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st: Re: reg3 command and first-stage estimations
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st: Re: reg3 command and first-stage estimations
From Kit Baum <baum@bc.edu>
To statalist@hsphsun2.harvard.edu
Subject st: Re: reg3 command and first-stage estimations
Date Wed, 5 Sep 2007 06:47:09 -0400
Please provide the exact command you are using. What you describe below doesn't make a lot of sense---if y = f (e,k) and both of those regressors are endogenous, then you have a three-equation
system. E.g.
y = b0 + b1 e + b2 k + err (requires two instruments by order condition)
e = c0 + c1 w + err (for instance)
k = d0 + d1 d + err (for instance)
The FSRs will be the regression of each of the endog on all of the exog. E.g.
. webuse klein
. reg3 (consump profits wagepriv) ( profits invest) ( wagepriv capital1), first
The only difference between the point and interval estimates of this consump equation and those of
ivreg2 consump (profits wagepriv=invest capital1)
is that the reg3 estimates apply the exclusion restrictions that capital1 does not appear in the 2d eqn and invest does not appear in the 3d eqn. Those are testable hypotheses (as you can see by
using the first option on ivreg2) and in this case the exclusion restrictions are rejected by the data (and the equations are misspecified, leading to misspecification error in the equation of
interest as well).
If the equations for e and k in your context are not truly simultaneous---i.e. if they do not contain y--then there is no need to use 3SLS to estimate the system. If you fear that e and k are
correlated with the error in the y eqn, use 2SLS (IV). You will find that 2SLS and 3SLS yield the same point and interval estimates if you relax the exclusion restrictions in the 2d and 3d eqns.
Kit Baum, Boston College Economics and DIW Berlin
An Introduction to Modern Econometrics Using Stata:
On Sep 5, 2007, at 2:33 AM, statalist-digest wrote:
I'm using the -reg3- command because I want to simultaneously estimate some equations, since I expect the error term across these equations to be correlated. I'm also using instruments for the
regressors, because I have an endogeneity problem. I use the option exog(varlist) and endog(varlist) to specify which are the endogenous regressors and which are the instruments. The dependent
variable is y, the regressors are e and k and the instruments are w and d. Stata provides the first-stage estimation for the endongenous regressors (e,k). But, what I don't understand is why it
also provides this first-stage estimation for the dependent variable (y). I had never seen that before. Is it something particular to seemlingly unrelated regressions? Or is something that I'm
doing wrong in Stata?
* For searches and help try:
* http://www.stata.com/support/faqs/res/findit.html
* http://www.stata.com/support/statalist/faq
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How to do FFT Analysis to EEG signals Using Matlab
6 Answers
Edited by
Wayne King
on 17 Nov 2012
Accepted answer
No products are associated with this question.
How to do FFT Analysis to EEG signals Using Matlab
I'm looking for FFT analysis to EEG or ECG signals in MATLAB.
I have tested some codes. Nevertheless I don't know how to obtain a normalized result.
I have read in the documentation that I need to normalize by sample length and sampling rate as well as the window energy, but I don't know if MATLAB functions applied a normalization before...
This is my code for ECG analysis of RR intervals:
psdperiodo = psd(Hwelch,RR,'nfft',nfft/2,'Fs',freqinterpol,'SpectrumType','onesided');
PSD = psdperiodo.data /(size(RR,1)*(freqinterpol * size(psdperiodo.data,1)));
% find the indexes corresponding to the VLF, LF, and HF bands
iVLF= find((psdperiodo.frequencies>VLFmin) & (psdperiodo.frequencies<=VLFmax));
iLF = find((psdperiodo.frequencies>LFmin) & (psdperiodo.frequencies<=LFmax));
iHF = find((psdperiodo.frequencies>HFmin) & (psdperiodo.frequencies<=HFmax));
% calculate raw areas (power under curve), within the freq bands (ms^2)
0 Comments
I do not see the need for this line:
psdest = psdperiodo.data /(size(psdperiodo.data,1));
All the normalization for the PSD estimate is taken care of inside of spectrum.welch. That includes dividing by the window energy, which depends on the window used, e.g. Hamming, Bartlett, etc.
I think you should use the one-sided estimate not the two-sided based on the way you are using avgpower(). For example, if you want to do
aVLF = avgpower(psdest,[VLFmin VLFmax]);
for a two-sided estimate, you really need to do
Fs = 1000;
t = 0:0.001:1-0.001;
x = cos(2*pi*100*t)+randn(size(t));
psd2 = psd(spectrum.periodogram,x,'SpectrumType','twosided','Fs',1000);
psd1 = psd(spectrum.periodogram,x,'Fs',1000);
avgpower(psd2,[98 102])+avgpower(psd2,[898 902])
avgpower(psd1,[98 102])
Or use the 'centerdc' option
psd2 = psd(spectrum.periodogram,x,'SpectrumType','twosided','Fs',1000,'centerDC',true);
avgpower(psd2,[-102 -98])+avgpower(psd2,[98 102])
The way you are using it, you are missing the energy in a real-valued signal which is necessarily "mirrored" for the negative and positive frequencies.
0 Comments
Edited by
Wayne King
on 16 Nov 2012
You should format your code when you post. Since you are using spectrum.welch, you should use the avgpower() method to calculate the average power in specific frequency intervals.
For example:
Fs = 1000;
t = 0:0.001:1-0.001;
x = cos(2*pi*100*t)+randn(size(t));
hs = spectrum.welch;
hs.SegmentLength = 200;
psdest = psd(hs,x,'Fs',Fs);
Now to determine the average power in an interval around 100 Hz and compare that to the total power over the Nyquist range.
avgpower(psdest,[90 110])/avgpower(psdest)
0 Comments
Thank you for your answer. Sorry for the format... For the normalization of the result is it necessary to divide "psdest" by sample length and sampling rate in matlab ? Thank you for your help
0 Comments
The nonparametric PSD estimates in MATLAB like the periodogram and Welch estimator already "normalize" the result to create the PSD estimate. For example, in the periodogram with the default
rectangular window, the magnitude squared DFT values are "normalized" by dividing by the length of the input and the sampling rate as you state in your post, although for a one-sided PSD estimate,
the frequencies other than 0 and the Nyquist are multiplied by 2 for energy conversion. You can see this worked out explicitly here:
In the case of the Welch estimate, it's more complicated since you have to divide also by the energy of the window.
0 Comments
Thank you for your answer. I have also read the post. But I have two more questions : Should I calculate psd in oneesided or two sided option in my case ? How could I know it ?
And the second one, if I understand correctly your explanations my code should be as following :
psdperiodo = psd(Hwelch,RR,'nfft',nfft/2,'Fs',freqinterpol,'SpectrumType','twosided');
% selected in twosided option
psdest = psdperiodo.data /(size(psdperiodo.data,1));
% divided by the window energy
% Values of interest in pourcent
aVLF = avgpower(psdest,[VLFmin VLFmax])/avgpower(psdest)*100;
aLF = avgpower(psdest,[LFmin LFmax])/avgpower(psdest)*100;
aHF = avgpower(psdest,[HFmin HFmax])/avgpower(psdest)*100;
% Values of interest in absolute values
aVLF = avgpower(psdest,[VLFmin VLFmax]);
aLF = avgpower(psdest,[LFmin LFmax]);
aHF = avgpower(psdest,[HFmin HFmax]);
Are you agree with my code ? Thanks a lot
0 Comments
Edited by
Ouamar ferhani
on 18 Nov 2012
Thank you for your answer and explanations!
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Archives of the Caml mailing list > Message from skaller
Date: -- (:)
From: skaller <skaller@u...>
Subject: Re: [Caml-list] Coinductive semantics
On Wed, 2006-01-18 at 13:58 +0100, Hendrik Tews wrote:
> skaller <skaller@users.sourceforge.net> writes:
> > Nobody is interested in final coalgebras in Set^op.
> Why not? This is really the key point of misunderstanding
> I think. I'm not disputing your claim, I'm asking why not?
> Perhaps they should be?
> Coalgebras in Set^op are for all intents and purposes identical
> to algebras in Set. If you want to study them, study them as
> algebras in Set. You will see nothing new if you look at these
> objects as coalgebras in Set^op. That's what duality means.
> Looking at an object through a mirror you see precisely what you
> can see looking at the object itself.
Perhaps my analysis is naive. But consider a simpler case
of products and sums. They're dual concepts, are they not?
In Ocaml we have representations of both, each can be used
with reasonable utility -- there is a degree of symmetry,
associated with the duality. It feels good!
Contrast to C, which has products, but the union construction
isn't a sum. And the many other 'popular' languages with
this weakness.
Sometimes it seems looking in the mirror is good.
It's what we want. We don't want something new!
> > Go out, read the papers on
> > the Co-Birkhoff theorem!
> That's a pretty big ask of someone who isn't a
> category theorist isn't it? Most mathematicians
> can't understand category theory .. and I'm just
> an ordinary programmer :)
> Well, you could try. I guess, that already the introductions
> contain enough information for what you are interested in: the
> duality of the Birkhoff and the Co-Birkhoff theorem. In any case,
> if you don't even try, your speculations about the contents of
> these papers remain wild guesses.
I often do try.. but seemed like a good idea to read Adameck first:
Still this is quite heavy going for me.
Incidentally .. if you look in Wikipedia for 'coalgebra' you may
be a bit disappointed.
John Skaller <skaller at users dot sf dot net>
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Analysis of Planar Curves
Parametric Equations
Representing Cartesian coordinate curves using explicit and implicit forms. Representing curves using parametric equations which define x and y in terms of a third variable. Includes examples of
parametric equations for a circle, ellipse, and projectile fired at an angle.
Course Material Related to This Topic:
Nine questions which involve finding equations in rectangular coordinates for those given in parametric form, or putting a rectangular equation in parametric form.
Course Material Related to This Topic:
Finding implicit forms for parameterized curves. Uses examples from the previous section of the notes.
Course Material Related to This Topic:
Polar Coordinate Curves
Definition, with examples of circles and a horizontal line defined in polar coordinates.
Course Material Related to This Topic:
Parametric Representation of a Curve
Using parametric equations to define a curve in two or three dimensions and properties of parametric equations.
Course Material Related to This Topic:
Polar Coordinate Graphs
Three multi-part questions which involve converting rectangular coordinates to polar coordinates, converting polar equations to rectangular equations, and graphing curves given in polar coordinates.
Course Material Related to This Topic:
Polar Curves
Sketching a curve in polar coordinates, and labeling the quadrants, endpoints, tangent slopes, and angles for the curve.
Course Material Related to This Topic:
Sketching a curve given in polar coordinates and finding the area swept by a line segment as one of the endpoints moves along this curve.
Course Material Related to This Topic:
Area Inside a Polar Curve
Sketching a curve defined in polar coordinates and finding the area inside it.
Course Material Related to This Topic:
Polar Coordinates
Course Material Related to This Topic:
Sketching a curve given in polar coordinates and finding points of intersection between that and other curves.
Representing a circle using both rectangular and polar coordinates.
Integrals in Polar Coordinates: Lunar Eclipse
Setting up and evaluating an integral to represent the uncovered area of the two moons involved in a lunar eclipse on another planet.
Course Material Related to This Topic:
Polar Coordinates: Spiral
Sketching a spiral defined in polar coordinates, counting the times it crosses the x-axis, and finding the area of specific regions of the spiral.
Course Material Related to This Topic:
Lines in Polar Coordinates
Finding an equation in polar coordinates and the appropriate range of theta for a line given in rectangular coordinates.
Course Material Related to This Topic:
Polar Plotter
Applet for showing the graph of a function defined in polar coordinates.
Course Material Related to This Topic:
Curves in Two Dimensions
Applet for plotting curves defined in rectangular or parametric form.
Course Material Related to This Topic: | {"url":"http://ocw.mit.edu/high-school/mathematics/exam-prep/parametric-polar-vector-functions/analysis-of-planar-curves/","timestamp":"2014-04-16T07:31:05Z","content_type":null,"content_length":"55610","record_id":"<urn:uuid:12d168ba-073a-44fe-a00e-1ae39ad58b33>","cc-path":"CC-MAIN-2014-15/segments/1398223205375.6/warc/CC-MAIN-20140423032005-00271-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Hello! This is a general Math question. Based on my past experience, am I ready for Calculus?
• one year ago
• one year ago
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@ParthKohli, without a doubt.
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you are! haha-
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Hmm, thanks for flattering. I want more answers with a detail.
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And I doubt if you guys know my past experience :P
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Parth, I will tutor you on Calculus for free if you want a good introduction and help.
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calculus is only an expansion of algebra, guess that includes your answer already.
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@ParthKohli To learn calculus you should be well versed in Logarithms, Trigonometry, and Sets. Have you learned all these?
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I felt Calc I was like 75% Algebra and 20% Trig...pretty sure it will be easy for someone like you
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Thank you. Yes, @ash2326
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Calculus is MOSTLY Algebra, just FYI. ;-)
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Speaking from direct experience.
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Then you are ready:D
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Parth, do you have knowledge in Trig, Algebra I-II, and working with functions?
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I went to a website and learned all these concepts very thoroughly. What is the starting in Calculus? Limits?
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math is hard keke ayiyi
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@Compassionate I do.
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It depends on the school. At MIT, we started with Integrals first and defined limits based on that.
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Calculus will come next year in my syllabus lol :P
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You're ready, Parth. If you wouldn't mind taking a few minutes out of your time today, and send me what you know and give me some examples. I can take that information and place you at a level.
:) And like I said, "I can tutor you."
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And when I say that it is mostly, I mean that you'll REALLY need to be good with it. Everything from factoring, to trig identities, to partial fractions, to series and sequences, to things like
this: |dw:1342968425801:dw|
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Derivative is small change in \(y\) - known as \(dy\) - over the small change in \(x\) - known as \(dx\).
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You see this stuff over and over in Calculus
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@mathdumbo That's a nice way:D
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Integral is a number of which something is a derivative of(as far as I know).
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not a number but function.. :P
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Average slope: \[m=\frac{\triangle y}{\triangle x}=\frac{y_2-y_1}{x_2-x_1}\] Turns into... \[m=\frac{d y}{d x}=\text{instantaneous slope}\]
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So, \[\int \limits {2xdx } = x^2\]?
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Yes, @agentx5. I saw a video on Khanacademy on the intuition of differentiation.
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And I believe that you are allowed to 'skip' some of the topics in Mathematics; that's how beautiful it is :)
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Parth, how would you solve this: \[{2 - 5x \sqrt23}{-3x \sqrt2}\]
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^ You need to have a background with working on radicals.
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I believe that we should stick to the question, and yes I do know that :)
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There is a lot of really small division in calculus.
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I don't think we can simplify that.
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And by small, I mean integrals.
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It was just an example.
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I have heard that we can calculate "nth" root of any number by differentiation. Am I correct ?
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You can move the constant out front: \(\large\int \limits {2x \ \ dx } = 2\int x \ \ dx = 2[\frac{x^{1+1}}{1+1}] + C = \cancel{2} * \frac{x^2}{\cancel{2}} + C = x^2 + C\)
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Parth, don't forget the C
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Yep, because the derivative of any constant is 0, and \(x + 0 = x\).
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The "C" means some constant. Why put it on indefinite integrals? Because \(\frac{d}{dx}\) constant = 0
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You got it lol :-D
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I am just starting it ;)
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Thank you, all! You are a great help!
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What concept should I start with?
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Any resource?
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parth,I learn it on MIT , and Khanacedemy
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That is a phenomenal reference.
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also , this awesome book
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Hah! Paul's Notes! I find it a little difficult to start with the basics on that.
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P.S. I learned my logs and trig on there. :)
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Try E-book "Calculus for dummies"
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That book help me a lot
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Parth try this one! ^_^ \[\huge \lim_{x \rightarrow 0} \frac{1}{x-1}\] and \[\huge \lim_{x \rightarrow 1} \frac{1}{x-1}\] and \[\huge \lim_{x \rightarrow \infty} \frac{1}{x-1}\] Do it visually
:-D (draw a graph with arrows & stuff)
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Try it @ParthKohli :-D It's not a hard problem, it just makes you think and once you get the concepts down it just goes back to Algebra tricks for the most part. hint: What happens if the x = 1
to the denominator?
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I find these as the answers: 1) \(-1\) 2) \(\infty \) 3) \(\infty\) You use L'Hopital's Rule.
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You may do it visually by taking the numbers closer to the number which it is approaching.
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Aww come on draw it! ^_^ Make a sketch
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Hmm. There's a thing that I am missing: CONIC SECTIONS.
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Would I be able to do Calculus without conic sections?
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Incorrect @ParthKohli |dw:1342970148488:dw| Correct answers are: \(\huge \lim_{x \rightarrow 0} \frac{1}{x-1} = -1\) \(\huge \lim_{x \rightarrow 1} \frac{1}{x-1} = \text{Does NOT exist.}\) \(\
huge \lim_{x \rightarrow \infty} \frac{1}{x-1} = 0\) You cannot use the l'Hospital's rule unless after simplification you still have a \(\large \frac{\pm\ \infty}{\pm\ \infty}\) or \(\large \frac
{0}{0}\) form when you go to substitute.
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This is why sketching/graphing things visually is so helpful in Calculus :-D
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Hmm—you should have included that I am a beginner too :)
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I had to get lunch, sry about the delay
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Hey no worries, we all have to learn to fly sometime right? ^_^
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Isn't 'Does NOT Exist' just infinity?
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Actually, no, take a closer look. The limit is as if you're taking your right and left hand and tracing the function from both points towards whatever the limit is approaching. In the case of the
first one, it's clearly approaching -1 from both the right and left. But in the second one, it goes on way up to infinity, and the other way down to negative infinity. It creates a kind of a
paradox, it can't be both and yet it is. So the limit simply doesn't exist. Here's an example where the limit is infinity: \[\huge \lim_{x \rightarrow \infty} x^2 = \infty\] |dw:1342970876711:dw|
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\(\huge \lim_{x \rightarrow 1} | \frac{1}{x-1} | = \infty\) |dw:1342971089127:dw|
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^_^ Make more sense now?
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I, for once, believe in visual techniques :)
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Amen brotha! ;-P
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Heh! That was so clear :')
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And don't forget you can't just use l'Hostpital's rule whenever you feel like it, it's got to meet the prerequisites of the theorem that I mentioned above :-)
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beanplot {beanplot}
Beanplot: A Boxplot/Stripchart/Vioplot Alternative
Plots beans to compare the distributions of different groups; it draws one bean per group of data. A bean consists of a one-dimensional scatter plot, its distribution as a density shape and an
average line for the distribution. Next to that, an overall average for the whole plot is drawn per default.
beanplot(..., bw = "SJ-dpi", kernel = "gaussian", cut = 3, cutmin = -Inf,
cutmax = Inf, grownage = 10, what = c(TRUE, TRUE, TRUE, TRUE),
add = FALSE, col, axes = TRUE, log = "auto", handlelog = NA,
ll = 0.16, wd = NA, maxwidth = 0.8, maxstripline = 0.96,
method = "stack", names, overallline = "mean", beanlines = overallline,
horizontal = FALSE, side = "no", jitter = NULL, beanlinewd = 2,
frame.plot = axes, border = NULL, innerborder = NA, at = NULL,
boxwex = 1, ylim = NULL, xlim = NULL, show.names = NA)
data which to perform the beanplot on. This data can consist of dataframes, vectors and/or formulas. For each formula, a dataset can be specified with data=[dataset], and a subset can be
specified with subset=[subset]. If subset/data arguments are passed, but there are not enough subset/data arguments, they are reused. Additionally, na.action, drop.unused.levels and xlev can be
passed to model.frame in the same way. Also, parameters for axis and title can be passed.
the bandwidth (method) being used, used by density. In case of a method, the average computed bandwidth is used.
see density.
the beans are cut beyond cut*bw
the low-ends of the beans are cut below mincut*bw. Defaults to cut.
the high-ends of the beans are cut beyond maxcut*bw. Defaults to cut.
the width of a bean grows linearly with the count of points, until grownage is reached.
a vector of four booleans describing what to plot. In the following order, these booleans stand for the total average line, the beans, the bean average, and the beanlines. For example, what=c(0,0
,0,1) produces a stripchart
if true, do not start a new plot
the colors to be used. A vector of up to four colors can be used. In the following order, these colors stand for the area of the beans (without the border, use border for that color), the lines
inside the bean, the lines outside the bean, and the average line per bean. Transparent colors are supported. col can be a list of color vectors, the vectors are then used per bean, and reused if
if false, no axes are drawn.
use log="y" or log="" to force a log-axis. In case of log="auto", a log-transformation is used if appropriate
if handlelog then all the calculations are done using a log-scale. By default this is determined using the log parameter.
the length of the beanline per point found.
the linear transformation that determines the width of the beans. By default determined using maxwidth, and returned.
the maximum width of a bean.
the maximum length of a beanline.
the method used when two points on a bean are the same. "stack", "overplot" and "jitter" are supported.
a vector of names for the groups.
the method used for determining the overall line. Defaults to "mean", "median" is also supported.
the method used for determining the average bean line(s). Defaults to "mean", "median" and "quantiles" are also supported.
if true, the beanplot is horizontal
the side on which the beans are plot. Default is "no", for symmetric beans. "first", "second" and "both" are also supported.
passed to jitter as amount in case of method="jitter".
the width used for the average bean line
if true, plots a frame.
the color for the border around a bean. NULL for par("fg"), NA for no border. If border is a vector, it specifies the color per bean, and colors are reused if necessary.
a color (vector) for the border inside the bean(s). Especially useful if side="both". Use NA for no inner border. Colors are reused if necessary. The inner border is drawn as the last step in the
drawing process.
the positions at which a bean should be drawn.
a scale factor applied to all beans. Compatible with boxplot.
the range to plot.
the range to plot the beans at.
if true, plots the names as axis labels
Most parameters are compatible with boxplot and stripchart. For compatibility, arguments with the name "formula" or "x" are used as data. However, data or formulas do not need to be named "x" or
"formula". The function handles (combinations of) dataframes, vectors and/or formulas.
The bandwith (bw) used.
The bean width (wd) used.
Kampstra, P. (2008) Beanplot: A Boxplot Alternative for Visual Comparison of Distributions. Journal of Statistical Software, Code Snippets, 28(1), 1-9. URL http://www.jstatsoft.org/v28/c01/
In case of more than 5000 values per bean, the autodetection of log fails. In such cases, use log="" or log="y".
#mostly examples taken from boxplot:
par(mfrow = c(1,2))
boxplot(count ~ spray, data = InsectSprays, col = "lightgray")
beanplot(count ~ spray, data = InsectSprays, col = "lightgray", border = "grey", cutmin = 0)
boxplot(count ~ spray, data = InsectSprays, col = "lightgray")
beanplot(count ~ spray, data = InsectSprays, col = "lightgray", border = "grey", overallline = "median")
boxplot(decrease ~ treatment, data = OrchardSprays,
log = "y", col = "bisque", ylim = c(1,200))
beanplot(decrease ~ treatment, data = OrchardSprays,
col = "bisque", ylim = c(1,200))
par(mfrow = c(2,1))
mat <- cbind(Uni05 = (1:100)/21, Norm = rnorm(100),
T5 = rt(100, df = 5), Gam2 = rgamma(100, shape = 2))
par(las=1)# all axis labels horizontal
boxplot(data.frame(mat), main = "boxplot(*, horizontal = TRUE)",
horizontal = TRUE, ylim = c(-5,8))
beanplot(data.frame(mat), main = "beanplot(*, horizontal = TRUE)",
horizontal = TRUE, ylim = c(-5,8))
par(mfrow = c(1,2))
boxplot(len ~ dose, data = ToothGrowth,
boxwex = 0.25, at = 1:3 - 0.2,
subset = supp == "VC", col = "yellow",
main = "Guinea Pigs' Tooth Growth",
xlab = "Vitamin C dose mg",
ylab = "tooth length", ylim = c(-1, 40), yaxs = "i")
boxplot(len ~ dose, data = ToothGrowth, add = TRUE,
boxwex = 0.25, at = 1:3 + 0.2,
subset = supp == "OJ", col = "orange")
legend("bottomright", bty="n", c("Ascorbic acid", "Orange juice"),
fill = c("yellow", "orange"))
allplot <- beanplot(len ~ dose+supp, data = ToothGrowth,
what=c(TRUE,FALSE,FALSE,FALSE),show.names=FALSE,ylim=c(-1,40), yaxs = "i")
beanplot(len ~ dose, data = ToothGrowth, add=TRUE,
boxwex = 0.6, at = 1:3*2 - 0.9,
subset = supp == "VC", col = "yellow",border="yellow2",
main = "Guinea Pigs' Tooth Growth",
xlab = "Vitamin C dose mg",
ylab = "tooth length", ylim = c(3, 40), yaxs = "i",
bw = allplot$bw, wd = allplot$wd, what = c(FALSE,TRUE,TRUE,TRUE))
beanplot(len ~ dose, data = ToothGrowth, add = TRUE,
boxwex = 0.6, at = 1:3*2-0.1,
subset = supp == "OJ", col = "orange",border="darkorange",
bw = allplot$bw, wd = allplot$wd, what = c(FALSE,TRUE,TRUE,TRUE))
legend("bottomright", bty="n", c("Ascorbic acid", "Orange juice"),
fill = c("yellow", "orange"))
par(mfrow = c(1,2))
boxplot(len ~ dose, data = ToothGrowth,
boxwex = 0.25, at = 1:3 - 0.2,
subset = supp == "VC", col = "yellow",
main = "Guinea Pigs' Tooth Growth",
xlab = "Vitamin C dose mg",
ylab = "tooth length", ylim = c(-1, 40), yaxs = "i")
boxplot(len ~ dose, data = ToothGrowth, add = TRUE,
boxwex = 0.25, at = 1:3 + 0.2,
subset = supp == "OJ", col = "orange")
legend("bottomright", bty="n",c("Ascorbic acid", "Orange juice"),
fill = c("yellow", "orange"))
beanplot(len ~ reorder(supp, len, mean) * dose, ToothGrowth,
side = "b", col = list("yellow", "orange"), border = c("yellow2",
"darkorange"), main = "Guinea Pigs' Tooth Growth",
xlab = "Vitamin C dose mg", ylab = "tooth length", ylim = c(-1,
40), yaxs = "i")
legend("bottomright", bty="n",c("Ascorbic acid", "Orange juice"),
fill = c("yellow", "orange"))
#Example with multiple vectors and/or formulas
par(mfrow = c(2,1))
beanplot(list(all = ToothGrowth$len), len ~ supp, ToothGrowth, len ~ dose)
title("Tooth growth length (beanplot)")
#Trick using internal functions to do this with other functions:
mboxplot <- function(...){
graphics::boxplot(beanplot:::getgroupsfromarguments(), ...)
mstripchart <- function(..., method = "overplot", jitter = 0.1, offset = 1/3,
vertical = TRUE, group.names, add = FALSE,
at = NULL, xlim = NULL, ylim = NULL,
ylab = NULL, xlab=NULL, dlab = "", glab = "",
log = "", pch = 0, col = par("fg"), cex = par("cex"),
axes = TRUE, frame.plot = axes) {
method, jitter, offset, vertical, group.names, add,
at, xlim, ylim, ylab, xlab, dlab, glab, log, pch, col, cex,
axes, frame.plot)
mstripchart(list(all = ToothGrowth$len), len ~ supp, ToothGrowth, len ~ dose,
xlim = c(0.5,6.5))
title("Tooth growth length (stripchart)")
Documentation reproduced from package beanplot, version 1.1. License: GPL-2 | {"url":"http://www.inside-r.org/packages/cran/beanplot/docs/beanplot","timestamp":"2014-04-16T23:00:29Z","content_type":null,"content_length":"68705","record_id":"<urn:uuid:ecf3a7f4-7c5e-4ea6-ab0f-c8399ffa646b>","cc-path":"CC-MAIN-2014-15/segments/1397609525991.2/warc/CC-MAIN-20140416005205-00400-ip-10-147-4-33.ec2.internal.warc.gz"} |
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S is a set containing 9 different numbers. T is a set contai
Author Message
S is a set containing 9 different numbers. T is a set contai [#permalink] 29 Sep 2010, 05:25
vanidhar Difficulty:
Manager 35% (medium)
Status: GMAT Preperation Question Stats:
Joined: 04 Feb 2010 51%
Posts: 106 (01:56) correct
Concentration: Social 48% (01:01)
Entrepreneurship, Social
Entrepreneurship wrong
GPA: 3 based on 160 sessions
WE: Consulting (Insurance) S is a set containing 9 different numbers. T is a set containing 8 different numbers, all of which are members of S. which of the following statements cannot be true?
