for a two-level system. (c) The signal from each of these diagrams alone has a phase-twisted lineshape. (d) Adding these four diagrams generates an absorptive fifth-order
Schematic of the six beam interferometer for 3D-IR spectroscopy. (a) interferometer layout, (b) phase matching geometry, and (c) balanced detection.
Absorption spectrum of in ; the sample path length is .
Phasing fifth-order signals. (a) Each pair of beams that creates a coherence in the sample also creates (b) a spatial grating in the focus. (c) A small pinhole, diameter, diffracts some of these
gratings in the direction of the detector, converting the spatial grating to an interferogram. (d) The phases from the Fourier transforms of these interferograms can be used to calculate the phase of
the 3D-IR spectrum.
(a) The experimental absorptive 3D spectrum of gives five peaks . (b) Simulations based on the cumulant expansion accurately reproduce the data. (c) The five peaks come from the various pathways up
and down the vibrational ladder. Above each column in gray are the relative amplitude and sign of each peak accounting for the various pathways through population states and harmonic scaling of the
transition dipole moments.
The intensity of the measured signal is linear with concentration, . Direct fifth-order signal should be linear with , whereas cascaded third-order signals should scale like . This indicates that
cascaded third-order signals do not contribute to the 3D-IR.
The double-sided Feynman diagrams for cascaded third-order signals. The left column gives the diagrams for direct fifth-order processes which generate the five peaks in the 3D-IR spectrum. The middle
column gives the sequential cascaded processes in which one chromophore emits a field which acts as a pump for another third-order process on another chromophore. The right column gives the parallel
cascades, in which a third-order process on one chromophore stimulates the emission of a third-order process on another chromophore. [(a) and (b)] For these two peaks cascaded processes exist to
generate signals. [(c) and (d)] For these two peaks there is a sequential cascade (but no parallel cascade) which could potentially generate a signal at these frequencies. It will be suppressed by
the frequency mismatch between the and coherences, which are circled. (e) There is no cascade which can generate this peak because it requires walking up the vibrational ladder.
A comparison of the amplitude of the five detected peaks with the analytical results assuming a harmonic scaling of the transition dipole moments, simulation, and experiment. The simulation of the
signals uses the cumulant expansion and the same time grid as the experiment. The coherence pathways correspond to reading the bra-ket pairs going up the double-sided Feynman diagrams of Fig. 5.
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Newtown Square Math Tutor
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Does the first singular cohomology of an ACM surface vanish?
up vote 3 down vote favorite
Hi everybody, I am interested in the following:
Let $I\subset S=\mathbb{C}[x_0,\ldots ,x_n]$ be a graded ideal such that $\operatorname{depth}(S/I)\geq 3$, and let $X^h$ denote the analytic space associated to $X=\operatorname{Proj}(S/I)$.
Is it true that $H_{Sing}^1(X^h)= 0$?
The answer is yes if $X$ is smooth: In fact, in this case, if $H_{Sing}^1(X^h)\neq 0$, then the Hodge decomposition would give $H^1(X,O_X)=H_{S_+}^2(S/I)_0\neq 0$, a contradiction to the fact that $\
operatorname{depth}(S/I)\geq 3$.
However, what can we say if $X$ is singular?
ag.algebraic-geometry ac.commutative-algebra at.algebraic-topology
Yes, by $H_{Sing}^1$ I mean singular cohomology over $\mathbb{C}$. I usually denote by $H^1(X^h,\mathbb{C})$ the sheaf cohomology of the constant sheaf associated to $\mathbb{C}$, however one can
show that $H_{Sing}^1\cong H^1(X^h,\mathbb{C})$, so it does not matter. $H_{S_+}^i(S/I)$ means local cohomology with support in the ideal $S_+=(x_0,\ldots ,x_n)$. $H_{S_+}^i(S/I)_0$ means its
degree $0$ part, and $\operatorname{depth}(S/I)\geq 3$ means $H_{S_+}^i(S/I)=0$ for all $i=0,1,2$. – Matteo Varbaro Dec 19 '11 at 10:05
I have a suggestion, but I'm not sure that it works. Take a smooth surface $S$ with $H^1>0$ in some projective space and project it generically to ${\mathbb P}^3$ to obtain a surface $X$.
1 Intuitively, I would say that $H^1(X)\ge H^1(S)$, but I don't know how to prove this. (Recall that a generic projection is a birational map with finite fibers and the sigular locus of $X$ is a
double curve with triple points that are triple also for the surface. So you can think of $X$ as being obtained form $S$ by identifying points on a curve with an involution). – rita Dec 21 '11 at
@Sandor: isn't a hypersurface ACM? I used to think it is, but I guess I got this wrong. – rita Dec 22 '11 at 6:31
To Rita and Sàndor: yes, a hypersurface is certainly ACM. – Angelo Dec 22 '11 at 7:18
I see, of course a hypersurface is ACM, I got stock with the smooth surface with the $H^1\neq 0$ which isn't... Sorry. – Sándor Kovács Dec 22 '11 at 10:03
show 1 more comment
2 Answers
active oldest votes
Addendum I wrote this up thinking that the question was something different. As Angelo pointed out, this does not answer the actual question. I will leave this here just in case someone
finds the computation useful. So this is a proof, that $H^1(X,\mathscr O_X)=0$. Not exactly what the question was, although it still implies that $Gr_F^0H^1(X,\mathbb C)=0$ where $F$ is
Deligne's Hodge filtration. :( end of Addendum
Using the notations of the question, in addition let $Y=\mathrm{Spec}(S/I)$ be the affine cone over $X$, $P\in Y$ the vertex, and $U=Y\setminus \{P\}$. Finally, let $\mathrm{depth}(S/I)=d\
geq 3$. First of all we have a long exact sequence:
$$ \dots \to H^i(Y,\mathscr O_Y) \to H^i(U,\mathscr O_U) \to H^{i+1}_P(Y,\mathscr O_Y) \to H^{i+1}(Y,\mathscr O_Y) \to \dots. $$ Since $Y$ is affine, this implies that for $i>0$, $$ H^i(U,\
up vote 1 mathscr O_U) \simeq H^{i+1}_P(Y,\mathscr O_Y) $$ and hence $$ H^i(U,\mathscr O_U)=0 \tag{$\star$} $$ for $0< i < d-1$.
down vote
Proposition $\quad\ H^i(U,\mathscr O_U) \simeq \bigoplus_{n\in\mathbb Z} H^i(X, \mathscr O_X(n)) $
Proof $U$ is an $\mathbb A^1$-bundle over $X$. In fact, it is easy to see that $U\simeq \mathrm{Spec}_X ( \oplus _{n\in \mathbb Z} \mathscr O_X(n))$ with a projection $\pi:U\to X$. It
follows that $\pi_*\mathscr O_U\simeq \oplus _{n\in \mathbb Z} \mathscr O_X(n)$ and $R^j\pi_*\mathscr O_U=0$ for $j>0$. Then the claimed isomorphism follows from the simple special case of
the Leray spectral sequence when all $R^j$'s with $j>0$ are $0$.
2 How does this imply vanishing of the first singular cohomology group? – Angelo Dec 22 '11 at 7:20
Oops, at some point I switched to thinking that the question was whether $H^1(X,\mathscr O_X)=0$.... This still says something, but certainly less. – Sándor Kovács Dec 22 '11 at 9:44
Concerning the (attempted) counterexample of Rita, I think that I can prove that the answer to my question is positive also if $X$ is a complete intersection. – Matteo Varbaro Dec 23 '11
at 16:42
Actually, I was interested in the question because I think I can show that a certain algebraic property of $I$ holds true if and only if the first singular cohomology of $X$ vanishes,
under the assumption depth$(S/I)\geq 3$. Since I have not much feeling with singular cohomology, I wanted to be sure that the vanishing of this singular cohomology wasn't well known or
easy, as it is under the smoothness assumption. In any case I stay very interested in the fact, and I am happy to have captured your attentions, thank you to think to the problem! –
Matteo Varbaro Dec 23 '11 at 16:48
add comment
If you are willing to assume that $X$ is locally a complete intersection, then the result you want will follow from a theorem due to A.Ogus:
Theorem. Suppose $X\subset\mathbb{P}^n_\mathbb{C}$ is a local complete intersection of dimension $d=n-r$ with $d-r\geq1$. Then $\mathrm{H}^1(X,\mathbb{C})=0$.
up vote 0 In your question $X$ is an ACM hypersurface and the cone has depth $\geq3$. So the cone has dimension $\geq3$, which means $d:=\dim X\geq2$. But since $X$ is a hypersurface we see $d=n-1\
down vote geq2$, i.e., with notation of theorem, $r=1$ and $d-r=d-1=n-2\geq1$ which is the condition needed in the theorem.
The above theorem is Theorem 4.9, page 1106 in On the formal neighborhood of a subvariety of projective space, Amer. J. Math. 97 (1975), no.4, p.p. 1085-1107.
Is $r$ the codimension of $X$? In my question $X$ is not a hypersurface! As I said in one comment I can prove what I asked whenever $X$ is a complete intersection, so this is a
particular case. However, if $X$ is a hypersurface, then it is automatically a locally complete intersection. In any case thank you for indicating me the paper of Ogus, I will read it!
– Matteo Varbaro Dec 23 '11 at 18:35
Yes, $r$ is the codimension of $X$. I think the title of your question led me to think that $X$ was a hypersurface. – Mahdi Majidi-Zolbanin Dec 23 '11 at 18:52
I see. In the title I put ACM surface because it was shorter. This is the case in which dim$(S/I)$ $=$ depth$(S/I)=3$: I cannot answer the question neither in this special case. –
Matteo Varbaro Dec 24 '11 at 9:19
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Not the answer you're looking for? Browse other questions tagged ag.algebraic-geometry ac.commutative-algebra at.algebraic-topology or ask your own question.
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No matter where your power comes from, however, all of the equipment and devices that use power—whether from rooftop solar panels, the public grid, rechargeable batteries, or gas-powered engines—are
generally based on some basic principles. Devices range from flashlights and phones to automobiles, televisions, and computers.
This week's lesson will spark your skills in electrical circuitry. Not only will you learn the basics of series and parallel circuits, but you will also figure out how your own home and other powered
devices are wired to work.
Circuit Basics
Circuits at the California Energy Commission's Energy Story site. This page explains the very basics of a circuit, and it also outlines how to make a working, simple circuit using a penlight bulb and
a flashlight battery. If possible, complete this circuit experiment in your classroom.
Next, get some more in-depth training with an Automotive Series course that covers Electrical Circuits. This training module is presented through a sequence of 31 Web pages. During your training,
make illustrated flash cards or diagrams that will help you remember the key terms and concepts.
As you move through the module, you will learn about Electrical Circuit Requirements and Basic Circuit Construction. What are a circuit's five basic components? What exactly is a "load"?
Starting on page seven, you will get a good introduction to Ohm's Law, which explains the relationship between voltage, current, and resistance. The Ohm's Law Formula shows the mathematical
relationship of those variables. Using the Ohm's Law Symbols Shortcut, you will learn an easy, graphical way to remember that formula. Go ahead and create your own handheld symbols shortcut using
something that can fit into your pocket, such as a bottle cap or a cardboard disc.
Continuing through the module, explore the details illustrating the Types of Circuits. For each type–Series, Parallel, and Series-Parallel–the module demonstrates some examples by changing the
formula's variables, including voltage drop calculations. Refer to your formula shortcut disc as you examine the circuit samples. On your own or with a partner, develop diagrams of two different
examples and accompanying formulas for each circuit type.
Make the Connection
Drag-and-Drop applet site. In order to properly view and use the applets, you will need use the right browser. (For example, Internet Explorer works fine, but Firefox may not.) If you cannot click
and drag the set of symbols on the right labeled "drag me," then that browser will not work for these activities.
Complete each of these exercises:
• Drag-and-Drop Electrical Circuits 0: Equations will help you better understand Ohm's Law, along with some other common equations used in current electricity. Drag the correct letters into the
boxes to create the equations and mouse over letters to see what each represents.
• Skip Drag-and-Drop Electrical Circuits 1, which requires a specific textbook.
Now, draft the floor plan of your own home. Include at least the bottom level .(You may also include additional floors if your home has them.) The floor plan should include any outdoor electrical
uses, such as porch or security lights. You may want to work with classmates to develop a legend for the floor plans, agreeing upon what symbol represents an electrical outlet, a light bulb, and so
forth. Investigate the voltage needs for each use and fill in the circuit layout on the floor plan. Write out accurate Ohm's Law formulas next to each use location. When your layout is done, swap
with a classmate to review each other's floor plan for quality and accuracy. Discuss your reviews with each other.
If you have time and want to learn more about electrical circuits and their real-world applications, check out the Interactive Java Tutorials illustrating electricity and magnetism at the Molecular
Expressions site.
Here, you can browse through and explore the variety of relevant applets. These come in forms that provide simple examples of electromagnetic fields, such as Magnetic Fields and Compass Orientation,
How a Metal Detector Works, Faraday's Magnetic Field Induction Experiment, and Lenz's Law, for example.
There are also a number of tutorials related to capacitance, through which you can learn about Factors Affecting Capacitance, Charging and Discharging a Capacitor, and Lightning: An Example of a
Natural Capacitor.
Newspaper Activities
Browsing through issues of the e-edition, find two or more examples of electrical circuits in use. For example, you may find a photo taken in one of your local parks that includes a streetlight. Or,
maybe you read a story about a business or homeowner installing solar panels. Whatever examples you find, create a detailed diagram for each that illustrates the complete circuit accurately. You may
need to contact the city, business, or whomever, as possible, to get the details about the example's components. Make sure to include the appropriate and accurate Ohm's Law formulas with your
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How To: Read Capacitor Values
Capacitors are categorised in terms of their capacitance, voltage and construction - so it is often necessary to tell different similar-looking capacitors apart based on their value markings.
Electrolytic capacitors are marked in terms of micro Farads (uF) - these are easy to interpret.
For example, the capacitor below is a 100uF capacitor, because it is marked as 100uF.
If a ceramic capacitor only has two digits (XY), then this value is XY pico Farads (pF). For example, the ceramic capacitor shown below has a value of 20 pF.
If a ceramic capacitor has three digits (XYZ), then the value of the capacitor is equal to:
XY * 10 ^ Z pico Farads
In the case below, XY is 22 and Z is 3. Thus, the value is:
22 * 10 ^ 3 pico Fards = 22,000 pF.
We can see from this example that, effectively, Z is the number of zeroes after XY.
In the example below the value of the capacitor is give as: 104J100V. This indicates that:
104 is the capacitance code
J is the tolerance code
100V is the voltage rating
Let's concentrate on the capacitance code.
Capacitor values are given in micro (u), nano (n) and pico (p) Farads. To convert between these magnitude prefixes, consider that:
1 Farad is the same as
1 000 000 micro Farads is the same as
1 000 000 000 nano Farads
1 000 000 000 000 pico Farads
1,000 pF = 1 nF
1,000 nF = 1 uF
Therefore, a capacitance code of 104 = 100,000 pF = 100 nF = 0.1 uF.
0 comments:
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Solids of Revolution
February 13th 2009, 08:11 AM
Solids of Revolution
Really need help, need to check my answers. These are parts of a review so..
#1 Region M is bounded by the x-axis, x=4 and the graph of $LN(x)$
is the interval 1 to 4?
a) find the volume of the solid created by revolving M around y=-4
I got 70.958 as the area
b) find the volume of the solid created by revolving M around y=7
c) let region M be the base of a solid whose cross sections perpendicular to the x-axis are isosceles right triangles, find the volume of the solid.
This is the one I don't understand. Please help me setup this problem.
d) let region M be the base of a solid whose cross sections perpendicular to the x-axis are circles, find the volume of the solid.
not sure how to do this one either.
#2 Region Q is bounded by the y-axis, the line y=5 and the curve e(x)
Let region R be the base of a solid whose cross sections perpendicular to the x-axis are semi-circles, find the volume of the solid.
Is the interval for the integral 0 to 1.609? And then what do I do? Need help setting it up.
#3 Region T is bounded by the x-axis and the graphs of $\sqrt{x}$ and $-0.2x^2+6$
This one is really confusing me. All I know is the interval 0 to 4.415. I don't understand how you'd solve it on the y-axis. The x-axis was easier to understand.
a) find the volume of the solid created by revolving T around the y-axis
b) around x=-3
c) around x=10
d) let T be the base of a solid whose cross sections perpendicular to the y-axis are squares, find the volume of the solid.
February 13th 2009, 11:42 AM
Using the washer method, we have an outer radius of $R(x) = \ln x - (-4) = \ln x + 4$ and inner radius $r(x) = 0 - (-4) = 4$. You have the right interval, so
$V = \pi\int_1^4\left[(R(x))^2 - (r(x))^2\right]\,dx$
$= \pi\int_1^4\left[(\ln x + 4)^2 - 4^2\right]\,dx$
$= \pi\int_1^4\left[(\ln x)^2 + 8\ln x\right]\,dx$
Is this how you started? I get a slightly different answer.
b) find the volume of the solid created by revolving M around y=7
On $[1,\,4],\;\ln x < 7,$ so your outer radius is $R(x) = 7 - 0 = 7$ and your inner radius $r(x) = 7 - \ln x$. So you have
$V = \pi\int_1^4\left[(R(x))^2 - (r(x))^2\right]\,dx = \pi\int_1^4\left[7^2 - (7 - \ln x)^2\right]\,dx$
Again, I get a different answer.
c) let region M be the base of a solid whose cross sections perpendicular to the x-axis are isosceles right triangles, find the volume of the solid.
This is the one I don't understand. Please help me setup this problem.
Which part don't you understand? The region $M$ is the base of the solid; the solid will be "sticking out" of the plane (above the xy-plane, that is). If you cut the solid (with a cut
perpendicular to the x-axis), the cross section is an isosceles triangle. This means that the base of each triangle has length $\ln x$.
The formula you need is $V = \int_a^b A(x)\,dx,$ where $A(x)$ is the area of the cross section at $x$.
d) let region M be the base of a solid whose cross sections perpendicular to the x-axis are circles, find the volume of the solid.
not sure how to do this one either.
I'm not sure how the solid can have $M$ as a base and still have circular cross sections perpendicular to the x-axis. Semicircles, perhaps?
#2 Region Q is bounded by the y-axis, the line y=5 and the curve e(x)
Let region R be the base of a solid whose cross sections perpendicular to the x-axis are semi-circles, find the volume of the solid.
Is the interval for the integral 0 to 1.609? And then what do I do? Need help setting it up.
I assume you mean region $Q$ is the base; you haven't specified $R$. Your interval is correct (approximately). Each semicircle has a base that extends from the top of the region to the bottom of
the region. So the diameter of each is $5 - e^x$. Use the formula I gave above.
#3 Region T is bounded by the x-axis and the graphs of $\sqrt{x}$ and $-0.2x^2+6$
This one is really confusing me. All I know is the interval 0 to 4.415.
Not quite. $x=4.415$ is roughly the intersection of the two curves, but the region goes beyond that (look at a graph). You want to solve $-0.2x^2 + 6 = 0$ for the upper bound.
I recommend using the washer method and integrating with respect to $y$. In that case, your lower bound is $y = 0,$ and your upper bound is approximately $\sqrt{4.415}\approx2.101\text{.}$ Solve
both equations for $x$ (in terms of $y$; you only need the positive root for the second one), and set up your integral.
For the parts involving revolution around horizontal axes, the shell method would be easiest, but you could do it with washers by splitting the interval at the intersection of the curves and
using two integrals.
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Results 1 - 10 of 23
- In Proc. 12th European Symposium on Algorithms, volume 3221 of Lecture Notes Comput. Sci , 2004
"... ..."
- In: VMCAI’11. LNCS , 2011
"... Abstract. We define several abstract semantics for the static analysis of finite precision computations, that bound not only the ranges of values taken by numerical variables of a program, but
also the difference with the result of the same sequence of operations in an idealized real number semantic ..."
Cited by 14 (3 self)
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Abstract. We define several abstract semantics for the static analysis of finite precision computations, that bound not only the ranges of values taken by numerical variables of a program, but also
the difference with the result of the same sequence of operations in an idealized real number semantics. These domains point out with more or less detail (control point, block, function for instance)
sources of numerical errors in the program and the way they were propagated by further computations, thus allowing to evaluate not only the rounding error, but also sensitivity to inputs or
parameters of the program. We describe two classes of abstractions, a non relational one based on intervals, and a weakly relational one based on parametrized zonotopic abstract domains called affine
sets, especially well suited for sensitivity analysis and test generation. These abstract domains are implemented in the Fluctuat static analyzer, and we finally present some experiments. 1
- in Proceedings of the 17th Symposium on Computer Arithmetic, P. Montuschi and E. Schwarz, Eds., Cape Cod , 2005
"... The fused multiply accumulate instruction (fused-mac) that is available on some current processors such as the Power PC or the Itanium eases some calculations. We give examples of some
floating-point functions (such as ulp(x) or Nextafter(x, y)), or some useful tests, that are easily computable usin ..."
Cited by 10 (3 self)
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The fused multiply accumulate instruction (fused-mac) that is available on some current processors such as the Power PC or the Itanium eases some calculations. We give examples of some floating-point
functions (such as ulp(x) or Nextafter(x, y)), or some useful tests, that are easily computable using a fused-mac. Then, we show that, with rounding to the nearest, the error of a fused-mac
instruction is exactly representable as the sum of two floating-point numbers. We give an algorithm that computes that error. 1
- IEEE TRANSACTIONS ON COMPUTERS, 2010. 9 HTTP://DX.DOI.ORG/10.1145/1772954.1772987 10 HTTP://DX.DOI.ORG/10.1145/1838599.1838622 11 HTTP://SHEMESH.LARC.NASA.GOV/NFM2010/PAPERS/NFM2010_14_23.PDF 12
HTTP://DX.DOI.ORG/10.1007/978-3-642-14203-1_11 13 HTTP://DX. , 2011
"... High confidence in floating-point programs requires proving numerical properties of final and intermediate values. One may need to guarantee that a value stays within some range, or that the
error relative to some ideal value is well bounded. This certification may require a time-consuming proof fo ..."
Cited by 8 (3 self)
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High confidence in floating-point programs requires proving numerical properties of final and intermediate values. One may need to guarantee that a value stays within some range, or that the error
relative to some ideal value is well bounded. This certification may require a time-consuming proof for each line of code, and it is usually broken by the smallest change to the code, e.g., for
maintenance or optimization purpose. Certifying floating-point programs by hand is, therefore, very tedious and error-prone. The Gappa proof assistant is designed to make this task both easier and
more secure, due to the following novel features: It automates the evaluation and propagation of rounding errors using interval arithmetic. Its input format is very close to the actual code to
validate. It can be used incrementally to prove complex mathematical properties pertaining to the code. It generates a formal proof of the results, which can be checked independently by a lower level
proof assistant like Coq. Yet it does not require any specific knowledge about automatic theorem proving, and thus, is accessible to a wide community. This paper demonstrates the practical use of
this tool for a widely used class of floating-point programs: implementations of elementary functions in a mathematical library.
, 2001
"... We present an algorithm for implementing correctly rounded exponentials in double-precision floating point arithmetic. This algorithm is based on floating-point operations in the widespread
IEEE-754 standard, and is therefore more ecient than those using multiprecision arithmetic, while being fully ..."
Cited by 6 (2 self)
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We present an algorithm for implementing correctly rounded exponentials in double-precision floating point arithmetic. This algorithm is based on floating-point operations in the widespread IEEE-754
standard, and is therefore more ecient than those using multiprecision arithmetic, while being fully portable. It requires a table of reasonable size and IEEE-754 double precision multiplications and
additions. In a preliminary implementation, the overhead due to correct rounding is a 2:3 times slowdown when compared to the standard library function.
"... We introduce several algorithms for accurately evaluating powers to a positive integer in floating-point arithmetic, assuming a fused multiply-add (fma) instruction is available. For bounded,
yet very large values of the exponent, we aim at obtaining correctly-rounded results in round-to-nearest mod ..."
Cited by 4 (0 self)
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We introduce several algorithms for accurately evaluating powers to a positive integer in floating-point arithmetic, assuming a fused multiply-add (fma) instruction is available. For bounded, yet
very large values of the exponent, we aim at obtaining correctly-rounded results in round-to-nearest mode, that is, our algorithms return the floating-point number that is nearest the exact value.
- Journal of Logic and Algebraic Programming. Special Issue on Practical Development of Exact Real Number Computation , 1999
"... this article we will describe a model known as significance arithmetic. We will give details of the implementation in Mathematica along with several examples that illustrate the design goals and
differences over conventional fixed precision floating point systems. Wolfram Research Inc, Champai ..."
Cited by 3 (0 self)
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this article we will describe a model known as significance arithmetic. We will give details of the implementation in Mathematica along with several examples that illustrate the design goals and
differences over conventional fixed precision floating point systems. Wolfram Research Inc, Champaign, Illinois, U. S. A. Department of Mathematics, University of Bologna, Italy Introduction Knuth
has expressed a desiderata of floating point arithmetic as follows [Knuth 1998]: It would be nice if we could give our input data for each problem in an unnormalized form which expresses how much
precision is assumed, and if the output would indicate just how much precision is known in the answer. This enhancement would assist those who do not wish to undertake a rigorous analysis of
computational error. Significance arithmetic is one approach for providing such a facility. It provides local error monitoring. The computati
, 2012
"... Abstract—Floating-point arithmetic is known to be tricky: roundings, formats, exceptional values. The IEEE-754 standard was a push towards straightening the field and made formal reasoning about
floating-point computations possible. Unfortunately, this is not sufficient to guarantee the final result ..."
Cited by 3 (0 self)
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Abstract—Floating-point arithmetic is known to be tricky: roundings, formats, exceptional values. The IEEE-754 standard was a push towards straightening the field and made formal reasoning about
floating-point computations possible. Unfortunately, this is not sufficient to guarantee the final result of a program, as several other actors are involved: programming language, compiler,
architecture. The CompCert formally-verified compiler provides a solution to this problem: this compiler comes with a mathematical specification of the semantics of its source language (ISO C90) and
target platforms (ARM, PowerPC, x86-SSE2), and with a proof that compilation preserves semantics. In this paper, we report on our recent success in formally specifying and proving correct CompCert’s
compilation of floating-point arithmetic. Since CompCert is verified using the Coq proof assistant, this effort required a suitable Coq formalization of the IEEE-754 standard; we extended the Flocq
library for this purpose. As a result, we obtain the first formally verified compiler that provably preserves the semantics of floating-point programs. Index Terms—floating-point arithmetic; verified
compilation; formal proof; floating-point semantic preservation; I.
- In Proceedings of the 21st Annual ACM Symposium on Applied Computing , 2006
"... We provide sufficient conditions that formally guarantee that the floating-point computation of a polynomial evaluation is faithful. To this end, we develop a formalization of floatingpoint
numbers and rounding modes in the Program Verification System (PVS). Our work is based on a well-known formali ..."
Cited by 3 (1 self)
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We provide sufficient conditions that formally guarantee that the floating-point computation of a polynomial evaluation is faithful. To this end, we develop a formalization of floatingpoint numbers
and rounding modes in the Program Verification System (PVS). Our work is based on a well-known formalization of floating-point arithmetic in the proof assistant Coq, where polynomial evaluation has
been already studied. However, thanks to the powerful proof automation provided by PVS, the sufficient conditions proposed in our work are more general than the original ones.
- Intl. J. High Speed Comput , 1991
"... ..."
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Boca Raton Calculus Tutor
Find a Boca Raton Calculus Tutor
...I am pursuing a graduate degree focused in Statistics. I can also tutor basic programming in C++, Python, or Matlab. I am curious, like to help others, and always eager to keep learning.
24 Subjects: including calculus, reading, physics, statistics
...References available on request.Algebra is a subject which is critical that a student do well in. The ability to master this subject will greatly affect the students performance in all
subsequent math classes. The concepts in this course build upon one another.
51 Subjects: including calculus, chemistry, statistics, geometry
...In order to do well, students must UNDERSTAND the concepts presented not merely “get the answer.” I have had great success teaching the concepts as well as the mechanics in a fun and
interesting way. You’re never too old or too young to laugh while learning math! Algebra 2 delves deeper into the concepts and skills introduced in Algebra 1 as well as introducing analytical
7 Subjects: including calculus, geometry, algebra 1, algebra 2
...I have many hours of experience tutoring Discrete Mathematics subjects on a one-on-one basis, including Logic, Set Theory, Probability, Combinatorics, Order Theory, Relations, Functions, and
more. I try to make learning Discrete Mathematics fun by showing students how logic can be learned throug...
27 Subjects: including calculus, English, reading, algebra 1
Computer Administrator with over 15 years IT experience. University Professor teaching computer-related courses including Microsoft Office, database systems, computer security, Unix, computer
networking, programming and spreadsheet modeling. Also hold a Bachelor's degree in Electrical Engineering ...
16 Subjects: including calculus, geometry, algebra 1, algebra 2
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Closed Set of Elements
Date: 06/30/2001 at 20:22:13
From: Craig Murai
Subject: Algebra
Can someone please explain to me the meaning of the word 'CLOSED' used
in the following sentence that I came across in a math text?
'... set of complex numbers is CLOSED under addition... '.
Date: 07/01/2001 at 12:14:19
From: Doctor Ian
Subject: Re: Algebra
Hi Craig,
A set of elements is closed under an operation if, when you apply the
operation to elements of the set, you always get another element of
the set.
For example, the whole numbers are closed under addition, because if
you add two whole numbers, you always get another whole number - there
is no way to get anything else.
But the whole numbers are _not_ closed under subtraction, because you
can subtract two whole numbers to get something that is not a whole
number, e.g.,
2 - 5 = -3
The integers are closed under multiplication (if you multiply two
integers, you get another integer), but they are _not_ closed under
division, since you can divide two integers to get a rational number
that isn't an integer.
The rationals, however, are closed under addition, subtraction,
multiplication, and division.
So the statement that 'the complex numbers are closed under addition'
means that if you add two complex numbers together, you are guaranteed
to get a complex number as the sum.
Does this help?
- Doctor Ian, The Math Forum
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WME@Kimpton Middle
symbols for numbers, or numerals. In the first place, it was necessary for humans to invent symbols for numerals.
You may ask who invented the numerals that the whole world use today for mathematics. That is a long story. For sure different parts of the world had used different numerals and it is still the case
Please refer to this Wiki page for more information.
Early numerials often do not have a symbol for zero. It turns out that the concept of zero is not intuitive. In ancient India,
they used an empty space for zero, or just a simple dot.
Later it spread West to the Middle East and East to China and evolved into the modern 0.
Other numerals exist. For example, the Roman and the Egyptian numerals are displayed on the right. Clicking on the images leads to more information. Visit this page to see ancient Chinese numerals.
Often we refer to the numerals we use as digits. With a single digit, we can write down just ten different numbers! We still have to figure out how to write down larger numbers.
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to Electromagnetics
1 Introduction to Electromagnetics
Electromagnetic fields are caused by electric charges at rest and in motion. Positive and negative electric charges are sources of the electric fields and moving electric charges yielding a current
is the source of magnetic fields. Time-varying electric and magnetic fields are coupled in an electromagnetic field radiating from the source.
Figure 1 Positive and negative electric charges are sources of an electric field
Electromagnetic fields are divided into four different quantities:
• the magnetic flux density B with the unit T (Tesla or volt-second per square meter)
• the magnetic field intensity H with the unit A/m (Ampere per meter)
• the electric field intensity E with the unit V/m (Volt per meter)
• the electric flux density D with the unit C/m^2 (Coulomb per square meter)
A time-varying E and D will give rise to B and H, and vice versa where the relation depends on the properties of the medium. Far enough from the source the magnetic field, H, will be perpendicular to
the electric field, E, and both are normal to the direction of propagation, as shown in the following figure:
Figure 2 A time-varying electric field, E, will give rise to a perpendicular magnetic field, H, and vice versa. Far enough from the source it will become a uniform plane wave and the ratio between E
and H will be the intrinsic impedance of the medium.
Far enough from the source, the wave-front, which will become almost spherical, can be seen as an almost plane front if the sphere is large enough. Then we have a uniform plane wave where the ratio
between the electric field and the magnetic field, called the wave impedance:
is a constant named the intrinsic impedance of the medium, h . The electromagnetic theory is also based on three universal constants:
• the velocity of an electromagnetic wave in free space c (the speed of light » 3^.10^8 m/s)
• the permittivity of free space e [0] (» 8.854^.10^-12 F/m)
• the permeability of free space m [0 ](= 4p ^.10^-7 H/m)
These constants are related by [8]:
The permittivity is a proportionality constant between the electric flux density D and the electric field intensity E, in free space as:
and the permeability is the proportionality constant between the magnetic flux density B and the magnetic field intensity H, in free space as:
From these constants, the intrinsic impedance of free space can be calculated as [8]:
Next: Maxwell's Equations
EMC of Telecommunication Lines
A Master Thesis from the Fieldbusters © 1997
Joachim Johansson and Urban Lundgren
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Overview and topics
A growing population of mathematicians, computer scientists, and engineers use computers to construct and verify proofs of mathematical results. Among the various approaches to this activity, a
fruitful one relies on interactive theorem proving. When following this approach, researchers have to use the formal language of a theorem prover to encode their mathematical knowledge and the
proofs they want to scrutinize. The mathematical knowledge often contains two parts: a static part describing structures and a dynamic part describing algorithms. Then proofs are made in a style
that is inspired from usual mathematical practice but often differs enough that it requires some training. A key ingredient for the mathematical practitioner is the amount of mathematical
knowledge that is already available in the system's library.
The Coq system is an interactive theorem prover based on Type Theory. It was recently used to study the proofs of advanced mathematical results. In particular, it was used to provide a
mechanically verified proof of the four-colour theorem and it is now being used in a long term effort, called Mathematical Components to verify results in group theory, with a specific focus on
the odd order theorem, also known as the Feit-Thompson theorem. These two examples rely on a structured library that covers various aspects of finite set theory, group theory, arithmetic, and
The aim of this school is to give mathematicians and mathematically inclined researchers the keys to the Coq system and the Mathematical Components library. The topics covered are:
□ Formal proof techniques
□ Structuration of libraries
□ Encoding of common mathematical structures
□ Formal description of algorithms
□ An overview of advanced projects:
☆ The odd order theorem
☆ Constructive algebraic topology
☆ Kepler's conjecture,
☆ Differential calculus,
☆ Foundational investigations.
The school's contents will be organized as a balanced schedule between lectures and laboratory sessions where participants will be invited to work on their own computer and try their hands on a
progressive collection of exercises.
The current list of speakers is:
□ Georges Gonthier (Microsoft Research)
□ Thomas C. Hales (University of Pittsburgh)
□ Julio Rubio (Universidad de La Rioja)
□ Bas Spitters (Radboud Universiteit, Nijmegen)
□ Vladimir Voevodsky (Institute for advanced study, Princeton)
□ Yves Bertot (INRIA)
□ Assia Mahboubi (INRIA)
□ Laurence Rideau (INRIA)
□ Pierre-Yves Strub (MSR-INRIA common laboratory)
□ Enrico Tassi (INRIA)
□ Laurent Théry (INRIA)
Associated sites
For more information about Coq you should have a look at this site; for more information about the math components library and the ssreflect approach you should have a look at this site. This
last page also gives indications on how to join a user mailing list.
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Syntax stats_mw_test::= Description of the illustration stats_mw_test.gif Purpose
A Mann Whitney test compares two independent samples to test the null hypothesis that two populations have the same distribution function against the alternative hypothesis that the two distribution
functions are different.
The STATS_MW_TEST does not assume that the differences between the samples are normally distributed, as do the STATS_T_TEST_* functions. This function takes three arguments and a return value of type
VARCHAR2. expr1 classifies the data into groups. expr2 contains the values for each of the groups. The function returns one value, determined by the third argument. If you omit the third argument,
the default is TWO_SIDED_SIG. The meaning of the return values is shown in the table that follows.
Table 7-7 STATS_MW_TEST Return Values
│ Return Value │ Meaning │
│ STATISTIC │ The observed value of Z │
│ U_STATISTIC │ The observed value of U │
│ ONE_SIDED_SIG │ One-tailed significance of Z │
│ TWO_SIDED_SIG │ Two-tailed significance of Z │
STATS_MW_TEST computes the probability that the samples are from the same distribution by checking the differences in the sums of the ranks of the values. If the samples come from the same
distribution, then the sums should be close in value.
STATS_MW_TEST Example
Using the Mann Whitney test, the following example determines whether the distribution of sales between men and women is due to chance:
(cust_gender, amount_sold, 'STATISTIC') z_statistic,
(cust_gender, amount_sold, 'ONE_SIDED_SIG') one_sided_p_value
FROM sh.customers c, sh.sales s
WHERE c.cust_id = s.cust_id;
----------- -----------------
-1.4011509 .080584471
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Vibration Analysis and Signal Processing in LabVIEW
1. Introduction
Sound and vibration transducers produce complex time series waveforms, within which are many specific signatures. It is important to understand these different vibration signatures and how to
properly extract them for trending analysis. With proper signature information, it becomes possible to tabulate specific metrics which can drive plant maintenance or production schedules. There are a
variety of different types of signal complexities, corresponding to different sound and vibration phenomena as represented in Figure 1 from top to bottom:
1. Some signals have a long time duration but narrow bandwidth such as rub & buzz noise.
2. Some signals have a short time duration but wide bandwidth such as impacts or transients.
3. Some signals have a short time duration and narrow bandwidth such as decayed resonance.
4. Some signals have a time-varying bandwidth such as an imbalanced shaft generating noise dependent on RPM or machine speed.
Figure 1. Various vibration signal complexity types
The first step in any sound and vibration application is to understand the system you are trying to monitor and the sound and vibration signals present in it. After this has been defined, the next
step is to choose the correct algorithm for extracting the signal feature of interest from the raw signal. Figure 2 shows the different types of signals and the algorithm that National Instruments
provides for extracting those features.
Figure 2. Different analysis algorithms and which signals they are ideal for use with.
National Instruments provides a range of algorithms such as standard frequency analysis; order analysis for monitoring rotating components such as a gearbox; time-frequency analysis for time varying
sound and vibration signals; quefrency analysis for detecting harmonics; and wavelet and model based analysis for transient detection. This complete collection of algorithms provides users the
ability to properly analyze and monitor their specific machine or device.
2. Frequency Analysis
Frequency analysis is the most commonly used method for analyzing a vibration signal. The most basic type of frequency analysis is an FFT, or Fast Fourier Transform, which converts a signal from the
time domain into the frequency domain. The product of this conversion is a power spectrum and shows the energy contained in specific frequencies of the overall signal. This is quite useful for
analyzing stationary signals whose frequency components do not change over time.
Despite its common use, there are many downfalls to just using frequency analysis because its results, such as a power spectrum or total harmonic distortion, contain only the frequency information of
the signal. They do not contain time information. This means that frequency analysis is not suitable for signals whose frequencies vary over time. This idea can even be extended further to imply that
there are an infinite number of signals that could produce the same power spectrum. As an example, in Figure 3 we have two chirp signals. The top signal’s frequencies increase with time while the
bottom signal’s frequencies decrease with time. Although the frequency behavior of the two signals is different, their frequency spectra (left) computed by the FFT are identical because the energy at
individual frequencies in each signal is the same. There are a number of completely different signals that can produce this same spectrum.
Figure 3. A signal and its reverse both produce the same frequency spectrum because there is no time data associated with the signals.
The second limitation of the FFT is that it cannot detect transients or short spikes in the signal. Transients are sudden events that last for a short time in a signal and usually have low energy and
a wide frequency band. When transients are transformed into the frequency domain, their energy is spread over a wide range of frequencies. Since transients have low energy, you might not be able to
recognize their existence in the frequency domain. In Figure 4, two similar signals can be seen, with Signal 2 containing a transient. Despite the presence of the transient, the power spectra of both
signals are identical because the energy of this transient is spread over a wide range of frequencies.
Figure 4. The transient in Signal 2 cannot be seen in the frequency analysis of the signal because its energy is spread over a wide range of frequencies.
3. Order Analysis
When performing vibration analysis many sound and vibration signal features are directly related to the running speed of a motor or machine such as imbalance, misalignment, gear mesh, and bearing
defects. Order analysis is a type of analysis geared specifically towards the analysis of rotating machinery and how frequencies change as the rotational speed of the machine changes. It resamples
raw signals from the time domain into the angular domain, aligning the signal with the angular position of the machine. This negates the effect of changing frequencies on the FFT algorithm, which
normally cannot handle such phenomena.
To better understand this analysis, examine the power spectrum in Figure 5. There are two large peaks in this power spectrum. The first peak at 60 Hz corresponds to the shaft rotational speed of a
machine. The second peak, which is the 4^th harmonic of the rotational speed, corresponds to the bearings of the machine. If we would like to monitor the health of the bearings it is important that
we follow this 4^th harmonic.
Figure 5. Power spectrum of a rotating wind turbine gearbox at 60 Hz.
However, as the speed changes downward to 50 Hz, the 4^th harmonic of the power spectrum shifts downward. The peaks in a power spectrum of a rotating device are all related to the fundamental
rotational speed of that device. So, even if the FFT is able to clearly analyze the data and show the power spectrum for the machine, it is not capable of easily tracking speed driven harmonics.
Figure 6. Power spectrum a rotating wind turbine gearbox at 50 Hz.
In order analysis, instead of taking the FFT of the time domain data, the signal is first resampled into the angular domain. Resampling combines the speed measurements taken from a tachometer on the
machine with the vibration measurements and interpolates the vibration measurements into a data point per fraction of angular rotation. The vibration measurements are now in the angular domain as
compared to the former time domain. Once in the angular domain, an FFT can be performed on the angular domain vibration measurement to produce what is known as an order spectrum. Figure 7 shows the
order spectra of the same shaft at both 60 Hz (top) and 50 Hz (bottom).
Figure 7. Order spectra of the wind turbine rotating at 60 Hz (top) and 50 Hz (bottom).
Notice that the 4^th harmonic is no longer in terms of frequency but harmonics, or orders of the fundamental rotational speed of the machine, where the first order corresponds to one times the
rotational speed of the machine and the fourth order corresponds to four times the rotational speed. The 4^th harmonic no longer shifts as the rotational speed of the machine changes, making it much
easier to monitor the harmonics of a rotating system.
4. Time Frequency Analysis
One of the drawbacks of frequency analysis was that, with no time domain data associated with the signal, it was only useful for static signals. Time-Frequency Analysis (sometimes called Joint
Time-Frequency Analysis or JTFA) allows a work around to this problem. Time-frequency Analysis is the process of taking multiple FFT’s of small portions of data, or rather data that was taken over a
short period of time. If the FFT’s are taken of small enough portions of data the frequencies will not have had time to change, these FFT’s can then be combined to see how the power spectrum of a
signal changes over time. Time-frequency Analysis results are usually displayed in a spectrogram, which shows how the energy of a signal is distributed in the time-frequency domain. A spectrogram is
an intensity graph with two independent variables: time and frequency. The x-axis is time, and the y-axis is frequency. The color intensity shows the power of the signal at the corresponding time and
As a simple example, consider a constant amplitude sound measurement whose frequency changes over time. This chirp sound has a frequency that linearly changes with time. Similar to the transient
impact of Figure 4, the FFT has trouble distinguishing between frequency and time. However, the spectrogram clearly shows how the frequency changes with time. If you recall we earlier examined two
chirp signals with frequency analysis and noted that they produced the same power spectrum, making the two signals indistinguishable. If we instead use Time-Frequency Analysis of the signals we can
see how they differ as demonstrated in Figure 8.