Followers: 1 A. The mean of S is equal to the mean of T
Kudos [?]: 6 [0], given: 15 B. The median of S is equal to the median of T
C. The range of S is equal to the range of T
D. The mean of S is greater than the mean of T
E. The range of S is less than the range of T
my question : what if the extra number is zero ?? ???
Spoiler: OA
Re: cannot be true . mean median range [#permalink] 29 Sep 2010, 05:47
This post received
Expert's post
The range of a set is the difference between the largest and smallest elements of a set.
Consider the set S to be {-4, -3, -2, -1, 0, 1, 2, 3, 4} -->
A. Mean of S = mean of T --> remove 0 from set S, then the mean of T still would be 0;
B. Median of S = Median of T --> again remove 0 from set S, then the median of T still would be 0;
Bunuel C. Range of S = range of T --> again remove 0 from set S, then the range of T still would be 8;
Math Expert D. Mean of S > mean of T --> remove 4, then the mean of T would be negative -0.5 so less than 0;
Joined: 02 Sep 2009 E. Range of S < range of T --> the range of a subset cannot be more than the range of a whole set: how can the difference between the largest and smallest elements of
a subset be
Posts: 17321
Followers: 2875
than the difference between the largest and smallest elements of a whole set.
Kudos [?]: 18401 [4] , given:
2350 Answer: E.
Hope it helps.
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Re: S is a set containing 9 different numbers. [#permalink] 22 Jan 2013, 00:20
fozzzy wrote:
Abhii46 S is a set containing 9 different numbers. T is a set containing 8 different numbers, all of which are members of S. which of the following statements cannot be true?
Manager A) The mean of S is equal to the mean of T
B) The median of S is equal to the median of T
Joined: 12 Mar 2012 C) The range of S is equal to the range of T
D) The mean of S is greater than the mean of T
Posts: 95 E) The range of S is less than the range of T
Location: India
Detailed explanation will be appreciated. Thanks!
Concentration: Technology,
Strategy Mean of both the sets can be equal. Let us suppose the mean of any 8 number is 10 then the 9th number could also be 10 and mean remains the same.
GMAT 1: 710 Q49 V36 In the same way median can also be same. In set of 9 numbers median will be the 5th number when arranged in ascending order and in set T it will be the mean of 4th
and 5th number.
GPA: 3.2
If we take S = { 1, 2, 3, 4 ,5, 6, 7, 8, 9 } and T as { 1, 2, 3, 4, 6, 7, 8, 9 } then median in both the cases will be 5.
WE: Information Technology
(Computer Software) From the above eg range is same in both the cases.
Followers: 9 If the number which is not the part of set T is greater than mean of T then the mean of set S will be greater than that of set T
Kudos [?]: 260 [0], given: 22 Range of S will always be greater than or equal to range of T because of an additional number. If that number is greater than the greatest number in set T the range
will be more, if that number is smaller than the smallest number in set T again the range will be more since now the new number is the smallest number. If that number
lies in between then the range will be equal.
If you like my explanation please give a kudo.
Re: cannot be true . mean median range [#permalink] 22 Jan 2013, 04:40
Bunuel wrote:
The range of a set is the difference between the largest and smallest elements of a set.
Consider the set S to be {-4, -3, -2, -1, 0, 1, 2, 3, 4} --> mean=median=0 and range=8.
fozzzy A. Mean of S = mean of T --> remove 0 from set S, then the mean of T still would be 0;
B. Median of S = Median of T --> again remove 0 from set S, then the median of T still would be 0;
Director C. Range of S = range of T --> again remove 0 from set S, then the range of T still would be 8;
D. Mean of S > mean of T --> remove 4, then the mean of T would be negative -0.5 so less than 0;
Joined: 29 Nov 2012 E. Range of S < range of T --> the range of a subset can not be more than the range of a whole set: how can the difference between the largest and smallest elements
of a subset be more than the difference between the largest and smallest elements of a whole set.
Posts: 936
Answer: E.
Followers: 11
Hope it helps.
Kudos [?]: 157 [0], given: 543
So this is a property of sets?
Click +1 Kudos if my post helped...
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Re: cannot be true . mean median range [#permalink] 22 Jan 2013, 05:22
Expert's post
fozzzy wrote:
Bunuel wrote:
The range of a set is the difference between the largest and smallest elements of a set.
Consider the set S to be {-4, -3, -2, -1, 0, 1, 2, 3, 4} --> mean=median=0 and range=8.
A. Mean of S = mean of T --> remove 0 from set S, then the mean of T still would be 0;
B. Median of S = Median of T --> again remove 0 from set S, then the median of T still would be 0;
C. Range of S = range of T --> again remove 0 from set S, then the range of T still would be 8;
D. Mean of S > mean of T --> remove 4, then the mean of T would be negative -0.5 so less than 0;
E. Range of S < range of T --> the range of a subset can not be more than the range of a whole set: how can the difference between the largest and smallest elements
of a subset be more than the difference between the largest and smallest elements of a whole set.
Answer: E.
Hope it helps.
Math Expert
So this is a property of sets?
Joined: 02 Sep 2009
Sure. The range of a subset cannot be greater than the range of the whole set.
Posts: 17321
Followers: 2875
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years. NEW!!!;
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PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6.
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Math Expert Re: S is a set containing 9 different numbers. T is a set contai [#permalink] 03 Jun 2013, 02:09
Joined: 02 Sep 2009 Expert's post
Posts: 17321
Followers: 2875
Alexmsi Re: S is a set containing 9 different numbers. T is a set contai [#permalink] 03 Jun 2013, 10:51
Intern @Bunuel:
Joined: 19 Apr 2012 I agree with you that the range of subsequent set is less than the whole set, but it can also be the same ? For exmaple:
Posts: 28 100 14 13 2 whole set.
100 14 2 subsequent set.
Followers: 0
All mebers of the subsequent set are also members of the whole set. But the range are in both cases the same. So it must be as follows: The range of the subsequent
Kudos [?]: 2 [0], given: 8 set can be equal or less than that of the whole set ?
Math Expert Re: S is a set containing 9 different numbers. T is a set contai [#permalink] 04 Jun 2013, 02:53
Joined: 02 Sep 2009 Expert's post
Posts: 17321
Followers: 2875
Re: S is a set containing 9 different numbers. T is a set contai [#permalink] 20 Nov 2013, 08:12
Solve for an "easier" problem and make up an example to see what's going on in the problem.
Set S = {1,2,3}
Set T = {2,3}
Okay so if we can get an answer that is true, we can eliminate that answer choice.
a) hmm, how to get the mean equal each other? Oh, just remove the 2 instead of 1.
Joined: 20 Jun 2011 b) same
c) same
Posts: 46 d) remove 3 instead of 1
Followers: 1 At this point we can choose e) and move on, but to be sure just test some numbers again.
Kudos [?]: 6 [0], given: 1 e)
range set S = 3-1 = 2
range set T = 3-2 = 1
range set T = 3-1 = 1
So, this can never be true. --> Bingo
This is basically the same method as Bunuel posted above, but for me it sometimes works better if I have a simpler set to work with.
gmatclubot Re: S is a set containing 9 different numbers. T is a set contai [#permalink] 20 Nov 2013, 08:12 | {"url":"http://gmatclub.com/forum/s-is-a-set-containing-9-different-numbers-t-is-a-set-contai-101912.html?fl=similar","timestamp":"2014-04-19T22:19:47Z","content_type":null,"content_length":"219405","record_id":"<urn:uuid:b9961127-7615-4cd9-94fb-f7e81e586db4>","cc-path":"CC-MAIN-2014-15/segments/1397609537754.12/warc/CC-MAIN-20140416005217-00464-ip-10-147-4-33.ec2.internal.warc.gz"} |
this is really into stuff and i fell really dumb that i cant figure this stuff out....
What is the net force on a Merecedes convertible traveling along a straight road at a steady speed of 110 km/hr? Answer in N
Where do i start?
If it takes 1 N to push horizontally on your book to make it slide at a constant velocity, how much force of friction acts on the book? Answer in N
If the book keeps moving at that constant speed i figured that i would be 0 but i was wrong... what am i doing wrong?
Three identical blocks are pulled on a horizontal frictionless surface. If tension in the rope held by the hand is T = 60 N, what is the tension in the other ropes?
Where do i start? | {"url":"http://www.physicsforums.com/showthread.php?t=88448","timestamp":"2014-04-20T05:56:03Z","content_type":null,"content_length":"25183","record_id":"<urn:uuid:fb22fa5d-8fde-4de4-84fb-f7f67eb9aa23>","cc-path":"CC-MAIN-2014-15/segments/1398223206120.9/warc/CC-MAIN-20140423032006-00596-ip-10-147-4-33.ec2.internal.warc.gz"} |
What is “Continuum mechanics” ?
Continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles.
The French mathematician Augustin-Louis Cauchy was the 1st to formulate such models in the 19th century, but research in the area continues today.
Modeling an object as a continuum assumes that the substance of the object completely fills the space it occupies. Modeling objects in this way ignores the fact that matter is made of atoms, and so
isn’t continuous; however, on length scales much greater than that of inter-atomic distances, such models are highly accurate. Fundamental physical laws such as the conservation of mass, the
conservation of momentum, and the conservation of energy may be applied to such models to derive differential equations describing the behavior of such objects, and some information about the
particular material studied is added through a constitutive relation.
Continuum mechanics deals with physical properties of solids and fluids which are independent of any particular coordinate system in which they are observed. These physical properties are then
represented by tensors, which are mathematical objects that have the required property of being independent of coordinate system. These tensors can be expressed in coordinate systems for
computational convenience.
Materials, such as solids, liquids and gases, are composed of molecules separated by “empty” space. On a macroscopic scale, materials have cracks and discontinuities. However, certain physical
phenomena can be modeled assuming the materials exist as a continuum, meaning the matter in the body is continuously distributed and fills the entire region of space it occupies. A continuum is a
body that can be continually sub-divided into infinitesimal elements with properties being those of the bulk material.
The validity of the continuum assumption may be verified by a theoretical analysis, in which either some clear periodicity is identified or statistical homogeneity and ergodicity of the
microstructure exists. More specifically, the continuum hypothesis/assumption hinges on the concepts of a representative volume element (sometimes called “representative elementary volume”) and
separation of scales based on the Hill–Mandel condition. This condition provides a link between an experimentalist’s and a theoretician’s viewpoint on constitutive equations (linear and nonlinear
elastic/inelastic or coupled fields) as well as a way of spatial and statistical averaging of the microstructure.
When the separation of scales does not hold, or when one wants to establish a continuum of a finer resolution than that of the RVE size, one employs a statistical volume element, which, in turn,
leads to random continuum fields. The latter then provide a micromechanics basis for stochastic finite elements (SFE). The levels of SVE and RVE link continuum mechanics to statistical mechanics. The
RVE may be assessed only in a limited way via experimental testing: when the constitutive response becomes spatially homogeneous.
Specifically for fluids, the Knudsen number is used to assess to what extent the approximation of continuity can be made.
Continuum mechanics models begin by assigning a region in three dimensional Euclidean space to the material body being modeled. The points within this region are called particles or material points.
Different configurations or states of the body correspond to different regions in Euclidean space. The region corresponding to the body’s configuration at time is labeled.
For the mathematical formulation of the model, is also assumed to be twice continuously differentiable, so that differential equations describing the motion may be formulated.
Continuum mechanics deals with deformable bodies, as opposed to rigid bodies. A solid is a deformable body that possesses shear strength, sc. a solid can support shear forces. Fluids, on the other
hand, don’t sustain shear forces. For the study of the mechanical behavior of solids and fluids these are assumed to be continuous bodies, which means that the matter fills the entire region of space
it occupies, despite the fact that matter is made of atoms, has voids, and is discrete. Therefore, when continuum mechanics refers to a point or particle in a continuous body it does not describe a
point in the interatomic space or an atomic particle, rather an idealized part of the body occupying that point.
The distribution of internal contact forces throughout the volume of the body is assumed to be continuous. Therefore, there exists a contact force density or Cauchy traction field that represents
this distribution in a particular configuration of the body at a given time. It isn’t a vector field because it depends not only on the position of a particular material point, but also on the local
orientation of the surface element as defined by its normal vector.
In continuum mechanics a body is considered stress-free if the only forces present are those inter-atomic forces required to hold the body together and to keep its shape in the absence of all
external influences, including gravitational attraction. Stresses generated during manufacture of the body to a specific configuration are also excluded when considering stresses in a body.
Therefore, the stresses considered in continuum mechanics are only those produced by deformation of the body, sc. only relative changes in stress are considered, not the absolute values of stress.
In the case of gravitational forces, the intensity of the force depends on, or is proportional to, the mass density of the material, and it is specified in terms of force per unit mass. These two
specifications are related through the material density by the equation. Similarly, the intensity of electromagnetic forces depends upon the strength (electric charge) of the electromagnetic field.
In certain situations, not commonly considered in the analysis of the mechanical behavior or materials, it becomes necessary to include two other types of forces: these are body moments and couple
stresses. Body moments, or body couples, are moments per unit volume or per unit mass applied to the volume of the body. Couple stresses are moments per unit area applied on a surface. Both are
important in the analysis of stress for a polarized dielectric solid under the action of an electric field, materials where the molecular structure is taken into consideration (e.g. bones), solids
under the action of an external magnetic field, and the dislocation theory of metals.
Materials that exhibit body couples and couple stresses in addition to moments produced exclusively by forces are called polar materials. Non-polar materials are then those materials with only
moments of forces. In the classical branches of continuum mechanics the development of the theory of stresses is based on non-polar materials.
A change in the configuration of a continuum body results in a displacement. The displacement of a body has two components: a rigid-body displacement and a deformation. A rigid-body displacement
consists of a simultaneous translation and rotation of the body without changing its shape or size. Deformation implies the change in shape and/or size of the body from an initial or undeformed
configuration to a current or deformed configuration.
The motion of a continuum body is a continuous time sequence of displacements. Thus, the material body will occupy different configurations at different times so that a particle occupies a series of
points in space which describe a pathline.
It is convenient to identify a reference configuration or initial condition which all subsequent configurations are referenced from. The reference configuration need not be one that the body will
ever occupy. Often, the configuration at is considered the reference configuration,. The components of the position vector of a particle, taken with respect to the reference configuration, are called
the material or reference coordinates.
When analyzing the deformation or motion of solids, or the flow of fluids, it is necessary to describe the sequence or evolution of configurations throughout time. One description for motion is made
in terms of the material or referential coordinates, called material description or Lagrangian description.
In the Lagrangian description the position and physical properties of the particles are described in terms of the material or referential coordinates and time. In this case the reference
configuration is the configuration at. An observer standing in the referential frame of reference observes the changes in the position and physical properties as the material body moves in space as
time progresses. The results obtained are independent of the choice of initial time and reference configuration,. This description is normally used in solid mechanics.
Physical and kinematic properties, i.e. thermodynamic properties and velocity, which describe or characterize features of the material body, are expressed as continuous functions of position and
time, i.e..
Related Sites for Continuum mechanics
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Linear dependent vectors
September 24th 2009, 12:53 PM
Linear dependent vectors
Consider a linear transformation T from R^n to R^p and some linearly dependent vectors v1, v2, ..., vm in R^n. Are the vectors T(v1), T(v2), ..., T(vm) necessarily linearly dependent? How can you
September 24th 2009, 04:36 PM
Yes. Let $v_m= a_1v_1 + ...+a_{m-1}v_{m-1}$ then $T(v_m)=a_1T(v_1)+...+a_{m-1}T(v_{m-1})$ which shows $T(v_i)$ are linerly dependent | {"url":"http://mathhelpforum.com/advanced-algebra/104109-linear-dependent-vectors-print.html","timestamp":"2014-04-17T01:34:38Z","content_type":null,"content_length":"4807","record_id":"<urn:uuid:b7252bd4-5605-472a-a6aa-88be1de4712f>","cc-path":"CC-MAIN-2014-15/segments/1397609526102.3/warc/CC-MAIN-20140416005206-00204-ip-10-147-4-33.ec2.internal.warc.gz"} |
The Earth Question (Latitude)
October 31st 2007, 05:22 PM #1
May 2007
The Earth Question (Latitude)
The circumference of a circle of latitude is two thirds of the circumference of the equator. What is the latitude?
A diagram would be nice
Equator: $2\pi\;r_{e}$
New Lattitude: $2\pi\;r_{new}\;=\;\frac{2}{3}\;2\pi\;r_{e}$
A right triangle: $\cos(lattitude)\;=\;\frac{r_{new}}{r_{e}}$
October 31st 2007, 07:09 PM #2
MHF Contributor
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Diophantine Equation to Find Perfect Square Values
Date: 03/19/2008 at 20:58:19
From: Peter
Subject: try to find integers to a polynomial
I would like to know how to prove whether there are positive integers
for long polynomials that would form a perfect square, such as 4x^4 +
x^3 + 2x^2 + x + 1, and x^4 + x^3 + x^2 + x + 1. I don't want an
answer but I just need help on how I would approach this kind of
I thought that I would set the polynomial equal to a variable that is
squared and try to solve the equation in terms of that variable to get
the answer, but I later realized that if this was the case then there
could be an infinite amount of answers but I know that there aren't an
infinite amount of answers. I was unable to factor out the long
polynomial. Thanks in advance.
Date: 03/21/2008 at 20:00:23
From: Doctor Vogler
Subject: Re: try to find integers to a polynomial
Hi Peter,
Thanks for writing to Dr. Math. When you're looking for solutions
that are integers (or positive integers), then you call that a
Diophantine equation. Diophantine equations are not easy to solve.
Certain kinds (like linear ones) we can solve without too much work.
Others (like two-variable quadratic ones) we can solve but they take a
lot of work. Still others we can't solve at all.
But you can learn a lot of math by learning to solve Diophantine
equations, and I personally think they're a lot of fun.
You mentioned two particular Diophantine equations, and described a
whole group of them. That group is teetering right there on the edge
of what today's mathematicians can solve.
But the two you described are special in two ways: First of all, the
polynomial has even degree. Second, the leading coefficient is a
square (namely, 1 or 4). Because of this, we can solve those two
equations. Here's how it goes.
First notice that if n and m are integers, then either n^2 = m^2, or
n^2 and m^2 differ by at least 2n - 1. Why is that? Because the
closest perfect square to n^2 is (n-1)^2.
Now let's complete the square on your polynomial. Suppose that x and
y are both integers, and
y^2 = 4x^4 + x^3 + 2x^2 + x + 1.
Then the right side is close to the polynomial
(2x^2 + 1/4*x + 31/64)^2.
In order to keep things integers, we'll multiply both sides of the
equation by 64^2. Then your equation is equivalent to
(64y)^2 = (128x^2 + 16x + 31)^2 + 3104x + 3135.
Now we apply the statement I made earlier using m = 64y and
n = 128x^2 + 16x + 31.
That is, either n^2 = m^2, which means that 3104x + 3135 = 0 (and I'll
let you explain why no integer satisfies this equation), or n^2 and
m^2 differ by at least 2n - 1. That means that
abs(3104x + 3135) >= 256x^2 + 32x + 61.
(That's "the absolute value of.") But this is impossible if x > 13 or
x < -12. And you can just check the x values in between. You get a
square only for x=0 and x=1. So all integer solutions are
(x, y) = (0, 1), (0, -1), (1, 3), and (1, -3).
You can do something similar for the other polynomial you gave. I'll
leave it up to you.
Unfortunately, when the leading coefficient is not a square, or if the
degree of the polynomial is odd, then this method won't work.
But sometimes if there are *no* solutions, then you can prove this
using modular arithmetic.
Mod, Modulus, Modular Arithmetic
On the other hand, if the degree is 2, then even if the leading
coefficient is not a square, there are ways to find all integer
solutions. And if the degree is 3, then you have an elliptic curve,
and there are methods (some of which are rather complicated) to find
all integer solutions. But in other cases (odd degree higher than 3,
or non-square leading coefficients with any degree higher than 3), the
problem is *much* harder.
If you have any questions about this or need more help, please write
back and show me what you have been able to do, and I will try to
offer further suggestions.
- Doctor Vogler, The Math Forum
Date: 03/22/2008 at 15:25:53
From: Peter
Subject: try to find integers to a polynomial
Thank you for your response. It was very helpful, and I learned a lot
more about these types of problems. | {"url":"http://mathforum.org/library/drmath/view/72081.html","timestamp":"2014-04-19T15:54:30Z","content_type":null,"content_length":"9304","record_id":"<urn:uuid:e9db02fe-253e-4d71-8a9e-4350b1e94feb>","cc-path":"CC-MAIN-2014-15/segments/1397609537271.8/warc/CC-MAIN-20140416005217-00537-ip-10-147-4-33.ec2.internal.warc.gz"} |
Wolfram Demonstrations Project
Lindgren's Two-Triangles-to-One Dissection
Lindgren used a quadrilateral slide to transform one triangle (as a trapezoid with a top edge of length 0) to a trapezoid with the same angles.
A quadrilateral slide (or Q-slide) transforms one quadrilateral to another with the same angles. The notion was introduced by Lindgren (1956).
Harry Lindgren (1912-1992) was born in England and emigrated to Australia in 1935. He was for many years the world's leading expert in geometric dissections (Frederickson 1997, 183).
G. N. Frederickson,
Dissections: Plane & Fancy
, New York: Cambridge University Press, 2002 pp. 41–42. | {"url":"http://demonstrations.wolfram.com/LindgrensTwoTrianglesToOneDissection/","timestamp":"2014-04-21T12:22:40Z","content_type":null,"content_length":"44008","record_id":"<urn:uuid:e54302b3-eb94-4a4a-b82d-e8f3e288edcb>","cc-path":"CC-MAIN-2014-15/segments/1397609539776.45/warc/CC-MAIN-20140416005219-00559-ip-10-147-4-33.ec2.internal.warc.gz"} |
Equivalence relations on the integers
October 25th 2010, 06:46 AM
Equivalence relations on the integers
I'm trying to solve
"If R is a relation on Z so that aRb if b=a+3 then how many equivalence relations, S on Z contain R?"
I had originally thought none, but then found that if S is a=b (mod 3) then this is an equivalence containing R.
So now I don't know where to go next. Is this the only relation? So how do I show that? Can someone point me in the right direction?
October 25th 2010, 07:06 AM
If aSb is a=b (mod 3), then S is indeed the least equivalence relation containing R. Obviously, it has three equivalence classes. Note that for two equivalence relations S' and S'', S' is a
subset of S'' iff every equivalence class of S' is a subset of some equivalence class of S''. So, to get an equivalence relation containing S, we can join either two of the the tree or all three
October 25th 2010, 07:38 AM
Ah, ok I hadn't thought to combine classes like that. Thanks.
I'm still not completely sure how we know that these (5) relations are the only ones?
October 25th 2010, 09:16 AM
First, S is the least equivalence relation containing R, i.e., every equivalence relation containing R also contains S. Second, the claim I wrote above is a biconditional; in particular, if S' is
an equivalence relation containing S, then its equivalence classes are unions of the equivalence classes of S. | {"url":"http://mathhelpforum.com/discrete-math/160911-equivalence-relations-integers-print.html","timestamp":"2014-04-18T04:46:11Z","content_type":null,"content_length":"5118","record_id":"<urn:uuid:f733e1f5-0141-41f8-8148-78034ed05fe3>","cc-path":"CC-MAIN-2014-15/segments/1397609532480.36/warc/CC-MAIN-20140416005212-00046-ip-10-147-4-33.ec2.internal.warc.gz"} |
Zipper monad
From HaskellWiki
(Difference between revisions)
(Updating with better code) (Also in category idioms)
← Older edit Newer edit →
Line 232: Line 232:
traverse t tt = evalState (unT tt) (t, Top) traverse t tt = evalState (unT tt) (t, Top)
</haskell> </haskell>
+ [[Category:Idioms]]
Revision as of 01:21, 18 April 2006
The TravelTree Monad is a monad proposed and designed by Paolo Martini (xerox), and coded by David House (davidhouse). It is based on the State monad and is used for navigating around data
structures, using the concept of TheZipper.
As the only zipper currently available is for binary trees, this is what most of the article will be centred around.
1 Definition
newtype Travel t a = Travel { unT :: State t a }
deriving (Functor, Monad, MonadState t)
type TravelTree a = Travel (Loc a) (Tree a) -- for trees
Computations in
are stateful.
are defined as follows:
data Tree a = Leaf a | Branch (Tree a) (Tree a)
data Cxt a = Top
| L (Cxt a) (Tree a)
| R (Tree a) (Cxt a)
deriving (Show)
type Loc a = (Tree a, Cxt a)
for an explanation of the
2 Functions
2.1 Moving around
There are four main functions for stringing together
left, -- moves down a level, through the left branch
right, -- moves down a level, through the right branch
up, -- moves to the node's parent
top -- moves to the top node
:: TravelTree a
All four return the subtree at the new location.
2.2 Mutation
There are also functions available for changing the tree:
getTree :: TravelTree a
putTree :: Tree a -> TravelTree a
modifyTree :: (Tree a -> Tree a) -> TravelTree a
These are direct front-doors for State's
, and all three return the subtree after any applicable modifications.
2.3 Exit points
To get out of the monad, use
traverse :: Tree a -> TravelTree a -> Tree a
Again, this is just a front-door for
, with an initial state of
is the
passed in.
3 Examples
The following examples use as the example tree:
t = Branch (Branch (Branch (Leaf 1) (Leaf 2))
(Leaf 3))
(Branch (Leaf 4)
(Leaf 5))
3.1 A simple path
This is a very simple example showing how to use the movement functions:
leftLeftRight :: TravelTree a
leftLeftRight = do left
Result of evaluation:
*Tree> t `traverse` leftLeftRight
Leaf 2
3.2 Tree reverser
This is a more in-depth example showing
, but is still rather contrived as it's easily done without the zipper (the zipper-less version is shown below).
The algorithm reverses the tree, in the sense that at every branch, the two subtrees are swapped over.
revTree :: Tree a -> Tree a
revTree t = t `traverse` revTree' where
revTree' :: TravelTree a
revTree' = do t <- getTree
case t of
Branch _ _ -> do left
l' <- revTree'
r' <- revTree'
putTree $ Branch r' l'
Leaf x -> return $ Leaf x
-- without using the zipper:
revTreeZipless :: Tree a -> Tree a
revTreeZipless (Leaf x) = Leaf x
revTreeZipless (Branch xs ys) = Branch (revTreeZipless ys) (revTreeZipless xs)
Result of evaluation:
*Tree> revTree $ Branch (Leaf 1) (Branch (Branch (Leaf 2) (Leaf 3)) (Leaf 4))
Branch (Branch (Leaf 4) (Branch (Leaf 3) (Leaf 2))) (Leaf 1)
3.2.1 Generalisation
Einar Karttunen (musasabi) suggested generalising this to a recursive tree combinator:
treeComb :: (a -> Tree a) -- what to put at leaves
-> (Tree a -> Tree a -> Tree a) -- what to put at branches
-> (Tree a -> Tree a) -- combinator function
treeComb leaf branch = \t -> t `traverse` treeComb' where
treeComb' = do t <- getTree
case t of
Branch _ _ -> do left
l' <- treeComb'
r' <- treeComb'
putTree $ branch l' r'
Leaf x -> return $ leaf x
is then easy:
revTreeZipper :: Tree a -> Tree a
revTreeZipper = treeComb Leaf (flip Branch)
It turns out this is a fairly powerful combinator. As with
, it can change the structure of a tree. Here's another example which turns a tree into one where siblings are sorted, i.e. given a
, if
are leaves, then the value of
is less than or equal to that of
. Also, if one of
is a
and the other a
, then
is the
sortSiblings :: Ord a => Tree a -> Tree a
sortSiblings = treeComb Leaf minLeaves where
minLeaves l@(Branch _ _) r@(Leaf _ ) = Branch r l
minLeaves l@(Leaf _) r@(Branch _ _ ) = Branch l r
minLeaves l@(Branch _ _) r@(Branch _ _ ) = Branch l r
minLeaves l@(Leaf x) r@(Leaf y ) = Branch (Leaf $ min x y)
(Leaf $ max x y)
Result of evaluation:
*Tree> sortSiblings t
Branch (Branch (Leaf 3) (Branch (Leaf 1) (Leaf 2))) (Branch (Leaf 4) (Leaf 5))
4 Code
Here's the Zipper Monad in full:
module Zipper where
-- A monad implementing The Zipper.