Figure 8. Joint Time-Frequency Analysis of a two chirp signals. One with frequency increasing over time (left) and the other with frequency decreasing over time (right).
Because time-frequency analysis represents a signal in the time-frequency domain, the results shown in the spectrogram reveal how the frequency components of a signal change over time. It can
therefore be seen that Time-Frequency Analysis is suitable for analyzing time-varying signals.
Some signals might have a narrow frequency band and last for a short time duration. These signals can have a good concentration in the time-frequency domain. Noise signals usually are distributed in
the entire time-frequency domain. So the time-frequency representation might be able to improve local signal-to-noise ratio in the time-frequency domain. That means you might recognize the existence
of a signal that might not be recognized in another domain.
A common use case of Time-Frequency Analysis is production testing for speakers. In typical production testing, speakers typically play a log chirp from 10Hz to 20kHz. Operators listen to the speaker
and judge the quality of the speakers.
You can use Time-Frequency analysis algorithms to analyze the sound generated by a speaker to automate a speaker quality test, replacing a “human” analyzer. In Figure 9, you can see the good speaker
generates the expected frequency components (log-chirp) with the exception of a few harmonics and yields a “clean” spectrogram. Conversely, the spectrogram of the failed speaker contains many
abnormal components.
Figure 9. Joint Time-Frequency Analysis of a bad speaker, with many abnormal components, compared to that of a good speak with a “clean” spectrogram
5. Quefrency Analysis
A Cepstrum Analysis, also called quefrency analysis, is the FFT of the log of a vibration spectrum. “Cepstrum” gets its name by reversing the first four letters of “spectrum”. The independent
variable on the x-axis of an FFT or power spectrum is frequency. The independent variable of a cepstrum is called “quefrency”. The name quefrency is derived from frequency by replacing the first
three letters of “frequency” with the second three letters of “frequency”.
Quefrency is a measure of time but not in the sense of time domain. While a frequency spectrum or FFT reveals the periodicity of a time domain measurement signal, the cepstrum reveals the periodicity
of a spectrum. A cepstrum is often referred to as a spectrum of a spectrum. Figure 10 depicts the relationship between a spectrum and a cepstrum.
Figure 10. The relationship between a spectrum and a cepstrum.
Cepstrum Analysis is especially useful for detecting harmonics. Harmonics are periodic components in a frequency spectrum and are common in machine vibration spectra. With the cepstrum, it is
possible to detect vibration harmonics such as those exhibited by a faulty roller bearing.
A roller element bearing is composed of an outer ring, an inner ring, and several roller element balls. When a failure develops in the outer or inner ring, the measured vibration signal will exhibit
larger frequency energy around the fault frequency of the inner or outer race. These characteristic frequencies are related to the geometries of the bearing including the number of balls, size of
races, and the rotational speed of the machine.
Figure 11 provides an example of bearing faults and cepstrum analysis. The power spectrum of the bearing vibration signal with an outer ring fault has a spectrum peak at 90Hz along with several
harmonics. The power spectrum of the bearing vibration signal with an inner ring fault has a spectrum peak at 120Hz along with several harmonics. There is also a significant 90Hz peak in the power
spectrum of a good bearing. In other words, it is not always possible to differentiate between good bearings and faulty bearings with a power spectrum alone. Looking at all three power spectrum in
Figure 11, harmonics are visible, just not easily noted without the aid of additional analysis.
A cepstrum is a good way to detect harmonics in the spectrum. The cepstrum of the bearing with a fault in its outer ring has an obvious peak at 11.2ms corresponding to harmonics of 90Hz. The cepstrum
of the bearing with a fault in its inner ring has an obvious peak at 8.3ms corresponding to 120Hz. The cepstrum of the good bearing does not have obvious peaks.
Figure 11. Power spectra and cepstrums of a bearing with an outer ring fault (top), an inner ring fault (middle), and a no fault (bottom)
6. Wavelet Analysis
Wavelet analysis is appropriate for characterizing machine vibration signatures with narrow band-width frequencies lasting for a short time period. For example, a cooling tower during a speed change
may produce a transient vibration measurement signal from its bearings, footing, shaft or other mechanical components. Another area where wavelet analysis is used is testing and monitoring of low RPM
gear boxes such as those in wind turbines and locomotives.
Wavelets are used as the reference in wavelet analysis and are defined as signals with two properties: admissibility and regularity. Admissibility means that a wavelet reference, or mother wavelet,
must have a band-pass-limited spectrum. Admissibility also means that wavelets must have a zero average in the time domain which implies that wavelets must be oscillatory. Regularity means that
wavelets have some smoothness and concentration in both the time and frequency domains, which means that wavelets are oscillatory and compact signals.
As comparison, sine waves oscillate along the time axis forever in time without any decay, which means they are not compact. In other words, sine waves do not have any concentration in the time
domain. On the other hand, sine waves have extreme concentration in frequency domain. Sine waves have maximum resolution in frequency domain but no resolution in time domain.
Wavelets have limited bandwidth in the frequency domain and compact bandwidth in the time domain. So, wavelets have a good concentration and resolution trade-off between the time and frequency
domain. Figure 12 depicts the differences between a sine wave and a wavelet in both time and frequency domains.
Figure 12. The frequency domain of a sine wave is very compact while the time domain is not. A wavelet is compact in both the time and frequency domain.
Wavelet Analysis then makes use of thousands of predefined wavelets. The vibration signal is then run through pattern matching algorithms which compares the signal to the known library of wavelets
representing different phenomena such as knocks and spikes of different frequencies, amplitudes and durations. The pattern matching algorithms will then return a coefficient indicating the “goodness”
of the match. A high coefficient indicates a good wavelet match and thus can be used to indicate a transient or noise impulse.
An example use of wavelets is the detection of engine knock in a diesel engine. Diesel engines can develop engine knock based on poor fuel, improper timing, or low engine compression. Engine knock
can result in poor fuel efficiency, excessive engine vibration, or damage to the piston.
Measuring the combustion signature with an accelerometer or dynamic pressure transducer allows for examination of the combustion waveform. However, as engine knock is a non-stationary transient
event, it is hard to distinguish in the time waveform or the typical FFT spectrum. However, the wavelet filter isolates the knock signature so that it is more easily detected in the time domain.
Figure 13 shows the time waveform of the diesel combustion cylinder of a normal engine and that of an engine with engine knock malfunctions. With the use of the wavelet filter, the engine knock is
isolated and available in the time domain. Limits can then be used in the time domain to identify and count engine knocks. As the frequency of engine knocks increases, corrective action may be
Figure 13. In the time domain the good motor is indistinguishable from the faulty motor, but using wavelet analysis the faults readily become apparent.
Wavelets are also finding use in monitoring of industrial gearbox such as those in helicopter or wind energy applications. Wavelets in these applications enhance the impact phenomena of cracked,
broken, and missing gear teeth.
7. Model Based Analysis
Model based analysis compares the vibration signal to a linear model of the signal and returns the error between the two which makes it useful for detecting transients. Autoregressive modeling
analysis is the use of a linear model, the AR model. The AR model represents any sample in a time series as the combination of the past samples in the same time series. The linear combination ignores
any noise and transients in the signal. When comparing a new measurement signal to the AR model, the modeling error corresponds to the noise and transients not recorded in the linear combination
Autoregressive Model analysis is useful for detecting transients in a machine vibration signal. Such transients can occur when a machine changes states, experiences variances in load, or begins to
develop a fault vibration that is non-periodic. The difference between the current vibration measurement and the AR model, also known as the modeling error, indicates transient vibrations in the
measured signal, Figure 14.
Figure 14. AR modeling error indicating transients in the original measured signal
8. Conclusion
By capturing dynamic measurements from operating machines such as vibration, electrical power, and dynamic pressure; it is possible to extract key component signature features. With this feature
information, it is possible to tabulate specific metrics which drive plant maintenance and production schedules.
Component features are best extracted from sound and vibration signals when the appropriate signal analysis technique is used. By understanding the time-frequency characteristics of the raw signal,
the algorithms that are most important can be identified. Figure 2, as listed earlier, provides a guideline to the selection of analysis methods for a range of time-frequency characteristics of the
sound and vibration signature.
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Math Forum Discussions
Math Forum
Ask Dr. Math
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Search All of the Math Forum:
Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.
Topic: Binary or Logical Selection : Given an array A and a binary
orlogical vector S, how do I select the rows of A when S = 1 or True ?
Replies: 2 Last Post: Sep 24, 2012 12:46 PM
Messages: [ Previous | Next ]
Re: Binary or Logical Selection : Given an array A and a binary
orlogical vector S, how do I select the rows of A when S = 1 or True ?
Posted: Sep 24, 2012 2:31 AM
On Sun, 23 Sep 2012 22:00:15 +0200, clicliclic wrote:
> Beau Webber schrieb:
>> Given a 3 column array A of x,y,z numbers, defining a set of spatial
>> coordinates, and a binary or logical selection vector S (with as many
>> elements as rows in A) :
>> How do I express the operation of selecting an array R with the set of
>> those co-ordinates in A where S is 1 (or true) ?
>> In Apl I write this as R <- S /[1] A .
>> But a referee of a paper is asking that I express this in standard
>> maths, and I have not yet managed to find an appropriate mathematical
>> expression.
> I can imagine that referees don't like mathematical explanations in
> terms of APL - unless yours is a paper about APL (does the language
> still exist?). But I can't think of an elegant way to express this in
> standard vector notation either. This newsgroup is only concerned with
> the development and application of computer-algebra systems like Maxima
> or Maple, I am therefore widening the audience to include sci.math.
> Can anybody help the original poster?
Perhaps write something like following, in which the form p_i_j
stands for p_{i_j} and the parenthesized lists stand for vectors
of points p_i.
Suppose A = (p_1, p_2, ... p_n). Let R = (p_i_1, p_i_2, ... p_i_k)
where i_j < i_{j+1} for all the j, and i \in { i_j } iff S_i = 1.
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Lesson 2.1 Key Terms
to draw a geometric figure around another figure so that the two are in contact but do not intersect
a hole (usually in wood) with the top part enlarged so that a screw or bolt will fit into it and lie below the surface
(1) A hole consisting of two depths so that the top of the head of an inserted bolt or screw will be flush with the outside surface. (2) A bit for enlarging the upper part of a hole.
the length of a straight line passing through the center of a circle and connecting two points on the circumference
A regular oval shape, traced by a point moving in a plane so that the sum of its distances from two other points is constant, or resulting when a cone is cut by an oblique plane which does not
intersect the base.
A modeling process that creates a three-dimensional form by defining a closed two-dimensional shape and a length.
The numerical value of the ratio of the circumference of a circle to its diameter of approximately 3.14159.
a polyhedron with two congruent and parallel faces (the bases) and whose lateral faces are parallelograms
Solid Modeling
A type of 3D CAD modeling that represents the volume of an object, not just its lines and surfaces. This allows for analysis of the object's mass properties.
Cartesian Coordinate System
a coordinate system for which the coordinates of a point are its distances from a set perpendicular lines that intersect at the origin of the system
Computer-Aided Design
Software that allows you to create engineering, architectural, and scientific designs
Geometric Constraint
Constant, non-numerical relationships between the parts of a geometric figure. Examples include parallelism, perpendicularity, and concentricity.
Numeric Constraint
A number value, or algebraic equation that is used to control the size or location of a geometric figure.
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Homework Help
Posted by Lou on Monday, April 13, 2009 at 8:35pm.
The mass of the CuSO4 x 5H2O sample is 1.664 g. The mass of the recovered copper is 1.198 g.
Calculate the percent copper in pure copper sulfate pentahydrate(this is the theoretical percent copper). One mole of copper has a mass of 63.55 g, and by addition, one mole of CuSO4 x 5H2O has a
mass of 249.5 g.
• chem - DrBob222, Monday, April 13, 2009 at 8:42pm
So what's the question? Percent Cu in the sample?
(1.198/1.664)*100 = ??
Theoretical yield =
(mass Cu/mass hydrate)*100 = (63.55/249.5)*100 = ??
• chem - Lou, Monday, April 13, 2009 at 8:57pm
The question is asking for percent of copper in the sample( so the first step shown above), but when I had to calculate the experimental percent I did the same thing, and i got 72%( is that the
correct number of significant figures)?
I also wanted to know when I have to calculate the error of the % of copper what number do I use as true value and actual value?
• chem - Lou, Monday, April 13, 2009 at 10:25pm
The mass of the CuSO4 x 5H2O sample is 1.664 g. The mass of the recovered copper is 1.198 g.
1. Compute the experimental percent by mass of copper in the sample:
1.1664/1.998x 100=71.81%
2. Calculate the percent copper in pure copper sulfate pentahydrate(this is the theoretical percent copper). One mole of copper has a mass of 63.55 g, and by addition, one mole of CuSO4 x 5H2O
has a mass of 249.5 g:
63.55/249.5x100= 25.47%
3. In order to calculate the error of the % Cu and the percent error of the % Cu I know I am suppose to use 1.198 as the experimental value, but I am not sure what to use for the true value.
• chem - Lou, Monday, April 13, 2009 at 10:26pm
The mass of the CuSO4 x 5H2O sample is 1.664 g. The mass of the recovered copper is 1.198 g.
1. Compute the experimental percent by mass of copper in the sample:
1.1664/1.998x 100=71.81%
2. Calculate the percent copper in pure copper sulfate pentahydrate(this is the theoretical percent copper). One mole of copper has a mass of 63.55 g, and by addition, one mole of CuSO4 x 5H2O
has a mass of 249.5 g:
63.55/249.5x100= 25.47%
3. In order to calculate the error of the % Cu and the percent error of the % Cu I know I am suppose to use 1.198 as the experimental value, but I am not sure what to use for the true value.
• chem - DrBob222, Monday, April 13, 2009 at 10:43pm
Answered above.
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Galilean transformation
Topic: Galilean transformation
Related Topics
In the News (Sun 13 Apr 14)
Lorentz transformation - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-11-07)
Under these transformations, the speed of light is the same in all reference frames, as postulated by special relativity.
The Lorentz transformation is a group transformation that is used to transform the space and time coordinates (or in general any four-vector) of one inertial reference frame, $S, into those of
another one, S", with S" traveling at a relative speed of {v} to S along the x-axis.$
The Lorentz transformations were first published in 1904, but their formalism was at the time imperfect.
www.newlenox.us /project/wikipedia/index.php/Lorentz_transformation_equations (792 words)
Galilean transformation - Wikipedia, the free encyclopedia
The Galilean transformation is used to transform between the coordinates of two coordinate systems in a constant relative motion in Newtonian physics.
Unlike the Galilean transformation, the relativistic Lorentz transformation can be shown to apply at all velocities so far measured, and the Galilean transformation can be regarded as a
low-velocity approximation to the Lorentz transformation.
Under the Erlangen program, the space-time (no longer spacetime) of nonrelativistic physics is described by the symmetry group generated by Galilean transformations, spatial and time translations
and rotations.
en.wikipedia.org /wiki/Galilean_transformation (368 words)
Lorentz transformation - Wikipedia, the free encyclopedia
This is in contrast to the more intuitive Galilean transformation, which is sufficient at non-relativistic speeds (i.e.
The Lorentz transformation is a group transformation that is used to transform the space and time coordinates (or in general any four-vector) of one inertial reference frame, S, into those of
another one, S', with S' traveling at a relative speed of v to S along the x-axis.
Beneath the Foundations of Spacetime The Lorentz transformation can be derived with moving rulers in such a way that the astonishing connection between space and time can be clearly understood.
en.wikipedia.org /wiki/Lorentz_transformation (705 words)
Invariant Galilean Transformations (FAQ) On All Laws
In particular, the Principle of Relativity is embodied in the form of the Galilean transformation, which relates the original x, y, z, t to x', y', z', t' by the transform equations x'=x-vt, y'=y,
z'=z, t'=t in the simplified case where attention is focused only on transforming the x-axis, and not y and z.
As the transform equations say, the relationship of t', x' to t, x is based on the relative velocity between the two systems, but neither the original (eq-99) equation nor the M observer equation
is about a relationship between coordinate systems or observers.
Their process, which says (x'+vt') is the transform of x, says that (x'+vt') is the moving system location of x, but it can't be because x is moving further in the negative direction from the
moving viewpoint.
www.cs.uu.nl /wais/html/na-dir/physics-faq/criticism/galilean-invariance.html (6572 words)
Math Forum - Ask Dr. Math
You're quite right: the Galilean transformation is a simple thing expressed in complex (or at least unfamiliar) terminology, and it doesn't work right for frames traveling near the speed of light.
The Galilean transformation, underneath its disguise, is old, familiar stuff, _not_ relativistic physics - which is why it doesn't work right (in relativistic terms) near the speed of light.
The Galilean transformation is: (1) x' = x - vt (2) t' = t where v is the relative speed between two reference frames (x, t) and (x',t').
mathforum.org /library/drmath/view/51487.html (1058 words)
Science Fair Projects - Principle of relativity
In Galilean relativity, reference frames are related to each other in an intuitive way: to transform the velocity of an object from one frame to another, the vector representing the velocity of
the object is added to the vector representing the velocity difference between the two reference frames.
Einstein saw, as did his contemporaries, that if one assumes that both the Maxwell equations are valid, and that Galilean transformation is the appropriate transformation, then it should be
possible to measure velocity absolutely.
Einstein saw that if one assumes that the Lorentz transformations are the appropriate transformations for transforming between inertial reference frames, then that constitutes a principle of
relativity that is compatible with the Maxwell equations.
www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Galilean-Newtonian_relativity (944 words)
Galilean transformation -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-07)
The Galilean transformation is used to transform between the coordinates of two coordinate systems in constant relative motion in (Click link for more info and facts about Newtonian physics)
Newtonian physics.
The equations below, although apparently obvious, break down at speeds that approach the (The speed at which light travels in a vacuum; the constancy and universality of the speed of light is
recognized by defining it to be exactly 299,792,458 meters per second) speed of light.
Unlike the Galilean transformation, the (Click link for more info and facts about relativistic) relativistic (Click link for more info and facts about Lorentz transformation) Lorentz
transformation can be shown to apply at all velocities so far measured, and the Galilean transformation can be regarded as a low-velocity approximation to the Lorentz transformation.
www.absoluteastronomy.com /encyclopedia/G/Ga/Galilean_transformation.htm (521 words)
Comments on Problem Set #4 (Site not responding. Last check: 2007-11-07)
The principle of Galilean relativity states that the laws of motion are the same as viewed from any inertial frame of reference.
The Galilean transformation has the property that when it is applied to Newton's laws of motion, the result is unchanged (that is, you get Newton's laws of motion).
The Galilean transformation (referred to as the ``classical transformation'' in Einstein and Infeld) turns out to be incompatible with the postulates of Special Relativity.
www.physics.nyu.edu /courses/V85.0020/node83.html (1534 words)
Staircase Wit
In addition, we notice that the rotational transformations maintain the orthogonality of the coordinate axes, whereas the lack of an invariant measure for the Galilean transformations prevents us
from even assigning a definite meaning to “orthogonality” between the time and space coordinates.
Since the velocity transformations leave the laws of physics unchanged, Minkowski reasoned, they ought to correspond to some invariant physical quantity, and their determinants ought to be unity.
It is certainly true that we are led toward the Lorentz transformations as soon as we consider the group of velocity transformations and attempt to identify a physically meaningful invariant
corresponding to these transformations.
www.mathpages.com /rr/s1-07/1-07.htm (3507 words)
Notes on Relativity
A coordinate transformation is the set of equations for converting the space and time coordinates of an event as measured in one reference frame to the coordinates measured in another frame.
Galilean transformation is implied by the writings of Galileo, although he did not write the equations.
The Lorentz transformation is equivalent to length contraction and time dilation (which had been proposed by Fitzgerald and by Lorentz earlier).
phys-astro.sonoma.edu /people/faculty/tenn/P314/RelativityNotes.html (421 words)
Newtonian physics - Wikipedia
In pre-Einstein relativity (known as Galilean relativity), time is considered an absolute in all reference frames.
The space-time coordinates of an event in Galilean-Newtonian relativity are governed by the set of formulas which defines a group transformation known as the Galilean transformation:
The set of formulas defines a group transformation known as the Galilean transformation (informally, the Galilean transform).
nostalgia.wikipedia.org /wiki/Newtonian_physics (398 words)
The Theory of Relativity: An Error of the Transformation of Coordinates *
the principle of existence and of transformation of coordinates: there are no coordinates and no transformation of coordinates in general, and there exist the coordinates and transformation of the
coordinates of the object only.
The Galilean transformation relates the coordinates of the point $M$ in the systems $S$ and $E$: $x_{M} = Vt + x^{\prime }_{M}$ where $V$ is the velocity of motion of the system $E$ relative to
the system $S$ in the positive direction of the axis $Ox$ ($V < c$).
Introduction (insertion) of the Galilean transformation into the equation for the front of the light beam means equality between the coordinates:
wbabin.net /physics/theoryrel.htm (597 words)
Lorentz transformation (Site not responding. Last check: 2007-11-07)
The Lorentz transformation, named after its discoverer, a Dutch physicist and mathematician Hendrik Antoon Lorentz (1853-1928), forms the basis for the special theory of relativity, that has been
introduced to remove contradictions between the theories of electromagnetism and classical mechanics.
If c is taken to be infinite, the Galilean transformation is recovered, such that it may be indentified as a limiting case.
The Lorentz transformation is a group transformation that is used to transform the space and time coordinates (or in general any four-vector) of one inertial reference frame, $S$, into those of
another one, $S"$, with $S"$ traveling at a relative speed of ${\mathbf u}$ to $S$.
www.city-search.org /lo/lorentz-transformation.html (882 words)
Lorentz Transformation Equations (Site not responding. Last check: 2007-11-07)
In the introduction I mentioned that classical mechanics required the use of Galilean Transformation equations to transform the results in one inertial frame of reference into another inertial
However, as was already shown, this transformation becomes less and less accurate as the velocity of the body approaches the speed of light.
The equations for transforming into a moving frame of reference (x prime, y prime, z prime, and t prime coordinates) are on the left.
ffden-2.phys.uaf.edu /212_fall2003.web.dir/Eddie_Trochim/Lorentztransform.htm (393 words)
Encyclopedia: Galilean-transformation (Site not responding. Last check: 2007-11-07)
The Galilean symmetries (interpreted as active transformations): An active transformation is one which actually changes the physical state of a system and makes sense even in the absence of a
coordinate system whereas a passive transformation is merely a change in the coordinate system of no physical significance.
In mathematics, a Lie algebra is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds.
representation theory of the Galilean group, Poincaré group In nonrelativistic quantum mechanics, an account can be given of the existence of mass and spin as follows: The spacetime symmetry
group of nonrelativistic mechanics is the Galilean group.
www.nationmaster.com /encyclopedia/Galilean_transformation (1381 words)
Invariant Galilean Transformations (FAQ) On All Laws (Site not responding. Last check: 2007-11-07)
Below, in the sci.math subject, we see that all sci.math respondents agree with the basic "controversial" position of this faq: every coordinate is transformed, whether a supposed "constant" or
What does sci.math have to say about x0'=x0-vt? The crackpots' positions/arguments were put to sci.math in such a way that at least two or three who posted re- sponses thought it was your faq-er
who was on the idiot's side of the questions.
A linear transformation, A, on the space is a method of corr- esponding to each vector of the space another vector of the space such that for any vectors U and V, and any scalars a and b, A(aU+bV)
= aAU + bAV.
omicron.felk.cvut.cz /FAQ/articles/a3503.html (6587 words)
Therefore, the Lorentz transformation equations must reduce to the following Galilean transformation equations in the limit of small velocities.
The Galilean transformation equations and their implications are discussed in greater detail on page 4 in
principle of relativity.) Hint: The Galilean transformations pertain to two arbitrary inertial frames, so any law that is invariant under a Galilean transformation takes on the same form in all
inertial frames.
physics.tamuk.edu /~hewett/ModPhy1/Unit1/SpecialRelativity/RelativeView/LorentzTransform/GalileanTransform/GalileanTransform.html (491 words)
The Ultimate Galilean invariance Dog Breeds Information Guide and Reference
Galilean invariance is a principle which states that the fundamental laws of physics are the same in all inertial (uniform-velocity) frames of reference.
Specifically, the term Galilean invariance today usually refers to this principle as applied to Newtonian mechanics, under which all lengths and times remain unaffected by a change of velocity,
which is described mathematically by a Galilean transformation.
At the low relative velocities characteristic of everyday life, Lorentz invariance and Galilean invariance are nearly the same, but for relative velocities close to that of light they are very
www.dogluvers.com /dog_breeds/Galilean_invariance (197 words)
Galilean transformation: Definition and Links by Encyclopedian.com - All about Galilean transformation (Site not responding. Last check: 2007-11-07)
Galilean transformation: Definition and Links by Encyclopedian.com - All about Galilean transformation
For example, if the frames in Fig.11.1 were exactly lined up at t = t' = 0, then the coordinate of the explosion in frame O would be equal to the coordinate x' as measured by O' plus the distance
that the frames had moved relative to each other in that time interval:
Eqs[11.1] and [11.2] constitute the so-called Galilean transformations relating coordinates as measured in two different frames.
www.encyclopedian.com /ga/Galilean-transformation.html (931 words)
[No title]
Galilean transformations Although we still accept MaxwellÕs equations today in the same form as in the 19th century, their interpretation is completely different.
transforms as a contravariant 4-vector transforms as a covariant 4-vector.
This is one reason for discarding the Lorentz transformation with F = -1, considered earlier, because such a transformation on the light cone would take us from a timelike ordering of events to a
spacelike one, or vice versa.
www3.baylor.edu /Physics/open_text/classical/ch13.2003.doc (4041 words)
The Collapse of the Lorentz Transformation
The aim of the Lorentz transformations (1) is to calculate the relationships between the lengths and time units between a frame supposedly at rest and another frame in motion, assuming that the
same velocity of light is measured in both frames.
In fact, the Lorentz transformations predicts only the transformation that gives an “average” velocity of light equal to c, which means that the velocity of light is slower in the forward
direction and faster in the backward direction, in the moving frame, just as illustrated in equation 17.
After a century, it is astonishing to discover that the Lorentz transformation, that requires a distortion between the X and Y axis, does not lead to a constant (one-way) velocity of light when
"measured" in the moving frame.
www.newtonphysics.on.ca /lorentz/lorentz.html (4367 words)
[No title] (Site not responding. Last check: 2007-11-07)
Galilean transformation (Common sense) 2.4 Consequences of Einstein’s Postulates 2.4.1 The relativity of time We consider a timing device which ticktack by light flashed from a bulb and reflected
by a mirror.
Thus, transforming to the reference frame of O’, EMBED Equation.3 Where x,y,z are the coordinate at the reference frame of O, and x’,y’,z’ are the coordinate at reference frame of O’.
The equation of the Lorentz transformation, derived from these treatments, are EMBED Equation.3 Where x is the rest (proper) distance measured by O at the reference frame of O, and x’ is the
distance measured by O’ at the reference frame of O’.
vega.icu.ac.kr /~ois/download/ICE2251/ICE2251_2.doc (1992 words)
Einstein's Theory of Relativity - Scientific Theory or Illusion?
A similar transformation can also be derived when the axes of these two systems are at a certain angle, that is when they are not parallel.
The above mentioned transformation is called Galilean transformation in honor of the founder of mechanics.
So, at the transformation of coordinates, the equation for the inertial law has remained the same, which means that with Galilean transformation is maintained the invariability of the equation for
acceleration in the case of an inertial system.
users.net.yu /~mrp/chapter2.html (365 words)
Special Relativity Explained by Diagrams (Site not responding. Last check: 2007-11-07)
In the first section, we present the Galilean relativity and the associated space-time diagrams to illustrate various events.
The position of the light wave front along the x axis in the K-frame is given by x=ct, where c is the speed of light in this frame.
K' moves at constant speed (v) along the x axis relative to the K-frame, and t = t' = 0 when O' coincides with O and event E is happening somewhere in space at a given moment in time.
www.colvir.net /prof/richard.beauchamp/rel-an/rela.htm (1588 words)
Is a Crystal-clear Theory Preferable to Dogmatic Riddles?
Essentially ==he takes Galilean transformations, and then does a change of ==variables to put it in the form of the Lorentz transformation, ==then claims to have derived the Lorentz transformation
from the ==Galilean transformation.
Essentially = he takes Galilean transformations, and then does a change of = variables to put it in the form of the Lorentz transformation, = then claims to have derived the Lorentz transformation
from the = Galilean transformation.
Essentially he takes = Galilean transformations, and then does a change of variables to put it in = the form of the Lorentz transformation, then claims to have derived the = Lorentz transformation
from the Galilean transformation.
www.pych-one.com /new-6421194-4388.html (8125 words)
Try your search on: Qwika (all wikis)
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de Bruijn, On the number of positive integers ≤ x and free of prime factors ≥ y, Indag
Results 1 - 10 of 21
- Mathematics of computation , 1996
"... Abstract. We call an integer semismooth with respect to y and z if each of its prime factors is ≤ y, and all but one are ≤ z. Such numbers are useful in various factoring algorithms, including
the quadratic sieve. Let G(α, β)bethe asymptotic probability that a random integer n is semismooth with res ..."
Cited by 22 (1 self)
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Abstract. We call an integer semismooth with respect to y and z if each of its prime factors is ≤ y, and all but one are ≤ z. Such numbers are useful in various factoring algorithms, including the
quadratic sieve. Let G(α, β)bethe asymptotic probability that a random integer n is semismooth with respect to n β and n α. We present new recurrence relations for G and related functions. We then
give numerical methods for computing G,tablesofG, and estimates for the error incurred by this asymptotic approximation. 1.
- ALGORITHMIC NUMBER THEORY , 2008
"... ..."
- Illinois J. Math , 1967
"... n + 1,..., n + k be consecutive composite numbers. Then for each i, 1 s i s k there is a p i, p i I n + i pi # pi for 1 1 ~ 1 2. 1 2 He also expressed the conjecture in a weaker form stating
that any set of k consecutive composite numbers need to have at least k prime factors. We first show that eve ..."
Cited by 9 (0 self)
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n + 1,..., n + k be consecutive composite numbers. Then for each i, 1 s i s k there is a p i, p i I n + i pi # pi for 1 1 ~ 1 2. 1 2 He also expressed the conjecture in a weaker form stating that any
set of k consecutive composite numbers need to have at least k prime factors. We first show that even in this weaker form the conjecture goes far beyond what is known about primes at present. First
we define a few number theoretic functions. Denote by o (n, k) the number of distinct prime factors of (n + 1)... (n + k). f 1 (n) is the smallest integer k so that for every 1 s t,4- k v (n, t) Z-t
but v (n, k + 1) = k. f0 (n) is the largest integer k for which v (n, k) k k. Clearly f0 (n) i f1 (n) and we shall show that infinitely often f0 (n)> f1 (n). Following Grimm let f2 (n) be the largest
integer k so that for each 1-. i!9k there is a p i ln + i, pi 1 1 pi if i 1 / 12. 2 Denote by P(m) the greatest prime factor of m. f.(n) is the
- Mathematics of Computation , 2004
"... Abstract. Ψ(x, y) denotes the number of positive integers ≤ x and free of prime factors>y. Hildebrand and Tenenbaum gave a smooth approximation formula for Ψ(x, y) in the range (log x) 1+ɛ
Cited by 3 (0 self)
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Abstract. Ψ(x, y) denotes the number of positive integers ≤ x and free of prime factors>y. Hildebrand and Tenenbaum gave a smooth approximation formula for Ψ(x, y) in the range (log x) 1+ɛ <y ≤
x,whereɛ is a fixed positive number ≤ 1/2. In this paper, by modifying their approximation formula, we provide a fast algorithm to approximate Ψ(x, y). The computational complexity of this algorithm
is O ( � (log x)(log y)). We give numerical results which show that this algorithm provides accurate estimates for Ψ(x, y) andisfaster than conventional methods such as algorithms exploiting
Dickman’s function. 1.
, 2006
"... Let P(n) denote the largest prime divisor of n, andlet Ψ(x,y) be the number of integers n ≤ x with P(n) ≤ y. Inthispaper we present improvements to Bernstein’s algorithm, which finds rigorous
upper and lower bounds for Ψ(x,y). Bernstein’s original algorithm runs in time roughly linear in y. Our fi ..."
Cited by 3 (2 self)
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Let P(n) denote the largest prime divisor of n, andlet Ψ(x,y) be the number of integers n ≤ x with P(n) ≤ y. Inthispaper we present improvements to Bernstein’s algorithm, which finds rigorous upper
and lower bounds for Ψ(x,y). Bernstein’s original algorithm runs in time roughly linear in y. Our first, easy improvement runs in time roughly y 2/3. Then, assuming the Riemann Hypothesis, we show
how to drastically improve this. In particular, if log y is a fractional power of log x, which is true in applications to factoring and cryptography, then our new algorithm has a running time that is
polynomial in log y, and gives bounds as tight as, and often tighter than, Bernstein’s algorithm.
"... Abstract. We adapt the Maier matrix method to the polynomial ring Fq[t], and prove analogues of results of Maier [4] and Shiu [10] concerning the distribution of primes in short intervals. 1.
Cited by 3 (2 self)
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Abstract. We adapt the Maier matrix method to the polynomial ring Fq[t], and prove analogues of results of Maier [4] and Shiu [10] concerning the distribution of primes in short intervals. 1.
, 2000
"... Using simple examples and informal discussions this article surveys the key ideas and major advances of the last quarter century in integer factorization. ..."
Cited by 2 (0 self)
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Using simple examples and informal discussions this article surveys the key ideas and major advances of the last quarter century in integer factorization.
"... We find infinitely many pairs of coprime integers, a and q, such that the least prime j a (mod q) is unusually large. In so doing we also consider the question of approximating rationals by
other rationals with smaller and coprime denominators. * The second author is partly supported by an N.S.F. gr ..."
Cited by 2 (0 self)
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We find infinitely many pairs of coprime integers, a and q, such that the least prime j a (mod q) is unusually large. In so doing we also consider the question of approximating rationals by other
rationals with smaller and coprime denominators. * The second author is partly supported by an N.S.F. grant 1. Introduction For any x ? x 0 and for any positive valued function g(x) define R(x) = e
fl log x log 2 x log 4 x=(log 3 x) 2 ; L(x) = exp(log x log 3 x= log 2 x) and E g (x) = exp \Gamma log x=(log 2 x) g(x) \Delta : Here log k x is the k-fold iterated logarithm, fl is Euler's constant,
and x 0 is chosen large enough so that log 4 x 0 ? 1. The usual method used to find large gaps between successive prime numbers is to construct a long sequence S of consecutive integers, each of
which has a "small" prime factor (so that they cannot be prime); then, the gap between the largest prime before S and the next, is at least as long as S. Similarly if one wishes to find an arithm...
"... . Let N = P R where P is a prime not dividing R. We show how a special class of functions f : ZN ! Z can be used to help obtain P given N . The requirements of f are that it be non-trivial and
that f(x) = f(x mod P ). Such a function does not \see" R. Hence the name partially oblivious. 1. Intr ..."
Cited by 1 (0 self)
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. Let N = P R where P is a prime not dividing R. We show how a special class of functions f : ZN ! Z can be used to help obtain P given N . The requirements of f are that it be non-trivial and that f
(x) = f(x mod P ). Such a function does not \see" R. Hence the name partially oblivious. 1. Introduction It is not known how to eciently factor a large integer N . Currently, the algorithm with best
asymptotic complexity is the Number Field Sieve (see [6] ). For numbers below a certain size (currently believed to be about 100 decimal digits), either the Quadratic Sieve [12] or Lenstra's Elliptic
Curve Method (ECM) [7] are faster. Which of these algorithms to use depends on the size of N and of the smallest prime factor of N . When the size of the smallest factor is suciently smaller than p N
, ECM is the fastest of the three. This note describes a speedup of ECM under special conditions. Suppose N = P R, where P is a prime not dividing R. We assume the size, in bits, of P is know...
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Re: [Axiom-developer] Re: hyperdoc
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Re: [Axiom-developer] Re: hyperdoc
From: root
Subject: Re: [Axiom-developer] Re: hyperdoc
Date: Fri, 14 Jan 2005 22:34:33 -0500
> That's one of the things I'll be interested to see how CATS deals with
> - there (as I understand it, please correct me if I'm wrong Tim) the
> goal is not to be able to swap inputs and outputs, but to have as
> nearly as possible the same question posed, and see if the systems
> agree mathematically?
In fact, the idea behind CATS is that we take a well-defined math
area, choose some problems with a known solutions and write up the
problems and their solutions. Then, for each CAS, there is a section
that sets up the problem and an encoding of the expected answer so
the testing can be automated. There is no assumption that every
CAS is capable of handling every test case, nor any assumption
about common structure/function/etc.
CATS is trying to suggest a way to pool the mathematical expertise
needed to solve a problem and present it so that each CAS can
consider the problem and know they have a correct solution. Of
course, the issue of a "correct" solution is not always clear and
often a point of debate but that's why the mathematical explanation
section exists.
See Jeffrey, D.J and Norman, A.C
"Not seeing the roots for the branches: multivalued functions in
computer algebra" SIGSAM Bulletin (Association for Computing
Machinery) Vol 38, Number 3; Sept 2004 Issue 149 pp57-66
for the kind of analysis that might be expected.
Arthur Norman probably has a copy of the paper on the web somewhere.
[Prev in Thread] Current Thread [Next in Thread]
• Re: [Axiom-developer] Re: hyperdoc, (continued)
• Re: [Axiom-developer] Re: hyperdoc, root, 2005/01/14
• Re: [Axiom-developer] Re: hyperdoc, root, 2005/01/14
• Re: [Axiom-developer] Re: hyperdoc, C Y, 2005/01/14
• Re: [Axiom-developer] Re: hyperdoc, root <=
• Re: [Axiom-developer] Re: hyperdoc, C Y, 2005/01/14
• Re: [Axiom-developer] Re: hyperdoc, Bob McElrath, 2005/01/14
• Re: [Axiom-developer] Re: hyperdoc, Gabriel Dos Reis, 2005/01/14
• RE: [Axiom-developer] Re: hyperdoc, Bill Page, 2005/01/14
• Re: [Axiom-developer] Re: hyperdoc, Gabriel Dos Reis, 2005/01/14
• Re: [Axiom-developer] Re: hyperdoc, Bob McElrath, 2005/01/15
• Re: [Axiom-developer] Re: hyperdoc, Martin Rubey, 2005/01/17
• Re: [Axiom-developer] Re: hyperdoc, Bob McElrath, 2005/01/18
• Re: [Axiom-developer] Re: hyperdoc, Martin Rubey, 2005/01/18
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Isometrical (definition)
I`so*met"ric (?), I`so*met"ric*al (?), a. [Iso- + Gr. measure.]
Pertaining to, or characterized by, equality of measure.
2. Crystallog.
Noting, or conforming to, that system of crystallization in which the three axes are of equal length and at right angles to each other; monometric; regular; cubic. Cf. Crystallization.
Isometric lines Thermodynamics, lines representing in a diagram the relations of pressure and temperature in a gas, when the volume remains constant. -- Isometrical perspective. See under Perspective
. -- Isometrical projection, a species of orthographic projection, in which but a single plane of projection is used. It is so named from the fact that the projections of three equal lines, parallel
respectively to three rectangular axes, are equal to one another. This kind of projection is principally used in delineating buildings or machinery, in which the principal lines are parallel to three
rectangular axes, and the principal planes are parallel to three rectangular planes passing through the three axes.
© Webster 1913.
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Hansel Giveaway... Win a 100$ dress!
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205 comments:
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91. Follow us on Bloglovin
92. Follow us on Twitter
Follow on Twitter as @Framboesa
93. 'Like' our Facebook Page
FB as Alda Lopes Moreira
94. my fav is option 1
GFC. Skye
FB. Skye Kumi
Twitter. SkyeKumi
Email. cloudlovely@hotmail.it
95. follow on gfc
GFC. Skye
FB. Skye Kumi
Twitter. SkyeKumi
Email. cloudlovely@hotmail.it
96. retweet
GFC. Skye
FB. Skye Kumi
Twitter. SkyeKumi
Email. cloudlovely@hotmail.it
97. follow hansel on fb
GFC. Skye
FB. Skye Kumi
Twitter. SkyeKumi
Email. cloudlovely@hotmail.it
98. follow hansel on twitter
GFC. Skye
FB. Skye Kumi
Twitter. SkyeKumi
Email. cloudlovely@hotmail.it
99. follow you on bloglovin
GFC. Skye
FB. Skye Kumi
Twitter. SkyeKumi
Email. cloudlovely@hotmail.it
100. follow you on twitter
GFC. Skye
FB. Skye Kumi
Twitter. SkyeKumi
Email. cloudlovely@hotmail.it
101. follow you on fb
GFC. Skye
FB. Skye Kumi
Twitter. SkyeKumi
Email. cloudlovely@hotmail.it
102. I love the second one!
Thank you for this opportunity!
103. retweeted!
104. I 'Like' Hansel's Facebook Page
Samantha Tedesco
105. I Follow Hansel on Twitter
106. I follow you via gfc: samantha.tedesco
107. I follow you on bloglovin
108. I follow you on twitter: @samanthatedesco
109. I follow you via Fb: Samantha Tedesco
110. Option 1 is my favorite
111. I retweeted as @LuluFleg
112. GFC: Lulu
113. Option 2 is my fav ..
Email: sania dot akbar at live dot com
114. I retweeted abt giveaway: @saniaakbar
115. 'Like' Hansel's Facebook Page
FB: Sania Akbar
116. Follow Hansel on Twitter
117. GFC: Sania
118. Follow on Bloglovin: sania_1611@yahoo.com
119. Like Angela Squared on FB: Sania Akbar
120. Follow Angela Squared on Twittter: @saniaakbar
Thanks for giveaway :)
121. I like this: http://www.ilovehansel.com/hansel-details.php?g=297
erikamilani at hotmail dot it
122. Love the first one :)
123. Retweeted
124. Follow Hansel on Twitter
Follow on GFC: Sherry86
-Follow on Bloglovin
127. - Follow on Twitter
128. I like option 1
GFC Francesca Scirpoli
129. liked Hansel on fb Francesca Loveredhair
130. followed on twitter @puntimania
131. retweeted @puntimania
132. followed on blogovin
133. followed you on GFC Francesca Scirpoli
134. followed you on fb Francesca Loveredhair
135. I love option 2. That fabric is adorable!! LindseyAylward@yahoo.com
136. I followed on gfc as mamamunky. Lindseyaylward@yahoo.com
137. I liked them on fb. Lindseyaylward@yahoo.com
138. I followed them on twitter as mamamunky. Lindseyaylward@yahoo.com
139. I followed you on twitter as mamamunky. Lindseyaylward@yahoo.com
140. I liked you on fb. Lindseyaylward@yahoo.com
141. I love "option 3" dress :)
porcukorborso at gmail dot com
142. Liked Hansel's Facebook Page as Szabina Luzics
porcukorborso at gmail dot com
143. GFC follower as Szappanbubi
porcukorborso at gmail dot com
144. already a bloglovin follower via my email!
porcukorborso at gmail dot com
145. facebook fan as Szabina Luzics
porcukorborso at gmail dot com
146. option n.5
gloriatea at hotmail dot com
147. retweeted!
gloria_tea on twitter
gloriatea at hotmail dot com
148. 'Like' Hansel's Facebook Page
gloria Tea
gloriatea at hotmail dot com
- Follow Hansel on Twitter
gloriatea at hotmail dot com
- Follow u on GFC
gloria tea
gloriatea at hotmail dot com
-Follow u on Bloglovin
gloriatea at hotmail dot com
152. Love the print! Option 3 is a winner :)
153. RT the giveaway @LornaBB
154. Like Hansel on FB
155. Follow Hansel on Twitter @LornaBB
156. Follow you on GFC as Lorna
157. Follow you on Bloglovin'
158. Follow you on Twitter @LornaBB
159. Already like your FB page
160. Great dresses !!! (:
xx from
161. Like n°5.
MAIL ricciolidoro1910@libero.it
162. Retweeted. @Ricciolidoro191
MAIL ricciolidoro1910@libero.it
163. Liked Hansel's Facebook Page.
FB Rioccioli D'oro
MAIL ricciolidoro1910@libero.it
164. Following Hansel on Twitter.
TW: @Ricciolidoro191
MAIL ricciolidoro1910@libero.it
165. Following on GFC: seresta99
MAIL ricciolidoro1910@libero.it
166. Following on Bloglovin with the MAIL ricciolidoro1910@libero.it
167. Following on Twitter.
TW: @Ricciolidoro191
MAIL ricciolidoro1910@libero.it
168. Liked Facebook Page.
FB: Riccioli D'oro
MAIL ricciolidoro1910@libero.it
169. love option 1
GFC: Cute Girl
Facebook: Sae Kurosawa
Twitter: Tifafd
Email: Tifafd@libero.it
170. follow on gfc
GFC: Cute Girl
Facebook: Sae Kurosawa
Twitter: Tifafd
Email: Tifafd@libero.it
171. retweet
GFC: Cute Girl
Facebook: Sae Kurosawa
Twitter: Tifafd
Email: Tifafd@libero.it
172. follow hansel on fb
GFC: Cute Girl
Facebook: Sae Kurosawa
Twitter: Tifafd
Email: Tifafd@libero.it
173. follow hansel on twitter
GFC: Cute Girl
Facebook: Sae Kurosawa
Twitter: Tifafd
Email: Tifafd@libero.it
174. follow you on bloglovin
GFC: Cute Girl
Facebook: Sae Kurosawa
Twitter: Tifafd
Email: Tifafd@libero.it
175. follow you on twitter
GFC: Cute Girl
Facebook: Sae Kurosawa
Twitter: Tifafd
Email: Tifafd@libero.it
176. follow you on fb
GFC: Cute Girl
Facebook: Sae Kurosawa
Twitter: Tifafd
Email: Tifafd@libero.it
177. Hey girls xD
My fav dress is the first options .