-- http://haskell.org/haskellwiki/ZipperMonad
import Control.Monad.State
import Control.Arrow (first, second)
data Tree a = Leaf a | Branch (Tree a) (Tree a) deriving (Show, Eq)
data Cxt a = Top
| L (Cxt a) (Tree a)
| R (Tree a) (Cxt a)
deriving (Show)
type Loc a = (Tree a, Cxt a)
newtype Travel t a = Travel { unT :: State t a }
deriving (Functor, Monad, MonadState t)
type TravelTree a = Travel (Loc a) (Tree a)
-- Movement around the tree
-- move down a level, through the left branch
left :: TravelTree a
left = modify left' >> liftM fst get where
left' (Branch l r, c) = (l, L c r)
-- move down a level, through the left branch
right :: TravelTree a
right = modify right' >> liftM fst get where
right' (Branch l r, c) = (r, R l c)
-- move to a node's parent
up :: TravelTree a
up = modify up' >> liftM fst get where
up' (t, L c r) = (Branch t r, c)
up' (t, R l c) = (Branch l t, c)
-- move to the top node
top :: TravelTree a
top = modify (second $ const Top) >> liftM fst get
-- Mutation of the tree
-- modify the subtree at the current node
modifyTree :: (Tree a -> Tree a) -> TravelTree a
modifyTree f = modify (first f) >> liftM fst get
-- put a new subtree at the current node
putTree :: Tree a -> TravelTree a
putTree t = modifyTree $ const t
-- get the current node and its descendants
getTree :: TravelTree a
getTree = modifyTree id -- works because modifyTree returns the 'new' tree
-- Exit points
-- get out of the monad
traverse :: Tree a -> TravelTree a -> Tree a
traverse t tt = evalState (unT tt) (t, Top) | {"url":"http://www.haskell.org/haskellwiki/index.php?title=Zipper_monad&diff=3707&oldid=3705","timestamp":"2014-04-19T10:24:30Z","content_type":null,"content_length":"44079","record_id":"<urn:uuid:084ed1c1-a87f-4dcf-b033-53552eefb4de>","cc-path":"CC-MAIN-2014-15/segments/1398223204388.12/warc/CC-MAIN-20140423032004-00217-ip-10-147-4-33.ec2.internal.warc.gz"} |
Disney Urban Legends that REFUSE TO DIE!
04-11-2007, 10:11 AM #46
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Re: Disney Urban Legends that REFUSE TO DIE!
(I've posted this three times this year. Truly a legend that ceases to die.)
It's a little deeper than 3 - 4 feet, and not as deep as the 10 - 15 feet often cited. Here's a response to a post I did earlier that may prove enlightening:
Another Disneyland Urban Myth ripe for destruction:
Photo from MousePlanet
See those stairs from the bottom of the ROA to the dock? I count 10 steps. At 8" rise per step, that's 6.6 feet--to the top of the dock! Subtract about a foot (since the water doesn't go to the
top of the dock), and your looking at a depth of about 5 1/2 feet. That's pretty consistent all the way 'round.
There's not reason, of course, for the River to be any deeper.
Last edited by Steve DeGaetano; 04-11-2007 at 10:32 AM.
Re: Disney Urban Legends that REFUSE TO DIE!
Don't know if someone mentioned this. But the survey marker (spike) under the castle gates as being the center of DL when it was built never seems to go away.
I'm also tired of this myth of an old attraction called Superstar Limo.
Re: Disney Urban Legends that REFUSE TO DIE!
Re: Disney Urban Legends that REFUSE TO DIE!
You see? You've been suckered into a Disney urban legend that refuses to die!!
See those two very-above-ground show buildings circled in red? Those are the POTC show buildings. As you can see, they are above ground. That's why you can see them. Boats and pirates are inside.
(I wonder why people don't think they're underground when they ride IASW. Same concept.)
You know how you go into the Blue Bayou, and you climb up some steps? You go up, correct? Not down, as one might to enter a basement. And what do you see at your waterside table? Yep--POTC boats
going by.
The boats do go under a tunnel to get to the outer show building, outside the berm. Through a tunnel. This is no more "underground" than thinking you go underground when you pass beneath the
park's entrance tunnels.
But, maybe you do think that.
Either way, the ride is most definitely not "underground." As much as your deceived senses tell you so (the imagineers should be proud!)
Re: Disney Urban Legends that REFUSE TO DIE!
What about the underground tunnel(s)?
Re: Disney Urban Legends that REFUSE TO DIE!
Have you ever been on POTC and looked up. It is absolutely humongous. It seems like the building is like 30 feet tall. That's because, it is underground! It might not be entirely underground
(because the entire building sticks out), but you are in the basement for sure. Why do you think you go down in the beginning, pass under the train tracks in the middle (i.e. the new Davy Jones
theme), and go back up to ground level at the end. You are telling me you are not underground for that but at ground level? That would imply that you go right through the train tracks, you also
sit at the same level as Jungle Cruise. Are you going to tell me next that HM is also not underground, when you clearly go down an elevator.
Here is an illustration:
(I wonder why people don't think they're underground when they ride IASW. Same concept.)
Because you never go down...you are entirely at ground level the whole time. The rides sits normal, while the train is elevated above it.
Re: Disney Urban Legends that REFUSE TO DIE!
Why does the last lift bring you about 20 feet up? It has to be undergorund at some point, just because you see a building doesn't mean that there is something underneith.
Re: Disney Urban Legends that REFUSE TO DIE!
Re: Disney Urban Legends that REFUSE TO DIE!
because when you drop, you drop to the lowest level of the ride, which would be the basement. and you do this so you can get underneath the RR tracks. You go underneath the tracks right before
you get into the Battle scene which is the second show building you see behind the first one. the first one is for Blue Bayou and cave scenes the second is the main potc show building. hope that
pshhh i guess
Re: Disney Urban Legends that REFUSE TO DIE!
The POTC dilemma here is really a case of semantics, what your particular definition of "underground" is. If you consider "below ground level" as being underground, then Uncle Tom has it right.
If your classification of "underground" means literally not visible from ground level, then Steve has it correct. Personally I think you are both right, as POTC does drop 35 or so feet below
ground level, but it's showbuilding rises above ground level
Re: Disney Urban Legends that REFUSE TO DIE!
The POTC dilemma here is really a case of semantics, what your particular definition of "underground" is. If you consider "below ground level" as being underground, then Uncle Tom has it right.
If your classification of "underground" means literally not visible from ground level, then Steve has it correct. Personally I think you are both right, as POTC does drop 35 or so feet below
ground level, but it's showbuilding rises above ground level
Yup, 'cause when you get on the boats you're not on the same level as the the tram loading area, you're probably closer to the same level as the RR tracks.
(I've posted this three times this year.
See those stairs from the bottom of the ROA to the dock? I count 10 steps. At 8" rise per step, that's 6.6 feet--to the top of the dock! Subtract about a foot (since the water doesn't go to the
top of the dock), and your looking at a depth of about 5 1/2 feet. That's pretty consistent all the way 'round.
There's not reason, of course, for the River to be any deeper.
Sorry to think that one of the two teenagers swimming from Tom Sawyers Island actually drowned in 5.5 feet of water... probably too scared to stand up / feel for the bottom.
Re: Disney Urban Legends that REFUSE TO DIE!
04-11-2007, 10:17 AM #47
04-11-2007, 10:19 AM #48
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04-11-2007, 11:40 AM #60 | {"url":"http://micechat.com/forums/disneyland-resort/60496-disney-urban-legends-refuse-die-4.html","timestamp":"2014-04-18T17:25:22Z","content_type":null,"content_length":"105918","record_id":"<urn:uuid:e5b51b10-e71d-4a6a-8945-a3c5691a459f>","cc-path":"CC-MAIN-2014-15/segments/1397609533957.14/warc/CC-MAIN-20140416005213-00470-ip-10-147-4-33.ec2.internal.warc.gz"} |
Achievement Test: An objective test that measures how much a student has learned or knows about a specific subject.
Aptitude Test: An objective test that measures one or more capabilities that involve spatial reasoning, manual dexterity, clerical perception, or other capacities to learn certain behaviors. Aptitude
test scores do not rely specifically on knowledge gained but may be enhanced through exposure to like activities.
Grade Equivalent: The average test score obtained by students classified at a given grade placement. For example, a grade equivalent score of 7.4 indicates that the student performed as well on the
test as an average student who has been in the seventh grade for four months.
Interest Inventory: An individual survey of a person’s preferences with regard to different factors related to work, such as environment, interpersonal activity, physical demand, education required,
Mean: The sum of a set of scores divided by the number of scores. In lay terms, it is a synonym for the average. The value of the mean can be strongly influenced by a few extreme scores.
Median: The middle point in a set of ranked scores. It is the value that has the same number of scores above it and below it in the distribution.
Mode: The score that occurs most frequently in a distribution, no matter whether it occurs - high, low, or in the middle of the distribution.
Normal Curve Equivalent Score: A scale designed to transform percentile rankings into equal units. NCEs are suitable for computing averages, because a difference of 5 NCE units represents the same
difference in achievement at any point on the NCE scale, which is not the case with percentiles. However, different NCE charts are needed for different intervals of the school year.
Percentile Rank: A type of converted score that expresses a student’s score relative to the group in percentile points. Indicates the percentage of students tested who made scores equal to or lower
than a given score. If a score of 82 has a percentile rank of 65, this means that 65 percent of the students who took the test had a score of 82 or lower than 82.
Reliability: The degree to which a student would obtain the same score if the test were readministered (assuming no further learning, practice effects, or other change). It is a measure of the
stability or consistency of scores.
Standardization: In test construction, refers to the process of trying the test out on a group of students to determine uniform or standard scoring procedures and methods of interpretation.
Standardized Test: Contains empirically selected materials, with specific directions for administration, scoring, and interpretation. Provides data on validity and reliability, and has adequately
derived norms.
Standard Score: Derived score that transforms a raw score in such a way that it has the same mean and the same standard deviation. The standard score scale is an equal-interval scale; that is, a
difference of five points has the same meaning throughout the scale.
Stanine: A weighted scale divided into nine equal units that represent nine levels of performance on any particular test. The stanine is a standard score. Thus the intervals between different points
on the scale are equal in terms of the number of correct test responses they represent. The mean is at stanine 5.
Validity: The extent to which a test measures what it is designed to measure. A test valid for one use may have little or no validity for another.
Download glossery (MSWord format) (.pdf format)
This documents can be viewed using Adobe Acrobat Reader.
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Strange behavior of correlation estimation
November 2, 2011
By arthur charpentier
The Gaussian vector is extremely interesting since it remains Gaussian when conditioning. More precisely, if
where the covariance matrix of the random vector was
i.e. it is the Schur complement of bloc
It is possible to derive explicitly the correlation of
> library(mnormt)
> SIGMA=matrix(c(1,.6,.7,.6,1,.8,.7,.8,1),3,3)
> SIGMA11=SIGMA[1:2,1:2]
> SIGMA21=SIGMA[3,1:2]
> SIGMA12=SIGMA[1:2,3]
> SIGMA22=SIGMA[3,3]
> S=SIGMA11-SIGMA12%*%solve(SIGMA22)%*%SIGMA21
> S[1,2]/sqrt(S[1,1]*S[2,2])
[1] 0.09335201
This concept is discussed in a paper on Conditional Correlation by Kunihiro Baba, Ritei Shibata and Masaaki Sibuya. But what about statistical inference ? In order to see what happens, let us run
some simulations, and condition with
> k=10
> correlV=rep(0,k)
> X=rmnorm(n=10000000,varcov=SIGMA)
> QZ=qnorm((0:k)/k)
> ZQ=cut(X[,3],QZ)
> LZ=levels(ZQ)
> correl=rep(NA,k)
> for(i in 1:k){
+ correl[i]=cor(X[ZQ==LZ[i],1:2])[1,2]
+ }
> barplot(correl,names.arg=LZ,col="light blue",ylim=range(correl))
or some percentiles,
> k=100
In the middle it looks fine (we can look at it more carefully and fit a spline curve), but for very large values of
Note that for all estimates (each bar), there are exactly the same number of observations since when conditioning, we keep only observations such that
(up to 1 million simulations, i.e. 100,000 conditional observations used to estimate the correlation). If we look at the third decile, it converges,
for the second one, it might be a bit slower,
but for the first decile... either it is extremely slow, or it has a strong bias. But I don't get why...
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The Twin Prime Hero - Issue 5: Fame - Nautilus
Yitang “Tom” Zhang spent the seven years following the completion of his Ph.D. in mathematics floating between Kentucky and Queens, working for a chain of Subway restaurants, and doing odd accounting
work. Now he is on a lecture tour that includes stops at Harvard, Columbia, Caltech, and Princeton, is fielding multiple professorship offers, and spends two hours a day dealing with the press.
That’s because, in April, Zhang proved a theorem that had eluded mathematicians for a century or more. When we called Zhang to see what he thought of being thrust into the spotlight, we found a shy,
modest man, genuinely disinterested in all the fuss.
Mathematicians don’t tend to get famous. That’s probably because it’s hard for the public to grasp what they do. The importance of the light bulb, penicillin, or DNA is easy to understand, and
Edison, Fleming, and Crick are household names. The likes of Euler, Riemann, or Dirichlet seem, by comparison, outside of mainstream awareness.
But mathematicians also star in some of the most dramatic stories in science. The lone, unrecognized genius laboring away on a groundbreaking theory over many years is more fiction than fact for most
of modern science—but not in mathematics. Andrew Wiles’ 1995 proof of Fermat’s last theorem, which had defied all efforts to prove it for more than 300 years, was made all the more dramatic by the
secrecy with which it was conducted. But Wiles was already ensconced in the elite circles of mathematics when he did his work.
Not only did Zhang work in relative secrecy, he was also a complete unknown. He had found work as a lecturer at the University of New Hampshire (UNH) in 1999, eight years after he finished his Ph.D.,
with the help of connections at his undergraduate institution, Peking University. Then, in April of this year, Zhang announced a proof that cracked open a century-old problem in mathematics, called
the twin prime conjecture. “It was like climbing Everest,” says Ayalur Krishnan, a math professor at CUNY’s Kingsborough community college.
The importance of the light bulb, penicillin, or DNA is easy to understand, and Edison, Fleming, and Crick are household names.
It’s well known that consecutive prime numbers become more widely separated from each other as they become larger. The twin prime conjecture stipulates that, despite this, there are infinitely many
pairs of prime numbers that are separated from each other by only two (for example, 11 and 13). Zhang proved a weaker variant of this conjecture: that there are infinitely many pairs of prime numbers
that are separated from each other by some fixed number that is greater than two, but less than 70 million.
Zhang’s success followed from the construction of a new kind of prime-number filter. Previous work had used a strong filter, or sieve, which discarded prime numbers that were too far apart from each
other. It allowed mathematicians to prove that there were always neighboring pairs of prime numbers closer together than some moving average. But they couldn’t prove these pairs were separated by
some constant finite difference. By making the sieve less selective, Zhang was able to reach the finite bound proof, representing the most significant progress to be made on the centuries-old twin
prime conjecture.
The result was a thunder strike for the pure mathematics community. Zhang submitted his work to the Annals of Mathematics, one of the most prestigious math journals in the world, with a typical
review time for a new paper of one or two years. Zhang’s paper was accepted in three weeks. Soon the fame machine was in full gear. Students stopped Zhang on the UNH campus to ask for his autograph.
“I had not expected it would cause such a big sensation,” Zhang commented at a conference. “I saw myself appearing in newspapers and magazines from India to Europe to Brazil.” The twin prime
conjecture began to be the unlikely subject of mainstream news stories, with 100 news headlines on the topic in May alone, according to Google Trends.
He is simultaneously driven and calm, with an ambition centered around a love of math.
Zhang’s work also set off a flurry of follow-up activity in the mathematics community, including a collaboration of a dozen or so mathematicians called the Polymath8 project, which is attempting to
reduce the maximum separation between primes in Zhang’s proof to something less than 70 million. As of late August, it had provisionally lowered it to 4,680. Terence Tao, a math professor at UCLA and
a member of the collaboration, described Zhang’s work as a technical breakthrough that should lead to more progress on other questions in analytic number theory, including the Goldbach conjecture,
which states that every even number is the sum of two primes.
Erica Klarreich, a Berkeley-based science writer who has a Ph.D. in mathematics and has written about Zhang, says his proof demonstrates the remarkable balance between order and randomness within the
prime numbers. “Prime numbers are anything but random—they are completely determined,” Klarreich says. “Nevertheless, they seem to behave in many respects like randomly-sprinkled numbers that
eventually display all possible clumps and clusters. Zhang’s work helps to put this conjectured picture of the primes on a solid footing. ”
Speaking from his office in New Hampshire, the soft-spoken Zhang replied to most questions with a quiet chuckle followed by a clipped sentence. He is simultaneously driven and calm, with an ambition
centered around a love of math. The most repeated phrase in our conversation was, “I am a peaceful man.”
When did you become interested in the twin prime conjecture?
That was when I was a child. That is a famous problem. And easy to understand for many educated people. I am interested in primes. Prime numbers are one of the biggest challenges to the intelligence
of human beings.
Why is the twin prime problem famous?
First, the conjecture has been known for a long time—at least 100 or maybe 200 years, maybe longer. Second, the statement is very simple. It is easy to understand for many people. Third, the problem
is believed to be very important.
What stopped you from finding work after your Ph.D?
During that period it was difficult to find a job in academics. That was a job market problem. Also, my advisor did not write me letters of recommendation. So I went to my friend’s house in Kentucky.
He was running some Subway restaurants. I was an assistant in his business for several years.
When you were working in Kentucky and Queens, did you think you’d ever work in mathematics again?
I believed so. I was waiting for the chance. I kept thinking about math during that time, every week.
When did you first try to solve the twin prime problem?
Around four years ago. I tried because at that time, in this field it was very hard, because some other professors, like Goldston and Pintz and Yildirim, they made significant progress towards this
problem. And at that time many people believed it would not be possible to solve it. And many people tried. And eventually I am successful. This is what I can say.
Why did you solve this problem and not somebody else?
I think the important reason is that I persisted for several years. I didn’t give up.
How long did you work on your conjecture?
I worked on my conjecture for four years. I worked on it seven days a week. I didn’t take any breaks almost. If we count the time I spent thinking about the problem, then it was more than 10 hours a
day. If you count sitting in front of a desk or using a computer, then it was five or six hours a day. I was also teaching two classes per semester at the time.
Did you show your work to your colleagues?
No, I didn’t. Even in the circle of my friends, although they are also mathematicians, they do not understand this problem very well. So I just mentioned I was working on it, but I didn’t show my
work to anybody. I don’t like to tell so much to other people before I get to the result. I am a shy person.
Andrew Wiles also told almost nobody about his work on Fermat’s Last Theorem.
If he had told the public about his research, he wouldn’t have been concentrating on the problem. That might have happened to me. Sometimes it’s important to be alone, so you can concentrate on what
you’re doing. If there are too many social obligations, that bothers me so much.
Did you experience any emotions when you realized you’d solved the problem?
Not so much. I am a very quiet person.
Were you excited?
A little. Not too much.
How has your wife reacted?
Before I published this paper, she had no idea about it. Now of course she’s happy. Still both of us are very peaceful. We are not too excited.
You must have gotten tired during your work. How did you rest?
I like listening to music and reading novels or fiction. That’s my way. I don’t like to watch TV. I listen to classical music only—all of the famous composers, Beethoven, Chopin, Brahms, Tchaikovsky,
Mozart, all of them. I also like walking. New Hampshire is a very good place to walk. Sometimes I get good ideas when I walk. There are small forests, creeks everywhere.
Is the most likely application of this result in cryptography?
Yes, this should be true.
Are you interested in cryptography?
Not so much.
How do pure mathematicians choose which problem to work on?
That depends. The different subjects have different choices. Geometry develops very quickly. There are many new problems every year or every 10 years. But in number theory there are lots of older
problems that exist for 100 or 200 years. Many of these problems are very easy to understand but very difficult to solve. They are more famous because they are easy to understand. Many people
understand these problems, not just mathematicians.
Are there problems that deserve to be famous but are not famous because they’re hard to understand?
For mathematicians, if a problem is important, it is famous. But in some subjects, like topology, differential geometry, it is famous only for mathematicians and not for other people, because the
statements are difficult to understand.
Would you describe yourself as famous now?
Are you comfortable speaking in public?
Now, it’s ok. It’s been several months now, and I’ve gotten used to speaking to the public. But I don’t enjoy it so much. I try to avoid the public. But I can’t avoid it completely right now.
You’ve said you don’t care about the money and honor. Why not?
Because of my personality. I am a quiet person. I like to concentrate on the math, on what I like. I do not care about the life conditions, like a good house, good cars, good clothing. This is my
personality. I don’t have a car right now. I have a townhouse, but it is in California, where my wife lives. In New Hampshire I rent an apartment. The most important thing is to concentrate on math
What do you think of Grigori Perelman, who turned down the Fields Medal?
I think it was because of his personality also. Maybe he himself was very proud of himself. Maybe he looked down on other people, even looked down on very important prizes and medals.
Would you accept a medal?
What would you do with the money?
Maybe the best way would be to give the money to my wife. Let her deal with this issue.
Do you ever think of the time that you worked in Subway?
Sometimes I think about it. I just think of it in a very peaceful way. It is past, so I don’t worry about that. Disappointment—that was the past.
Did you have any scientific heroes?
Maybe Gauss. When I was 10, I learned his name. He helped to inspire me to study mathematics.
Do you expect yourself to be an inspiration for students?
Yes, particularly in China. When I came to China I was treated like a hero. They took me to dinner, provided good housing, gave me personal gifts and concert tickets.
What are you working on now?
I am working on a problem related to the Goldbach conjecture.
Is your current research being slowed down by all the interviews?
A little bit.
Why do you do them?
I couldn’t avoid them completely. Say your friends ask you—that’s a problem. But I try to reduce the quantity as much as possible.
What would you say to a young student who wants to solve a problem?
Keep going. Do not easily give up.
Where do you suggest they find the motivation?
The most important motivation is to really love mathematics. | {"url":"http://nautil.us/issue/5/fame/the-twin-prime-hero","timestamp":"2014-04-20T01:11:55Z","content_type":null,"content_length":"40612","record_id":"<urn:uuid:092d5f3d-eb4f-4fb1-b0ee-2e1dd51f63a1>","cc-path":"CC-MAIN-2014-15/segments/1397609537804.4/warc/CC-MAIN-20140416005217-00070-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °
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Post a reply
Topic review (newest first)
2013-09-24 00:45:44
auyeungyat wrote:
I got stuck in thinking the infinity.
I thought like this:
Let infinity be x
2x=2 infinity
The main problem is:There isn't 2 times infinity!
How can it be proven?
infinity is like /dev/null
Whatever you write or add to it it still remains /dev/null
2013-09-24 00:33:08
Welcome to the forum.
I think most of that was established by Cantor and the mathematicians that followed. For one thing infinity is not a number, it is a concept. The rules of arithmetic do not
2013-09-24 00:04:39
I got stuck in thinking the infinity.
I thought like this:
Let infinity be x
2x=2 infinity
The main problem is:There isn't 2 times infinity!
How can it be proven?
Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °
auyeungyat wrote:I got stuck in thinking the infinity.I thought like this:Let infinity be x2x=2 infinityThe main problem is:There isn't 2 times infinity!How can it be proven?
I got stuck in thinking the infinity.I thought like this:Let infinity be x
infinity is like /dev/nullWhatever you write or add to it it still remains /dev/null
Hi;Welcome to the forum.I think most of that was established by Cantor and the mathematicians that followed. For one thing infinity is not a number, it is a concept. The rules of arithmetic do not | {"url":"http://www.mathisfunforum.com/post.php?tid=20033&qid=285196","timestamp":"2014-04-17T12:40:02Z","content_type":null,"content_length":"16680","record_id":"<urn:uuid:47128f16-968f-4ea6-9362-0300a6f48159>","cc-path":"CC-MAIN-2014-15/segments/1397609530131.27/warc/CC-MAIN-20140416005210-00119-ip-10-147-4-33.ec2.internal.warc.gz"} |
update pari-2.0.21.beta
Karim BELABAS on Fri, 27 Oct 2000 21:42:08 +0200 (MET DST)
[Date Prev] [Date Next] [Thread Prev] [Thread Next] [Date Index] [Thread Index]
Hi PARI lovers,
here's another update, mostly bugfix, but some of them involving new
implementations e.g:
1) improved bnf* numerical stability, due to a new hnfLLL [ try mathnf(,4),
it's much better than it used to be, and should replace the default (naive)
algorithm ]
2) rewritten bnfisprincipal, profiting from 1) above + improved
reduction/splitting strategy + taking advantage of known factorizations
instead of splitting gigantic ideals (most class group related functions
benefit heavily when class group is large).
3) factorization over Z[X] (van Hoeij's knapsack algorithm) or over finite
fields when the polynomial is defined over the prime field (Allomber's
Have fun,
P.S: This release is available on the cvs server as release-2-0-21. I've
updated the (PostScript and PDF) manuals on the megrez server. I haven't
updated the binaries (DOS, etc.) for this release.
P.S2: the detailed Changelog.
Done for version 2.0.21 (released 27/10/2000):
1- trap(gdiver2,a,1/0) <C-D> trap(,a,1/0); trap(gdiver2,a,1/0) --> crash
2- lllgramintern could reduce a wrong lattice (after precision problems)
XR 3- not enough GC in bezout()
4- use hnfmodid in ideallllred (work mod I \cap Z, not mod N(I))
IS 5- C++ compilation problem (compl is a C++ operator + casts)
6- factor(x^2 - 16810110*x + 62994937599000) --> division by 0
BA 7- Pocklington-Lehmer didn't stop factorisation after sqrt(N)
8- exp(log(Pi+x+y)*1.) --> SEGV (still gives an error, but a decent one)
9- factorff needed a prime field F_p where p was single precision
10- more digits than were significant could be printed (ex: precision(Pi,1))
11- rnfisnorm didn't accept vectors on Zk basis as argument
12- [cygwin] fixed timer (always returned 0)
IS 13- function prototypes (missing 'extern')
14- quadray(bnf, ...) didn't work when bnf.disc < 0. Could also return
a relative equation over an intermediate field
15- deficient argument checks: asinh(2*x-1), agm(1,1-x), zetakinit(2*x-1),
idealinv( , wrongtype), nfnewprec([2*x-1]), thueinit(non-monic or
degree 1), lcm(x, 0) rnfidealmul(2*x-1,0), galoisfixedfield(2*x-1),
qfbnupow(2*x-1, 0), polred([x,x]), bezout(Mod(x,x^2+1),0),
numtoperm(-1,0), polred(nf), idealhnf(nf,y+z), factorpadic(,y,),
nfelftreduce, rnfsteinitz, idealappr(not an nf, ...): SEGV
16- check variable numbers in algtobasis: nfalgtobasis(nfinit(P(y), x))
17- factorcantor(x^4 - x^3 - 2*x - 1, 3) --> SEGV [typo!]