Following all the things you said.
Or keep in touch in my blog.
178. i love # 1 (and #4 too)
179. retweet
180. liked on fb-annushka solovey
181. following on twitter
182. blog follower via gfc-ne-knopka
183. Following on Bloglovin-#245
liked AngelaSquared on fb-annushka solovey
185. following @AngelaSquared-annakulu1
186. Option 1! It's nearly summer and I love T-shirt dresses!
Pop by and visit my blog Taken By Surprise! xx
187. I love option 5 .
188. Following Hansel on Twitter @blogulcuhainute
189. GFC follower: Cami87
190. Bloglovin follower
191. Great! I love option 4
192. 'Like' Hansel's Facebook Page: Ananda Rock
193. Follow Hansel on Twitter: @AnandaRock1
194. Follow us on GFC: Ananda Rock
195. Follow us on Bloglovin: Ananda Rock
196. Follow us on Twitter: @AnandaRock1
197. 'Like' our Facebook Page: Ananda Rock
198. retweet:https://twitter.com/AngelaSquared/status/256432802319437825
199. I love Option 4!
z853www at hotmail dot com
200. I retweeted as @babybraddy.
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Trigonometric Ratios
2.1: Trigonometric Ratios
Created by: CK-12
This activity is intended to supplement Trigonometry, Chapter 1, Lesson 3.
ID: 9534
Time required: 35 minutes
Activity Overview
Students will use a graphing calculator to discover the relationship between the trigonometric functions: sine, cosine and tangent and the side length ratios of a right triangle.
Topic: Trigonometric Functions
• Solve any right triangle given an angle and the length of an opposite or adjacent side.
• Use technology to obtain the sine, cosine, or tangent of any angle.
Teacher Preparation and Notes
Students will use the graphing calculator to discover the relationship between the trigonometric functions: sine, cosine and tangent and the side length ratios of a right triangle. Prior to beginning
the activity, students should download the CabriJr file TRIG to their graphing calculators.
This activity is designed as an introduction to the world of trigonometry. Students will explore the trigonometric ratios (sine, cosine, tangent) of a right triangle.
• This activity requires students to drag a point in CabriJr. If students are not familiar with this function of the CabriJr application, extra time should be taken to explain this.
• This activity is intended to be teacher-led. You may use the following pages to present the material to the class and encourage discussion. Students will follow along using their handhelds,
although the majority of the ideas and concepts are only presented in this document; be sure to cover all the material necessary for students’ total comprehension.
• The student worksheet is intended to guide students through the main ideas of the activity. It also serves as a place for students to record their answers. Alternatively, you may wish to have the
class record their answers on separate sheets of paper, or just use the questions posed to engage a class discussion.
• To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/Software/Detail?id=258#.
• To download the calculator file, go to http://www.education.ti.com/calculators/downloads/US/Activities/Detail?id=9534 and select TRIG.
Associated Materials
In this activity, students will explore:
• Sine, cosine and tangents of angles
• Side length ratios of a right triangle
• Finding the missing side of a triangle when an angle and side are given.
Note: Students will need to exit the application in order to find the decimal values for the ratios and the trigonometric values for angle A. One option is for students to go through the TRIG
application first, fill in all the ratios on the worksheet, and copy down the measures of angle A for each triangle. Then they would go back and complete the rest of the worksheet. Alternatively,
students could work in pairs with one using the application and the other performing the calculations.
Problem 1 – Trigonometric Ratios
Students will open the TRIG file using the CabriJr application. Given a triangle, students will find the relationship between the ratios of the side lengths and the trigonometric functions.
Record the following ratios and trigonometric values to two decimal places.
$\frac{BC}{AC} & = \frac{4.0}{5.9}, \frac{AC}{AB} = \frac{5.9}{7.1}, \frac{BC}{AB} = \frac{4.0}{7.1}\\\\\text{Sin} \ A & = \underline{0.56}, \text{Cos} \ A = \underline{0.83}, \text{Tan} \ A = \
Note: Make sure all student calculators are in DEGREE mode.
Make sure students find the decimal values below and compare the answers with the trig values.
$\frac{BC}{AC} &= 0.67\\\frac{AC}{AB} &= 0.83\\\frac{BC}{AB} &= 0.56$
Students are asked to repeat this process for two more different triangles by moving point B to a different location (to grab a point, press the ALPHA button and use the arrow keys to move it.)
Answers will vary. Students should check their answers with the trig values. Answers should match the trig identities.
Based upon your answers hypothesize which ratio goes with each trigonometric function.
$\text{Sin} \ A = \frac{BC}{AB} && \text{Cos} \ A = \frac{AC}{AB} && \text{Tan} \ A = \frac{BC}{AC}$
A good acronym to use to help remember these relationships is SOHCAHTOA
$\sin A &=\frac{\text{Opposite}}{\text{Hypotenuse}}\\ \cos A &= \frac{\text{Adjacent}}{\text{Hypotenuse}}\\ \tan A &=\frac{\text{Opposite}}{\text{Adjacent}}$
Problem 2 – Trigonometry, What Is It Good For?
One of the uses of trigonometry is finding missing side lengths of a triangle.
To find the length of side $BC$
Now solve for $BC$
$24\sin27 &= BC\\24 \cdot 0.45 &= BC\\10.90 &= BC$
To find the length of side $AC$
$\cos48 =\frac{AC}{19}$
Now solve for $AC$
$19\cos48 = AC\\19 \cdot 0.67 = AC\\12.71 = AC$
To find the length of side $AC$
Now solve for $AC$
$b\tan50 &= 10\\ b = \frac{10}{\tan50} &= \frac{10}{1.19} = 8.4$
1. $AC = 14.62$
2. $BC = 19.51$
3. $AC = 49.52$
4. $AC = 95.09$
Files can only be attached to the latest version of None
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Combinatorial Model Categories
Posted by Urs Schreiber
Over the last years Jeff Smith has been thinking about – and notably talking about – but not publishing about – the idea and the theory of a certain type of model category called
While he himself didn’t publish, the theory was found to be very useful and bits and pieces of it appear now more or less scattered in the works of other authors.
As far as I can tell the earliest published reference of the central recognition result for combinatorial model categories – now widely known as Jeff Smith’s theorem – is Tibor Beke’s Sheafifiable
homotopy model categories.
Apart from helping find various new interesting model category structures, this and related (unpublished) results made Dan Dugger understand that the local projective model structures on simplicial
presheaves which he had been studying exhaust precisely all (simplicially enriched) combinatorial model categories:
(Dan Dugger’s theorem) combinatorial model categories are precisely (up to Quillen equivalence) the
of the global projective model structures on functors $Func(C^{op},SSet)_{proj}$ from a small category $C$ to the category of simplicial sets.
I can’t know if it’s close to the way it happend, but if you read Jacob Lurie’s book Higher Topos Theory backwards, starting with the very last three propositions of the appendix, you can read it as
the result of taking Dan Dugger’s theorem in Jeff Smith’s theory and finding its intrinsic model independent version:
Simplicial combinatorial model categories are precisely the models for locally presentable $(\infty,1)$-categories: those that are localizations – i.e. reflective $(\infty,1)$-subcategories – of
$(\infty,1)$-categories of $(\infty,1)$-presheaves.
(Among these the left exact localizations are precisely the $\infty$-stack $(\infty,1)$-toposes.)
In total this provides us with the optimal situation where on the one hand we have a comprehensive abstract nonsense picture that tells us what is going on globally, while on the other hand we have
highly developed concrete models and tools for realizing this abstract nonsense: the theory of combinatorial model categories.
Accordingly, we are eagerly awaiting Jeff Smith’s book to appear, whose upcoming existence keeps being hinted at in the literature. While that is not available yet, I thought it might be worthwhile
to start compiling the available material in a useful coherent fashion at one place. First steps in this direction are at combinatorial model category and Bousfield localization of model categories.
One very useful input I already got on this was very kindly from Denis-Charles Cisinki, who pointed out the semi-published document by Clark Barwick: On left and right model categories and left and
right Bousfield localization. Parts of that I have used in the construction of these entries. (All nonsense and other imperfections, of which there is still plenty, is my fault.) Much more needs to
be done.
Posted at November 25, 2009 5:44 PM UTC
Re: Combinatorial Model Categories
Thank you! I’m sure this will be a very useful reference; I’ve often wished that the important results on combinatorial model categories weren’t scattered across so much of the literature. I also
like the way of thinking of simplicial combinatorial model categories as a particularly nice sort of presentation of locally presentable $(\infty,1)$-categories.
I think it’s also worth reminding people that since basically all model categories are Quillen equivalent to simplicial combinatorial ones, basically all model categories can also be regarded as
presentations of locally presentable $(\infty,1)$-categories. It’s just that the simplicial combinatorial ones are somewhat easier to work with formally, e.g. small object arguments and localizations
happen more easily.
Posted by: Mike Shulman on November 25, 2009 8:39 PM | Permalink | PGP Sig | Reply to this
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Convex Non-Negative Matrix Factorization in the Wild
Christian Thurau, Kristian Kersting and Christian Bauckhage
In: ICDM 2009, 6-9 Dec 2009, Miami, USA.
Non-negative matrix factorization (NMF) has recently received a lot of attention in data mining, information retrieval, and computer vision. It factorizes a non-negative input matrix V into two
non-negative matrix factors V = WH such that W describes "clusters" of the datasets. Analyzing genotypes, social networks, or images, it can be beneficial to ensure V to contain meaningful "cluster
centroids", i.e., to restrict W to be convex combinations of data points. But how can we run this convex NMF in the wild, i.e., given millions of data points? Triggered by the simple observation that
each data point is a convex combination of vertices of the data convex hull, we propose to restrict W further to be vertices of the convex hull. The benefits of this convex-hull NMF approach are
twofold. First, the expected size of the convex hull of, for example, n random Gaussian points in the plane is , i.e., the candidate set typically grows much slower than the data set. Second,
distance preserving low-dimensional embeddings allow one to compute candidate vertices efficiently. Our extensive experimental evaluation shows that convex-hull NMF compares favorably to convex NMF
for large data sets both in terms of speed and reconstruction quality. Moreover, we show that our method can easily be applied to large-scale, real-world data sets, in our case consisting of 1.6
million images respectively 150 million votes on World of Warcraft guilds.
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the first resource for mathematics
A first course in the numerical analysis of differential equations.
(English) Zbl 0841.65001
Cambridge Texts in Applied Mathematics. Cambridge: Cambridge Univ. Press. 400 p. £17.95; $ 27.95/pbk; £55.00; $ 74.95/hbk (1995).
There exist few textbooks only which are exclusively dedicated to the numerical solution of ordinary and partial differential equations although this field is of significant importance for courses on
numerical analysis. The present book gives a rigorous account of the respective fundamentals. In the exposition it strives to maintain a balance between theoretical, algorithmic and applied aspects
of the subject. This is not quite an easy task but the author, a specialist in the field and an experienced teacher excellently realizes the forementioned aims.
The book covers a broad range of material. The first 100 pages are dedicated to the numerical solution of ordinary differential equations including explicit and implicit Runge-Kutta methods,
multistep methods including error control devices, solution of stiff equations and iteration for solving nonlinear algebraic systems. The next 160 pages are used to present finite difference and
finite element methods for discretizing the Poisson equation and a variety of algorithms for solving the resulting large algebraic systems as Gaussian elimination for banded systems, iterative
methods (the alternating directions implicit method and, unfortunately, the conjugate gradient method in the form of a remark only), multigrid techniques, fast Poisson solvers (Hockney method, fast
Fourier transform, and, as a remark, odd-even reduction). Evolution type equations (parabolic and hyperbolic) are considered the next 80 pages which are followed by an appendix containing
fundamentals in linear algebra, interpolation and quadrature and ordinary differential equations (20 pages). Each chapter is concluded with useful comments and a bibliography as well as a collection
of exercises.
This book can be highly recommended as a basis for courses in numerical analysis. Since the emphasis does not lie in presenting deeper mathematical proofs but in providing mainly the unavoidable
mathematics for a thorough understanding of the numerical methods it is equally well suited for students in science and engineering. The price for the paperback edition seems to be reasonably
calculated also for the normally smaller budget of students.
65-01 Textbooks (numerical analysis)
65Lxx Numerical methods for ODE
65Mxx Numerical methods for initial value problems (IVP) of PDE
65Nxx Numerical methods for boundary value problems (BVP) of PDE
65F05 Direct methods for linear systems and matrix inversion (numerical linear algebra)
65F10 Iterative methods for linear systems
65H10 Systems of nonlinear equations (numerical methods)
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Linfield, PA Algebra 2 Tutor
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Topic: hypergeom() gives Inf
Replies: 1 Last Post: Jun 30, 2012 3:06 AM
Messages: [ Previous | Next ]
hypergeom() gives Inf
Posted: Jun 29, 2012 11:49 AM
I am using hypergeom function in Matlab to find 1F1. The arguments of hypergeom are:
hypergeom(1.5,4,973). I get "Inf" for it. So, I tried to write the code for hypergeom function. Here is the code:
total = 0;
a = 1.5;b = 4;z = 973.2763;
for k = 1:103 %theoretically k goes upto infinity
num = 1;
den = 1;
for i = 1:k
num = num*(a+i-1);
den = den*(b+i-1);
total = total + ((num/den)*(z^k)/factorial(k));
I get Inf when I increase limit of k to 104. Unfortunately, the series in this case is non-converging. i.e as I increase the limit of k, the value "total" increases. How should I go about it??
Meanwhile, I tried a simple converging series with hypergeom(1,1,2). MATLAB gave me 7.3891. I used the code written above to calculate it myself. I got 6.3891 when the last limit of k = 172 i.e.k =
1:172. As soon as I increased k to 173, I got NaN. So, I want to know How does the function hypergeom in MATLAB work?? Does it also work for diverging series, as in my case.
Date Subject Author
6/29/12 kumar vishwajeet
6/30/12 Re: hypergeom() gives Inf Bruno Luong
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Calculations of Observations
A. Latitude
1. Latitude by Octant and Meridian Altitude of the Sun
The angle that Lewis recorded for this observation, however, was not the sun’s altitude; it was the supplement of the double altitude of the sun’s lower limb. This means that the angle he observed
had to be subtracted from 180° then divided by two to find the “observed” altitude of the sun’s lower limb.^1 From the “observed” altitude it still was necessary to subtract the effect of refraction,
add the effect of parallax and add the sun’s semidiameter; the “sum” of these three operations was the altitude of the sun’s center. Having mathematically determined the true altitude of the sun’s
center it was then necessary to algebraically subtract the sun’s declination from it. Having completed that operation, subtracting the result from 90° gives the calculated latitude (for the
Using the index error for the octant that Lewis gave on May 22, 1804 and which he reaffirmed while at Fort Clatsop (see Fort Clatsop Miscellany), his observed angle yields a latitude of 46°09'12" N.
The actual latitude of Camp Chopunnish determined from information in the courses and distances and from early sources who identified the camp’s location is 46°14'31"--a difference of 5'19". This
difference of 5'19" is larger than any of the differences between the actual latitude and the latitude recalculated from observations that Lewis made in 1805. The difference might result from a
changed index error or from the fact that Lewis had not taken an observation with the octant since August 1805. In any case, the difference is not much greater than for many observations taken for
latitude at sea at that time.
2. Latitude from Sun’s Altitude, Declination and Hour Angle
Equal Altitudes observations normally provide all that data, but the observation for May 25 was not followed by an observation the next day, so the chronometer’s rate-of-going is uncertain. The Equal
Altitudes observations for June 5 and June 6, however appear to provide the data necessary to determine the rate. At noon on June 5 the chronometer was 4h 33m 11s slow on Local Mean Time and on June
6 it was 5h 50m 22s slow--a loss of 1h 17m 11s in 24 hours! The question is, did the chronometer stop or change rate between the two observations or did it run at a steady rate?
This exceptionally high rate-of-loss, at first, doesn’t inspire much confidence in a calculation for latitude from chronometer’s time. Nevertheless, by plotting the chronometer error for June 5 and 6
and projecting its rate-of-loss of 1h17m per 24 hours back to May 25, one discovers that the line passes close to the chronometer’s error at noon on May 25--provided 12 hours are subtracted from that
error (that is, at a rate-of-loss of about 1h17m per day, the chronometer had lost more than 14 hours in eleven days). This is an unbelievably high daily rate-of-loss, but the near-concordance of
times suggests that the chronometer had been running at nearly a steady rate during this period of time. It would be unwise to make any calculation from the Equal Altitudes observation on May 25
because the projected line, although passing close to the chronometer’s error at noon determined from that observation, still misses by too far to warrant that calculation. It does appear reasonable,
however, to attempt a calculation from the time given by PM observation on June 5 and that of the AM observation on June 6.
Unfortunately, the calculations, once made, show that Lewis took both the Equal Altitudes observations at a time when the sun was nearly east or west. Therefore, the last step of the equation for
determining the latitude as given in Robert Patterson’s Astronomical Notebook, Problem 1, “falls apart.” The problem is that the arc-cosine needed to complete the calculation is too near unity to
give reliable results.
Fortunately, Lewis took two observations for Magnetic Declination between the PM Equal Altitude observation on June 5 and the AM Equal Altitudes observation on June 6. Even though these observations
for Magnetic Declination shortly followed the PM observation on June 5, the sun would have moved farther to the north of west away from the 0°-90°-problem. The calculations for the first Magnetic
observation produce a latitude of 46°10'37"N, seeming to confirm the octant latitude. The second Magnetic observation, however, produces a latitude of 46°20'41”N. The average of these two
calculations is 46°15'39”N, but can an average from such divergent results be trusted?
As noted above in the Summary on page 3, above, the latitude of Camp Chopunnish can be approximately determined on the Lewis and Clark Map of 1806 (Moulton, Atlas, map 123) at about 46°15' North.
--Robert N. Bergantino, 11/04
1. Lewis first would divide the observed angle by 2, then subtract the result from 90°. Both his method and that given above produce the same answer, though the one above is procedurally more
Funded in part by the Idaho Governor's Lewis and Clark Trail Committee.
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Sugar Land Geometry Tutor
Find a Sugar Land Geometry Tutor
...I have extensive knowledge of Anatomy and Physiology. Anatomy and Physiology is an enthralling subject for me- I loved helping my fellow classmates in high school when I took Anatomy and
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39 Subjects: including geometry, English, reading, chemistry
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11 Subjects: including geometry, calculus, statistics, algebra 1
...To achieve that, I try to break the information down into smaller pieces to make it easier for students to comprehend. Also, I show students how to figure out what is the most important
information so that they can more quickly master the material. I love watching a student make progress and see the "lightbulb" go off as they realize they actually get it!
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Euler Characteristic of a disk
December 8th 2012, 01:25 PM
Euler Characteristic of a disk
I need to find the integral of gaussian curvature of a surface given by the graph over disk x^2+y^2<=2. I know that the integral of gaussian curvature= 2pi*(euler characteristic).
The Euler characterisic of the disk x^2+y^2<=1 is 1. So would the Euler characterisic of my disk: x^2+y^2<=2 also be 1?
December 8th 2012, 04:13 PM
Re: Euler Characteristic of a disk
Yes since they're homeomorphic.
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Summary: Submitted to the Journal of Computational Intelligence in Finance (8/4/99)
Conventional modeling techniques for option pricing have systematic biases resulting from the assumption of
constant volatility (homoskedasticity) for the price of the underlying asset. Nevertheless, practitioners seldom
use stochastic volatility models since the latter require making unverifiable assumptions about the price process.
A different approach consists of ``letting the data speak for itself'', i.e. to make a few general assumptions about
the process to be modeled, and to exploit the information available from the prices of traded options. In this
paper we develop of a non-parametric model for specifying the volatility of the underlying asset based on
Feedforward Neural Networks and a Bayesian learning approach. We then develop of an option-pricing model
based on this volatility specification. Numerical experiments are presented for the case of the USD/DEM
options, accompanied by a graphical analysis of the resulting smiles.
Options are some of the most important financial instruments traded in the markets today. The technical aspects of option-
pricing theory and practice have attracted the attention of mathematicians, statisticians, physicists and computer scientists.
According to Rubinstein [1985] and Eales et al. [1990], the celebrated Black & Scholes option-pricing formula has
systematic and persistent biases. These biases depend upon both the "time to maturity" and the "option moneyness ratio"
which is the relationship between the spot and the strike price. Among several plausible reasons accounting for these biases is
the one introduced in Hull and White [1987] where the conventional models bias is shown to be consistent with the theory of
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Bacteria Growth
February 3rd 2009, 02:02 PM #1
Jan 2009
Bacteria Growth
The number of bacteria in a culture is increasing according to the law of exponential growth. There are 125 bacteria in the culture after 2 hours and 350 bacteria after 4 hours.
How do I find the the initial population?
$P = P_o e^{kt}$
$\frac{350 = P_o e^{4k}}{125 = P_o e^{2k}}$
$2.8 = e^{2k}$
solve for $k$, then go back to the easier equation and solve for $P_o$
February 3rd 2009, 03:26 PM #2
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Fractional Scale
Determining the Fractional Scale of a Map or Scale Model
The fractional scale of a scale model is the ratio between two sets of
dimensions (as between those of a model car and its equivalent actual
car). The formula for determining the fractional scale of a map or scale
model is:
1 Scale = _____ AS/MS Where: 1 = In a fractional scale, the numerator is always "1" AS = Actual Size MS = Equivalent Model Size
"AS" and "MS" should be in the same units, e.g., cm, so that the units will cancel. Example: If a model car is 20cm long, and the actual car is 3m long, then what is the scale of the model car?
Step 1. Convert meters to centimeters: 3m = 300cm Step 2. Calculate scale: 1 __________ = 1/15 300cm/20cm
Therefore, one unit on the model is equivalent to 15 of the same units in the real world. Every feature on the model is 1/15 of actual size; every feature on the actual car is 15 times larger than
the same model feature.
Copyright © 1997-2012 by Walter Sanford. All rights reserved.
Comments/Suggestions? | Online Geoscience-Related Activities | Geosystems in FCPS
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Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.
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0.6875 us cups in ounces
You asked:
0.6875 us cups in ounces
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Talking in binary
My brother and I figured out how to send finger shapes to each other (but we promised not to do any bad ones). Here's how we do it:
First, we tell each other how to hold our hands (we haven't figured out a binary system for that yet)
The first 10 digits tells us what fingers, from left to right, are up and which are down. 1 is up, 0 is down.
The next 8 digits tells us if there are any gaps, from left to right, between our fingers. 1 means gap, 0 means no gap. (Do not count the gap between your hands)
The next 10 digits tells us if any of the fingers are curled. 1 means curled, 0 means not curled.
The next 2 digits tells us if the palms are curled. 1 means curled, 0 means not curled.
Not sure if this makes us the biggest nerds ever... we'll see!
So, here's a few examples:
Both of your palms should be away from you for this one.
Live long and prosper!
Your left hand should be facing you and tipped such that your pinky is parallel to the floor. On your right hand, your palm should be facing down.
Yay cello! (Only two zeroes, weird...)
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1.2 Speed and Acceleration
Speed and Acceleration
Critical Questions:
• What happens in a car when you push on the gas pedal or step on the brake?
As a physics teacher, I blame a lot of my problems on cars. People spend hours every day in cars of one kind or another, and they’ve developed strong ideas about the relationships between the gas
pedal, the engine, and the car’s movement. I will soon try to convince you to think differently about those relationships, but for now I’m going to make use of what you already know.
Speed is the easy one. If you want to know how fast a car is going, just glance at its speedometer. Unfortunately for high school physics students, however, it is quite easy to complicate even such a
simple concept as speed. This practice goes all the way back to pre-Socratic Greece, when the philosopher Zeno asked how a flying arrow could both occupy a space and yet also be in motion, a question
which eventually forced mathematicians to give up and invent calculus. Nevertheless, for the purposes of this site, “speed” (or “velocity”) means nothing more than how fast a thing is travelling.
We’re going to have to be a bit more careful about acceleration, though. Not only is the physics definition of acceleration slightly different from the everyday one, it also represents our first
tricky concept — one that you might find difficult to wrap your head around.
There are three kinds of acceleration. The first is the one you already know about, which is what happens to your car when you push down on the accelerator: the car speeds up.
The second kind of acceleration is the one that non-physicists might call deceleration. We don’t use that word in physics; slowing down is not any special kind of acceleration, it’s just speeding up
The third kind of acceleration occurs when something changes direction. That’s right, on this site we are going to say that a car rounding a bend in the highway is accelerating, even if its
speedometer stays pointing at exactly 44 miles per hour.
The reason for this is that when physicists talk about velocity, they are also very much interested in the direction of that velocity. For an air traffic controller, a plane travelling at 900 km/h is
rather different from one going the other direction; the same is usually true in physics. (In fact, from now on I will often use the word “velocity” instead of “speed” just to remind you that
direction is important.) And if we define acceleration as any change of velocity, then it makes sense that changing direction counts as acceleration. There is, in fact, a very good reason for lumping
these three types of acceleration together, but I don’t want to start talking about it yet.^[1]
So at this point, we have two connections to a car: speed is the number the speedometer is pointing at; acceleration is how quickly the speedometer needle is moving (or how quickly the car is
changing direction).
Another important idea to discuss here is that of “constant” acceleration. One trick to understanding this concept is simply to replace the word ‘acceleration’ with its definition: change of
velocity. To imagine an object with a constant acceleration, then, we just have to imagine an object whose velocity is constantly changing. The simplest example of this would be an object whose speed
is increasing over time; if the object keeps moving faster and faster at a consistent rate, we can say it has a constant acceleration.
The other trick that can help you feel comfortable about this idea is to fully separate in your mind the concepts of velocity and acceleration. For example: imagine two cars, a red one and a blue
one. The red car is ahead of the blue car, moving at a constant velocity. The blue car is moving at a slower speed, but it has a forward acceleration.
Can the blue car ever pass the red car? Of course. Aren’t you glad I didn’t put numbers into that question and ask you when the two cars would meet if one left from Los Angeles and the other left
from Cleveland? Frankly, so am I. But try this: imagine the blue car’s speedometer steadily climbing up, and confidently tell yourself that it has a constant acceleration, while the red car has a
constant velocity. And if you want to get really good at this, imagine different combinations of accelerations and velocities, and watch the results play out in your imagination.
So what happens when something doesn’t have a constant acceleration? This is something we’re all familiar with — it happens every time we change the pressure we’re applying to the gas pedal. With a
small push, you get a low acceleration (the speedometer needle moves slowly), and if you push harder, you can increase your acceleration or at least spin your tires. Acceleration is a change of
velocity; the technical term for a change of acceleration is jerk (seriously). In the real world, cases of constant acceleration are actually very rare. Things are jerking all over the place,
literally and figuratively. But in order to understand jerk, you need to first understand acceleration and what causes it, so we will usually be dealing only with situations of constant acceleration,
and if I’m ever describing an object’s motion and forget to say whether its acceleration is constant or not, you can just go ahead and assume that it is.
There is another assumption I’m going to make for a while (right up until Chapter 10), which is that the things we’ll be talking about are all moving at normal, everyday speeds. This is an assumption
that people didn’t even know they were making throughout most of history, but in 1905 we learned just how weird things can get when you start moving faster. That is the year when Einstein published
his Special Theory of Relativity. His work showed that in situations of very high speeds or energies, bizarre things happen: time takes longer to pass by, lengths expand and contract, and the very
fabric of space changes its shape.
But as I said, all of that can wait until Chapter 10. For now, please relax and enjoy the comfortable simplicity of the regular or “non-relativistic” velocities.
Big Ideas:
• Velocity is the number the speedometer points at, although direction is also important.
• Acceleration is any change of velocity, either in amount or direction.
• Constant acceleration occurs when an object’s velocity changes constantly (e.g. the speedometer needle moves at a constant speed).
Recent Comments
• Henry on 1.3 Falling Objects
• Daryll on 2.8 Friction and Air Resistance
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addition and subracting
April 22nd 2010, 04:06 AM
addition and subracting
hi, ive just been reading through this website on differentiation, anyway ive just come across subtraction and addition of functions, KryssTal : Introduction to Calculus theres the link, if you
scroll down about 1/4 of the way down you will see the section i am talking about, can anyone explain this a bit easier i cant seem to grasp the concept, especially when i was following an
example; (i) For the function y = 4x2 + 2x + 3, the derivative is dy/dx = 8x + 2
now i cant see where the 2 has come from, i get the 8x part but not the second bit.
please help.
April 22nd 2010, 04:32 AM
if you know that for
$y(x) = ax^n$
$dy/dx = a n x^{n-1}$
think about $y(x) = 5x$
here a = 5 and n = 1
now do the substitution with the formula I gave you...
April 22nd 2010, 04:34 AM
Is this you first your first experience with Calculus?
To differentiate basic functions such as the one above, you multiply the coefficient by the power, and then reduce the power by one.
For example:
The derivative of $4x^2$ is $(4 \times 2)x^{2-1} = 8x^1 = 8x$
As for the second bit:
$2x = 2x^1$, so differentiating this gives you $(2 \times 1)x^0 = 2(1) = 2$.
Hope this clear is up a bit. A great place to check out if you want to learn the basics of calculus is Khan Academy on Youtube.
YouTube - khanacademy's Channel
April 23rd 2010, 11:41 AM
yeah it does help, thanks. just a couple of queeries though..
so if in such a problem if theres an x with no power then the outcome is going to be just a number without the x ? also you ended up with 2(1) where did you get the 1 from?
and yeah it is my first experience hence the uncertanties.
also in the origional post, y = 4x2 + 2x + 3 now where does the 3 go?
April 24th 2010, 12:41 PM
Yes it would just be the constant without the x. I got the 1 because $x^0 = 1$
If you differentiate a constant it goes to zero, you could think of it like this:
$3 = x^0$, so differentiating it would mean that you multiply by zero.
April 25th 2010, 02:35 AM
thats great thanks for your help craig much appreciated
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Summary: Solution to the King's Problem in Prime Power Dimensions
P. K. Aravind
Physics Department, Worcester Polytechnic Institute, Worcester, MA 01609, U. S.A.
Reprint requests to Prof. P.K. A.; E-mail: paravind@wpi.edu
Z. Naturforsch. 58a, 85 92 (2003); received November 5, 2002
It is shown how to ascertain the values of a complete set of mutually complementary observables of
a prime power degree of freedom by generalizing the solution in prime dimensions given by Englert
and Aharonov [Phys. Lett. A284, 1 5 (2001)].
Key words: Mutually Unbiased Bases; King's Problem; Quantum State Estimation.
1. Introduction
The King's Problem [1] is the following: A physi-
cist is trapped on an island ruled by a mean king who
promises to set her free if she can give him the an-
swer to the following puzzle. The physicist is asked to
prepare a d-state quantum system in any state of her
choosing and give it to the king, who measures one
of several sets of mutually unbiased observables (this
term will be defined below) on it. Following this, the
physicist is allowed to make a control measurement on
the system, as well as any other systems it may have
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[FOM] Are Proofs in mathematics based on sufficient evidence?
Monroe Eskew meskew at math.uci.edu
Fri Jul 16 06:30:46 EDT 2010
On Thu, Jul 15, 2010 at 2:21 PM, Vaughan Pratt <pratt at cs.stanford.edu> wrote:
> The problem comes at the step "join FC". Postulate 5 ruled out
> hyperbolic geometry but not elliptical. On the sphere, if the distance
> from F to C is more than half the circumference the segment FC will be
> the complete line FC less the segment drawn by Euclid in the diagram.
> This screws up the angles in the rest of the argument.
This can be viewed as an instance of a hidden assumption. Hilbert's
axioms for plane geometry provide the needed assumptions-- the
incidence axiom which says that two lines meet only in one point. One
can use Hilbert's axioms to perfect Euclid's proofs, without
introducing formal rules of derivation. If we wish, we can take them
into formal FOL and use a standard formal proof system for FOL. Or we
can work informally but rigorously as we do in ordinary mathematics.
The problem is not a lack of formal derivation rules; it is a lack of
ordinary validity because of the use of hidden assumptions.
More information about the FOM mailing list
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Radiation Reaction and Energy Conservation
Next: Integrodifferential Equations of Motion Up: Radiation Reaction Previous: The Death of Classical Contents
We know that
is (nonrelativistic) Newton's 2nd Law for a charged particle being accelerated by a (for the moment, non-electromagnetic) given external force. The work energy theorem dictates how fast the particle
can gain kinetic energy if this is the only force acting.
However, at the same time it is being acted on by the external force (and is accelerating), it is also radiating power away at the total rate:
(the Larmor formula). These are the two pieces we've thus far treated independently, neglecting the one to obtain the other.
However, in order for Newton's law to correctly lead to the conservation of energy, the work done by the external force must equal the increase in kinetic energy plus the energy radiated into the
field. Energy conservation for this system states that:
or the total work done by the external force must equal the change in the total energy of the charged particle (electron) plus the energy that appears in the field. If we rearrange this to:
and consider the electron only, we are forced to conclude that there must be another force acting on the electron, one where the total work done by the force decreases the change in energy of the
electron and places the energy into the radiated field. We call that force radiation reaction force.
Thus (rewriting Newton's second law in terms of this force):
defines the radiation reaction force that must act on the particle in order for energy conservation to make sense. The reaction force has a number of necessary or desireable properties in order for
us to not get into ``trouble''^19.2.
• We would like energy to be conserved (as indicated above), so that the energy that appears in the radiation field is balanced by the work done by the radiation reaction force (relative to the
total work done by an external force that makes the charge accelerate).
• We would like this force to vanish when the external force vanishes, so that particles do not spontaneously accelerate away to infinity without an external agent acting on them.
• We would like the radiated power to be proportional to
• Finally, we want the force to involve the ``characteristic time''
Let's start with the first of these. We want the energy radiated by some ``bound'' charge (one undergoing periodic motion in some orbit, say) to equal the work done by the radiation reaction force in
the previous equation. Let's start by examining just the reaction force and the radiated power, then, and set the total work done by the one to equal the total energy radiated in the other, over a
suitable time interval:
for the relation between the rates, where the minus sign indicates that the energy is removed from the system. We can integrate the right hand side by parts to obtain
Finally, the motion is ``periodic'' and we only want the result over a period; we can therefore pick the end points such that
One (sufficient but not necessary) way to ensure that this equation be satisfied is to let
This turns Newton's law (corrected for radiation reaction) into
This is called the Abraham-Lorentz equation of motion and the radiation reaction force is called the Abraham-Lorentz force. It can be made relativistic be converting to proper time as usual.
Note that this is not necessarily the only way to satisfy the integral constraint above. Another way to satisfy it is to require that the difference be orthogonal to required is that the total
integral be zero, and short of decomposing the velocity trajectory in an orthogonal system and perhaps using the calculus of variations, it is not possible to make positive statements about the
necessary form of
This ``sufficient'' solution is not without problems of its own, problems that seem unlikely to go away if we choose some other ``sufficient'' criterion. This is apparent from the observation that
they all lead to an equation of motion that is third order in time. Now, it may not seem to you (yet) that that is a disaster, but it is.
Suppose that the external force is zero at some instant of time
Recalling that
Let us note that the radiation reaction force in almost all cases will be very small compared to the external force. The external force, in addition, will generally be ``slowly varying'', at least on
a timescale compared to smooth (continuously differentiable in time), slowly varying, and small enough that
Under these circumstances, we can assume that
This latter equation has no runaway solutions or acausal behavior as long as
We will defer the discussion of the covariant, structure free generalization of the Abraham-Lorentz derivation until later. This is because it involves the use of the field stress tensor, as does
Dirac's original paper -- we will discuss them at the same time.
What are these runaway solutions of the first (Abraham-Lorentz) equation of motion? Could they return to plague us when the force is not small and turns on quickly? Let's see...
Next: Integrodifferential Equations of Motion Up: Radiation Reaction Previous: The Death of Classical Contents Robert G. Brown 2013-01-04
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From the really great book by David Kahaner, Cleve Moler and Stephen Nash -
"Numerical Methods and Software", Prentice-Hall, 1989:
"Suppose that a satellite in space is taking photographs of Jupiter to be sent back to earth.
The satellite digitizes the picture by subdividing it into tiny squares called pixels or picture elements.
Each pixel is represented by a single number that records the average light intensity in that square.
If each photograph were divided into 500 x 500 pixels, it would have to send 250,000 numbers to earth for each picture.
This would take a great deal of time and would limit the number of photographs that could be transmitted.
It [is sometimes] possible to approximate this matrix with a matrix which requires less storage."
Here's a Martian image of a rock called "Yogi" sent back by the Sojourner rover.
The image is stored as a 256 x 264 matrix M with entries between 0 and 1.
The matrix M has rank 256.
Here's a plot of the singular values for M.
Since the singular values are decaying so rapidly, you can expect that there will be a good lower rank approximation to M.
That is, for a relatively small k, you should have
Have a look at various lower rank approximations to M.
The rank 36 approximation looks fairly good.
What's the advantage of using a lower rank approximation for M?
To send the matrix M you need to send 256 x 264 = 67584 numbers.
To send the rank 36 approximation to M you need only send
• the first 36 singular values,
• the first 36 hanger vectors, each of which has 256 entries,
• the first 36 aligner vectors, each of which has 264 entries.
So in total you need to send only 36(1+256+264)=18756 numbers.
1. How many numbers would be required to send the rank 49 approximation to M ?
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data.tables and sweep function
up vote 3 down vote favorite
Using a data.table, which would be the fastest way to "sweep" out a statistic across a selection of columns?
Starting with (considerably larger versions of ) DT
p <- 3
DT <- data.table(id=c("A","B","C"),x1=c(10,20,30),x2=c(20,30,10))
DT.totals <- DT[, list(id,total = x1+x2) ]
I'd like to get to the following data.table result by indexing the target columns (2:p) in order to skip the key:
id x1 x2
[1,] A 0.33 0.67
[2,] B 0.40 0.60
[3,] C 0.75 0.25
r data.table
add comment
1 Answer
active oldest votes
I believe that something close to the following (which uses the relatively new set() function) will be quickest:
DT <- data.table(id = c("A","B","C"), x1 = c(10,20,30), x2 = c(20,30,10))
total <- DT[ , x1 + x2]
rr <- seq_len(nrow(DT))
for(j in 2:3) set(DT, rr, j, DT[[j]]/total)
# id x1 x2
# [1,] A 0.3333333 0.6666667
# [2,] B 0.4000000 0.6000000
# [3,] C 0.7500000 0.2500000
FWIW, calls to set() takes the following form:
# set(x, i, j, value), where:
# x is a data.table
# i contains row indices
up vote 2 # j contains column indices
down vote # value is the value to be assigned into the specified cells
My suspicion about the relative speed of this, compared to other solutions, is based on this passage from data.table's NEWS file, in the section on changes in Version 1.8.0:
o New function set(DT,i,j,value) allows fast assignment to elements
of DT. Similar to := but avoids the overhead of [.data.table, so is
much faster inside a loop. Less flexible than :=, but as flexible
as matrix subassignment. Similar in spirit to setnames(), setcolorder(),
setkey() and setattr(); i.e., assigns by reference with no copy at all.
M = matrix(1,nrow=100000,ncol=100)
DF = as.data.frame(M)
DT = as.data.table(M)
system.time(for (i in 1:1000) DF[i,1L] <- i) # 591.000s
system.time(for (i in 1:1000) DT[i,V1:=i]) # 1.158s
system.time(for (i in 1:1000) M[i,1L] <- i) # 0.016s
system.time(for (i in 1:1000) set(DT,i,1L,i)) # 0.027s
Thanks for the answer. I've upgraded to data.table 1.8.0, and successfully ran the test code above. I do get an elaborate warning (won't fit in here) about coercion to double when
both numerator and denominators are integer columns from data.tables. I'll edit the question to that effect. – M.Dimo Apr 11 '12 at 19:40
I'm having a tough time with edits today: No line feed. Anyway, here's the code: for(j in 2:p){ set( dt , allrows , j , dt[[j]] / denom[[2]] ) } and for both dt and denom, columns 2
to p are integer. The warning I get is – M.Dimo Apr 11 '12 at 19:47
"Warning message: In set(dt, allrows, j, dt[[j]]/denom[[2]]) : Coerced 'double' RHS to 'integer' to match the column's type; may have truncated precision. Either change the target
column to 'double' first (by creating a new 'double' vector length 16863 (nrows of entire table) and assign that; i.e. 'replace' column), or coerce RHS to 'integer' (e.g. 1L, NA_[real
|integer]_, as.*, etc) to make your intent clear and for speed. Or, set the column type correctly up front when you create the table and stick to it, please." – M.Dimo Apr 11 '12 at
You should pay close attention to those warnings. Try out the following to see why. D1 <- data.table(1:5); class(D1[[1]]); set(D1, 1:5, 1L, 0.33); D1. Compare that with what you
1 probably wanted to see: D2 <- data.table(as.numeric(1:5)); class(D2[[1]]); set(D2, 1:5, 1L, 0.33); D2. And then go about converting the class of the columns you are going to sweep to
numeric rather than integer (either in their initial construction, or after the fact by something like D1[[1]] <- as.numeric(D1[[1]])). (There might be more efficient ways to do that
last operation.) – Josh O'Brien Apr 11 '12 at 21:39
1 @Josh And when plonking (either with set() or :=) the new column into the column slot (instead of passing 1:nrow as i), the coercion warning goes away (coercion is only necessary when
updating subsets of the column). – Matt Dowle Apr 12 '12 at 5:42
show 6 more comments
Not the answer you're looking for? Browse other questions tagged r data.table or ask your own question.