18- matsupplement(non-exact entries): fixed zero test (--> |x| < eps)
19- precision(I*1.) --> +oo (should be realprecision)
20- global() could check uninitialized memory
21- default(prettyprinter,"non-existent-file") ---> SIGPIPE and GP crash
22- gamma(1+O(3^2)+x) --> SIGFPE
23- typo in cxgamma (gamma(-4.1+Pi*I/2) --> exponent overflow)
24- Odos/paricfg.h defined PARIINFO incorrectly
25- inconsistenty t_POLMOD specification: Mod(x,y)+16 --> invalid object
26- bnfsunit error (impossible I-->S) when large class group
27- qfminim([;],...) / qfperfection([;]) --> SEGV
28- idealpow(,,,1) only reduced at the end, not after each multiplication
29- typo in zbrent (always used bisection)
BA 30- cotan(x) wasn't accepted
31- wrong internal prototype for thetanullk (prec not taken into account)
32- various bugs in gphelp -detex (??\a)
33- trap + ^C + next would interrupt \r or read()
IZ 34- external prettyprinter incompatible with colors
35- idealval used much larger numbers than necessary
36- sqrtint very inefficient (computed with full precision all along)
37- bnfisprincipal couldn't deal with some non-Galois fields
38- listcreate allowed creating longer lists than GP could later handle
39- INPUT colors was "leaking out" (affected status messages)
40- idealpowred(, power > 2^32) --> wrong result
41- when lines > 0, output driver didn't reset properly after overflow
42- huge precision losses in bnfinit computations (--> truncation error)
43- mateigen([1,2;3,4]) wouldn't work at default precision
44- leftover debugging statements in gphelp -to_pod
45- zeta(x) didn't always use the precision of x (contrary to the other
trans. functions)
46- some Warnings from MIPSPro 7.2.1 compiler (contributed by PM)
BA 47- gdiv(SER,POL) --> SEGV when POL has lgef < lg
48- check relative pol is monic before calling a rnf* function
XR 49- cleaned up bnrL1 character computations
XR 50- round 4: possible problems when increasing p-adic precision
51- stack corruption in mppgcd(huge t_FRAC, huge t_FRAC)
BA 52- abs(t_SER) not allowed
53- nfdisc(non monic polynomial,, disc factorization) returned wrong answer
54- rnfidealnormrel(rnfinit(nfinit(y^2-2),x),2) --> SIGBUS
55- hardcoded paths in a few scripts (tex2mail, gpflog, make_vi_tags)
BA 1- more efficient quadratic Hensel lift
2- made ya optional in polinterpolate
BA 3- more efficient factorff when pol belongs to Fp[X] (based on Fp_isom)
4- catch SIGFPE
5- made poltchebi efficient
6- write all real zeroes in exponential format (0e28) in format 'g'
7- bnfsunit: try to compute S-units even when the class group is large
reduce class group generators
8- 0.e N now inputs a real 0 of decimal exponent N (was N-defaultprecision)
9- try to avoid errors due to precision loss (while computing archimedian
components) in bnfnewprec/bnfmake
10- restore \o2 to previous meaning, use \o3 for alternate prettyprinter
11- normalized output of idealfactor (sort factors)
12- better TeXization (\left / \right)
13- allow color changes in error messages (when sample input is given)
14- format for bnf[9] (new isprincipal)
15- cleaned up isprincipal + bnfinit (small_norm, getfu, class_group_gen)
XR 16- added flag for D>0 in quadhilbert: if non-zero, try more modulii
17- idealred: reduce huge ideals as Z-module first (using lllintpartial)
18- TeXmacs interface
1- library function zerovec
2- van Hoeij's algo. for modular factors recombination (factor over Z[X])
3- algdep for p-adic numbers
IZ 4- default values for 'colors' (light/dark) and 'prettyprinter' (tex2mail)
5- hnflll implementation (old one was preliminary and didn't work at all)
GN 6- squfof implementation
7- numerical derivation
XR 8- GP interface to hensel_lift functions (polhensellift)
1- quadrayimagwei, preliminary implementation to quadray(D < 0)
Karim Belabas email: Karim.Belabas@math.u-psud.fr
Dep. de Mathematiques, Bat. 425
Universite Paris-Sud Tel: (00 33) 1 69 15 57 48
F-91405 Orsay (France) Fax: (00 33) 1 69 15 60 19
PARI/GP Home Page: http://www.parigp-home.de/ | {"url":"http://pari.math.u-bordeaux.fr/archives/pari-dev-0010/msg00019.html","timestamp":"2014-04-20T23:47:57Z","content_type":null,"content_length":"10271","record_id":"<urn:uuid:75428c72-24a7-47d4-bfe7-a8196a0dbc83>","cc-path":"CC-MAIN-2014-15/segments/1398223205137.4/warc/CC-MAIN-20140423032005-00509-ip-10-147-4-33.ec2.internal.warc.gz"} |
Lecture 31: Parametric Equations
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PROFESSOR: OK. Now, today we get to move on from integral formulas and methods of integration back to some geometry. And this is more or less going to lead into the kinds of tools you'll be using in
multivariable calculus. The first thing that we're going to do today is discuss arc length. Like all of the cumulative sums that we've worked on, this one has a storyline and the picture associated
to it, which involves dividing things up. If you have a roadway, if you like, and you have mileage markers along the road, like this, all the way up to, say, sn here, then the length along the road
is described by this parameter, s. Which is arc length. And if we look at a graph of this sort of thing, if this is the last point b, and this is the first point a, then you can think in terms of
having points above x1, x2, x3, etc. The same as we did with Riemann sums.
And then the way that we're going to approximate this is by taking the straight lines between each of these points. As things get smaller and smaller, the straight line is going to be fairly close to
the curve. And that's the main idea. So let me just depict one little chunk of this. Which is like this. One straight line, and here's the curved surface there. And the distance along the curved
surface is what I'm calling delta s, the change in the length between, so this would be s2 - s1 if I depicted that one.
So this would be delta s is, say s. si - si - 1, some increment there. And then I can figure out what the length of the orange segment is. Because the horizontal distance is delta x. And the vertical
distance is delta y. And so the formula is that the hypotenuse is delta x ^2 + delta y ^2. Square root. And delta s is approximately that. So what we're saying is that delta s ^2 is approximately
this. So this is the hypotenuse. Squared. And it's very close to the length of the curve.
And the whole idea of calculus is in the infinitesimal, this is exactly correct. So that's what's going to happen in the limit. And that is the basis for calculating arc length. I'm going to rewrite
that formula on the next board. But I'm going to write it in the more customary fashion. We've done this before, a certain amount. But I just want to emphasize it here because this handwriting is a
little bit peculiar. This ds is really all one thing. What I really mean is to put the parenthesis around it. It's one thing. It's not d * s, it's ds. It's one thing. And we square it. But for
whatever reason people have gotten into the habit of omitting the parentheses. So you're just going to have to live with that. And realize that this is not d of s ^2 or anything like that. And
similarly, this is a single number, and this is a single number. Infinitesimal. So that's just the way that this idea here gets written in our notation. And this is the first time we're dealing with
squares of infinitesimals. So it's just a little different. But immediately the first thing we're going to do is take the square root.
If I take the square root, that's the square root of dx ^2 + dy ^2. And this is the form in which I always remember this formula. Let's put it in some brightly decorated form. But there are about 4,
5, 6 other forms that you'll derive from this, which all mean the same thing. So this is, as I say, the way I remember it. But there are other ways of thinking of it. And let's just write a couple of
them down. The first one is that you can factor out the dx. So that looks like this. 1 + (dy / dx)^2. And then I factored out the dx. So this is a variant. And this is the one which actually we'll be
using in practice right now on our examples. So the conclusion is that the arc length, which if you like is this total sn - s0, if you like, is going to be equal to the integral from a to b of the
square root of 1 + (dy / dx)^2 dx.
In practice, it's also very often written informally as this. The integral ds. So the change in this little variable s, and this is what you'll see notationally in many textbooks. So that's one way
of writing it, and of course the second way of writing it which is practically the same thing is square root of 1 + f ' ( x^2) dx. Mixing in a little bit of Newton's notation. And this is with y = f
(x). So this is the formula for arc length. And as I say, I remember it this way. But you're going to have to derive various variants of it. And you'll have to use some arithmetic to get to various
formulas. And there will be more later. Yeah, question.
PROFESSOR: OK, the question is, is f ' (x)^2 = f '' (x). And the answer is no. And let's just see what it is. So, for example, if f ( x) = x ^2, which is an example which will come up in a few
minutes, then f ' (x) = 2x and f ' ( x )^2 = = (2x), which is 4x ^2. Whereas f '' ( x) is equal to, if I differentiate this another time, it's equal to 2. So they don't mean the same thing. The same
thing over here. You can see this dy / dx, this is the rate of change of y with respect to x. The quantity squared. So in other words, this thing is supposed to mean the same as that. Yeah. Another
PROFESSOR: So the question is, you got a little nervous because I left out these limits. And indeed, I did that on purpose because I didn't want to specify what was going on. Really, if you wrote it
in terms of ds, you'd have to write it as starting at s0 and ending at sn to be consistent with the variable s. But of course, if you write it in terms of another variable, you put that variable in.
So this is what we do when we change variables, right? We have many different choices for these limits. And this is the clue as to which variable we use.
PROFESSOR: Correct. s0 and sn are not the same thing as a and b. In fact, this is xn. And this x0, over here. That's what a and b are. But s0 and sn are mileage markers on the road. They're not the
same thing as keeping track of what's happening on the x axis. So when we measure arc length, remember it's mileage along the curved path.
So now, I need to give you some examples. And my first example is going to be really basic. But I hope that it helps to give some perspective here. So I'm going to take the example y = m x, which is
a linear function, a straight line. And then y ' would = m, and so ds is going to be the square root of 1 + (y ') ^2 dx. Which is the square root of 1 + m ^2 dx. And now, the length, say, if we go
from, I don't know, let's say to 10, let's say. Of the graph is going to be the integral from to 10 of the square root of 1 + m ^2 dx. Which of course is just 10 square root of 1 + m ^2. Not very
surprising. This is a constant. It just factors out and the integral from to 10 of dx is 10.
Let's just draw a picture of this. This is something which has slope m here. And it's going to 10. So this horizontal is 10. And the vertical is 10 m. Those are the dimensions of this. And the
Pythagorean theorem says that the hypotenuse, not surprisingly, let's draw it in here in orange to remind ourselves that it was the same type of orange that we had over there, this length here is the
square root of 10 ^2 + (10m)^2. That's the formula for the hypotenuse. And that's exactly the same as this. Maybe you're saying duh, this is obvious. But the point that I'm trying to make is this. If
you can figure out these formulas for linear functions, calculus tells you how to do it for every function. The idea of calculus is that this easy calculation here, which you can do without any
calculus at all, all of the tools, the notations of differentials and limits and integrals, is going to make you be able to do it for any curve. Because we can break things up into these little
infinitesimal bits. This is the whole idea of all of the methods that we had to set up integrals here. This is the main point of these integrals.
Now, so let's do something slightly more interesting. Our next example is going to be the circle, so y = square root of 1 - x ^2. If you like, that's the graph of a semicircle. And maybe we'll set it
up here this way. So that the semicircle goes around like this. And well start it here at x = 0. And we'll go over to a. And we'll take this little piece of the circle. So down to here. If you like.
So here's the portion of the circle that I'm going to measure the length of. Now, we know that length. It's called arc length. And I'm going to give it a name, I'm going to call it alpha here. So
alpha's the arc length along the circle. Now, let's figure out what it is. First, in order to do this, I have to figure out what y ' is. Or, if you like, dy / dx. Now, that's a calculation that we've
done a number of times. And I'm going to do it slightly faster. But you remember it gives you a square root in the denominator. And then you have the derivative of what's inside the square root.
Which is - 2x. But then there's also 1/2, because in disguise it's really (1 - x ^2)^2 1/2. So we've done this calculation enough times that I'm not going to carry it out completely. I want you to
think about what it is. It turns out to - x up here, because the 1/2 and the 2 cancel.
And now the thing that we have to integrate is this arc length element, as it's called. ds. And that's going to be the square root of 1 + (y ') ^2 dx. And so I'm going to have to carry out the
calculation, some messy calculation here. Which is that this is 1 + ( - x / square root of 1 - x ^2) ^2. So I have to figure out what's under the square root sign over here in order to carry out this
calculation. Now let's do that.
This is 1 + x ^2 / 1 - x ^2. That's what this simplifies to. And then that's equal to, over a common denominator, (1 - x ^2). 1 - x ^2 + x^2. And there is a little bit of simplification now. Because
the 2x ^2's cancel. And we get 1 / 1 - x^2. So now I get to finish off the calculation by actually figuring out what the arc length is. And what is it? Well, this alpha is equal to the integral from
to a of ds. Well, it's going to be the square root of what I have here. This was a square, this is just what was underneath the square root sign. This is 1 + (y ') ^2. Have to take the square root of
that. So what I get here is dx / the square root of 1 - x ^2.
And now, we recognize this. The antiderivative of this is something that we know. This is the inverse sine. Evaluated at 0 and a. Which is just giving us the inverse sine of a, because the inverse
sine of = 0. So alpha = the inverse sine of a. That's a very fancy way of saying that sine alpha = a. That's the equivalent statement here. And what's going on here is something that's just a little
deeper than it looks. Which is this. We've just figured out a geometric interpretation of what's going on here. That is, that we went a distance alpha along this arc. And now remember that the radius
here is 1. And this horizontal distance here is a. This distance here is a. And so the geometric interpretation of this is that this angle is alpha radians. And sine alpha = a. So this is consistent
with our definition previously, our previous geometric definition of radians. But this is really your first true definition of radians. We never actually, people told you that radians were the arc
length along this curve. This is the first time you're deriving it. This is the first time you're seeing it correctly done. And furthermore, this is the first time you're seeing a correct definition
of the sine function.
Remember we had this crazy way, we we defined the exponential function, then we had another way of defining the ln function as an integral. Then we defined the exponential in terms of it. Well, this
is the same sort of thing. What's really happening here is that if you want to know what radians are, you have to calculate this number. If you've calculated this number then by definition if sine is
the thing whose alpha radian amount gives you a, then it must be that this is sine inverse of a. And so the first thing that gets defined is the arc sine. And the next thing that gets defined is the
sine afterwards. This is the way the foundational approach actually works when you start from first principles. This arc length being one of the first principles. So now we have a solid foundation
for trig functions. And this is giving that connection. Of course, it's consistent with everything you already knew, so you don't have to make any transitional thinking here. It's just that this is
the first time it's being done rigorously. Because you only now have arc length.
So these are examples, as I say, that maybe you already know. And maybe we'll do one that we don't know quite as well. Let's find the length of a parabola. This is Example 3. Now, that was what I was
suggesting we were going to do earlier. So this is the function y = x ^2. y ' = 2x. And so ds = the square root of 1 + (2x) ^2 dx. And now I can figure out what a piece of a parabola is. So I'll draw
the piece of parabola up to a, let's say, starting from 0. So that's the chunk. And then its arc length, between and a of this curve, is the integral from to a of square root of 1 + 4x ^2 dx.
OK, now if you like, this is the answer to the question. But people hate looking at answers when they're integrals if they can be evaluated. So one of the reasons why we went through all this
rigamarole of calculating these things is to show you that we can actually evaluate a bunch of these functions here more explicitly. It doesn't help a lot, but there is an explicit calculation of
this. So remember how you would do this. So this is just a little bit of review. What we did in techniques of integration. The first step is what? A substitution. It's a trig substitution. And what
is it?
PROFESSOR: So x equals something tan theta. I claim that it's 1/2 tan, and I'm going to call it u. Because I'm going to use theta for something else in a couple of days. OK? So this is the
substitution. And then of course dx = 1/2 sec ^2 u du , etc. So what happens if you do this? I'll write down the answer, but I'm not going to carry this out. Because every one of these is horrendous.
But I think I worked it out. Let's see if I'm lucky. Oh yeah. I think this is what it is. It's a 1/4 ln 2x + square root of 1 + 4x ^2 + 1/2 x ( square root of 1 + 4x ^2). Evaluated at a and 0.
So yick. I mean, you know.
PROFESSOR: Why I did I make it 1/2? Because it turns out that when you differentiate. So the question is, why there 1/2 there? If you differentiate it without the 1/2, you get this term and it looks
like it's going to be just right. But then if you differentiate this one you get another thing. And it all mixes together. And it turns out that there's more. So it turns out that it's 1/2.
Differentiate it and check. So this just an incredibly long calculation. It would take fifteen minutes or something like that. But the point is, you do know in principle how to do these things.
PROFESSOR: Oh, he was talking about this 1/2, not this crazy 1/2 here. Sorry.
PROFESSOR: Yeah, OK. So sorry about that. Thank you for helping. This factor of 1/2 here comes about because when you square x, you don't get tan ^2. When you square 2x, you get 4x ^2 and that
matches perfectly with this thing. And that's why you need this factor here. Yeah. Another question, way in the back.
PROFESSOR: The question is, when you do this substitution, doesn't the limit from to a change. And the answer is, absolutely yes. The limits in terms of u are not the same as the limits in terms of
a. But if I then translate back to the x variables, which I've done here in this bottom formula, of x = and x = a, it goes back to those in the original variables. So if I write things in the
original variables, I have the original limits. If I use the u variables, I would have to change limits. But I'm not carrying out the integration, because I don't want to. So I brought it back to the
x formula. Other questions.
OK, so now we're ready to launch into three-space a little bit here. We're going to talk about surface area. You're going to be doing a lot with surface area in multivariable calculus. It's one of
the really fun things. And just remember, when it gets complicated, that the simplest things are the most important. And the simple things are, if you can handle things for linear functions, you know
all the rest. So there's going to be some complicated stuff but it'll really only involve what's happening on planes. So let's start with surface area. And the example that I'd like to give, this is
the only type of example that we'll have, is the surface of rotation. And as long as we have our parabola there, we'll use that one. So we have y = x ^2, rotated around the x axis.
So let's take a look at what this looks like. It's the parabola, which is going like that. And then it's being spun around the x axis. So some kind of shape like this with little circles. It's some
kind of trumpet shape, right? And that's the shape that we're. Now, again, it's the surface. It's just the metal of the trumpet, not the insides. Now, the principle for figuring out what the formula
for area is, is not that different from what we did for surfaces of revolution. But it just requires a little bit of thought and imagination. We have a little chunk of arc length along here. And
we're going to spin that around this axis. Now, if this were a horizontal piece of arc length, then it would spin around just like a shell. It would just be a surface. But if it's tilted, if it's
tilted, then there's more surface area proportional to the amount that it's tilted. So it's proportional to the length of the segment that you spin around.
So the total is going to be ds, that's one of the factors here. Maybe I'll write that second. That's one of the dimensions. And then the other dimension is the circumference. Which is 2 pi, in this
case y. So that's the end of the calculation. This is the area element of surface area. Now, when you get to 18.02, and maybe even before that, you'll also see some people referring to this area
element when it's a curvy surface like this with a notation d S. That's a little confusing because we have a lower case s here. We're not going to use it right now. But the lower case s is usually
arc length. The upper case s is usually surface area. So. Also used for dA. The area element. Because this is a curved area element. So let's figure out this example.
So in the example, is equal to x ^2 then the situation is, we have the surface area is equal to the integral from, I don't know, to a if those are the limits that we wanted to choose. Of 2 pi x ^2,
right? Because y = x ^2 ( the square root of 1 + 4x ^2) dx. Remember we had this from our previous example. This was ds from previous. And this, of course, is 2 pi y. Now again, the calculation of
this integral is kind of long. And I'm going to omit it. But let me just point out that it follows from the same substitution. Namely, x = 1/2 tan u. Is going to work for this integral. It's kind of
a mess. There's a tan ^2 here and the sec ^2. There's another secant and so on. So it's one of these trig integrals that then takes a while to do.
So that just is going to trail off into nothing. And the reason is that what's important here is more the method. And the setup of the integrals. The actual computation, in fact, you could go to a
program and you could plug in something like this and you would spit out an answer immediately. So really what we just want is for you to have enough control to see that it's an integral that's a
manageable one. And also to know that if you plugged it in, you would get an answer. When I actually do carry out a calculation, though, what I want to do is to do something that has an answer that
you can remember. And that's a nice answer. So that turns out to be the example of the surface area of a sphere. So it's analogous to this 2 here. And maybe I should remember this result here. Which
was that the arc length element was given by this. So we'll save that for a second.
So we're going to do this surface area now. So if you like, this is another example. The surface area of a sphere. This is a good example, and one, as I say, that has a really nice answer. So it's
worth doing. So first of all, I'm not going to set it up quite the way I did in Example 2. Instead, I'm going to take the general sphere, because I'd like to watch the dependence on the radius. So
here this is going to be the radius. It's going to be radius a. And now, if I carry out the same calculations as before, if you think about it for a second, you're going to get this result. And then,
the rest of the arithmetic, which is sitting up there in the case, a = 1, will give us that ds = what? Well, maybe I'll just carry it out. Because that's always nice. So we have 1 + x ^2 / a ^2 - x ^
2. That's 1 + (y ') ^2. And now I put this over a common denominator. And I get a ^2 - x ^2. And I have in the numerator a ^2 - x ^2 + x^2. So the same cancellation occurs. But now we get an a ^2 in
the numerator.
So now I can set up the ds. And so here's what happens. The area of a section of the sphere, so let's see. We're going to start at some starting place x1, and end at some place x2. So what does that
look like? Here's the sphere. And we're starting at a place x1. And we're ending at a place x2. And we're taking more or less the slice here, if you like. The section of this sphere. So the area's
going to equal this. And what is it going to be? Well, so I have here 2 pi y. I'll write it out, just leave it as y for now. And then I have ds. So that's always what the formula is when you're
revolving around the x axis. And then I'll plug in for those things. So 2 pi, the formula for y is square root a ^2 - x ^2. And the formula for ds, well, it's the square root of this times dx. So
it's the square root of a ^2 / a ^2 - x ^2 dx. So this part is ds. And this part is y.
And now, I claim we have a nice cancellation that takes place. Square root of a ^2 = a. And then there's another good cancellation. As you can see. Now, what we get here is the integral from x1 to
x2, of 2 pi a dx, which is about the easiest integral you can imagine. It's just the integral of a constant. So it's 2 pi a ( x2 - x1). Let's check this in a couple of examples. And then see what
it's saying geometrically, a little bit.
So what this is saying, so special cases that you should always check when you have a nice formula like this, at least. But really with anything in order to make sure that you've got the right
answer. If you take, for example, the hemisphere. So you take 1/2 of this sphere. So that would be starting at 0, sorry. And ending at a. So that's the integral from to a. So this is the case x1 = 0.
x2 = a. And what you're going to get is a hemisphere. And the area is (2 pi a ) a. Or in other words, 2 pi a ^2. And if you take the whole sphere, that's starting at x1 = - a, and x2 = a, you're
getting (2 pi a) ( a - (- a)). Which is 4 pi a ^2. That's the whole thing. Yeah, question.
PROFESSOR: The question is, would it be possible to rotate around the y axis? And the answer is yes. It's legal to rotate around the y axis. And there is, if you use vertical slices as we did here,
that is, well they're sort of tips of slices, it's a different idea. But anyway, it's using dx as the integral of the variable of integration. So we're checking each little piece, each little strip
of that type. If we use dx here, we get this. If you did the same thing rotated the other way, and use dy as the variable, you get exactly the same answer. And it would be the same calculation.
Because they're parallel. So you're, yep.
PROFESSOR: Can you do service area with shells? Well, ah shell shape. The short answer is not quite. The shell shape is a vertical shell which is itself already three-dimensional, and it has a
thickness. So this is just a matter of terminology, though. This thickness is this dx, when we do this rotation here. And then there are two other dimensions. If we have a curved surface, there's no
other dimension left to form a shell. But basically, you can chop things up into any bits that you can actually measure. That you can figure out what the area is. That's the main point.
Now, I said we were going to, we've just launched into three-dimensional space. And I want to now move on to other space-like phenomena. But we're going to do this. So this is also a preparation for
18.02, where you'll be doing this a tremendous amount. We're going to talk now about parametric equations. Really just parametric curves. So you're going to see this now and we're going to interpret
it a couple of times, and we're going to think about polar coordinates. These are all preparation for thinking in more variables, and thinking in a different way than you've been thinking before. So
I want you to prepare your brain to make a transition here. This is the beginning of the transition to multivariable thinking.
We're going to consider curves like this. Which are described with x being a function of t and y being a function of t. And this letter t is called the parameter. In this case you should think of it,
the easiest way to think of it is as time. And what you have is what's called a trajectory. So this is also called a trajectory. And its location, let's say, at time 0, is this location here. Of (0,
y ( 0)), that's a point in the plane. And then over here, for instance, maybe it's (x ( 1), y (1)). And I drew arrows along here to indicate that we're going from this place over to that place. These
are later times. t = 1 is a later time than t = 0. So that's just a very casual, it's just the way we use these notations.
Now let me give you the first example, which is x = a cos t, y = a sin t. And the first thing to figure out is what kind of curve this is. And to do that, we want to figure out what equation it
satisfies in rectangular coordinates. So to figure out what curve this is, we recognize that if we square and add. So we add x ^2 to y^2, we're going to get something very nice and clean. We're going
to get a^2 cos ^2 t + a ^2 sin ^2 t. Yeah that's right, OK. Which is just a ^2 (cos^2 + sin ^2), or in other words a^2. So lo and behold, what we have is a circle. And then we know what shape this is
And the other thing I'd like to keep track of is which direction we're going on the circle. Because there's more to this parameter then just the shape. There's also where we are at what time. This
would be, think of it like the trajectory of a planet. So here, I have to do this by plotting the picture and figuring out what happens. So at t = 0, we have (x, y) is equal to, plug in here (a cos
0, a sin 0). Which is just a * 1 + a * 0, so a0. And that's here. That's the point (a, 0). We know that it's the circle of radius a. So we know that the curve is going to go around like this somehow.
So let's see what happens at t = pi / 2. So at that point, we have (x,y) = ( a cos pi / 2, a sin pi / 2). Which is (0, a), because sine of pi / 2 = 1. So that's up here. So this is what happens at t
= 0. This is what happens at t = pi / 2. And the trajectory clearly goes this way. In fact, this turns out to be t = pi, etc. And it repeats at t = 2 pi. So the other feature that we have, which is
qualitative feature, is that it's counterclockwise. No the last little bit is going to be the arc length. Keeping track of the arc length. And we'll do that next time. | {"url":"http://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2006/video-lectures/lecture-31-parametric-equations/","timestamp":"2014-04-19T17:03:45Z","content_type":null,"content_length":"78732","record_id":"<urn:uuid:a8a798b2-4c6d-419d-b1db-d6303d225ce5>","cc-path":"CC-MAIN-2014-15/segments/1398223207046.13/warc/CC-MAIN-20140423032007-00262-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Sampling Standard Deviation
Suppose that the average weekly earnings for employees in general automotive repair shops is $450, and that the standard deviation for the weekly earnings for such employees is $50. A sample of 100
such employees is selected at random.
Find the standard deviation of the sampling distribution of the means of average weekly earnings for samples of size 100.
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Suppose xxxx xxx average xxxxxx earnings xxx xxxxxxxxx xx general xxxxxxxxxx xxxxxx shops is xxxxx xxx xxxx xxx xxxxxxxx deviation xxx xxx xxxxxx earnings for such xxxxxxxxx is $50. x sample of xxx
xxxx employees xx selected xx random.
xxxx the standard xxxxxxxxx of xxx sampling distribution of the xxxxx of xxxxxxx xxxxxx earnings for xxxxxxx of size xxxx
xxx mean xx xxx sampling distribution of the average xxxxxx earnings = 450
xxx xxxxxxxx xxxxxxxxx of xxx xxxxxxxx distribution of the average xxxxxx earnings x xx x xxxxxxxxx x 5
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xxxxxxx xxxx the xxxxxxx xxxxxx earnings xxx employees xx general automotive xxxxxx xxxxx is xxxxx and that the xxxxxxxx xxxxxxxxx for the weekly earnings for such xxxxxxxxx is $50. A sample of 100
xxxx employees is selected xx xxxxxxx
xxxx xxx xxxxxxxx xxxxxxxxx xx xxx sampling xxxxxxxxxxxx xx xxx means xx xxxxxxx xxxxxx xxxxxxxx xxx xxxxxxx of xxxx xxxx
The mean xx the xxxxxxxx xxxxxxxxxxxx xx xxx xxxxxxx weekly xxxxxxxx x xxx
The standard xxxxxxxxx of xxx sampling xxxxxxxxxxxx of the xxxxxxx xxxxxx xxxxxxxx x 50 x xxxxxxxxx = 5
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Check plagiarism for $2.00 | {"url":"http://www.homeworkmarket.com/content/sampling-standard-deviation","timestamp":"2014-04-20T11:29:07Z","content_type":null,"content_length":"58298","record_id":"<urn:uuid:6d37245e-1483-4a63-9b16-015e5ecab067>","cc-path":"CC-MAIN-2014-15/segments/1398223206147.1/warc/CC-MAIN-20140423032006-00593-ip-10-147-4-33.ec2.internal.warc.gz"} |
the first resource for mathematics
Single-machine scheduling to minimize total convex resource consumption with a constraint on total weighted flow time.
(English) Zbl 1251.90197
Summary: In this paper, we consider single-machine scheduling problem in which processing time of a job is described by a convex decreasing resource consumption function. The objective is to minimize
the total amount of resource consumed subject to a constraint on total weighted flow time. The optimal resource allocation is obtained for any arbitrary job sequence. The computational complexity of
the general problem remains an open question, but we present and analyze some special cases that are solvable by using polynomial time algorithms. For the general problem, several dominance
properties and some lower bounds are derived, which are used to speed up the elimination process of a branch-and-bound algorithm proposed to solve the problem. A heuristic algorithm is also proposed,
which is shown by computational experiments to perform effectively and efficiently in obtaining near-optimal solutions. The results show that the average percentage error of the proposed heuristic
algorithm from optimal solutions is less than 3%.