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Reply to comment
If you've never heard of cubic Hamiltonian graphs before then take a look at Christopher Manning's wonderful cubic Hamiltonian graph builder. No, really, do! We too had never heard of them and now we
think they are the bee's knees!
But what is a cubic Hamiltonian graph, you ask? A graph, of course, is just a bunch of points (vertices) connected by lines (edges). A cubic graph is a graph where every vertex has 3 edges – that is,
each vertex is connected to exactly three others in the graph. And a Hamiltonian graph is a graph which has a closed loop of edges (a cycle) that visits each vertex in the graph once and only once,
(this is called a Hamiltonian cycle). So a cubic Hamiltonian graph is a graph where each vertex is joined to exactly three others and the graph has a cycle visiting each vertex exactly once.
What has made us so excited about cubic Hamiltonian graphs is watching Manning's cubic Hamiltonian graph builder in action. The builder starts from the Hamiltonian cycle in the graph. This loop of
edges accounts for two out of the three edges for every vertex in the graph. The builder then starts adding in the third edge for each vertex, knitting the graph together before your eyes. The
creation of a torus is particularly beautiful to watch.
Manning uses something called the LCF notation to build the graphs. This ingenious notation succinctly describes the structure of cubic Hamiltonian graphs by describing how you add the extra edge to
each vertex by counting backwards or forwards around the Hamiltonian cycle. For example our torus has the LCF notation [10]^150. This means that every vertex is joined to another 10 edges along the
cycle, and this process is repeated 150 times, once for each vertex, to complete the graph. (The 1st vertex is joined to the 11th, the 2nd to the 12th, the 3rd to the 13th, and so on...)
A graph represented by the LCF notation [10,7,4,-4,-7,10,-4,7,-7,4]^2 starts with a cycle of 20 vertices – inside the square brackets is a list of instructions for 10 edges, and these are repeated
twice, giving the total number of extra edges, and therefore vertices, as 20. In this graph the 1st vertex is joined to the 11th (a distance of 10 edges), the 2nd to the 9th (a distance of 7), the
3rd to the 7th (a distance of 4), the 4th to the 20th (a distance of -4, or counting backwards 4 edges), the 5th to the 18th (distance of -7), and so on. This graph is knitted into a dodecahedron.
Watching Manning's program knit together these graphs is beautiful to watch. But this mathematics has many important uses as well. Not only are Hamiltonian cycles important mathematically, they also
have many useful applications. You can read more about graphs on Plus, and about the role Hamiltonian cycles play in bell ringing and DNA analysis. And you can read more about Christopher Manning's
work on his blog.
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|
Reply to comment
If you've never heard of cubic Hamiltonian graphs before then take a look at Christopher Manning's wonderful cubic Hamiltonian graph builder. No, really, do! We too had never heard of them and now we
think they are the bee's knees!
But what is a cubic Hamiltonian graph, you ask? A graph, of course, is just a bunch of points (vertices) connected by lines (edges). A cubic graph is a graph where every vertex has 3 edges – that is,
each vertex is connected to exactly three others in the graph. And a Hamiltonian graph is a graph which has a closed loop of edges (a cycle) that visits each vertex in the graph once and only once,
(this is called a Hamiltonian cycle). So a cubic Hamiltonian graph is a graph where each vertex is joined to exactly three others and the graph has a cycle visiting each vertex exactly once.
What has made us so excited about cubic Hamiltonian graphs is watching Manning's cubic Hamiltonian graph builder in action. The builder starts from the Hamiltonian cycle in the graph. This loop of
edges accounts for two out of the three edges for every vertex in the graph. The builder then starts adding in the third edge for each vertex, knitting the graph together before your eyes. The
creation of a torus is particularly beautiful to watch.
Manning uses something called the LCF notation to build the graphs. This ingenious notation succinctly describes the structure of cubic Hamiltonian graphs by describing how you add the extra edge to
each vertex by counting backwards or forwards around the Hamiltonian cycle. For example our torus has the LCF notation [10]^150. This means that every vertex is joined to another 10 edges along the
cycle, and this process is repeated 150 times, once for each vertex, to complete the graph. (The 1st vertex is joined to the 11th, the 2nd to the 12th, the 3rd to the 13th, and so on...)
A graph represented by the LCF notation [10,7,4,-4,-7,10,-4,7,-7,4]^2 starts with a cycle of 20 vertices – inside the square brackets is a list of instructions for 10 edges, and these are repeated
twice, giving the total number of extra edges, and therefore vertices, as 20. In this graph the 1st vertex is joined to the 11th (a distance of 10 edges), the 2nd to the 9th (a distance of 7), the
3rd to the 7th (a distance of 4), the 4th to the 20th (a distance of -4, or counting backwards 4 edges), the 5th to the 18th (distance of -7), and so on. This graph is knitted into a dodecahedron.
Watching Manning's program knit together these graphs is beautiful to watch. But this mathematics has many important uses as well. Not only are Hamiltonian cycles important mathematically, they also
have many useful applications. You can read more about graphs on Plus, and about the role Hamiltonian cycles play in bell ringing and DNA analysis. And you can read more about Christopher Manning's
work on his blog.
|
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Math Forum Discussions
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Topic: Chapter 3--Everybody Counts
Replies: 11 Last Post: Mar 18, 1995 1:02 AM
Messages: [ Previous | Next ]
Re: Chapter 3--Everybody Counts
Posted: Mar 18, 1995 1:02 AM
>[snipped in several places]
>1. What really is "mathematical power" and how do your students get it?
> [The definition given in this reform document differs markedly from
> those given in the various Standards documents, in my opinion]
I believe that "mathematical power" is the power to use mathematical
language to articulate relationships, patterns, etc. that one observes in
their dealing with the environment. For example: My son (now 18yrs. old)
playes a game called Mancala, I'm sure many of you know it. He is
unbeatable! All of our family members, electrical engineers included, have
made it their mission to beat him at this game and cannot. It does not
matter whether he begins the game or plays second. It does not matter the
number of stones that the players begin with. When questioned about what
he does (thinks) as he plays he cannot articulate what he is thinking. If
he had "mathematical power" he would be able provide the mathematical
explaination as to what he is thinking when he plays. Since he has taken
mathematics up through pre-calc. It is particularly discouraging to find
that with all that math he cannot use the language to articulate his
thinking. I see his math knowledge to be as useless as his knowledge of
spanish. He has memorized "dialogues" and does not have the power to
generate "sentences" of his own. In otherwords he has never been taken to
the application/synthesis level in math.
People with "math power" can create their own "sentences" about the world
and its relationships.
>3. Comment on the statement: "As computers become more powerful, the
> need for mathematics will decline."
I would rather say, as computers become more powerful the need for
arithmetic will decline, the need for mathematics will increase.
>4. Why is it that mathematics education in the United States resists
> change in spite of the many forces that are revolutionizing the nature
> and role of mathematics itself?
Our reliance on "sub standard" standardized tests!
>5. Why do you suppose that 50% of school teachers leave the profession
> every seven years?
Cut backs in funding are eliminating 2.5 million from our school district
budget for next year. 90% of these cuts will be staff.
Eileen Abrahamson
Edw. Neill Elemetary
13409 Upton Ave. So.
Burnsville, MN 55337
Date Subject Author
3/14/95 Chapter 3--Everybody Counts Ronald A Ward
3/14/95 Re: Chapter 3--Everybody Counts Steve Means
3/14/95 Re: Chapter 3--Everybody Counts Tad Watanabe
3/14/95 Re: Chapter 3--Everybody Counts Andre TOOM
3/14/95 Re: Chapter 3--Everybody Counts Michael Paul Goldenberg
3/15/95 Re: Chapter 3--Everybody Counts Andre TOOM
3/15/95 Re: Chapter 3--Everybody Counts Michael Paul Goldenberg
3/14/95 Re: Chapter 3--Everybody Counts Jack Roach
3/15/95 Re: Chapter 3--Everybody Counts Tad Watanabe
3/15/95 Re: Chapter 3--Everybody Counts Herbert Kasube
3/15/95 Re: Chapter 3--Everybody Counts Mary K. Hannigan
3/18/95 Re: Chapter 3--Everybody Counts Eileen Abrahamson
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Matches for:
The Malliavin calculus was developed to provide a probabilistic proof of Hörmander's hypoellipticity theorem. The theory has expanded to encompass other significant applications.
The main application of the Malliavin calculus is to establish the regularity of the probability distribution of functionals of an underlying Gaussian process. In this way, one can prove the
existence and smoothness of the density for solutions of various stochastic differential equations. More recently, applications of the Malliavin calculus in areas such as stochastic calculus for
fractional Brownian motion, central limit theorems for multiple stochastic integrals, and mathematical finance have emerged.
The first part of the book covers the basic results of the Malliavin calculus. The middle part establishes the existence and smoothness results that then lead to the proof of Hörmander's
hypoellipticity theorem. The last part discusses the recent developments for Brownian motion, central limit theorems, and mathematical finance.
A co-publication of the AMS and CBMS.
Graduate students and research mathematicians interested in probability, the Malliavin calculus, and stochastic partial differential equations.
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flow chart
Definition of Flow Chart
● A Flow Chart represents a sequence of operations or algorithms by using diagrams.
More about Flow Chart
• It is advisable to draw a flow chart before writing a computer program.
Example of Flow Chart
• See the following flow chart.
The flow chart shows the flow of data.
In the above flow chart, it shows the daily time table of a school boy.
Solved Example on Flow Chart
The flow chart shows the probability of getting a sum, which is a multiple of 5 when two dice are rolled. If Robert gets the sum as 8, according to the flow chart, what will Robert do next?
A. Roll the dice again.
B. He subtracts 3 from 8 to proceed.
C. He adds 2 to 8 to proceed.
D. He gets the probability as 1/2.
Correct Answer: A
Step 1: It is given that the flow chart shows the probability of getting a sum that is a multiple of 5 (5, 10, 15…).
Step 2: Robert gets the sum as 8, which is not a multiple of 5.
Step 3: So, Robert has to roll the dice again.
Related Terms for Flow Chart
● Carroll Diagram
● Scatter Diagram
● Tree Diagram
● Venn Diagram
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Integration by parts
$\int_0^\infty 2\alpha x^{2 }e^{-\alpha x^2}dx$ we can here use the substitution $t= x^2 \,\, \, dx = \frac{1}{2}t^{\frac{-1}{2}}\, dt$ so we get the following : $\alpha \int^{\infty}_0 \, t^{1/2}e^
{-\alpha t }\, dt$ Now this is solved by the Laplace transform or gamma function which converges for $\alpha >0$ $\alpha \int^{\infty}_0 \, t^{{1/2}}e^{-\alpha t }\, dt = \alpha \frac{\Gamma {(\frac
{3}{2})}}{\alpha^{\frac{3}{2}}}= \frac{\Gamma(\frac{1}{2})}{2\sqrt{\alpha }}= \frac{\sqrt{\pi}}{2\sqrt{\alpha }}$ --------------------------------------------------------------- Now for the integrand
: $\int_0^\infty x^{2\alpha}e^{-\alpha x^2}dx$ here if we use the same sub we get : $\int^{\infty}_0 \, t^{\alpha -\frac{1}{2}}e^{-\alpha t }\, dt = \frac{\Gamma {(\alpha +\frac{1}{2})}}{2\alpha^{\
alpha +\frac{1}{2}}}$
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Thornton, CO Algebra 2 Tutor
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Math Forum Discussions
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Topic: Convergence
Replies: 0
Posted: May 9, 2001 10:33 AM
The issue has nothing to do with infinity or 1/infinity. It
has to do with the definition of real numbers and what it
means for a sequence (or series) to converge.
Here's a sketch of the ideas as I see them.
1. Define the real numbers; for the sake of argument, as
Cauchy sequences of rationals.
2. Define when two real numbers are equal: Suppose A=(an)
and B=(bn) (Cauchy sequences). Then A = B if the terms of
the sequence (an-bn) can be made less than any epsilon>0 by
making n big enough.
3. Define the arithmetic of the reals: addition,
multiplication, inequalities, absolute value.
4. Define L = Lim An if |L - An| can be made less than any
epsilon>0 by making n big enough (L and the An are reals
here -- see 3 above).
5. Define the infinite sum: Sum(An) = L if L is the limit
of the partial sums of the An (see 4 above).
That all there is! No mystery. The wonderful thing about
the real analysis developed by Weierstrasse, Cauchy et. al.
is that it FREED US FROM THE TERROR OF THE INFINITE! You
never have to mention infinity anywhere in definitions 1-5
above. It is very worthwhile to read Aristotle on the
distinction between the "actual" infinite (which is scary
and not quite believable) and the "potential" infinite,
which is what modern (post 18th century) analysis is all
about. Forget "nonstandard analysis" which is based on
formalism and, as they say, seeks to explain the obscure by
the more obscure.
--- Mark
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Manhattan GMAT Challenge Problem of the Week – 12 November 2012
Here is a new Challenge Problem! If you want to win prizes, try entering our Challenge Problem Showdown. The more people that enter our challenge, the better the prizes!
The positive difference of the fourth powers of two consecutive positive integers must be divisible by
A. one less than twice the larger integer
B. one more than twice the larger integer
C. one less than four times the larger integer
D. one more than four times the larger integer
E. one more than eight times the larger integer
You could solve this problem algebraically, or you could solve by picking numbers. Let’s try the latter method first, as it winds up being faster and easier mentally.
Write the fourth powers of the first three positive integers:
Now, for the first two fourth powers, the “positive difference of the fourth powers of two consecutive positive integers” would be this:
15 is divisible by 3 and by 5 (as well as by 1 and by 15). Since the “larger integer” is 2, eliminate answers:
(A) one less than twice the larger integer = 2×2 – 1 = 3 = fine
(B) one more than twice the larger integer = 2×2 + 1 = 5 = fine
(C) one less than four times the larger integer = 4×2 – 1 = 7 = wrong
(D) one more than four times the larger integer = 4×2 + 1 = 9 = wrong
(E) one more than eight times the larger integer = 8×2 + 1 = 17 = wrong
You’re left with (A) and (B). Now use 2 and 3 as your consecutive integers:
65 is divisible by 5 and by 13 (as well as by 1 and by 65). Test (A) and (B):
(A) one less than twice the larger integer = 2×3 – 1 = 5 = fine
(B) one more than twice the larger integer = 2×3 + 1 = 7 = wrong
The answer must be (A).
As for the algebraic proof, here it is:
“The positive difference of the fourth powers of two consecutive positive integers”
If you expand the
Now expand the right side:
So you get:
Thus, as long as n is an integer, the expression on the left is always divisible by 2n – 1, which corresponds to choice A.
The correct answer is A.
Special Announcement: If you want to win prizes for answering our Challenge Problems, try entering our Challenge Problem Showdown. Each week, we draw a winner from all the correct answers. The winner
receives a number of our our Strategy Guides. The more people enter, the better the prize. Provided the winner gives consent, we will post his or her name on our Facebook page.
If you liked this article, let Manhattan GMAT know by clicking Like.
2 comments
Be careful of term "larger integer" which should be "n".
If you starts as (n+1)^4-(n)^4, the fomular will mislead you (B).
Great point Jinwoo. This is what I did when I was trying to solve this question. But, I am still not sure why it did not work if we took the numbers as n+1 and n ; considering n+1 as the
greater consecutive integer. Any thoughts please.
Ask a Question or Leave a Reply
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What is 1 NIBBLE?
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Date: 01 Feb 2010 Group: Computers Category: Software
What is 1 NIBBLE?
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Author: ranganathan 01 Feb 2010 Member Level: Gold Points : 3 (Rs 1) Voting Score: 2
In computers everything is represented in terms of bits in terms of 1's or 0's. A bit represents a single 1 or 0.
A nibble consists of 4 bits or it constitutes to a hexadecimal unit.
A byte consists of 8 bits which is also called an octet. Nibble can be also termed as half byte or semioctet.
"11110000" this can be said as a byte and half of this is nibble that is "1111".
Author: arjun c c 01 Feb 2010 Member Level: Bronze Points : 1 Voting Score: 0
nibble is a word used in binary number system. it is a combination of 4 bits.
Author: 02 Feb 2010 Member Level: Bronze Points : 1 Voting Score: 0
nibble means 4bits
in computers all the memory is stored in terms of bits(1,0)
and 4 bits are called one nibble
Author: Arwin Pereira 02 Feb 2010 Member Level: Bronze Points : 1 Voting Score: 0
1 byte devided by 2 is equal to 1 nibble.
1 byte = 8 bit so 1 nibble = 4 bit
Author: anshul pareek 03 Feb 2010 Member Level: Bronze Points : 1 Voting Score: 0
computer can only understand binary language that is 0 and 1.
in computer these digits are called 1 bit more specifically.
when computer get either 0 or 1 it means it is 1 bit. therefore groups of 4 bits are called 1 nibble.
Author: anshul pareek 03 Feb 2010 Member Level: Bronze Points : 1 Voting Score: 0
computer can only understand binary language that is 0 and 1.
in computer these digits are called 1 bit more specifically,
when computer get either 0 or 1 it means it is 1 bit. therefore groups of 4 bits are called 1 nibble.
Author: Ramachandran Ravi 03 Feb 2010 Member Level: Bronze Points : 1 Voting Score: 0
Computer Understands Binary Digits (Bits 0 & 1) only.
Four Bits constitute 1 Nibble
Eight Bits constitute a Byte
Author: Jayaramachandaran 18 Feb 2010 Member Level: Silver Points : 2 Voting Score: 0
hi this is jayaram,
*Machine understands only the language which consists of 0's and 1's .
*Then they call each 0 and 1 as bit.
*They call the series of 4 (combinations of 4 bits i,e. 0000,0001,0010,0011,0100,0101,0110,0111,1000,1001,1010,1011,1100,1101,1110,1111)bits as 1 Nibble and 8 bits as 1
byte.sometimes they call 1 byte and 1 nibble as word.
*this gives the clear idea of NIBBLE.
Author: vijay 18 Feb 2010 Member Level: Bronze Points : 1 Voting Score: 0
Nibble is a unit of computer system.
It is a bunch of 4 bits like 1001,1101,0101......so it is a half of byte.
Mostly we don't use it but we'll use it in IPv6.
Author: gokul 19 Feb 2010 Member Level: Bronze Points : 1 Voting Score: 0
NIBBLES:combinations of 4 bit is called nibble.... bits are nothing but 0's and 1's.... 4 bits possibilities are 1111,1110,0000,0001,0010,1100,1000,0100,..)
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On the theory of correlation for any number of variables, treated by a new system of notation
Results 1 - 10 of 13
- Communications of the ACM , 1995
"... We examine a graphical representation of uncertain knowledge called a Bayesian network. The representation is easy to construct and interpret, yet has formal probabilistic semantics making it
suitable for statistical manipulation. We show how we can use the representation to learn new knowledge by c ..."
Cited by 299 (13 self)
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We examine a graphical representation of uncertain knowledge called a Bayesian network. The representation is easy to construct and interpret, yet has formal probabilistic semantics making it
suitable for statistical manipulation. We show how we can use the representation to learn new knowledge by combining domain knowledge with statistical data. 1 Introduction Many techniques for
learning rely heavily on data. In contrast, the knowledge encoded in expert systems usually comes solely from an expert. In this paper, we examine a knowledge representation, called a Bayesian
network, that lets us have the best of both worlds. Namely, the representation allows us to learn new knowledge by combining expert domain knowledge and statistical data. A Bayesian network is a
graphical representation of uncertain knowledge that most people find easy to construct and interpret. In addition, the representation has formal probabilistic semantics, making it suitable for
statistical manipulation (Howard,...
- Journal of Labor Economics , 2006
"... We thank Art Goldberger for helpful pointers to the literature. Estimating the Returns to College Quality with Multiple ..."
- http://arxiv.org/abs/cs.AI/0308002 v3 , 2004
"... Interactions are patterns between several attributes in data that cannot be inferred from any subset of these attributes. While mutual information is a well-established approach to evaluating
the interactions between two attributes, we surveyed its generalizations as to quantify interactions between ..."
Cited by 25 (4 self)
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Interactions are patterns between several attributes in data that cannot be inferred from any subset of these attributes. While mutual information is a well-established approach to evaluating the
interactions between two attributes, we surveyed its generalizations as to quantify interactions between several attributes. We have chosen McGill’s interaction information, which has been
independently rediscovered a number of times under various names in various disciplines, because of its many intuitively appealing properties. We apply interaction information to visually present the
most important interactions of the data. Visualization of interactions has provided insight into the structure of data on a number of domains, identifying redundant attributes and opportunities for
constructing new features, discovering unexpected regularities in data, and have helped during construction of predictive models; we illustrate the methods on numerous examples. A machine learning
method that disregards interactions may get caught in two traps: myopia is caused by learning algorithms assuming independence in spite of interactions, whereas fragmentation arises from assuming an
interaction in spite of independence.
- J. Multivariate Anal , 2003
"... The second order properties of a process are usually characterized by the autocovariance function. In the stationary case, the parameterization by the partial autocorrelation function is
relatively recent. We extend this parameterization to the nonstationary case. The advantage of this function is t ..."
Cited by 9 (5 self)
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The second order properties of a process are usually characterized by the autocovariance function. In the stationary case, the parameterization by the partial autocorrelation function is relatively
recent. We extend this parameterization to the nonstationary case. The advantage of this function is that it is subject to very simple constraints in comparison with the autocovariance function which
must be nonnegative definite. As in the stationary case, this parameterization is well adapted to autoregressive models or to the identification of deterministic processes.
- Journal of Time Series Analysis
"... Abstract. The extension of stationary process autocorrelation coefficient sequence is a classical problem in the field of spectral estimation. In this note, we treat this extension problem for
the periodically correlated processes by using the partial autocorrelation function. We show that the theor ..."
Cited by 6 (4 self)
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Abstract. The extension of stationary process autocorrelation coefficient sequence is a classical problem in the field of spectral estimation. In this note, we treat this extension problem for the
periodically correlated processes by using the partial autocorrelation function. We show that the theory of the non-stationary processes can be adapted to the periodically correlated processes. The
partial autocorrelation function has a clear advantage for parameterization over the autocovariance function which should be checked for non-negative definiteness. In this way, we show that contrary
to the stationary case, the Yule–Walker equations (for a periodically correlated process) is no longer a tool for extending the first autocovariance coefficients to an autocovariance function. Next,
we treat the extension problem and present a maximum entropy method extension through the the partial autocorrelation function. We show that the solution maximizing the entropy is a periodic
autoregressive process and compare this approach with others.
"... Abstract. In 1922 R. A. Fisher introduced the modern regression model, synthesizing the regression theory of Pearson and Yule and the least squares theory of Gauss. The innovation was based on
Fisher’s realization that the distribution associated with the regression coefficient was unaffected by the ..."
Cited by 3 (3 self)
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Abstract. In 1922 R. A. Fisher introduced the modern regression model, synthesizing the regression theory of Pearson and Yule and the least squares theory of Gauss. The innovation was based on
Fisher’s realization that the distribution associated with the regression coefficient was unaffected by the distribution of X. Subsequently Fisher interpreted the fixed X assumption in terms of his
notion of ancillarity. This paper considers these developments against the background of the development of statistical theory in the early twentieth century.
"... Finding an unconstrained and statistically interpretable reparameterization of a covariance matrix is still an open problem in statistics. Its solution is of central importance in covariance
estimation, particularly in the recent high-dimensional data environment where enforcing the positive-definit ..."
Cited by 2 (0 self)
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Finding an unconstrained and statistically interpretable reparameterization of a covariance matrix is still an open problem in statistics. Its solution is of central importance in covariance
estimation, particularly in the recent high-dimensional data environment where enforcing the positive-definiteness constraint could be computationally expensive. We provide a survey of the progress
made in modeling covariance matrices from the perspectives of generalized linear models (GLM) or parsimony and use of covariates in low dimensions, regularization (shrinkage, sparsity) for
high-dimensional data, and the role of various matrix factorizations. A viable and emerging regressionbased setup which is suitable for both the GLM and the regularization approaches is to link a
covariance matrix, its inverse or their factors to certain regression models and then solve the relevant (penalized) least squares problems. We point out several instances of this regression-based
setup in the literature. A notable case is in the Gaussian graphical models where linear regressions with LASSO penalty are used to estimate the neighborhood of one node at a time (Meinshausen and
Bühlmann, 2006). Some advantages
"... In this paper, an old identity of G. U. Yule among partial correlation coefficients is recognized as being equal to the cosine law of spherical trigonometry. Exploiting this connection enables
us to derive some new (and potentially useful) relations among partial correlation coefficients. Moreover, ..."
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In this paper, an old identity of G. U. Yule among partial correlation coefficients is recognized as being equal to the cosine law of spherical trigonometry. Exploiting this connection enables us to
derive some new (and potentially useful) relations among partial correlation coefficients. Moreover, this observation provides new (dual) non-Euclidean geometrical interpretations of the Schur and
Levinson-Szegö algorithms.
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La Canada Flintridge Precalculus Tutor
Find a La Canada Flintridge Precalculus Tutor
...I tutored throughout high school (algebra, calculus, statistics, chemistry, physics, Spanish, and Latin) and tutored advanced math classes during college. Above all other things, I love to
learn how other people learn and to teach people new things in ways so that they will find the material int...
28 Subjects: including precalculus, chemistry, Spanish, physics
...I have taken the standard Caltech course in differential equations, the graduate course in differential equations, and my thesis is on differential equations of ions in an ion trap. I am a
graduating senior at Caltech in physics. I have taken the undergraduate course in Linear algebra.
26 Subjects: including precalculus, calculus, physics, algebra 2
I have always been asked for help in many subjects, from high school until graduating college as a Mechanical Engineer. Friends and family always asked for my help and I have had the patience to
show them step by step until they could do it on their own, which will be my goal with any student needing help. As a Mechanical Engineer I specialize in math and science at almost any level.
9 Subjects: including precalculus, calculus, algebra 1, algebra 2
Whatever the task at hand — whether it's SAT prep, a fourth-grade math test, a physics problem set, or an essay on Othello — my focus is always on the thinking and learning process involved. I
don't tell students what to do; I teach them how to do it. I've been professionally tutoring since I was 17, when the Princeton Review hired me to teach SAT classes in Michigan.
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Engineering Graphics–January 2010 Question Paper
Anna University
B.E./B.Tech. DEGREE EXAMINATIONS, JANUARY 2010
Regulations 2008
First Semester
Common to all branches
GE2111 ENGINEERING GRAPHICS
Time: Three Hours
Maximum: 100 Marks
Answer ALL Questions
(5 x 20 = 100 Marks)
1. (a) Draw a hyperbola when the distance between its focus an d directrix is 50 mm and eccentricity is 3/2. Also draw the tangent an d normal at a point 25 mm from the directrix. (20)
1. (b) Make free hand sketches of front, top and right side views of the 3D object shown below : (20)
Fig. 1.(b)
2. (a) A line PQ has its end P, 10 mm above th e HP and 20 mm in front of t he VP. The end Q is 35 mm in front of t he VP. The front view of t he line measures 75 mm. The distance between th e end
projectors is 50 mm. Draw the projections of t he line an d find its true length and its true inclinations with th e VP and HP. (20)
2. (b) Draw th e projections of a circle of 70 mm diameter resting on t he H.P. on a point A of t he circumference. The plane is inclined to the H.P. such that th e top view of it i s an ellipse of
minor axis 40 mm. The top view of t he diameter, through the point A is making an angle of 45◦ with t he V.P. Determine the inclination of the plane with th e H.P. (20)
3. (a) An equilateral triangular prism 20 mm side of base a nd 50 mm long rests with one of its shorter edges on HP such that the rectangular face containing th e edge on which the prism rests is
inclined at 30◦ to H.P. The shorter edge resting on HP is perpendicular to VP. (20)
3. (b) A square pyramid of base 40 mm a nd axis 70 mm long has one of its triangular faces on VP a nd th e edge of base contained by that face perpendicular to HP. Draw its projections. (20)
4. (a) A hexagonal prism of side of base 35 mm and axis length 55 mm rests with its base on H.P such that two of the vertical surfaces ar e perpendicular to V.P. It is cut by a plane inclined at 50◦
to H.P a nd perpendicular to V.P an d passing through a point on the axis at a distance 15 mm from th e top. Draw its front view, sectional top view and true shape of section. (20)
4. (b) Draw the development of the lateral surface of the lower portion of a cylinder of diameter 50 mm a nd axis 70 mm. The solid is cut by a section plane inclined at 40◦ to HP and perpendicular to
VP an d passing through the midpoint of the axis. (20)
5. (a) Draw th e isometric projection of t he object from th e views shown in Figure 5(a). (20)
5. (b) Draw the perspective projection of a cube of 25 mm edge, lying on a face on th e ground plane, with an edge touching the picture plane and all vertical faces equally inclined to th e picture
plane. The station point is 50 mm in front of the picture plane, 35 mm above th e ground plane and lies in a central plane which is 10 mm to t he left of the center of the cube. (20)
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Summary: In ComputerAided Verification, CAV '96 (Rajeev Alur and Thomas A. Henzinger, eds.), LNCS vol. 1102, 1991, 26--37, c
Symbolic Model Checking Using Algebraic Geometry
George S. Avrunin
Department of Mathematics, University of Massachusetts, Amherst, MA 010034515
Abstract. In this paper, I show that methods from computational algebraic ge
ometry can be used to carry out symbolic model checking using an encoding of
Boolean sets as the common zeros of sets of polynomials. This approach could
serve as a useful supplement to symbolic model checking methods based on Or
dered Binary Decision Diagrams and may provide important theoretical insights
by bringing the powerful mathematical machinery of algebraic geometry to bear
on the model checking problem.
1 Introduction
Symbolic model checking [8, 13] with Ordered Binary Decision Diagrams (OBDDs),
or variants of OBDDs, is a widely used and successful technique for verifying properties
of concurrent systems, both hardware and software. But there are many systems for
which the OBDDs are too large to make model checking feasible and, aside from a few
results like McMillan's theorem on bounded width circuits [13] or Bryant's theorem on
integer multiplication [5], there is little theoretical guidance to indicate precisely when
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Differential equations of fluid mechanics
2. Basic Differential Equations
The flux equations can be used to determine the net rate of increase of each quantity within a cube of volume ΔV= ΔxΔyΔz. We implicitly assume the limit where the dimensions approach zero.
Mass Flux
For mass, the net flux inflow through the face at x=x[o ]is ρu[x] kg-m^-2-s^-1. The outflow through the face at x=x[o]+Δx is
Therefore the net flow across the two faces, each of area ΔyΔz, is
Converting this to density, and adding in the flow through the other four faces,
This is known as the equation of continuity, and basically reflects the law of conservation of mass.
Momentum Flux
The net flow of x-directed momentum across the same two faces parallel to the y-z plane is
In addition there is a flow of x-directed momentum across the other four faces, given by
Putting this together with the other momentum components, and dividing by ΔV yields
This equation reflects the law of conservation of momentum.
Energy Flux
Following the same procedure for the third and final flux yields
For a monatomic molecule, the factor of 5 in the total energy term on the left side of equation (15) would be replaced by 3, and the factor of 7 on the right-hand side of the equation would be
replaced by 5. These terms represent the effects that are typically explained in terms of specific heat ratios, which we do not need to be concerned with. This equation reflects the law of
conservation of energy, and also implicitly defines adiabatic behavior, since no energy is added to or subtracted from the system as a whole.
Expanding the time derivative in equation (14), substituting the right-hand side of equation (11) for the left-hand side, and using the following vector identity
This is known as Euler's equation [Landau and Lifshitz page 3].
Pressure Differential Equation
Following a similar procedure, equation (15) can be transformed into a differential equation for pressure. The right-hand sides of both equations (11) and (14) are substituted for their left-hand
sides, and the following vector identity is also used:
The derivation is a bit messy, but otherwise straightforward. The result is:
This equation probably also has a name, but I haven't run across it. Equations (11), (17) and (19) are the basic equations we need to proceed. The next section makes several approximations based on
the assumption that the variation in all of the parameters are small compared to the static values, which finally leads to one primary goal of the analysis - the wave equations.
One-dimensional versions of equations (11), (17), and (19) can be found here.
It is interesting to note that this entire derivation could have been done in terms of expected values, using only the number of particles per unit volume, and the velocity statistics, without ever
making a connection to mass, momentum, or energy. Newton's 2nd law, relating force, mass, and acceleration, which is central to the usual derivation of sound waves, is not used at all in the above
derivation. The concepts of pressure and force are not necessary. In order to relate the results to the real world a connection to physical quantities is necessary, but the derivation itself is
almost entirely independent of Newtonian physics.
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User:Marshall Hampton
From OpenWetWare
I am a new member of OpenWetWare!
Contact Info
• Marshall Hampton
• University of Minnesota, Duluth
• Email: (first initial)hampton at d dot umn dot edu
• Blog: Neutral Drifts
I work in the Department of Mathematics and Statistics at the University of Minnesota, Duluth. I am also a member there in the Integrated Biosciences program. I learned about OpenWetWare from Julius
Lucks, and bioinformatics blogs, and I've joined because I would like to contribute to the awareness of the Sage computational platform..
• 2002, PhD, Mathematics, University of Washington
• 2000, MS, Mathematics, University of Washington
• 1994, BS in Mathematics and Physics, Stanford University
Research interests
1. Gene regulation and systems physiology in mammalian hibernators
2. Bioinformatics, particularly in AT-rich genomes of parasites such as P. falciparum
3. Celestial mechanics, dynamical systems, computational algebra and geometry
1. Hampton M and Andrews MT. . pmid:17459419.
My other papers are more mathematical in nature, and can be found on MathSciNet.
Useful links
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Component form of a vector
October 3rd 2008, 11:02 AM #1
Junior Member
Nov 2007
Component form of a vector
Hi, Im trying to find the component form of a vector obtained by rotating <0,1> through an angle of 2pi/3 radians. Can someone please help me with this problem? Thanks.
$\cos\left(\frac{2\pi}{3}\right)\vec{i} + \sin\left(\frac{2\pi}{3}\right)\vec{j}$
October 3rd 2008, 01:47 PM #2
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When we build large clusters, such as high performance clusters or any cluster with a large number of computing nodes, we begin to look in detail at the repair models for the system. You are probably
aware of the need to study power usage, air conditioning, weight, system management, networking, and cost for such systems. So you are also aware of how multiplying the environmental needs of one
computing node times the number of nodes can become a large number. This can be very intuitive for most folks. But availability isn't quite so intuitive. Deferred repair models can also affect the
intuition of the design. So, I thought that a picture would help show how we analyze the RAS characteristics of such systems and why we always look to deferred repair models in their design.
To begin, we have to make some assumptions:
• The availability of the whole is not interesting. The service provided by a big cluster is not dependent on all parts being functional. Rather, we look at it like a swarm of bees. Each bee can be
busy, and the whole swarm can contribute towards making honey, but the loss of a few bees (perhaps due to a hungry bee eater) doesn't cause the whole honey producing process to stop. Sure, there
may be some components of the system which are more critical than others, like the queen bee, but work can still proceed forward even if some of these systems are temporarily unavailable (the
swarm will create new queens, as needed). This is a very different view than looking at the availability of a file service, for example.
• The performability will might be interesting. How many dead bees can we have before the honey production falls below our desired level? But for very, very large clusters, the performability will
be generally good, so a traditional performability analysis is also not very interesting. It is more likely that a performability analysis of the critical components, such as networking and
storage, will be interesting. But the performability of thousands of compute nodes will be less interesting.
• Common root cause failures are not considered. If a node fails, the root cause of the failure is not common to other nodes. A good example of a common root cause failure is loss of power -- if we
lose power to the cluster, all nodes will fail. Another example is software -- a software bug which causes the nodes to crash may be common to all nodes.
• What we will model is a collection of independent nodes, each with their own, independent failure causes. Or just think about bees.
For a large number of compute nodes, even using modern, reliable designs, we know that the probability of all nodes being up at the same time is quite small. This is obvious if we look at the simple
availability equation:
Availability = MTBF / (MTBF + MTTR)
where, MTBF (mean time between failure) is MTBF[compute node]/N[nodes]
and, MTTR (mean time to repair) is > 0
The killer here is N. As N becomes large (thousands) and MTTR is dependent on people, then the availability becomes quite small. The time required to repair a machine is included in the MTTR. So as N
becomes large, there is more repair work to be done. I don't know about you, but I'd rather not spend my life in constant repair mode, so we need to look at the problem from a different angle.
If we make MTTR large, then the availability will drop to near zero. But if we have some spare compute nodes, then we might be able to maintain a specified service level. Or, some a practical
perspective, we could ask the question, "how many spare compute nodes do I need to keep at least M compute nodes operational?" The next, related question is, "how often do we need to schedule service
actions?" To solve this problem, we need a model.
Before I dig into the model results, I want to digress for a moment and talk about Mean Time Between Service (MTBS) and Mean Time Between System Interruption (MTBSI). I've blogged in detail about
these before, but to put there use in context here, we will actually use MTBSI and not MTBF for the model. Why? Because if a compute node has any sort of redundancy (ECC memory, mirrored disks, etc.)
then the node may still work after a component has failed. But we want to model our repair schedule based on how often we need to fix nodes, so we need to look at how often things break for two
cases. The models will show us those details, but I won't trouble you with them today.
The figure below shows a proposed 2000+ node HPC cluster with two different deferred repair models. For one solution, we use a one week (168 hour) deferred repair time. For the other solution, we use
a two week deferred repair time. I could show more options, but these two will be sufficient to provide the intuition for solving such mathematical problems.
We build a model showing the probability that some number of nodes will be down. The OK state is when all nodes are operational. It is very clear that the longer we wait to repair the nodes, the less
probable it is that the cluster will be in the OK state. I would say, that that with a two week deferred maintenance model, there is nearly zero probability that all nodes will be operational.
Looking at this another way, if you want all nodes to be available, you need to have a very, very fast repair time (MTTR approaching 0 time). Since fast MTTR is very expensive, accepting a deferred
repair and using spares is usually a good cost trade-off.
OK, so we're convinced that a deferred repair model is the way to go, so how many spare compute nodes do we need? A good way to ask that question is, "how may spares do I need to ensure that there is
a 95% probability that I will have a minumum of M nodes available?" From the above graph, we would accumulate the probability until we reached the 95% threshold. Thus we see that for the one week
deferred repair case, we need at least 8 spares and for the two week deferred repair case we need at least 12 spares. Now this is something we can work with.
The model results will change based on the total number of compute nodes and their MTBSI. If you have more nodes, you'll need more spares. If you have more reliable or redundant nodes, you need fewer
spares. If we know the reliability of the nodes and their redundancy characteristics, we have models which can tell you how many spares you need.
This sort of analysis also lets you trade-off the redundancy characteristics of the nodes to see how that affects the system, too. For example, we could look at the affect of zero, one, or two disks
(mirrored) per node on the service levels. I personally like the zero disk case, where the nodes boot from the network, and we can model such complex systems quite easily, too. This point should not
be underestimated, as you add redundancy to increase the MTBSI, you also increase the MTBS, which impacts your service costs. The engineer's life is a life full of trade-offs.
In conclusion, building clusters with lots of nodes (red shift designs) requires additional analysis beyond what we would normally use for critical systems with few nodes (blue shift designs). We
often look at service costs using a deferred service interval and how that affects the overall system service level. We also look at the trade-offs between per-node redundancy and the overall system
service level. With proper analysis, we can help determine the best performance and best cost for large, red shift systems.
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Good day my name is Dudley and I would like to know how do I calculate the opportunity cost
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When given that a party can produce two goods, take the quantity of the second and put it over the quantity of the first. You now have the opportunity cost of doing 1 unit of the first good, in
terms of the second. Bob can do 8 math problems or 4 science problems in one sitting. Put the 4 over the 8 to get 1/2. Bob could have done 1/2 science for every 1 math he chooses to do. That is
the opportunity cost. The reciprocal is then also true, He could have done 2 math for every 1 science that he does.
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opportunity cost = explicit costs + implicit cost explicit cost= payments to non owners of a firms for their resources. example, wages paid to labor, the cost of electricity. implicit cost=
opportunity costs of using resources owned by the firm. example, when you used your own building for the firm, you made no payment to anyone, then you gave up the opportunity to earn rental
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OC cost means that you have chosen to be on the Internet altough you could have been hanging out with your friends. When the PRODUCER wants to measure his OC of producing 2 goods (X1 and X2) he
can calculate it like this: \[OC= \frac{ DeltaX2 }{ DeltaX1 } = -\frac{ p1 }{ p2 }\] THe producer can see clearly how much does it cost to produce an EXTRA UNIT of the good X2, in other words,
how much units of the X1 good he has to give up so he could produce an extra unit of X2 good. Hope this helps.
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C# Some Math
06-21-2011 #1
Registered User
Join Date
Mar 2009
C# Some Math
Okay I am screwing around with web sockets and am trying to negotiate a connection but I have to do some math to generate a proper negotiation.
draft-ietf-hybi-thewebsocketprotocol-03 - The WebSocket protocol That is the RFC I am referencing.
Okay I have the following information:
Key1: 18x 6]8vM;54 *(5: { U1]8 z [ 8
Key2: 1_ tx7X d < nw 334J702) 7]o}` 0
Final: Tm[K T2u
First step is removing all non-numeric characters from key1 and key 2 leaving me with.
Key1: 1868545188
Key2: 1733470270
I used a simple regex for that [^[0-9]] that worked fine
Next I have to calculate the spaces in the original key1 and key 2 which I did sucesfully with a foreach statement.
Key1 spaces = 12
Key2 spaces = 10
Now heres the part I am not sure of and need help on. I need to divide the key1 number by the number of spaces
1868545188 / 12 155712099
1733470270 / 10 173347027
I need to express each of these output as Big Endian 32 bit numbers? Does that mean flip them? how would I do that in C#.
After that I concatenate the two numbers with the Final and get the md5 sum of that. Again how do I do this in c#. I am currently doing it this way:
MD5CryptoServiceProvider x = new MD5CryptoServiceProvider();
byte[] output = x.ComputeHash(input);
string MD5Output = ASCIIEncoding.UTF8.GetString(output);
If you guys could help me out I would appreciate it. Note the final outcome of those calculations should be a 16 byte 128 bit value, which in this case is: fQJ,fN/4F4!~K~MH
I need to express each of these output as Big Endian 32 bit numbers? Does that mean flip them? how would I do that in C#.
Use BitConverter.GetBytes, reverse the array and put it back to an int (or not, depending on what you want to do next):
int value;
if (BitConverter.IsLittleEndian)
byte[] temp = BitConverter.GetBytes(155712099);
value = BitConverter.ToInt32(temp);
After that I concatenate the two numbers with the Final and get the md5 sum of that. Again how do I do this in c#
See the System.Security.Cryptography.MD5 class
The programmer’s wife tells him: “Run to the store and pick up a loaf of bread. If they have eggs, get a dozen.”
The programmer comes home with 12 loaves of bread.
Originally Posted by brewbuck:
Reimplementing a large system in another language to get a 25% performance boost is nonsense. It would be cheaper to just get a computer which is 25% faster.
Yes, reversing the arrays for the two ints is required.
Just a small point, but if you're already looping through the chars of each key to count the spaces, you could use that same loop to check for numbers.
static int NumberFromKey(String key, out int spaces)
spaces = 0;
String str = String.Empty;
// why bother with regex if you're already looping to count spaces?
foreach (char c in key.ToCharArray())
if (c == ' ')
else if (Char.IsDigit(c))
str += c.ToString();
return int.Parse(str);
06-21-2011 #2
06-21-2011 #3
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Mar 2009
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