90B35 Scheduling theory, deterministic
[1] Elmaghraby, S. E.: Activity network, (1977)
[2] Kaspi, M.; Shabtay, D.: Convex resource allocation for minimizing the makespan in a single machine with job release dates, Computers and operations research 31, 1481-1489 (2004) · Zbl 1076.68018
· doi:10.1016/S0305-0548(03)00103-5
[3] Leyvand, Y.; Shabtay, D.; Steiner, G.; Yedidsion, L.: Just-in-time scheduling with controllable processing times on parallel machines, Journal of combinatorial optimization 19, 347-368 (2010) ·
Zbl 1188.90100 · doi:10.1007/s10878-009-9270-5
[4] Monma, C. L.; Schrijver, A.; Todd, M. J.; Wei, V. K.: Convex resource allocation problems on directed acyclic graphs: duality, complexity, special cases and extensions, Mathematics of operations
research 15, 736-748 (1990) · Zbl 0717.90080 · doi:10.1287/moor.15.4.736
[5] Nowicki, E.; Zdrzalka, S.: A survey of results for sequencing problems with controllable processing times, Discrete applied mathematics 26, 271-287 (1990) · Zbl 0693.90056 · doi:10.1016/0166-218X
[6] Shabtay, D.; Itskovich, Y.; Yedidsion, L.; Oron, D.: Optimal due date assignment and resource allocation in a group technology scheduling environment, Computers and operations research 37,
2218-2228 (2010) · Zbl 1231.90214 · doi:10.1016/j.cor.2010.03.012
[7] Shabtay, D.; Kaspi, M.: Minimizing the total weighted flow time in a single machine with controllable processing times, Computers and operations research 31, 2279-2289 (2004) · Zbl 1073.90017 ·
[8] Shabtay, D.; Kaspi, M.: Parallel machine scheduling with a convex resource consumption function, European journal of operational research 173, 92-107 (2006) · Zbl 1125.90023 · doi:10.1016/
[9] Shabtay, D.; Steiner, G.: A survey of scheduling with controllable processing times, Discrete applied mathematics 155, 1643-1666 (2007) · Zbl 1119.90022 · doi:10.1016/j.dam.2007.02.003 | {"url":"http://zbmath.org/?q=an:1251.90197","timestamp":"2014-04-21T15:18:34Z","content_type":null,"content_length":"24661","record_id":"<urn:uuid:50c8ebd7-09c0-4e26-9105-50d9ba934250>","cc-path":"CC-MAIN-2014-15/segments/1398223203841.5/warc/CC-MAIN-20140423032003-00417-ip-10-147-4-33.ec2.internal.warc.gz"} |
Maximizing a volume
July 8th 2009, 01:01 PM #1
Maximizing a volume
I find the following problem very hard!
A rectangular box has 3 faces in the planes given by $x=0$, $z=0$ and $y=0$. The vertex that is not in the latter planes is in the plane $4x+3y+z=36$. Determine the dimensions that maximizes the
volume of the box.
I tried to visualize the box by drawing a sketch and I think I couldn't. So I'm stuck on giving an expression of the volume of the box. I'd like some help for this task.
Precisely I've drew a rectangular parallelepiped in the xyz plane but I don't see how it could have a maximum volume so I think I drew it wrongly.
We'll assume that the box is suppose to lie in the first octant. This seems like a reasonable starting point.
If the vertex in the plane $4x+3y+z=36$ has coordinates $(x,y,z)$ then the volume of the box is $xyz$.
So the problem boils down to maximising the quantity $xyz$ subject to the conditions that $x$, $y$, $z$ are positive and $4x+3y+z=36$.
Now try the AM-GM inequality, namely $\frac{4x+3y+z}3\geq\root 3\of{4x.3y.z}$ with equality iff $4x=3y=z$.
You should now be able to deduce that the maximum volume is 144. Happy hunting.
Ok thanks a lot. I'll try it.
Err... I would have never thought to suppose that the box has to lie in the first octant. I hope the question will be clearer in the exam.
Instead of using the inequality, I used Lagrange's multipliers method. I found $x=3$, $y=4$ and $z=12$, hence $V=144$ as you pointed out.
I still have some difficulties to see why we can suppose the vertex to have the coordinates $(x,y,z)$. I don't know if I could solve a similar problem. (Hopefully yes).
July 8th 2009, 01:57 PM #2
Mar 2009
July 8th 2009, 07:08 PM #3
July 10th 2009, 01:52 PM #4 | {"url":"http://mathhelpforum.com/calculus/94662-maximizing-volume.html","timestamp":"2014-04-18T00:31:29Z","content_type":null,"content_length":"41199","record_id":"<urn:uuid:24ebc313-daa1-400e-8373-749fa8dc4219>","cc-path":"CC-MAIN-2014-15/segments/1397609532374.24/warc/CC-MAIN-20140416005212-00609-ip-10-147-4-33.ec2.internal.warc.gz"} |
volume formula [Archive] - Car Audio Forum - CarAudio.com
04-23-2003, 11:52 PM
whats the formula for finding the cubic feet of a box??
Randy Savage
04-23-2003, 11:58 PM
Originally posted by phazeone
whats the formula for finding the cubic feet of a box??
LxWxH/1728- outside measurements
For inside, its (L-2T)x(W-2T)x(H-2T)/1728
T= thickness of wood. | {"url":"http://www.caraudio.com/forums/archive/index.php/t-29376.html","timestamp":"2014-04-17T22:39:19Z","content_type":null,"content_length":"3235","record_id":"<urn:uuid:dd3fcbc3-a9c7-4779-ba20-871f5d8d9507>","cc-path":"CC-MAIN-2014-15/segments/1397609532128.44/warc/CC-MAIN-20140416005212-00634-ip-10-147-4-33.ec2.internal.warc.gz"} |
Syllabus - First Year Computer Science and Engineering - SNDT University, Mumbai
Detailed syllabus and recommended books. Links to appropriate online articles and videos will be given. This blog will also contain number of articles and videos.
Computer Science and Engineering - Knol Books - Catalogue
Sub-Directories of Articles/Knols in the Area of Computer Science and Engineering
Knol Sub-Directory - Computer Science, Engineering and Technology - Subjects
Knol Sub-Directory - New Knols - Computer Science, Engineering and Technology
Syllabus - May 2011
Applied Mathematics - I
Semester: I Lect: 4Hr
Branch: ENC / CST / IT Credit: 04
1. MATRICES
Types of Matrices. Adjoint of a matrix, Inverse of a matrix. Elementary transformations. Rank of a matrix. Reduction to a normal form. Partitioning of matrices. System of homogeneous and non –
homogeneous equations, their consistency and solutions. Linear dependence and independence of rows and columns of a matrix area in a real field. Eigen values and Eigen vectors. Cayley Hamilton
theoram, Minimal Polynominal – Derogatory and non derogatory matrices. Applications in Engg.
Differential equation of 1st order and 1st degree, Linear – equations. Bernoulli’s equations. Differential equation exact differential equations – integrating factors. Differential equations of
higher order. Differential operator D, Where f (D) y = X, {x = eax , sin(ax+b), Cos(ax+b), xm , eax f(x). Linear differential equations with constant and variable coefficients. (Cauchy Linear
Equations and Legendre’s Liner equations). Simple applications (Where the differential equation is given). Applications in Engg.
Successive differentiation, Leibnitz’s theorem ( without proof ) and applications, Rolle’s theorem, Lagrange’s and Cauchy’s Mean value theorem. Applications in Engg.
4. COMPLEX NUMBERS
Definition of complex numbers Cartesian, Polar and exponential form, De–Moiver’s theorem and roots of complex numbers. Hyperbolic functions Separation real and imaginary parts of circular &
Hyperbolic functions. Logarithm of complex numbers. Applications in Engg.
Reference Books:
1. P.N.Wartikar & J. N. Wartikar, Elements of Applied Mathematics, 1st edition, Pune Vidyarthi Griha Prakashan, 1995. (rs. 110/-)
2. B. S. Grewal, Higher Engineering Mathematics, 34th edition, Khanna Publishers, 1998. (Rs. 170).
3. Shanti Narayan, Matrices, 9th Edition, S. Chand, 1997. (Rs. 45/-)
4. Shanti Narayan, Differential Calculus, 14th Edition, S. Chand, 1996. (Rs. 60/-)
5. A. R. Vashishtha, Matrices, 27th Edition, Krishna Prakashan Mesdia(P) Ltd; 1996. (RS. 75/-)
6. Edwin Kreyszig, Advance Engg. Mathematics, 5th Edition, New Age International (P) Ltd; 1997. (Rs. 295/-)
Applied Mathematics - II
Semester: II
Lect: 4 Hr
Branch:ENC/CST/IT Credit: 04
Partial Differentiation: Definition, differentiation of composite and implicit functions, Euler’s theorem on Homogeneous functions, total differentiation of composite functions using partial
differentiation, errors and approximation, extreme values of functions of two variables, applications in engineering.
Vector Algebra And Vector Calcus: Product of three or more vectors, vector differentiation – rules and theorems on vector differentiation, scalar point functions and vector point function, gradient,
divergent and curl and applications solenoidal and irrotational fields, scalar potential of irrotational vectors, applications in engineering.
Differentiation Under Integral Sign: Theorems on differentiation under integral sign (without proof), Applications in engineering.
Integral Calculus: Curve tracing (only standard curves) Rectification (only arc length), double Integrals – Change of order of integration, double integration of polar coordinates, application of
single and double integration – mass and volume, triple integration, applications in engineering.
Error Functions – Beta And Gamma Functions: Error functions and its properties, simple problems based on it, beta and gamma functions, properties, relation between beta & gamma functions, duplication
formula and problems based on it, applications in engineering.
1. P.N.Wartikar & J. N. Wartikar, Elements of Applied Mathematics, 1st edition, Pune Vidyarthi Griha Prakashan, 1995. (rs. 110/-)
2. B. S. Grewal, Higher Engineering Mathematics, 34th edition, Khanna Publishers, 1998. (Rs. 170).
3. Shanti Narayan, Differential Calculus, 14th Edition, S. Chand, 1996. (Rs. 60/-)
4. Murry Spiega, Vector Analysis
5. Edwin Kreyszig, Advance Engg. Mathematics, 5th Edition, New Age
6. International (P) Ltd; 1997. (Rs. 295/-)
Applied Science – I
Semester: II Lect: 4 Hr
Branch:ENC/CST/IT Credit: 04
Section – I (Physics)
Physics of Semiconductors
Introduction to band theory, metals, semiconductors and insulators; charge carriers in semiconductors; conductivity and mobility of charge carriers; concepts of fermi level; fermi level in Intrinsic
and Entrinsic semiconductors; semiconductor junction diodes.
Introduction to Fiber Optic Communication
i] Propagation of light in an optical fiber; TIR, Angle of Acceptance; Numerical Aperture; Index Difference; Types of Fibers i) Step Index Fiber ii) Graded Index Fiber; Advantages of Optical Fiber,
Applications of Optical Fiber Communication System.
ii] Optical Sources
Introduction to Lasers; Terms Associated with Lasers; Theory of Ruby Lasers; He-Ne Laser, LED, Semiconductor Lasers.
iii] Photo Detectors
Minority Charge Carrier Injection, Photo Diodes – p-n, p-in Avalanche.
Characteristics of U. S. Waves, Magnetosrictive effect, Magnetosrictive Transducer, Piezoelectric effect, Piezo Quartz Crystal and transducer Applications of U. S. Waves – i) High power applications
such as ultrasonic cleaners and cavitation ii) Low power applications such as Non Destructive Testing Methods – flaw detectors, Ultrasonic Thickness Gauges, Sonar’s etc.
Super Conductors
Properties Characterizing Superconductors; Implications of Zero resistivity, Critical temp-Tc, critical magnetic field – Hc, Critical current Ic, Meissner effect, Penetration depth;
Types of superconductors; London’s equation; B.C.S. Theory, Josephson’s Effect and junctions, SQUID, Applications of Superconductors.
Introduction to Electromagnetic
Laws of Physics such as Gauss’s Law, Ampere’s Circuital Law, Solenoidal vector B, Faraday – Lenz’s Law expressed in terms of Maxwell’s equations, Modified form of Ampere’s Law.
1. R. K. Gaur, and S. L. Gupta, Engineering Physics, 7th Edition, Dhanpat Rai Publication Pvt. Ltd., 1997. (165/-)
2. B. L. Theraja, Modern Physics., S. Chand and Company Ltd., 1996. (Rs. 60/-)
3. S. G. Patgawkar, Applied Physics – I, 5th Edition, Technova Publication, 1999. (Rs. 75/-)
4. Arthur Beiser, Perspective of Modern Physics, McGraw Hill, 1997. (Rs. 400/-)
5. Charles Kittle, Solid State Physics, 7th Edition, John Wiley & Sons, 1996. (Rs. 254/-)
6. I. Wilson and J. F. B. Hawkes, Optoelectronics – An Introduction, 2nd Edition, PHI, 1999. (175/-)
Section – II (Chemistry)
7. Phase Rule
Phase Rule, Water System, Sulphur System, Phase Rule for two Component Alloy Systems, Eutectic System, Bismuth – Cadmium Eutectic System, Lead – Silver System – Simple Eutectic Formation.
8. Electrochemistry, Specific, Equivalent and Molar Conductance
Introduction, Kohlrausch’s Law of Independent Migration of Ions, Laws of Electrolysis, Transport Number, Conductometric Titration.
9. Spectroscopy
Electromagnetic radiation, Spectroscopy, Principle, Instrumentation and Applications of Microwave, IR and UV Visible Spectroscopy, Beer Lamber’s Law.
10. Atomic Structure & Atomic Properties
Rutherford’s Modes, Bohr’s Model, Aufbau’s Principle, Pauli’s Law, Hund’s Rule, Electronic Configuration Atomic Properties like Ionization potential electro negativity, electron affinity, Atomic
size, oxidation potential.
1. Glasstone Lewis, Physical Chemistry
2. C. N. Banwell, Fundamentals of Molecular Spectroscopy, 3rd Edition, Tata McGraw Hill, 1992. (Rs. 69/-)
3. Anand and Chatwal, Instrumental Methods of Analysis, Himalaya Publishing House, 1997. (Rs. 160/-)
Communication Skills
Semester: I Lect: 4 Hr
Branch: ENC/ CST/ IT Credit: 04
1. Communication
The process, channels and media, Oral and written communication,
Verbal and non-verbal communication, Body language,
Barriers to communication , Developing communication through techniques.
2. Writing Skills
Vocabulary building- use of synonyms, antonyms, homonyms, homophones,word formation, confused set of words.
Writing effective paragraphs-through illustration, example, argument, analysis,description and comparison, expansion of key sentences.
Business correspondence-Principles of correspondence,Form,Formats,
Types of letters-Application with bio-data, enquiries, replies to enquiries, claims, adjustments, sales.
3. Summarising Techniques
One word substitutes( noun, verb,adverb, adjective)
Reduction of sentence length, Reduction of paragraph length,
Paraphrasing longer units.
4. Oral Communication Practice
Group discussion, Extempore speaking- introducing a speaker,
introducing a topic, vote of thanks, offering condolence, making
an announcement, speech on given topic, oral instructions.
5. Meeting Documentation
Notices, Circulars,Agendas,Minutes of meetings
6. Report Writing
Basics-What is a report, Qualities of a good report,
Style of language in reports,Methods,Sequencing, Structures
Types of reports-analytical, feasibility, informative etc.
Non-formal short reports-letter reports, memorandum reports
7. Descriptive Writing
Simple description of an object often used by engineering students
Writing instructions on using an object or performing a process
Reference Books
1. Sushil Bahl, “Business Communication Today”, Response Books, 1996, Rs.125/-
2. Krishna Mohan, R.C. Sharma, “Business Correspondence and Report Writing”, 2nd ed., Tata McGraw Hill, 1997, Rs.110/-
3. Krishna Mohan, Meera Banerji, “Developing Communication Skills”, McMillan & India Ltd., 1997, Rs.88/-
4. E.H.Macgraw, “Basic Managerial Skills For All”, 4th ed., PHI, 1996, Rs.125/-
Basic Electronics
Semester: II Lect: 4 Hr
Branch: ENC/ CST/ IT Credit: 04
Modeling devices: Static characteristics of ideal two terminal and three terminal devices, small signal models of non-linear devices.
Semiconductor diodes, construction and characteristics, Static and dynamic resistance, temperature effects, Avalanche and zener diodes. Small signal models of diodes; some applications of diodes.
Specification of diodes, rectifiers ripple factor, rectification efficiency, regulation, and filters.
Bipolar junction transistor: Construction, characteristics. BJT as amplifier, CB, CE, CC configurations. Biasing circuits, dc analysis and stability factor, DC load line and ac load line.
Single stage transistor amplifiers (CB, CC, and CE). h-parameters, Small signal low frequency ac equivalent circuit, h parameter measurements
FET:- Construction, characteristics, amplifier. CS, CD and CG configurations. Biasing. Low frequency small signal ac equivalent circuit of JFET amplifiers.
Text Books / Reference Books
1. Boylstead & Nshelasky, “Electronic Devices & Circuit”, 6th edition, PHI. (Rs.295/-)
2. Milman Grabel, “Microelectronics”
3. V. K. Mehata, Principles of Electronics”, 7th edition. (Rs.210/-)
4. Bhargav Gupta, “Basic Electronics & Linear Circuit”. (Rs.120/-)
5. Kakani – Bhandari, “A Textbook of Electronics”.
2009-10 First Semester Subjects
AS-1 Chemistry
AS-2 Physics
CS - Computer Skills
EC - Electrical Circuits
ED - Engineering Drawing
EM-1 Engineering Mathematics
EW - Electronics and Mechanical Workshop
Applied Mathematics - I
Semester: I Lect: 4Hr
Branch: ENC / CST / IT Credit: 04
1. MATRICES
Types of Matrices. Adjoint of a matrix, Inverse of a matrix. Elementary transformations. Rank of a matrix. Reduction to a normal form. Partitioning of matrices. System of homogeneous and non –
homogeneous equations, their consistency and solutions. Linear dependence and independence of rows and columns of a matrix area in a real field. Eigen values and Eigen vectors. Cayley Hamilton
theoram, Minimal Polynominal – Derogatory and non derogatory matrices. Applications in Engg.
Differential equation of 1st order and 1st degree, Linear – equations. Bernoulli’s equations. Differential equation exact differential equations – integrating factors. Differential equations of
higher order. Differential operator D, Where f (D) y = X, {x = eax , sin(ax+b), Cos(ax+b), xm , eax f(x). Linear differential equations with constant and variable coefficients. (Cauchy Linear
Equations and Legendre’s Liner equations). Simple applications (Where the differential equation is given). Applications in Engg.
Successive differentiation, Leibnitz’s theorem ( without proof ) and applications, Rolle’s theorem, Lagrange’s and Cauchy’s Mean value theorem. Applications in Engg.
4. COMPLEX NUMBERS
Definition of complex numbers Cartesian, Polar and exponential form, De–Moiver’s theorem and roots of complex numbers. Hyperbolic functions Separation real and imaginary parts of circular &
Hyperbolic functions. Logarithm of complex numbers. Applications in Engg.
Reference Books:
1. P.N.Wartikar & J. N. Wartikar, Elements of Applied Mathematics, 1st edition, Pune Vidyarthi Griha Prakashan, 1995. (rs. 110/-)
2. B. S. Grewal, Higher Engineering Mathematics, 34th edition, Khanna Publishers, 1998. (Rs. 170).
3. Shanti Narayan, Matrices, 9th Edition, S. Chand, 1997. (Rs. 45/-)
4. Shanti Narayan, Differential Calculus, 14th Edition, S. Chand, 1996. (Rs. 60/-)
5. A. R. Vashishtha, Matrices, 27th Edition, Krishna Prakashan Mesdia(P) Ltd; 1996. (RS. 75/-)
6. Edwin Kreyszig, Advance Engg. Mathematics, 5th Edition, New Age International (P) Ltd; 1997. (Rs. 295/-)
Applied Mathematics - II
Semester: II
Lect: 4 Hr
Branch:ENC/CST/IT Credit: 04
Partial Differentiation: Definition, differentiation of composite and implicit functions, Euler’s theorem on Homogeneous functions, total differentiation of composite functions using partial
differentiation, errors and approximation, extreme values of functions of two variables, applications in engineering.
Vector Algebra And Vector Calcus: Product of three or more vectors, vector differentiation – rules and theorems on vector differentiation, scalar point functions and vector point function, gradient,
divergent and curl and applications solenoidal and irrotational fields, scalar potential of irrotational vectors, applications in engineering.
Differentiation Under Integral Sign: Theorems on differentiation under integral sign (without proof), Applications in engineering.
Integral Calculus: Curve tracing (only standard curves) Rectification (only arc length), double Integrals – Change of order of integration, double integration of polar coordinates, application of
single and double integration – mass and volume, triple integration, applications in engineering.
Error Functions – Beta And Gamma Functions: Error functions and its properties, simple problems based on it, beta and gamma functions, properties, relation between beta & gamma functions, duplication
formula and problems based on it, applications in engineering.
1. P.N.Wartikar & J. N. Wartikar, Elements of Applied Mathematics, 1st edition, Pune Vidyarthi Griha Prakashan, 1995. (rs. 110/-)
2. B. S. Grewal, Higher Engineering Mathematics, 34th edition, Khanna Publishers, 1998. (Rs. 170).
3. Shanti Narayan, Differential Calculus, 14th Edition, S. Chand, 1996. (Rs. 60/-)
4. Murry Spiega, Vector Analysis
5. Edwin Kreyszig, Advance Engg. Mathematics, 5th Edition, New Age
6. International (P) Ltd; 1997. (Rs. 295/-)
Applied Science – I
Semester: II Lect: 4 Hr
Credit: 04
Section – I (Physics)
Physics of Semiconductors
Introduction to band theory, metals, semiconductors and insulators; charge carriers in semiconductors; conductivity and mobility of charge carriers; concepts of fermi level; fermi level in Intrinsic
and Entrinsic semiconductors; semiconductor junction diodes.
Introduction to Fiber Optic Communication
i] Propagation of light in an optical fiber; TIR, Angle of Acceptance; Numerical Aperture; Index Difference; Types of Fibers i) Step Index Fiber ii) Graded Index Fiber; Advantages of Optical Fiber,
Applications of Optical Fiber Communication System.
ii] Optical Sources
Introduction to Lasers; Terms Associated with Lasers; Theory of Ruby Lasers; He-Ne Laser, LED, Semiconductor Lasers.
iii] Photo Detectors
Minority Charge Carrier Injection, Photo Diodes – p-n, p-in Avalanche.
Characteristics of U. S. Waves, Magnetosrictive effect, Magnetosrictive Transducer, Piezoelectric effect, Piezo Quartz Crystal and transducer Applications of U. S. Waves – i) High power applications
such as ultrasonic cleaners and cavitation ii) Low power applications such as Non Destructive Testing Methods – flaw detectors, Ultrasonic Thickness Gauges, Sonar’s etc.
Super Conductors
Properties Characterizing Superconductors; Implications of Zero resistivity, Critical temp-Tc, critical magnetic field – Hc, Critical current Ic, Meissner effect, Penetration depth;
Types of superconductors; London’s equation; B.C.S. Theory, Josephson’s Effect and junctions, SQUID, Applications of Superconductors.
Introduction to Electromagnetic
Laws of Physics such as Gauss’s Law, Ampere’s Circuital Law, Solenoidal vector B, Faraday – Lenz’s Law expressed in terms of Maxwell’s equations, Modified form of Ampere’s Law.
1. R. K. Gaur, and S. L. Gupta, Engineering Physics, 7th Edition, Dhanpat Rai Publication Pvt. Ltd., 1997. (165/-)
2. B. L. Theraja, Modern Physics., S. Chand and Company Ltd., 1996. (Rs. 60/-)
3. S. G. Patgawkar, Applied Physics – I, 5th Edition, Technova Publication, 1999. (Rs. 75/-)
4. Arthur Beiser, Perspective of Modern Physics, McGraw Hill, 1997. (Rs. 400/-)
5. Charles Kittle, Solid State Physics, 7th Edition, John Wiley & Sons, 1996. (Rs. 254/-)
6. I. Wilson and J. F. B. Hawkes, Optoelectronics – An Introduction, 2nd Edition, PHI, 1999. (175/-)
Section – II (Chemistry)
7. Phase Rule
Phase Rule, Water System, Sulphur System, Phase Rule for two Component Alloy Systems, Eutectic System, Bismuth – Cadmium Eutectic System, Lead – Silver System – Simple Eutectic Formation.
8. Electrochemistry, Specific, Equivalent and Molar Conductance
Introduction, Kohlrausch’s Law of Independent Migration of Ions, Laws of Electrolysis, Transport Number, Conductometric Titration.
9. Spectroscopy
Electromagnetic radiation, Spectroscopy, Principle, Instrumentation and Applications of Microwave, IR and UV Visible Spectroscopy, Beer Lamber’s Law.
10. Atomic Structure & Atomic Properties
Rutherford’s Modes, Bohr’s Model, Aufbau’s Principle, Pauli’s Law, Hund’s Rule, Electronic Configuration Atomic Properties like Ionization potential electro negativity, electron affinity, Atomic
size, oxidation potential.
1. Glasstone Lewis, Physical Chemistry
2. C. N. Banwell, Fundamentals of Molecular Spectroscopy, 3rd Edition, Tata McGraw Hill, 1992. (Rs. 69/-)
3. Anand and Chatwal, Instrumental Methods of Analysis, Himalaya Publishing House, 1997. (Rs. 160/-)
Communication Skills
Semester: I Lect: 4 Hr
Branch: ENC/ CST/ IT Credit: 04
1. Communication
The process, channels and media, Oral and written communication,
Verbal and non-verbal communication, Body language,
Barriers to communication , Developing communication through techniques.
2. Writing Skills
Vocabulary building- use of synonyms, antonyms, homonyms, homophones,word formation, confused set of words.
Writing effective paragraphs-through illustration, example, argument, analysis,description and comparison, expansion of key sentences.
Business correspondence-Principles of correspondence,Form,Formats,
Types of letters-Application with bio-data, enquiries, replies to enquiries, claims, adjustments, sales.
3. Summarising Techniques
One word substitutes( noun, verb,adverb, adjective)
Reduction of sentence length, Reduction of paragraph length,
Paraphrasing longer units.
4. Oral Communication Practice
Group discussion, Extempore speaking- introducing a speaker,
introducing a topic, vote of thanks, offering condolence, making
an announcement, speech on given topic, oral instructions.
5. Meeting Documentation
Notices, Circulars,Agendas,Minutes of meetings
6. Report Writing
Basics-What is a report, Qualities of a good report,
Style of language in reports,Methods,Sequencing, Structures
Types of reports-analytical, feasibility, informative etc.
Non-formal short reports-letter reports, memorandum reports
7. Descriptive Writing
Simple description of an object often used by engineering students
Writing instructions on using an object or performing a process
Reference Books
1. Sushil Bahl, “Business Communication Today”, Response Books, 1996, Rs.125/-
2. Krishna Mohan, R.C. Sharma, “Business Correspondence and Report Writing”, 2nd ed., Tata McGraw Hill, 1997, Rs.110/-
3. Krishna Mohan, Meera Banerji, “Developing Communication Skills”, McMillan & India Ltd., 1997, Rs.88/-
4. E.H.Macgraw, “Basic Managerial Skills For All”, 4th ed., PHI, 1996, Rs.125/-
Basic Electronics
Semester: II Lect: 4 Hr
Branch: ENC/ CST/ IT Credit: 04
Modeling devices: Static characteristics of ideal two terminal and three terminal devices, small signal models of non-linear devices.
Semiconductor diodes, construction and characteristics, Static and dynamic resistance, temperature effects, Avalanche and zener diodes. Small signal models of diodes; some applications of diodes.
Specification of diodes, rectifiers ripple factor, rectification efficiency, regulation, and filters.
Bipolar junction transistor: Construction, characteristics. BJT as amplifier, CB, CE, CC configurations. Biasing circuits, dc analysis and stability factor, DC load line and ac load line.
Single stage transistor amplifiers (CB, CC, and CE). h-parameters, Small signal low frequency ac equivalent circuit, h parameter measurements
FET:- Construction, characteristics, amplifier. CS, CD and CG configurations. Biasing. Low frequency small signal ac equivalent circuit of JFET amplifiers.
Text Books / Reference Books
1. Boylstead & Nshelasky, “Electronic Devices & Circuit”, 6th edition, PHI. (Rs.295/-)
2. Milman Grabel, “Microelectronics”
3. V. K. Mehata, Principles of Electronics”, 7th edition. (Rs.210/-)
4. Bhargav Gupta, “Basic Electronics & Linear Circuit”. (Rs.120/-)
5. Kakani – Bhandari, “A Textbook of Electronics”. | {"url":"http://knols-cse.blogspot.in/2012/02/syllabus-first-year-computer-science.html","timestamp":"2014-04-18T13:51:08Z","content_type":null,"content_length":"115104","record_id":"<urn:uuid:1939f442-af2b-4da5-a2c8-0126ddf1539b>","cc-path":"CC-MAIN-2014-15/segments/1397609533689.29/warc/CC-MAIN-20140416005213-00323-ip-10-147-4-33.ec2.internal.warc.gz"} |
Logic problem
06-12-2008 #1
Village id10t
Join Date
May 2008
Logic problem
Ok, i am busy learning recursion. I want a function that receives two int and add everyting from int a to int b. then it must return the sum
double Function(int Start, int End)
int sum=0;
Function(Start ,End);
else return sum;
I cant use sum because with each recursion sum gets int to zero. I dont want to pass a variable for sum to the function and i dont want to use a global variable... any suggestions for the noob?
Best way:
double foo(int begin, int end, double sum = 0)
if (begin == end)
return sum;
sum += begin++;
return foo(begin, end, sum);
Otherwise you have to mess with a static or global variable and this is not recommended.
For information on how to enable C++11 on your compiler, look here.
よく聞くがいい！私は天才だからね！ ^_^
Ok - so without using global varibales, this is the only way? what if I dont want to pass sum to the function, but only the 2 ints? is that possible?
It's possible via a static variable, but then you would have trouble because you have to reset that everytime. The static variables will retain its value over the function's entire lifetime.
However, by using an optional argument as the last, you don't have to pass in a third argument when calling the function, while the function will use the third argument itself when recursing.
For information on how to enable C++11 on your compiler, look here.
よく聞くがいい！私は天才だからね！ ^_^
What about
double foo(int start, int end)
if ( end < start )
return 0;
return start + foo(start + 1, end);
06-12-2008 #2
06-12-2008 #3
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06-12-2008 #4
06-12-2008 #5
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Pls I need help on these questions that are making me go crazy
September 25th, 2012, 12:49 PM #1
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1. Write a program in Java that takes as input two integers from the command-line, and
outputs true if they are divisible either by 7 or by 11, and outputs false otherwise.
2. Write a program in Java that takes as input a positive integer n from the command-line,
and outputs all the powers of 3 that are less than or equal to n. State the limitations, if
any, of your program.
3. Write a program in Java that computes the trigonometric function sin(x) given by the
sin(x) = x – x3/3! + x5/5! – x7/7! + · · · .
The program should take as input a number from the command-line, and output the sine
value of that number.
Welcome to the forums. While we would like to help, simply posting a homework assignment is what we call a 'homework dump', and makes it look as if you are asking someone to do this for you -
which is not what we are going to do. If you wish to make full use of the forums, I recommend posting what you have tried, and asking a specific question. I also recommend reading the forum rules
- I have had to move your post to a more appropiate forum because of this.
Code Tags | Java Tutorials | SSCCE | Getting Help | What Not To Do
Now I've got these codes for my number 3 but can I get someone to help me correct the error in my codes pls?
Here they are
import java.util.Scanner;
import java.Math;
public class trig
Scanner input = new Scanner(System.in);
public static void main(String[] args)
int angle, sineangle;
System.out.println("Enter the angle you wish to convert");
angle = input.nextInt();
sineangle = Math.sin(Math.toRadians(angle));
System.out.print("The sine is " + sineangle);
Last edited by helloworld922; September 27th, 2012 at 03:01 AM. Reason: please use [code] tags
What's wrong with the code?
Specifically, does it compile? If not, and you can't understand the compiler's messages, post them
Or does it compile but do something unintended when you run it? In that case describe the actual program behaviour.
[Edit] Also I think you may have misinterpreted the question. You aren't supposed to use the Math.sin() method, but rather you should use the series provided. Ie add up a whole bunch of terms in
1-x^3/3!+x^5/5!-... to get the sine.
Last edited by pbrockway2; September 27th, 2012 at 02:30 AM.
September 25th, 2012, 01:12 PM #2
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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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Math Help
how would you solve the problem if $n,p$ were any non-zero real numbers? (of course L'Hospital's Rule is not allowed!)
Last edited by simplependulum; December 30th 2009 at 06:53 PM.
here's a hint: we only need to show that $\forall a \in \mathbb{R} : \ \lim_{x\to1} \frac{x^a - 1}{x-1} = a.$ it's easy to prove it for rational numbers. now for any $n \in \mathbb{N}$ choose $q_n \
in \mathbb{Q}$ such that $q_n - \frac{1}{n} \leq a \leq q_n + \frac{1}{n}.$ then $\lim_{n\to\infty} q_n = a$ and ...
Yeah yeah, and then $\lim_{x\to1}\frac{x^a-1}{x^p-1}=\lim_{x\to1}\left\{\frac{x^a-1}{x-1}\cdot\frac{x-1}{x^p-1}\right\}=\frac{a}{p}$ By the way, was your point that if we can prove that the above
limit is true for any $a\in\mathbb{Q}$ and since any real number has a sequence of rational points which converges to it we are done? | {"url":"http://mathhelpforum.com/calculus/121579-nice-limit.html","timestamp":"2014-04-20T20:00:17Z","content_type":null,"content_length":"78731","record_id":"<urn:uuid:0640312f-5062-4e14-8486-74bf86bd3139>","cc-path":"CC-MAIN-2014-15/segments/1397609539066.13/warc/CC-MAIN-20140416005219-00016-ip-10-147-4-33.ec2.internal.warc.gz"} |
Math Help
October 21st 2009, 10:22 PM #1
Feb 2009
I like proving but this one, I got stuck because of time pressure. It was the last question and I was hurrying since I only got 5 mins left. I didn't get to finish my solution so I may have
gotten a few points off it.
here it goes:
1/cos^2B = [cosB sin2B - sinB cos2B]/cosB sinB
so what I did first was:
1/cos^2B = cos(B+2B)/cosB sinB
then I got stuck so I repeated it then came out with:
1/cos^2B = sinB - cosB/cosB sinB by subtracting the functions (idk if that's allowed) then I tried to break that down. I am still stuck.
anyone who can help me on how the sol'n goes??
Hello pathurr19
I like proving but this one, I got stuck because of time pressure. It was the last question and I was hurrying since I only got 5 mins left. I didn't get to finish my solution so I may have
gotten a few points off it.
here it goes:
1/cos^2B = [cosB sin2B - sinB cos2B]/cosB sinB
so what I did first was:
1/cos^2B = cos(B+2B)/cosB sinB
then I got stuck so I repeated it then came out with:
1/cos^2B = sinB - cosB/cosB sinB by subtracting the functions (idk if that's allowed) then I tried to break that down. I am still stuck.
anyone who can help me on how the sol'n goes??
I get a slightly different result.
$\frac{\cos B\sin2B-\sin B\cos2B}{\cos B \sin B}=\frac{\sin(2B-B)}{\cos B \sin B}$
$=\frac{\sin B}{\sin B\cos B }$
$=\frac{1}{\cos B }$
Last edited by Grandad; October 21st 2009 at 10:58 PM. Reason: Simpler method
Just like Grandad, I got different results.
$\frac{1}{cos^{2B}} = \frac{cosB sin2B - sinB cos2B}{cosB sinB}$
$\frac{1}{cos^{2B}} = \frac{cosB (2sinB cosB) - sinB (2cos^{2}B - 1)}{cosB sinB}$
$\frac{1}{cos^{2B}} = \frac{(2sinB cos^{2}B) - sinB (2cos^{2}B - 1)}{cosB sinB}$
$\frac{1}{cos^{2B}} = \frac{(2sinB cos^{2}B) - (2sinBcos^{2}B - sin B)}{cosB sinB}$
$\frac{1}{cos^{2B}} = \frac{sin B}{cosB sinB}$
$\frac{1}{cos^{2B}} = \frac{sin B}{cosB sinB}$
$\frac{1}{cos^{2B}} = \frac{1}{cosB}$
October 21st 2009, 10:46 PM #2
October 22nd 2009, 12:50 AM #3
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Physics Forums - View Single Post - Interpreting the Supernova Data
With lamba-CDM, I can put in some parameters and get a fit to get the lumpiness.
By the way, you can generate any function with a Fourier Series expansion. With only a few parameters, generally, you can get a pretty good fit. Your ability to put in parameters to an equation and
get an approximate fit uses a similar kind of math using spherical Bessel functions or some related idea. | {"url":"http://www.physicsforums.com/showpost.php?p=2877419&postcount=42","timestamp":"2014-04-17T18:31:30Z","content_type":null,"content_length":"7862","record_id":"<urn:uuid:159742ee-9827-4bb6-b7c0-56297d39886d>","cc-path":"CC-MAIN-2014-15/segments/1397609530895.48/warc/CC-MAIN-20140416005210-00214-ip-10-147-4-33.ec2.internal.warc.gz"} |
Reflections ( Read ) | Geometry
What if you noticed that a lake can act like a mirror in nature? Describe the line of reflection in the photo below. If this image were on the coordinate plane, what could the equation of the line of
reflection be? (There could be more than one correct answer, depending on where you place the origin.) After completing this Concept, you'll be able to answer this question.
Watch This
CK-12 Foundation: Chapter12ReflectionsA
Watch more about transformations and isometries by watching the last part of this video.
A transformation is an operation that moves, flips, or changes a figure to create a new figure. A rigid transformation is a transformation that preserves size and shape. The rigid transformations
are: translations, reflections (discussed here), and rotations. The new figure created by a transformation is called the image. The original figure is called the preimage. Another word for a rigid
transformation is an isometry. Rigid transformations are also called congruence transformations. If the preimage is $A$$A'$$A'$$A''$
A reflection is a transformation that turns a figure into its mirror image by flipping it over a line. Another way to describe a reflection is a “flip.” The line of reflection is the line that a
figure is reflected over. If a point is on the line of reflection then the image is the same as the original point.
Common Reflections
• Reflection over the $y-$ If $(x,y)$$y-$$(-x,y)$
• Reflection over the $x-$ If $(x,y)$$x-$$(x,-y)$
• Reflection over $x = a:$ If $(x, y)$$x = a$$(2a - x, y)$
• Reflection over $y = b:$ If $(x, y)$$y = b$$(x, 2b - y)$
• Reflection over $y = x$ If $(x, y)$$y = x$$(y, x)$
• Reflection over $y = -x$ If $(x, y)$$y = -x$$(-y, -x)$
Example A
Reflect the letter $''F''$$x-$
To reflect the letter $F$$x-$$x-$$y-$$x-$
Example B
Reflect $\triangle ABC$$y-$
To reflect $\triangle ABC$$y-$$y-$$x-$$y-$$y-$
$A(4,3) & \rightarrow A'(-4,3)\\B(7,-1) & \rightarrow B'(-7,-1)\\C(2,-2) & \rightarrow C'(-2,-2)$
Example C
Reflect the triangle $\triangle ABC$$A(4, 5), B(7, 1)$$C(9, 6)$$x = 5$
Notice that this vertical line is through our preimage. Therefore, the image’s vertices are the same distance away from $x = 5$$y-$$x = 0$$y-$
$A(4,5) & \rightarrow A'(6,5)\\B(7,1) & \rightarrow B'(3,1)\\C(9,6) & \rightarrow C'(1,6)$
Example D
Reflect square $ABCD$$y = x$
The purple line is $y = x$
$A(-1, 5) & \rightarrow A'(5, -1)\\B(0, 2) & \rightarrow B'(2, 0)\\C(-3, 1) & \rightarrow C'(1, -3)\\D(-4, 4) & \rightarrow D'(4, -4)$
Watch this video for help with the Examples above.
CK-12 Foundation: Chapter12ReflectionsB
Concept Problem Revisited
The white line in the picture is the line of reflection. This line coincides with the water’s edge. If we were to place this picture on the coordinate plane, the line of reflection would be any
horizontal line. One example could be the $x-$
A transformation is an operation that moves, flips, or otherwise changes a figure to create a new figure. A rigid transformation (also known as an isometry or congruence transformation) is a
transformation that does not change the size or shape of a figure. The new figure created by a transformation is called the image. The original figure is called the preimage. A reflection is a
transformation that turns a figure into its mirror image by flipping it over a line. The line of reflection is the line that a figure is reflected over.
Guided Practice
1. Reflect the line segment $\overline{PQ}$$P(-1, 5)$$Q(7, 8)$$y = 5$
2. A triangle $\triangle LMN$$\triangle L'M'N'$
3. Reflect the trapezoid TRAP over the line $y = -x$
1. Here, the line of reflection is on $P$$P'$$Q'$$x-$$Q$$y = 5$
$P(-1,5) & \rightarrow P'(-1,5)\\Q(7,8) & \rightarrow Q'(7,2)$
2. Looking at the graph, we see that the preimage and image intersect when $y = 1$
3. The purple line is $y = -x$
$T(2,2) & \rightarrow T'(-2, -2)\\R(4, 3) & \rightarrow R'(-3,-4)\\A(5, 1) & \rightarrow A'(-1,-5)\\P(1, -1) & \rightarrow P'(1,-1)$
1. Which letter is a reflection over a vertical line of the letter $b$
2. Which letter is a reflection over a horizontal line of the letter $b$
Reflect each shape over the given line.
3. $y-$
4. $x-$
5. $y = 3$
6. $x = -1$
Find the line of reflection of the blue triangle (preimage) and the red triangle (image).
Two Reflections The vertices of $\triangle ABC$$A(-5, 1), B(-3, 6)$$C(2, 3)$
10. Plot $\triangle ABC$
11. Reflect $\triangle ABC$$y = 1$$\triangle A'B'C'$
12. Reflect $\triangle A'B'C'$$y = -3$$\triangle A''B''C''$
13. What one transformation would be the same as this double reflection?
Two Reflections The vertices of $\triangle DEF$$D(6, -2), E(8, -4)$$F(3, -7)$
14. Plot $\triangle DEF$
15. Reflect $\triangle DEF$$x = 2$$\triangle D'E'F'$
16. Reflect $\triangle D'E'F'$$x = -4$$\triangle D''E''F''$
17. What one transformation would be the same as this double reflection?
Two Reflections The vertices of $\triangle GHI$$G(1, 1), H(5, 1)$$I(5, 4)$
18. Plot $\triangle GHI$
19. Reflect $\triangle GHI$$x-$$\triangle G'H'I'$
20. Reflect $\triangle G'H'I'$$y-$$\triangle G''H''I''$
21. What one transformation would be the same as this double reflection? | {"url":"http://www.ck12.org/geometry/Reflections/lesson/Reflections---Intermediate/","timestamp":"2014-04-21T02:11:17Z","content_type":null,"content_length":"138501","record_id":"<urn:uuid:0e2ae293-4ab8-4557-83ac-b1362b8af20d>","cc-path":"CC-MAIN-2014-15/segments/1397609539447.23/warc/CC-MAIN-20140416005219-00102-ip-10-147-4-33.ec2.internal.warc.gz"} |
Reachability analysis of Petri nets using symmetries
Results 1 - 10 of 33
, 1999
"... We describe a method for reducing the complexity of temporal logic model checking in systems composed of many parallel processes. The goal is to check properties of the components of a system
and then deduce global properties from these local properties. The main difficulty with this type of approac ..."
Cited by 2407 (62 self)
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We describe a method for reducing the complexity of temporal logic model checking in systems composed of many parallel processes. The goal is to check properties of the components of a system and
then deduce global properties from these local properties. The main difficulty with this type of approach is that local properties are often not preserved at the global level. We present a general
framework for using additional interface processes to model the environment for a component. These interface processes are typically much simpler than the full environment of the component. By
composing a component with its interface processes and then checking properties of this composition, we can guarantee that these properties will be preserved at the global level. We give two example
compositional systems based on the logic CTL*.
, 1996
"... A fundamental difficulty in automatic formal verification of finite-state systems is the state explosion problem -- even relatively simple systems can produce very large state spaces, causing
great difficulties for methods that rely on explicit state enumeration. We address the problem by exploiting ..."
Cited by 185 (8 self)
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A fundamental difficulty in automatic formal verification of finite-state systems is the state explosion problem -- even relatively simple systems can produce very large state spaces, causing great
difficulties for methods that rely on explicit state enumeration. We address the problem by exploiting structural symmetries in the description of the system to be verified. We make symmetries easy
to detect by introducing a new data type scalarset, a finite and unordered set, to our description language. The operations on scalarsets are restricted so that states are guaranteed to have the same
future behaviors, up to permutation of the elements of the scalarsets. Using the symmetries implied by scalarsets, a verifier can automatically generate a reduced state space, on the fly. We provide
a proof of the soundness of the new symmetry-based verification algorithm based on a definition of the formal semantics of a simple description language with scalarsets. The algorithm has been
implemented ...
, 1994
"... We show how to exploit symmetry in model checking for concurrent systems containing many identical or isomorphic components. We focus in particular on those composed of many isomorphic
processes. In many cases we are able to obtain significant, even exponential, savings in the complexity of model ch ..."
Cited by 166 (15 self)
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We show how to exploit symmetry in model checking for concurrent systems containing many identical or isomorphic components. We focus in particular on those composed of many isomorphic processes. In
many cases we are able to obtain significant, even exponential, savings in the complexity of model checking. 1 Introduction In this paper, we show how to exploit symmetry in model checking. We focus
on systems composed of many identical (isomorphic) processes. The global state transition graph M of such a system exhibits a great deal of symmetry, characterized by the group of graph automorphisms
of M. The basic idea underlying our method is to reduce model checking over the original structure M, to model checking over a smaller quotient structure M, where symmetric states are identified. In
the following paragraphs, we give a more detailed but still informal account of a "group-theoretic" approach to exploiting symmetry. More precisely, the symmetry of M is reflected in the group, Aut
"... PROD is a Pr/T-net reachability analysis tool that supports on-the-fly verification of linear time temporal properties with the aid of the stubborn set method. Branching time temporal properties
can be verified, too. ..."
Cited by 24 (1 self)
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PROD is a Pr/T-net reachability analysis tool that supports on-the-fly verification of linear time temporal properties with the aid of the stubborn set method. Branching time temporal properties can
be verified, too.
, 1997
"... In this paper we describe the use of symmetry for verification of transistor-level circuits by symbolic trajectory evaluation. We show that exploiting symmetry can allow one to verify systems
several orders of magnitude larger than otherwise possible. We classify symmetries in circuits as struct ..."
Cited by 23 (5 self)
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In this paper we describe the use of symmetry for verification of transistor-level circuits by symbolic trajectory evaluation. We show that exploiting symmetry can allow one to verify systems several
orders of magnitude larger than otherwise possible. We classify symmetries in circuits as structural symmetries, arising from similarities in circuit structure, data symmetries, arising from
similarities in the handling of data values, and mixed structural-data symmetries. We use graph isomorphism testing and symbolic simulation to verify the symmetries in the original circuit. Using
conservative approximations, we partition a circuit to expose the symmetries in its components, and construct reduced system models which can be verified efficiently. We have verified Static Random
Access Memory circuits with up to 1.5 Million transistors.
, 1996
"... As part of our continuing research on using Petri nets to support automated analysis of Ada tasking behavior, we have investigated the application of Petri net reduction for deadlock analysis.
Although reachability analysis is an important method to detect deadlocks, it is in general inefficient or ..."
Cited by 23 (6 self)
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As part of our continuing research on using Petri nets to support automated analysis of Ada tasking behavior, we have investigated the application of Petri net reduction for deadlock analysis.
Although reachability analysis is an important method to detect deadlocks, it is in general inefficient or even intractable. Net reduction can aid the analysis by reducing the size of the net while
preserving relevant properties. We introduce a number of reduction rules and show how they can be applied to Ada nets, which are automatically generated Petri net models of Ada tasking. We define a
reduction process and a method by which a useful description of a detected deadlock state can be obtained from the reduced net's information. A reduction tool and experimental results from applying
the reduction process are discussed.
, 1997
"... Coloured Petri nets are well suited to the modelling of symmetric systems. Model symmetries can be usefully exploited for the sake of analysis efficiency as well as for modelling convenience.
Cited by 23 (6 self)
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Coloured Petri nets are well suited to the modelling of symmetric systems. Model symmetries can be usefully exploited for the sake of analysis efficiency as well as for modelling convenience.
, 1993
"... Previously, we proposed a reduction technique [ID93] based on symmetries to alleviate the state explosion problem in automatic verification of concurrent systems. This paper describes the
results of testing the technique on a wide range of algorithms and protocols, including realistic multiprocessor ..."
Cited by 21 (6 self)
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Previously, we proposed a reduction technique [ID93] based on symmetries to alleviate the state explosion problem in automatic verification of concurrent systems. This paper describes the results of
testing the technique on a wide range of algorithms and protocols, including realistic multiprocessor synchronization algorithms and cache coherence protocols. Memory requirements were reduced by
amounts ranging from 83% to over 99%, and time requirements were often reduced as well. We also consider the effectiveness of the technique on different types of symmetries, such as symmetries in
identical system components and symmetries in data values.
, 2004
"... We describe a prototype extension of the Uppaal real-time model checking tool with symmetry reduction. The symmetric data type scalarset, which is also used in the Mur' model checker, was added
to Uppaal's system description language to support the easy static detection of symmetries. Our prototy ..."
Cited by 19 (4 self)
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We describe a prototype extension of the Uppaal real-time model checking tool with symmetry reduction. The symmetric data type scalarset, which is also used in the Mur' model checker, was added to
Uppaal's system description language to support the easy static detection of symmetries. Our prototype tool uses state swaps, described and proven sound earlier by Hendriks, to reduce the space and
memory consumption of Uppaal. Moreover, under certain assumptions the reduction strategy is canonical, which means that the symmetries are optimally used. For all examples that we experimented with
(both academic toy examples and industrial cases), we obtained a drastic reduction of both computation time and memory usage, exponential in the size of the scalar sets used.
- ACM TRANSACTIONS ON PROGRAMMING LANGUAGES AND SYSTEMS , 1998
"... This article describes a technique for analyzing relational specifications. The underlying idea is very simple. Both simulation and checking amount to finding models of a relational formula,
i.e., assignments for which the formula is true. For simulation the formula is the description of the operati ..."
Cited by 18 (10 self)
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This article describes a technique for analyzing relational specifications. The underlying idea is very simple. Both simulation and checking amount to finding models of a relational formula, i.e.,
assignments for which the formula is true. For simulation the formula is the description of the operation; for checking, the formula is the negation of an assertion about an operation. Models are
found by a generate-and-test strategy: the formula is repeatedly evaluated for a series of assignments until one is found for which the formula is true | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=124348","timestamp":"2014-04-18T11:40:17Z","content_type":null,"content_length":"36828","record_id":"<urn:uuid:27009a83-ef58-4b3c-8d7f-82943caf7153>","cc-path":"CC-MAIN-2014-15/segments/1397609533308.11/warc/CC-MAIN-20140416005213-00291-ip-10-147-4-33.ec2.internal.warc.gz"} |
MathGroup Archive: February 2009 [00253]
[Date Index] [Thread Index] [Author Index]
Re: FourierTransform
• To: mathgroup at smc.vnet.net
• Subject: [mg96190] Re: FourierTransform
• From: John Doty <jpd at whispertel.LoseTheH.net>
• Date: Mon, 9 Feb 2009 05:32:20 -0500 (EST)
• References: <gm1dks$3nk$1@smc.vnet.net> <gm3r8h$mev$1@smc.vnet.net> <gm6kvu$a39$1@smc.vnet.net> <gm99v7$95$1@smc.vnet.net> <200902041021.FAA18709@smc.vnet.net> <gmecf2$adc$1@smc.vnet.net>
Andrzej Kozlowski wrote:
> On 4 Feb 2009, at 11:21, John Doty wrote:
>> Jens-Peer Kuska wrote:
>>> Hi,
>>> this is called a distribution or generalized function
>>> and not a function and it is only defined
>>> inside of an integral as my Vladimirov
>>> http://www.amazon.de/Methods-Generalized-Functions-Analytical-Special/dp/0415273560/ref=sr_1_33?ie=UTF8&s=books-intl-de&qid=1233576829&sr=8-33
>>> say.
>> That restriction is Vladimirov's. We who actually apply generalized
>> functions to physics and engineering problems are not shy about using
>> them outside of integrals. This is the approach that Mathematica
>> implements, as you can see below.
>> A better reference is Bracewell:
>> www.amazon.com/Fourier-Transform-Its-Applications/dp/0073039381,
>> especially applicable to this question.
>> By the way, the term "distribution" seems designed to confuse the
>> innocent. Many applications of generalized functions also involve
>> probability, where the term "distribution" has a different and far
>> more
>> familiar meaning.
>> Most physicists and engineers will drop the "generalized" and simply
>> consider things like delta functions to be functions. They often have
>> the right properties to represent the behavior of real world objects,
>> when other notions of "function" don't.
> Sorry, but the last paragraph isn't that much of a recommendation.
> There is hardly any (pseudo)-mathematical nonsense that has not been
> believed to be true or valid by some engineer, occasionally with
> lamentable consequences. I remember that when, I was a math
> undergraduate, a "popular" mathematics journal published a long list
> of examples (with detailed references) of mathematical nonsense
> perpetrated by engineers, economists and some others (some quite
> hilarious).
Well, the classic of this genre is Berkeley's mocking demolition of
calculus. Should physicists and engineers have abandoned calculus after
Berkeley demonstrated that it was pseudo-mathematical nonsense?
One I recall from my undergraduate days was a university computer center
director who advocated automated scanning of student Fortran jobs on the
mainframe to detect time-wasting "infinite loops". He was ridiculed as a
disgrace to his profession: "doesn't he know about the halting
problem?". But the mockers got it wrong: he wasn't asking for a perfect
decision procedure, merely a practical one that could weed out the
easily decidable cases.
Of course, mathematicians perpetrate mathematical howlers, too. Euler's
assertion that a divergent series may be replaced by its genesis formula
is an example. The idea that mathematics advances by stepping from truth
to truth by proving theorems is easily seen to be a myth if you know
some history. Mathematics is much more interesting, creative, and useful
than that.
Jens' authority, Vladimirov, asserts that a delta function is only
defined inside an integral. But I've also heard a mathematician complain
"that's not what we mean by integration". And Jens earlier insisted on
"square integrable" functions. Different notions of "function" and
"integral" lead to different ideas about what is allowable.
Calculus, Fourier analysis, delta functions, renormalization, all
recognized as "mathematical nonsense" in their time, yet they proved to
be indispensable tools for effectively applying mathematics to real
world problems.
> I think some of it might have involved the use (or misuse)
> of generalized functions, though I am no longer sure. (I guess I could
> still trace the list, if you really wanted to see it ...but you could
> instead just search the archives of this forum ;-))
> The most intuitive yet rigorous theory of generalized functions that
> allows then to be treated as "functions" was created by Jean Francois
> Colombeau. He wrote a beautifully clear and quite short exposition of
> his theory entitled "Elementary Introduction to new Generalized
> Functions" (North-Holland 1985).
> The Colombeau genealized functions have values at all points, but they
> are "generalized numbers" (which include ordinary numbers). Thus the
> Dirac Delta is a function defined on R, whose value is 0 for x!=0 and
> a generalized number at x==0. Colombeau generalized functions have
> derivatives of all orders (which are themselves generalized functions)
> and classical distributions are precisely those generalized functions
> which, in a neighborhood of each point, are partial derivatives of
> continuous functions.
> Colombeau theory was the first one that solved satisfactorily the old
> problem of giving a satisfactory definition of multiplication of
> generalized functions (something that before Colombeau was considered
> impossible by many). Colembeau theory justified much of the heuristics
> that had been previously known to physicists but it also helped to
> uncover nonsense where there was nonsense to uncover.
The physical design of the communication network you used to broadcast
this assertion involved heavy use of applied mathematics results
obtained earlier than this using generalized functions. What is "white
noise" anyway? Mathematical (at least at the time its use became common)
and even physical nonsense, but an indispensable concept.
Mathematical objects are products of human imagination. That we can
obtain reliable knowledge of their properties is a profound mystery.
That they can effectively model real world objects is another profound
mystery. But they don't do so perfectly: any particular mathematical
model of reality will have (often poorly understood) limits to its
applicability. The justification for any applied mathematics model is
that you can verify it gets correct answers. The scientific method. The
problem with most bad applied math in my experience is the dogmatic use
of inapplicable methods.
> Last but not least, Mathematica's notion of a generalized function is
> based on Colombeau. To convince yourself look at the documentation for
> HeavisideTheta, in the section "Possible issues". I quote:
> Products of distributions with coincident singular support cannot be
> defined (no Colombeau algebra interpretation).
I would guess that this sort of formalization is more important to a CAS
than a human. Physical intuition is reasonably effective at weeding out
the nonsense here, otherwise these techniques would never have gained a
foothold. But a CAS has no physical intuition.
It is nice that Mathematica 7 seems to have improved here.
> I get the impression that not all engineers believe in this even now.
Many engineers are suspicious of mathematical abstraction. Mathematical
notions of "correct" and "true" don't map completely reliably into real
world situations. So, understanding the connection between the
abstraction and the physical situation is essential.
> Andrzej Kozlowski
>>> Regards
>>> Jens
>>> John Doty wrote:
>>>> Jens-Peer Kuska wrote:
>>>>> Hi,
>>>>> the Fourier transform over the interval x in (-Infinity,Infinity)
>>>>> converges only for quadratic integrable functions, i.e., functions
>>>>> where Integrate[Conjugate[f[x]]*f[x],{x,-Infinity,Infinity}]<
>>>>> Infinity
>>>>> This is not the case for Cosh[x], and so no Fourier transform
>>>>> exist.
>>>> Depends on what you mean by "function". Mathematica tries in its
>>>> pragmatic way to do what you might want here:
>>>> In[1]:= FourierTransform[t^2,t,w]
>>>> Out[1]= -(Sqrt[2 Pi] DiracDelta''[w])
>>>> t^2 is certainly not square integrable, but this is the kind of
>>>> useful
>>>> result scientists and engineers want.
>>>> Mathematica's support for "generalized functions" still has room for
>>>> improvement, but it has come a long way. The bizarre problems I
>>>> saw in
>>>> the past trying Fourier methods to perform fractional
>>>> differentiation
>>>> and integration
>>>> (http://forums.wolfram.com/mathgroup/archive/2000/Apr/msg00043.html)
>>>> seem no longer to be with us in Mathematica 7.
>> --
>> John Doty, Noqsi Aerospace, Ltd.
>> http://www.noqsi.com/
>> --
>> The axiomatic method of mathematics is one of the great achievements
>> of
>> our culture. However, it is only a method. Whereas the facts of
>> mathematics once discovered will never change, the method by which
>> these
>> facts are verified has changed many times in the past, and it would be
>> foolhardy to expect that changes will not occur again at some future
>> date. - Gian-Carlo Rota
John Doty, Noqsi Aerospace, Ltd.
In theory there is no difference between theory and practice. In
practice there is. -Yogi Berra
• Follow-Ups:
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Multiple positive solutions of semilinear elliptic equations involving concave and convex nonlinearities in ℝN
In this article, we investigate the effect of the coefficient f(z) of the sub-critical nonlinearity. For sufficiently large λ > 0, there are at least k + 1 positive solutions of the semilinear
elliptic equations
where 1 ≤ q < 2 < p < 2* = 2N/(N - 2) for N ≥ 3.
AMS (MOS) subject classification: 35J20; 35J25; 35J65.
semilinear elliptic equations; concave and convex; positive solutions
1 Introduction
For N ≥ 3, 1 ≤ q < 2 < p < 2* = 2N/(N - 2), we consider the semilinear elliptic equations
where λ > 0.
Let f and h satisfy the following conditions:
(f 1) f is a positive continuous function in ℝ^N and lim[|z| → ∞ ]f(z) = f[∞ ]> 0.
(f2) there exist k points a^1, a^2,..., a^k in ℝ^N such that
and f[∞ ]< f[max].
Semilinear elliptic problems involving concave-convex nonlinearities in a bounded domain
have been studied by Ambrosetti et al. [1] (h ≡ 1, and 1 < q < 2 < p ≤ 2* = 2N/(N- 2)) and Wu [2] and changes sign, 1 < q < 2 < p < 2*). They proved that this equation has at least two positive
solutions for sufficiently small c > 0. More general results of Equation (E[c]) were done by Ambrosetti et al. [3], Brown and Zhang [4], and de Figueiredo et al. [5].
In this article, we consider the existence and multiplicity of positive solutions of Equation (E[λ]) in ℝ^N. For the case q = λ = 1 and f(z) ≡ 1 for all z ∈ ℝ^N, suppose that h is nonnegative, small,
and exponential decay, Zhu [6] showed that Equation (E[λ]) admits at least two positive solutions in ℝ^N. Without the condition of exponential decay, Cao and Zhou [7] and Hirano [8] proved that
Equation (E[λ]) admits at least two positive solutions in ℝ^N. For the case q = λ = 1, by using the idea of category and Bahri-Li's minimax argument, Adachi and Tanaka [9] asserted that Equation (E[λ
]) admits at least four positive solutions in ℝ^N, where f(z) ≢ 1, f(z) ≥ 1 - C exp((-(2 + δ) |z|) for some C, δ > 0, and sufficiently small . Similarly, in Hsu and Lin [10], they have studied that
there are at least four positive solutions of the general case -Δu + u = f(z)v^p-1 + λh(z) v^q-1 in ℝ^N for sufficiently small λ > 0.
By the change of variables
Equation (E[λ]) is transformed to
Associated with Equation (E[ε]), we consider the C^1-functional J[ε], for u ∈ H^1 (ℝ^N),
where is the norm in H^1 (ℝ^N) and u[+ ]= max{u, 0} ≥ 0. We know that the nonnegative weak solutions of Equation (E[ε]) are equivalent to the critical points of J[ε]. This article is organized as
follows. First of all, we use the argument of Tarantello [11] to divide the Nehari manifold M[ε ]into the two parts and . Next, we prove that the existence of a positive ground state solution of
Equation (E[ε]). Finally, in Section 4, we show that the condition (f2) affects the number of positive solutions of Equation (E[ε]), that is, there are at least k critical points of J[ε ]such that
for 1 ≤ i ≤ k.
For the semilinear elliptic equations
we define the energy functional , and
where N[ε ]= {u ∈ H^1 (ℝ^N) \ {0} | u[+ ]≢ 0 and }. Note that
where N[∞ ]= {u ∈ H^1 (ℝ^N) \ {0} | u[+ ]≢ 0 and };
(ii) if f ≡ f[max], we define and
where N[max ]= {u ∈ H^1 (ℝ^N) \ {0} | u[+ ]≢ 0 and }.
Lemma 1.1
Proof. It is similar to Theorems 4.12 and 4.13 in Wang [[12], p. 31].
Our main results are as follows.
(I) Let Λ = ε^2(p-q)/(p-2). Under assumptions (f 1) and (h1), if
where ∥h∥[# ]is the norm in , then Equation (E[ε]) admits at least a positive ground state solution. (See Theorem 3.4)
(II) Under assumptions (f1) - (f2) and (h1), if λ is sufficiently large, then Equation (E[λ]) admits at least k + 1 positive solutions. (See Theorem 4.8)
2 The Nehari manifold
First of all, we define the Palais-Smale (denoted by (PS)) sequences and (PS)-conditions in H^1(ℝ^N) for some functional J.
Definition 2.1 (i) For β ∈ ℝ, a sequence {u[n]} is a (PS)[β]-sequence in H^1(ℝ^N) for J if J(u[n]) = β + o[n](1) and J'(u[n]) = o[n](1) strongly in H^-1 (ℝ^N) as n → ∞, where H^-1 (ℝ^N) is the dual
space of H^1(ℝ^N);
(ii) J satisfies the (PS)[β]-condition in H^1(ℝ^N) if every (PS)[β]-sequence in H^1(ℝ^N) for J contains a convergent subsequence.
Next, since J[ε ]is not bounded from below in H^1 (ℝ^N), we consider the Nehari manifold
Note that M[ε ]contains all nonnegative solutions of Equation (E[ε]). From the lemma below, we have that J[ε ]is bounded from below on M[ε].
Lemma 2.2 The energy functional J[ε ]is coercive and bounded from below on M[ε].
Proof. For u ∈ M[ε], by (2.1), the Hölder inequality and the Sobolev embedding theorem (1.1), we get
Hence, we have that J[ε ]is coercive and bounded from below on M[ε].
Then for u ∈ M[ε], we get
We apply the method in Tarantello [11], let
Lemma 2.3 Under assumptions (f1) and (h1), if 0 < Λ (= ε^2(p-q)/(p-2)) < Λ[0], then .
Proof. See Hsu and Lin [[10], Lemma 5].
Lemma 2.4 Suppose that u is a local minimizer for J[ε ]on M[ε ]and . Then in H^-1 (ℝ^N).
Proof. See Brown and Zhang [[4], Theorem 2.3].
Lemma 2.5 We have the following inequalities.
(iv) If , then J[ε](u) > 0 for each .
Proof. (i) It can be proved by using (2.2).
(ii) For any , by (2.2), we apply the Hölder inequality to obtain that
(iii) For any , by (2.3), we have that
(iv) For any , by (iii), we get that
Thus, if , we get that J[ε](u) ≥ d[0 ]> 0 for some constant d[0 ]= d[0](ε, p, q, S, ∥h∥[# ], f[max]).
For u ∈ H^1 (ℝ^N) \ {0} and u[+ ]≢ 0, let
Lemma 2.6 For each u ∈ H^1 (ℝ^N)\ {0} and u[+ ]≢ 0, we have that
(i) if , then there exists a unique positive number such that and J[ε](t^-u) = sup[t ≥ 0 ]J[ε](tu);
(ii) if 0 < Λ ( = ε^2(p-q)/(p-2)) < Λ[0 ]and , then there exist unique positive numbers such that and
Proof. See Hsu and Lin [[10], Lemma 7].
Applying Lemma 2.3 , we write , where
Lemma 2.7 (i) If 0 < Λ ( = ε^2(p-q)/(p-2)) < Λ[0], then ;
(ii) If 0 < Λ < qΛ[0]/2, then for some constant d[0 ]= d[0 ](ε, p, q, S, ∥h∥[#], f[max]).
Proof. (i) Let , by (2.2), we get
By the definitions of α[ε ]and , we deduce that .
(ii) See the proof of Lemma 2.5 (iv).
Applying Ekeland's variational principle and using the same argument in Cao and Zhou [7] or Tarantello [11], we have the following lemma.
Lemma 2.8 (i) There exists a -sequence {u[n]} in M[ε ]for J[ε];
(ii) There exists a -sequence {u[n]} in for J[ε];
3 Existence of a ground state solution
In order to prove the existence of positive solutions, we claim that J[ε ]satisfies the (PS)[β]-condition in H^1(ℝ^N) for , where Λ = ε^2(p-q)/(p-2) and C[0 ]is defined in the following lemma.
Lemma 3.1 Assume that h satisfies (h1) and 0 < Λ ( = ε^2(p-q)/(p-2)) < Λ[0]. If {u[n]} is a (PS)[β]-sequence in H^1(ℝ^N) for J[ε ]with u[n ]⇀ u weakly in H^1 (ℝ^N), then in H^-1 (ℝ^N) and , where
Proof. Since {u[n]} is a (PS)[β]-sequence in H^1(ℝ^N) for J[ε ]with u[n ]⇀ u weakly in H^1 (ℝ^N), it is easy to check that in H^-1(ℝ^N) and u ≥ 0. Then we have , that is, . Hence, by the Young
Lemma 3.2 Assume that f and h satisfy (f1) and (h1). If 0 < Λ ( = ε^2(p-q)/(p-2)) < Λ[0], then J[ε ]satisfies the (PS)[β]-condition in H^1(ℝ^N) for .
Proof. Let {u[n]} be a (PS)[β]-sequence in H^1(ℝ^N) for J[ε ]such that J[ε](u[n]) = β + o[n](1) and (1) in H^-1(ℝ^N). Then
where c[n ]= o[n](1), d[n ]= o[n](1) as n → ∞. It follows that {u[n]} is bounded in H^1(ℝ^N). Hence, there exist a subsequence {u[n]} and a nonnegative u ∈ H^1 (ℝ^N) such that in H^-1 (ℝ^N), u[n ]⇀
u weakly in H^1 (ℝ^N), u[n ]⇀ u a.e. in ℝ^N, u[n ]⇀ u strongly in for any 1 ≤ s < 2*. Using the Brézis-Lieb lemma to get (3.1) and (3.2) below.
Next, claim that
For any σ > 0, there exists r > 0 such that . By the Hölder inequality and the Sobolev embedding theorem, we get
Applying (f1) and u[n ]→ u in , we get that
Let p[n ]= u[n ]- u. Suppose p[n ]↛ 0 strongly in H^1 (ℝ^N). By (3.1)-(3.4), we deduce that
By Theorem 4.3 in Wang [12], there exists a sequence {s[n]} ⊂ ℝ^+ such that s[n ]= 1 + o[n](1), {s[n ]p[n]} ⊂ N[∞ ]and I[∞](s[n ]p[n]) = I[∞](p[n]) + o[n](1). It follows that
which is a contradiction. Hence, u[n ]→ u strongly in H^1(ℝ^N).
Remark 3.3 By Lemma 1.1, we obtain
By Lemma 2.8 (i), there is a -sequence {u[n]} in M[ε ]for J[ε]. Then we prove that Equation (E[ε]) admits a positive ground state solution u[0 ]in ℝ^N.
Theorem 3.4 Under assumptions (f1), (h1), if 0 < Λ ( = ε^2(p-q)/(p-2)) < Λ[0], then there exists at least one positive ground state solution u[0 ]of Equation (E[ε]) in ℝ^N. Moreover, we have that
Proof. By Lemma 2.8 (i), there is a minimizing sequence {u[n]} ⊂ M[ε ]for J[ε ]such that J[ε](u[n]) = α[ε ]+ o[n](1) and in H^-1 (ℝ^N). Since , by Lemma 3.2, there exist a subsequence {u[n]} and u
[0 ]∈ H^1 (ℝ^N) such that u[n ]→ u[0 ]strongly in H^1 (ℝ^N). It is easy to see that is a solution of Equation (E[ε]) in ℝ^N and J[ε](u[0]) = α[ε]. Next, we claim that . On the contrary, assume that
We get that
It follows that
which contradicts to α[ε ]< 0. By Lemma 2.6 (ii), there exist positive numbers such that and
which is a contradiction. Hence, and
By Lemma 2.4 and the maximum principle, then u[0 ]is a positive solution of Equation (E[ε]) in ℝ^N.
4 Existence of k + 1 solutions
From now, we assume that f and h satisfy (f1)-(f2) and (h1). Let w ∈ H^1 (ℝ^N) be the unique, radially symmetric, and positive ground state solution of Equation (E0) in ℝ^N for f = f[max]. Recall the
facts (or see Bahri and Li [13], Bahri and Lions [14], Gidas et al. [15], and Kwong [16]).
(ii) for any ε > 0, there exist positive numbers C[1], , and such that for all z ∈ ℝ^N
For 1 ≤ i ≤ k, we define
Clearly, . By Lemma 2.6 (ii), there is a unique number such that , where 1 ≤ i ≤ k.
We need to prove that
Lemma 4.1 (i) There exists a number t[0 ]> 0 such that for 0 ≤ t ≤t[0 ]and any ε > 0, we have that
(ii) There exist positive numbers t[1 ]and ε[1 ]such that for any t > t[1 ]and ε < ε[1], we have that
Proof. (i) Since J[ε ]is continuous in is uniformly bounded in H^1 (ℝ^N) for any ε > 0, and γ[max ]> 0, there is t[0 ]> 0 such that for 0 ≤ t ≤ t[0 ]and any ε > 0
(ii) There is an r[0 ]> 0 such that f (z) ≥ f[max]/2 for z ∈ B^N (a^i; r[0]) uniformly in i. Then there exists ε[1 ]> 0 such that for ε < ε[1]
Thus, there is t[1 ]>0 such that for any t > t[1 ]and ε < ε[1]
Lemma 4.2 Under assumptions (f1), (f2), and (h1). If 0 < Λ ( = ε^2(p-q)/(p-2)) < q Λ[0]/2, then
Proof. By Lemma 4.1, we only need to show that
We know that sup[t ≥0 ]I[max ](tw) = γ[max]. For t[0 ]≤ t ≤ t[1], we get
then , that is, uniformly in i.
Applying the results of Lemmas 2.6, 2.7(ii), and 4.2, we can deduce that
Since γ[max ]< γ[∞], there exists ε[0 ]> 0 such that
Choosing 0 < ρ[0 ]< 1 such that
where and f(a^i) = f[max]. Define K = {a^i | 1 ≤ i ≤ k} and . Suppose for some r[0 ]> 0.
Let Q[ε ]: H^1 (ℝ^N) \ {0} → ℝ^N be given by
where χ : ℝ^N → ℝ^N, χ (z) = z for |z| ≤ r[0 ]and χ (z) = r[0]z/|z| for |z| > r[0].
Lemma 4.3 There exists 0 < ε^0 ≤ ε[0 ]such that if ε < ε^0, then for each 1 ≤ i ≤ k.
Proof. Since
there exists ε^0 > 0 such that
Lemma 4.4 There exists a number such that if u ∈ N[ε ]and , then for any 0 < ε < ε^0.
Proof. On the contrary, there exist the sequences {ε[n]} ⊂ ℝ^+ and such that (1) as n → ∞ and for all n ∈ ℕ. It is easy to check that {u[n]} is bounded in H^1 (ℝ^N). Suppose u[n ]→ 0 strongly in L
^p (ℝ^N). Since
which is a contradiction. Thus, u[n ]↛ 0 strongly in L^p (ℝ^N). Applying the concentration-compactness principle (see Lions [17] or Wang [[12], Lemma 2.16]), then there exist a constant d[0 ]> 0 and
a sequence such that
Let , there are a subsequence {v[n]} and v ∈ H^1 (ℝ^N) such that v[n ]⇀ v weakly in H^1 (ℝ^N). Using the similar computation in Lemma 2.6, there is a sequence such that and
We deduce that a convergent subsequence satisfies . Then there are subsequences and such that weakly in H^1 (ℝ^N). By (4.2), then . Moreover, we can obtain that strongly in H^1 (ℝ^N) and .
Now, we want to show that there exists a subsequence such that z[n ]→ z[0 ]∈ K.
(i) Claim that the sequence {z[n]} is bounded in ℝ^N. On the contrary, assume that |z[n]| → ∞, then
which is a contradiction.
(ii) Claim that z[0 ]∈ K. On the contrary, assume that z[0 ]∉ K, that is, f(z[0]) < f[max]. Then using the above argument to obtain that
which is a contradiction. Since v[n ]→ v ≠ 0 in H^1 (ℝ^N), we have that
which is a contradiction.
Hence, there exists a number such that if u ∈ N[ε ]and , then for any 0 < ε < ε^0.
From (4.1), choosing such that
For each 1 ≤ i ≤ k, define
Lemma 4.5 If and J[ε ](u) ≤ γ[max ]+ δ[0]/2, then there exists a number such that for any .
Proof. We use the similar computation in Lemma 2.6 to get that there is a unique positive number
such that . We want to show that for some constant c > 0 (independent of u). First, since ,
and J[ε ]is coercive on M[ε], then for some constants c[1 ]and c[2 ](independent of u). Next, we claim that for some constant c[3 ]> 0 (independent of u). On the contrary, there exists a sequence
such that
By (2.3),
which is a contradiction. Thus, for some constant c > 0 (independent of u). Now, we get that
From the above inequality, we deduce that
Hence, there exists such that for
By Lemma 4.4, we obtain
Applying the above lemma, we get that
By Lemmas 4.2, 4.3, and Equation (4.3), there exists such that
Lemma 4.6 Given , then there exist an η > 0 and a differentiable functional l : B(0; η) ⊂ H^1(ℝ^N) → ℝ^+ such that for any v ∈ B(0;η) and
Proof. See Cao and Zhou [7].
Lemma 4.7 For each 1 ≤ i ≤ k, there is a -sequence in H^1(ℝ^N) for J[ε].
Proof. For each 1 ≤ i ≤ k, by (4.4) and (4.5),
Let be a minimizing sequence for . Applying Ekeland's variational principle, there exists a subsequence such that and
Using (4.7), we may assume that for sufficiently large n. By Lemma 4.6, then there exist an and a differentiable functional such that , and for . Let v[σ ]= σv with ║v║[H ]= 1 and . Then and
. From (4.8) and by the mean value theorem, we get that as σ → 0
Since we can deduce that for all n and i from (4.6), then strongly in H^-1 (ℝ^N) as n → ∞.
Theorem 4.8 Under assumptions (f1), (f[2]), and (h1), there exists a positive number λ*(λ* = (ε*)^-2) such that for λ > λ*, Equation (E[λ]) has k + 1 positive solutions in ℝ^N.
Proof. We know that there is a -sequence in H^1(ℝ^N) for J[ε ]for each 1 ≤ i ≤ k, and (4.5). Since J[ε ]satisfies the (PS)[β]-condition for , then J[ε ]has at least k critical points in for 0 < ε
< ε*. It follows that Equation (E[λ]) has k nonnegative solutions in ℝ^N. Applying the maximum principle and Theorem 3.4, Equation (E[λ]) has k + 1 positive solutions in ℝ^N.
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A Couple of Geometric Triangle Problems
August 24th 2011, 03:07 PM #1
May 2011
A Couple of Geometric Triangle Problems
1. One of the legs of a right triangle has the length of 4 cm. Express the length of the altitude (h) perpendicular to the hypotenuse (c) as a function of the length of the hypotenuse. (Label the
additional leg as b) Make a sketch to support the given information before working the problem.
2. The altitude perpendicular to the hypotenuse of a right triangle is 12 cm. Express the length of the hypotenuse (c) as a function of the perimeter (P). Make a sketch to support the given
information before working the problem.
Last edited by Ackbeet; August 24th 2011 at 04:57 PM.
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Operational Semantics
COMP 317: Semantics of Programming Languages
Operational Semantics
In these notes we look at the operational semantics of the language IMP. Our goal is to develop a semantics that describes how programs of the language are executed. In IMP, as in any other
imperative language, assignment is the most basic form of program: the other linguistic constructs (conditionals and loops) are simply ways of organizing assignments into series that are executed in
a particular order. For example, think of how (according to our intuitions) the following program is evaluated:
x := 0 ;
f := 1 ;
while x <= 2
x := x + 1 ;
f := f * x
On each pass through the loop, the value of x is increased by 1, and the body of the loop is gone through three times (while x has values 0, 1, then 2). Thus, we could "unfold" the program above to
the following series of assignments:
x := 0 ;
f := 1 ;
x := x + 1 ;
f := f * x ;
x := x + 1 ;
f := f * x ;
x := x + 1 ;
f := f * x
(one of the benefits of a formal semantics is that we could give a rigorous argument that these two programs are indeed equivalent).
Assignments, therefore, lie at the heart of imperative programming languages. They also lie at the heart of the semantics of imperative languages. In particular, they suggest that any semantics
should be based on the notion of state (or storage), by which we mean the particular values that are associated with a program's variables. When a program is being executed, we can think of the
computer that is running the program as being in a certain state. This state is determined by the values stored by the program's variables, and these values are updated by assignments to the
variables. For example, after running the program above, the computer will be in a state where the variable x has the value 3, and the variable f has the value 6. We can think of a state as being a
table that tells us the value associated with any given variable (since variables in IMP aren't declared, and have no scope, the state should tell us the value associated with any variable. This
would give us an infinitely long table, and it is more convenient to think of a state as a function that takes a variable as argument and returns the value stored in that variable. More formally, a
state is a function
Loc -> N
where Loc is the set of variable names, and N is the set of numbers (in IMP, all variables store numbers). For example, after running a program
x := 8 ;
y := x + 1 ;
z := y + 2
we obtain a state s, with
• s(x) = 8,
• s(y) = 9, and
• s(z) = 11.
(We don't know what values s gives to variables other than x, y or z, but presumably those values aren't changed by the program.)
The operational semantics we will give for IMP will describe how assignments (and other programs) update states. Before we do this, note that an assignment like
y := x + 1
depends on the state of the computer before the assignment is executed: the value assigned to y depends on the value of x in this "initial" state (e.g., if the value of x in the initial state is 8,
then y is assigned the value 9, and if x has the value 23, then y is assigned the value 24). We begin by describing how expressions such as x+1 are evaluated in a given state.
The Evaluation of Arithmetic Expressions
The value of an IMP expression such as
(x + 1) * (2*y + 1)
clearly depends upon the values of x and y. Since we want the semantics of (the symbol) "+" to be addition and the semantics of "*" to be multiplication, we should be able to get the value of such an
expression by "plugging in" the values of x and y. The values of these variables depends upon the state of the computer when the expression is evaluated: for example, if the state s gives the value
of x as 5 and the value of y as 3 (i.e., s(x) = 5 and s(y) = 3), then the value of the expression above is
(s(x) + 1) * (2 * s(y) + 1)
(5 + 1) * (2*3 + 1)
6 * 7
We capture these intuitions by defining a relation between expressions, states and numbers that says what the value of an expression is in a given state. We use the notation (e,s) --> n to indicate
that the IMP expression e has the value n when evaluated in the state s.
We define this relation by considering each possible form of the expression e (i.e., each of the options in the BNF definition of arithmetic expressions in IMP).
1. The first case is where the expression is a number n; clearly, the value of this expression is just n, and doesn't depend on the state:
(n,s) --> n
for all numbers n and states s.
2. The second case is where the expression is a variable x; in this case, the value does depend on the state, and in fact, the value should be the value of the variable in that state:
(x,s) --> s(x)
for all locations x and states s.
3. The third case is where the expression is built from two expressions e1 and e2 using the symbol +, i.e., the expression is of the form e1 + e2. In this case, the result should be the sum of the
values of e1 and e2, i.e., the result should be n1 + n2, where n1 is the value of e1 and n2 the value of e2 in the state s. We write this as the following implication:
If (e1,s) --> n1 and (e2,s) --> n2, then (e1+e2,s) --> n1+n2.
for all expressions e1 and e2, numbers n1 and n2, and states s.
4. The fourth case is where the expression is built from two expressions e1 and e2 using the symbol *, i.e., the expression is of the form e1 * e2. In this case, the result should be the product of
the values of e1 and e2, i.e., the result should be n1 * n2, where n1 is the value of e1 and n2 the value of e2 in the state s. We write this as the following implication:
If (e1,s) --> n1 and (e2,s) --> n2, then (e1*e2,s) --> n1*n2.
for all expressions e1 and e2, numbers n1 and n2, and states s.
5. The final case is where the expression is built from two expressions e1 and e2 using the symbol -, i.e., the expression is of the form e1 - e2. This is treated in the same way as for + and *, and
is left as an exercise.
For example, consider the expression
(x + 1) * (2*y + 1).
Let s be a state with s(x) = 5 and s(y) = 3. We show that the value of the above expression in the state s is 42.
• By (2), and our assumptions about s, we have (x,s) --> 5.
• By (1), we have (1,s) --> 1.
• Therefore, by (3), (x+1,s) --> 6.
• By (1), we have (2,s) --> 2.
• By (2), and our assumptions about s, we have (y,s) --> 3.
• Therefore, by (4), (2*y,s) --> 6.
• From this and the second of these bullet-points, by (3), (2*y + 1,s) --> 7.
• From this and the third of these bullet-points, by (3), ((x+1)*(2*y + 1),s) --> 42.
The clauses (1)-(5) above define the relation (e,s) --> n. Clauses (3)-(5) define the semantics of compound expressions (expressions built from other expressions) using the semantics (i.e., the
values) of their subexpressions. This is evident from the example of (x+1)*(2*y+1) above. When a semantics defines the meaning of expressions (whether these expressions be arithmetic expressions,
Boolean expressions or programs) in terms of the meanings of their subexpressions, we say the semantics is compositional.
Note that each of the five clauses (1)-(5) in the definition of the evaluation relation corresponds to one of the "options" (or "cases") in the BNF definition of <Aexp>. When a definition gives a
clause for each case, in a compositional way, we say the definition is inductive. Thus, for example, clauses (1)-(5) above give an inductive definition of the evaluation relation (e,s) --> n.
The Evaluation of Boolean Expressions
Boolean expressions are given an operational semantics in much the same way as arithmetic expressions. We define a relation (b,s) --> v that describes how Boolean expressions are evaluated; here, b
is a Boolean expression in <Bexp>, s is a state, and v is a truth-value (i.e., either true or false). The definition of this evaluation relation is inductive:
1. The first case is where the expression is true clearly, the value of this expression should just be true, and doesn't depend on the state:
(true,s) --> true
for all states s.
2. The second case is where the expression is false clearly, the value of this expression should just be false, and doesn't depend on the state:
(false,s) --> false
for all states s.
3. The third case is where the expression is built from two arithmetic expressions e1 and e2 using the symbol "="; i.e., the expression is of the form e1 = e2. The idea is that this Boolean
expression represents an equality test, and should be true if, and only if, the expressions e1 and e2 are "equal" (the quotes here are because we're not testing whether the expressions are equal
- they're equal only when they're identical as strings of characters - rather, we're testing whether their semantics - their values - are equal). Therefore, the result should be true if e1 and e2
have the same value in a given state, and false otherwise:
If (e1,s) --> n1 and (e2,s) --> n2, then (e1=e2,s) --> v,
where v is true if n1 = n2, and false otherwise.
for all expressions e1 and e2, numbers n1 and n2, and states s.
4. The case where the expression is built from two arithmetic expressions e1 and e2 using the symbol "<=" is treated similarly.
5. We now consider the third case where the expression is built from two Boolean expressions e1 and e2 using the symbol "and" (logical conjunction); i.e., the expression is of the form b1 and b2.
The idea is that this Boolean expression should be true just when both b1 and b2 are true and false otherwise:
If (b1,s) --> v1 and (b2,s) --> v2,
then (b1 and b2,s) --> v,
where v is true if v1 and v2 are both true, and false otherwise.
for all Boolean expressions b1 and b2, truth-values v1, v2 and v, and states s.
The remaining cases, corresponding to the cases in the BNF definition of <Bexp>, are left as an exercise.
The Evaluation of Programs
Again, we define the semantics of programs by inductively defining a relation (c,s) --> s', where c is a program (a term in <Com>) and s and s' are states. What we mean when we write (c,s) --> s' is
this: if we start in a state s and run the program c then we end up in the state s'.
1. The first case is where c is the program skip. The idea is that skip is a program that does nothing, and therefore doesn't change the state at all:
(skip,s) --> s
for all states s.
2. Assignments do change the state, by updating the value stored in a variable. If we evaluate an assignment x := e in a state s, then we obtain a new state that differs from s only in the value of
x, which is now set to be the value of e. Let's suppose that the value of e in s is the number n; we write s[n/x] for this new state (this can be pronounced "s where n is now stored in x. This
new state is, of course, a state, and therefore a function from variable names to numbers; it is formally described by the following two equations:
s[n/x](x) = n
s[n/x](y) = s(y), for all y different from x.
The first of these equations says that the value now stored in x is n (the value of e); the second equation says that only the value of x has been changed: all other variables have their original
values. It is this updated state, then, that is the result of evaluating the assignment in state s:
If (e,s) --> n, then (x := e,s) --> s[n/x]
for all states s, variables x and expressions e.
3. For sequential compositions, i.e., programs of the form c1 ; c2, we want to say that c1 is evaluated first, and then c2. If we start in a state s, evaluating c1 will lead to a new state, say s',
and it is in this state that we start to evaluate c2:
If (c1,s) --> s' and (c2,s') --> s'',
then (c1;c2,s) --> s''
for all states s, s' and s'', and programs c1 and c2.
4. Consider a program of the form
if b then c1 else c2 .
The evaluation of this program proceeds by first evaluating the Boolean expression b; if this evaluates to true, then c1 is evaluated:
If (b,s) --> true and (c1,s) -- s',
then (if b then c1 else c2,s) --> s'.
Similarly, if b evaluates to false, then c2 is evaluated:
If (b,s) --> false and (c2,s) -- s',
then (if b then c1 else c2,s) --> s'.
5. Finally, we define the semantics of loops, i.e., programs of the form
while b do c .
Again, evaluation proceeds by first evaluating b. If the result is false, then the body of the loop isn't evaluated; the program terminates immediately without changing the state:
If (b,s) --> false
then (while b do c,s) --> s.
On the other hand, if b evaluates to true, then the body c is evaluated, and then the process is repeated: b is evaluated again, and the loop is either exited or iterated depending on whether the
result is true or false. Effectively, if b evaluates to true, then the program
c ; while b do c
is evaluated. Therefore:
If (b,s) --> true and (c,s) --> s' and (while b do c,s') --> s''
then (while b do c,s) --> s''.
This says: when the condition b is true, then evaluate the body c, giving a new state s', then repeat the loop again: if this gives a final state s'', then that state is the result of evaluating
the loop.
This completes the definition of the evaluation relation for programs, and gives an operational semantics for IMP.
Some Issues Concerning the Operational Semantics
What do we get from the semantics?
The semantics describes how arithmetic and Boolean expressions are evaluated, and how programs update states. The semantics is given by the inductively defined relations (a,s) --> n for arithmetic
expressions, (b,s) --> v for Boolean expressions, and (c,s) --> s' for programs. Because these relations are formally defined, we can use them in giving rigorous arguments about programs (Winskel
gives some examples). We won't prove anything in these notes, as the proofs use some rather involved inductive arguments that go beyond the scope of this module (we'll see some rigorous arguments
once we've described the denotational and axiomatic semantics of IMP). However, the possibility of constructing formal correctness proofs for programs is certainly one of the main uses of the
semantics. Moreover, because the semantics is compositional, it is easy to modify or extend the semantics; some examples follow:
What about "lazy" and?
The semantics of logical conjunction ("and") requires both arguments to be evaluated: clause (5) in the definition of the evaluation relation for Boolean expressions gives the semantics of "b1 and b2
" by looking at the values of both b1 and b2. Many languages have a "lazy" logical conjunction (e.g., the operator "&&" of C or Java) which will first evaluate b1, and if this evaluates to false,
then b2 is not evaluated. This kind of lazy semantics could be captured by the following two rules:
If (b1,s) --> false, then (b1 and b2, s) --> false
If (b1,s) --> true and (b2,s) --> v, then (b1 and b2, s) --> v
Thus the expression b2 is only evaluated if b1 evaluates to true.
How do we get a more operational operational semantics?
The semantics as it stands doesn't say much about how programs are evaluated, in the sense that we simply have a relation (c,s) --> s' that tells us what the final state is after evaluating the
program c - it doesn't say anything about the intermediate states that we pass through in evaluating the program. For example, consider the program:
x := 0 ;
y := 1 ;
z := x + y
When we evaluate this program in a state s, we begin by evaluating the assignment x := 0, giving a new state s[0/x], and the "remainder"
y := 1 ;
z := x + y
still to be evaluated; the next assignment, y := 1, is then evaluated, giving a state s[0/x][1/y], and the remaining program
z := x + y
still to be evaluated; and so on until the program is fully evaluated. We can give an alternative semantics that describes the individual steps in a computation by defining a "one-step" evaluation
relation (c,s) -1-> (c',s') which says that if we evaluate a program c in a state s, then after one step of the evaluation, we obtain a new state s', and the program c' remains to be evaluated. For
example, we would like to have:
(x := 0 ; y := 1 ; z := x+y, s) -1-> (y := 1 ; z := x+y, s[0/x]).
Of course, some programs, such as single assignments, can be fully evaluated in just one step, in which case, we will write (c,s) -1-> s', which means "the program c is fully evaluated in one step,
starting in state s and giving a new state s' as a result". We would then be able to chain these relations together to obtain our original operational semantics for IMP; for example:
(x := 0 ; y := 1 ; z := x+y, s)
(y := 1 ; z := x+y, s[0/x])
(z := x+y, s[0/x][1/y])
Clearly, we would also like to have
(c0,s0) -1-> (c1,s1) -1-> ... -1-> (cn,sn) -1-> s
if and only if (c0,s0) --> s (this will, in fact, be the case, although we won't prove it: a proof is given in Winskel's book).
Before giving the one-step evaluation relation for programs, we give one-step evaluation relations for arithmetic expresions and for Boolean expressions. The idea is that arithmetic expressions are
evaluated in a number of steps, beginning with the leftmost innermost subexpression. For example, if the state s gives the value of x as 5 and the value of y as 3 (i.e., s(x) = 5 and s(y) = 3), then
the expression (x + 1) * (2*y + 1) would be evaluated as follows:
((x + 1) * (2*y + 1), s)
((5 + 1) * (2*y + 1), s)
(6 * (2*y + 1), s)
(6 * (2*3 + 1), s)
(6 * (6 + 1), s)
(6 * 7, s)
An inductive definition of this relation is:
1. For numbers n,
(n,s) -1-> n .
2. For variables x,
(x,s) -1-> s(x) .
3. For expressions a1 and a2:
If (a1,s) -1-> (a1',s) then (a1+a2,s) -1-> (a1'+a2,s) .
If (a1,s) -1-> m then (a1+a2,s) -1-> (m+a2,s) .
If (a2,s) -1-> (a2',s) then (m+a2,s) -1-> (m+a2',s)
If (a2,s) -1-> n then (a1+a2,s) -1-> m+n .
Here, m and n are numbers, i.e., fully evaluated expressions. These four rules say that the left subexpression (a1) is evaluated first, and the second (a2) is evaluated only once the first is
fully evaluated (to a number m); finally, when both a1 and a2 are fully evaluated (to the numbers m and n), the result of evaluating the expression is the number m+n.
The cases of expressions a1 - a2 and a1 * a2 are similar to the above.
A one-step evaluation relation for Boolean expressions can also be defined similarly.
Now the one-step evaluation relation for programs is defined inductively as follows:
1. The first case says that skip is evaluated in one step:
(skip,s) -1-> s .
2. Assignments are evaluated by first evaluating the expression on the left of the := operator:
If (a,s) -1-> (a',s) then (x := a, s) -1-> (x := a', s) .
Assignments are then evaluated in one step once the expression is fully evaluated to a number n:
If (a,s) -1-> n then (x := a, s) -1-> s[n/x] .
3. For sequential composition:
If (c1,s) -1-> (c1',s') then (c1;c2, s) -1-> (c1';c2, s') .
If (c1,s) -1-> s' then (c1;c2, s) -1-> (c2, s') .
4. Conditionals begin by evaluating the test:
If (b,s) -1-> (b',s) then (if b then c1 else c2, s) -1->, (if b' then c1 else c2, s) .
If (b,s) -1-> true then (if b then c1 else c2, s) -1-> (c1,s) .
If (b,s) -1-> false then (if b then c1 else c2, s) -1-> (c2,s) .
5. The case of while loops is left as an exercise.
What about side-effects?
In the first operational semantics and in the one-step operational semantics, the evaluation of expressions does not change the state. In this sense, IMP has no side-effects. Many languages, however,
do allow side-effects; for example, in C and Java, the expression x++ evaluates to the value of the variable x, but has the side-effect of incrementing the value stored in this variable. Thus, in a
state s with s(x) = 3, the expression
(2 * x++) + 1
would evaluate to 7, but the state would be updated to s[4/x]. It is straightforward to extend the syntax of IMP to allow expressions with side-effects; we extend the BNF syntax with
<Aexp> ::= ... | <Loc>++
and we give a semantics for this operator by adding the following rule to the inductive definition of the one-step evaluation relation for arithmetic expressions:
(x++, s) -1-> (s(x), s[(s(x)+1)/x]) .
1. Complete the definition of the evaluation relation for arithmetic expressions by writing out clause (5) as fully as possible.
2. The example of the evaluation of the arithmetic expression (x+1)*(2*y+1) is really a proof that ((x+1)*(2*y+1),s) --> 42 when s(x)=5 and s(y)=3. In the lectures, we saw how to write such proofs
in "tree form". Rewrite the example proof that ((x+1)*(2*y+1), s) --> 42) in tree form.
3. Complete the definition of the evaluation relation for Boolean expressions by writing out the required clauses (this differs from Exercise 1 in that you should also know what clauses are required
for an inductive definition). (And yes, the answer to this exercise has been given in the lectures; one point of this exercise is to test how good your lecture notes are.)
4. Use the tree-form style to discover the state that results from running the program
x := 1 ; while not(x <= 0) do x := x-1 .
5. Write out an inductive definition for the one-step evaluation relation for Boolean expressions. Does your definition give "lazy" logical operators? If not, give an alternative semantics that does
give lazy operators.
6. Complete the definition of the one-step evaluation relation for programs by giving the rule for while-loops.
7. Extend the syntax and operational semantics of IMP with an if-then construct, so that a program of the form
if b then c
is equivalent to
if b then c else skip
Show that these two programs are actually equivalent.
8. Extend the syntax and (one-step) semantics of IMP with a pre-increment operator ++<Loc> (the value of the expression ++x is the value of x plus one, and has the side-effect of incrementing x.
9. Some languages, such as Charity, are non-deterministic, in that the programmer can specify a non-deterministic choice of programs. Such programs are evaluated by choosing (randomly) one of the
given programs. We write c1<>c2 to denote a non-deterministic choice between programs c1 and c2; this program is evaluated by either evaluating c1 or by evaluating c2, with no way of knowing
beforehand which choice will be made. For example, the program
x := 0 <> x := 1
Will assign either 0 or 1 to x. Extend the syntax of IMP with a non-deterministic choice <Com> <> <Com>. Give an operational semantics for this by extending the inductive definition of the
evaluation relation (c,s) --> s'. Also, give an extension of the one-step semantics.
10. Some languages allow "assertions" that, if false, will cause a run-time error. If b is a Boolean expression, then the assertion {{b}} has no effect if b is true, but causes the program to fail if
b is false. An operational semantics for this can be given by extending the inductive definition of the evaluation relation with:
({{b}}, s) --> s' if and only if (b,s) --> true and s = s' .
Describe in words the behaviour of a program
({{b1}} ; c1) <> ({{b2}} ; c2) .
Review Questions
1. What do we mean when we say a definition is "inductive"?
2. Review the rules for "lazy" and. How do these rules actually determine an order of evaluation? Suppose we had the rule
If either (b1,s) --> false or (b2,s) --> false
then (b1 and b2, s) --> false .
What, if anything, does this say about the order of evaluation?
Grant Malcolm
Last modified: Tue Oct 8 00:02:15 BST 2002 | {"url":"http://cgi.csc.liv.ac.uk/~grant/Teaching/COMP317/opsem.html","timestamp":"2014-04-21T02:02:39Z","content_type":null,"content_length":"33314","record_id":"<urn:uuid:d80ab197-1948-4db2-b2d8-563e5223e8a6>","cc-path":"CC-MAIN-2014-15/segments/1397609539447.23/warc/CC-MAIN-20140416005219-00452-ip-10-147-4-33.ec2.internal.warc.gz"} |
MathGroup Archive: June 2006 [00117]
[Date Index] [Thread Index] [Author Index]
Re: Re: Re: Homotopic algorithm to solve a system of equations
• To: mathgroup at smc.vnet.net
• Subject: [mg66942] Re: [mg66914] Re: [mg66875] Re: Homotopic algorithm to solve a system of equations
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Sun, 4 Jun 2006 02:01:19 -0400 (EDT)
• References: <e5mhml$kjl$1@smc.vnet.net> <200606020809.EAA18052@smc.vnet.net> <200606030726.DAA17296@smc.vnet.net> <2A4A2745-0F59-4312-B649-95C84A1AF645@mimuw.edu.pl>
• Sender: owner-wri-mathgroup at wolfram.com
I got a bit confused writing the stuff below, as often happens when I
try to write faster than I can think. Although the confusion does not
relate to the main point, I thought it would better to correct it so
are the Errata. What got me confused it the fact is that there is
more than one way to view a differential equation as an algebraic
equation. When we think of as an equation in the space of jets, each
partial derivative is viewed as an independent variable. But there is
also another approach : a partial derivative like D[f[x,y],{x,3}] is
replaced by x^3, a mixed derivative D[f[x,y],{x,1},{y,1}] by x y etc.
Both approaches turn a system of partial differential equations into
a system of polynomial equations but I don't think there is any
relationship between them. In the case of the Gromov's h-principle it
is the first approach that is relevant, and in the case of the
analogy between Janet and Groebner bases the second.
Andrzej Kozlowski
Tokyo, Japan
On 3 Jun 2006, at 21:38, Andrzej Kozlowski wrote:
> Homotopy (which is actually what I do during day hours ;-)) has
> nothing in particular to do with differential equations, or rather,
> differential equations are just one of many different kinds of
> mathematical objects to which the methods of homotopy theory can be
> applied. To speak of "homotopy" you simply need two topological
> spaces (they can be the same) and the notion of continuous maps
> between them. Two such maps are said to be homotopic if one can be
> "continuously deformed" into the other. For example, an example
> homotopy between two maps f and g between a topological space X and
> an affine (topological) space Y would be just the map F(x,t)= (1-t)
> g(x)+ t f(x). F is a map from the Cartesian product of X and the
> unit interval to Y, but you can think of it as a continuous family
> of maps F( ,t), such that F( ,0)= g() and F( ,1) = f( ). F is a
> very simple example of a homotopy. The basic idea of homotopy
> methods in problems like the one discussed by Etienne is that to
> solve a certain problem about the map f you try to find a homotopy
> between it and another map g, for which the problem is much simpler
> to solve. If the problem is "invariant under homotopy" (which is
> the kind of problem that algebraic topologists like myself usually
> study) than solving if for the simpler map will automatically
> produce a solution for the more complicated map. However, the
> problem of solving a polynomial equation f==0 is obviously not
> homotopy invariant: the roots of the equation g==0 will obviously
> not be the same as the roots of a "homotopic" equation f==0. So
> what one needs to do is to take an explicit homotopy F and study
> its zero set i.e. the set of points (x,t) such that F[x,t)==0.
> More precisely, we want to find a path x(t) such that F(x(t),t)==0
> and x(0) is the well known root of g==0. In that case x(1) will
> be a root of f=0. If we can find such a path path x(t) than the
> problem of finding a root of f will be solved.
> It is in the process of finding the path x(t) that ordinary
> differential equations come in. If the homotopy is "regular" than
> indeed the problem can be reduced to that of solving a system of
> ordinary differential equations. But actually that is not
> necessary: one can simply work with the original homotopy F(x(t),t)
> ==0 and try to solve it using an interative method: for example the
> Newton-Raphson. So differential equations are really quite
> incidental here. Homotopy techniques can be enormously powerful,
> particularly in the global setting, when we are considering
> equations on manifolds. Actually, systems of partial differential
> equations can be viewed simply as a special case of systems
> algebraic equations: but they have to be viewed as polynomial
> equations in the so called space (or bundle) of jets. In
> particular, all the apparatus of polynomial equation solving,
> including Groebner basis, etc. can be applied to system of PDEs
> ( One can even argue that the famous Buchberger algorithm for
> computing Groebner basis was first discovered in the PDE setting by
> Janet in 1920, that is 45 years before Buchberger).
> Now, once we come to the bundle of jets: homotopy methods shows
> there true power. Probably the most spectacular achievement is the
> work of Gromov on the so called h-principle (h for homotopy): among
> the vast number of things that can be deduced form it is Smale's
> famous proof that the 2-dimensional sphere in three space can be
> turned inside out by a regular homotopy (it will have to intersect
> itself in the process of doing so).
> Anyway, this is taking me too far afield, but the best answer to
> your question is, I think, : not necessarily.
> Andrzej Kozlowski
> On 3 Jun 2006, at 16:26, Chris Chiasson wrote:
>> Are you talking about using differential equations to go from the
>> solution of a system of equations to which one _does_ know the
>> answer
>> to the solution of a system of equations to which one _does not_
>> (yet)
>> know the answer?
>> On 6/2/06, aTn <ayottes at dms.umontreal.ca> wrote:
>>> Ok, some of you wrote back to me wondering if such algorithms
>>> exist and
>>> how they work. Here is a quick explanation.
>>> The methods I'm referring to are know as "probability one globally
>>> convergent homotopy algorithms" (see L. Waston's great introductory
>>> paper) that were developped to help solve a system of equations
>>> (polynomial with real coefficients).
>>> The basic result used in these algorithms is a parametrized
>>> version of
>>> Sards theorem:
>>> Let U be an open set in R^m and let p: U x R^n x [0,1[ ---> R^n be a
>>> C^2 map. If p is transversal to zero (i.e. 0 is a regular value
>>> of p),
>>> then the set of points 'v' in the open set U such that p_v : R^n x
>>> [0,1[ --> R^n : (x,t) ---> p(v,x,t) is transversal to 0, is of
>>> measure
>>> one.
>>> The way to apply this result to equation solving is the following.
>>> Consider a finite set of n polynomials with n indeterminates (and
>>> real
>>> coefficients). To it corresponds a (smooth) function F:R^n--->
>>> R^n. The
>>> idea is to find an open set U in R^m, a C^2 map p: U x R^n x [0,1
>>> [ -->
>>> R^n which is transversal to zero, such that a solution of p_v(x,
>>> 0) = 0
>>> is known explicitly for all v in U, p_v(x,1)= F(x) for all v in
>>> U and
>>> all x in R^n, and finally for which p_v^(-1)(0) is bounded for
>>> all v in
>>> U. Then (by the previous theorem) you can find a path in p_a^(-1)(0)
>>> for some 'v' for which the differential of p_v is of full rank
>>> linking
>>> a zero of p_v(. , 0) to a zero of p_v(. , 1) = F. We want the
>>> differential to be of full rank so we can use algorithms such as the
>>> Newton method to solve successive problems along the chosen path.
>>> I'm looking for a package implementing known probability one
>>> globally
>>> convergent homotopy algorithms. I hope this message makes my
>>> question
>>> clearer.
>>> Regards,
>>> Etienne (aTn)
>>> P.S.: I typed fast, so feel free to point out mystakes I might have
>>> made :)
>> --
>> http://chris.chiasson.name/
• References: | {"url":"http://forums.wolfram.com/mathgroup/archive/2006/Jun/msg00117.html","timestamp":"2014-04-16T07:26:50Z","content_type":null,"content_length":"43439","record_id":"<urn:uuid:e7ceefd0-b4e9-4512-91dd-8536230cdd78>","cc-path":"CC-MAIN-2014-15/segments/1398223210034.18/warc/CC-MAIN-20140423032010-00028-ip-10-147-4-33.ec2.internal.warc.gz"} |
QM - Part 1 - Eigenvalues of the Hermitian Operator A are real
March 16th 2011, 04:35 AM
So for orthonormality we also take the double assumption but in the sense the eigen values AND eigenkets are NOT equal
No, for the reality proof, you have the double assumption. For the orthogonality proof, you ONLY assume that the eigenvalues differ. Since eigenkets are, by definition, nonzero, proving
orthogonality (which follows only from the Hermitian nature of the operator and the differing eigenvalues) will, in effect, prove that the eigenkets are not equal. But you're proving that. Assume
as little as you need to, and prove as much as you can!
$a' eq a'' AND |a' \rangle eq |a'' \rangle \implies a'-a'' eq 0 \implies \langle a'|a'' \rangle =0 \implies |a' \rangle and |a'' \rangle$ are orthonormal
Cheers :-) | {"url":"http://mathhelpforum.com/advanced-applied-math/174330-qm-part-1-eigenvalues-hermitian-operator-real-2-print.html","timestamp":"2014-04-20T17:36:29Z","content_type":null,"content_length":"8624","record_id":"<urn:uuid:32a458f2-c8a0-4cc2-9039-b0b23a761e96>","cc-path":"CC-MAIN-2014-15/segments/1397609538824.34/warc/CC-MAIN-20140416005218-00109-ip-10-147-4-33.ec2.internal.warc.gz"} |
Finding Resistance
1. The problem statement, all variables and given/known data
I need to find to resistance and am having problem as I do not know KCL's. All I know is the total resistance in series or parallel. I have attached the image. Find resistance b/w the 2 ends.
2. Relevant equations
If you have Resnik & Halliday open page 723(Vol2). and see figure 31-41. My question is same with r = 20ohm & R=10ohm.
3. The attempt at a solution | {"url":"http://www.physicsforums.com/showthread.php?p=4204859","timestamp":"2014-04-17T09:58:24Z","content_type":null,"content_length":"26512","record_id":"<urn:uuid:d0eff3bf-38b0-49e9-a229-6b2005eb6634>","cc-path":"CC-MAIN-2014-15/segments/1397609527423.39/warc/CC-MAIN-20140416005207-00418-ip-10-147-4-33.ec2.internal.warc.gz"} |
Graphing Polynomial Functions with a Graphing Calculator
6.14: Graphing Polynomial Functions with a Graphing Calculator
Difficulty Level:
At Grade
Created by: CK-12
To make a fair race between a dragster and a funny car, a scientist devised the following polynomial equation:
$f(x) = 71.682x -60.427x^2 + 84.710x^3 -27.769x^4 + 4.296x^5 - 0.262x^6$
Source: http://ceee.rice.edu/Books/CS/chapter3/data1.html
Watch This
James Sousa: Ex: Solve a Polynomial Equation Using a Graphing Calculator (Approximate Solutions)
In the Quadratic Functions chapter, you used the graphing calculator to graph parabolas. Now, we will expand upon that knowledge and graph higher-degree polynomials. Then, we will use the graphing
calculator to find the zeros, maximums and minimums.
Example A
Graph $f(x)=x^3+x^2-8x-8$
Solution: These instructions are for a TI-83 or 84. First, press $Y=$ENTER. Now, in $Y1$$x^\land 3+x^\land 2-8x-8$GRAPH.
To adjust the window, press ZOOM. To get the typical -10 to 10 screen (for both axes), press 6:ZStandard. To zoom out, press ZOOM, 3:ZoomOut, ENTER, ENTER. For this particular function, the window
needs to go from -15 to 15 for both $x$$y$$Xmin, Xmax, Ymin,$$Ymax$
Example B
Find the zeros, maximum, and minimum of the function from Example A.
Solution: To find the zeros, press $2^{nd}$TRACE to get the CALC menu. Select 2:Zero and you will be asked “Left Bound?” by the calculator. Move the cursor (by pressing the $\uparrow$$\downarrow$
ENTER. Then, it will ask “Right Bound?” Move the cursor just to the right of that zero. Press ENTER. The calculator will then ask “Guess?” At this point, you can enter in what you think the zero is
and press ENTER again. Then the calculator will give you the exact zero. For the graph from Example A, you will need to repeat this three times. The zeros are -2.83, -1, and 2.83.
To find the minimum and maximum, the process is almost identical to finding zeros. Instead of selecting 2:Zero, select 3:min or 4:max. The minimum is (1.33, -14.52) and the maximum is (-2, 4).
Example C
Find the $y-$
Solution: If you decide not to use the calculator, plug in zero for $x$$y$
$f(0) &= 0^3+0^2 - 8 \cdot 0 - 8\\&= -8$
Using the graphing calculator, press $2^{nd}$TRACE to get the CALC menu. Select 1:value. $X=$CLEAR to remove it. Then press 0 and ENTER. The calculator should then say “$Y=-8$
Intro Problem Revisit If you plug the equation $f(x) = 71.682x -60.427x^2 + 84.710x^3 -27.769x^4 + 4.296x^5 - 0.262x^6$$x = 6.15105$x, f(x) equals 1754.43. Thefore the maximum point of the function's
graph is (6.15105, 1754.43).
Guided Practice
Graph and find the critical values of the following functions.
1. $f(x)=-\frac{1}{3}x^4-x^3+10x^2+25x-4$
2. $g(x)=2x^5-x^4+6x^3+18x^2-3x-8$
3. Find the domain and range of the previous two functions.
4. Describe the types of solutions, as specifically as possible, for question 2.
Use the steps given in Examples $A, B$$C$
1. zeros: -5.874, -2.56, 0.151, 5.283
minimum: (-1.15, -18.59)
local maximum: (-4.62, 40.69)
absolute maximum: (3.52, 113.12)
2. zeros: -1.413, -0.682, 0.672
minimum: (-1.11, 4.41)
maximum: (0.08, -8.12)
3. The domain of #1 is all real numbers and the range is all real numbers less than the maximum; $(-\infty, 113.12]$
4. There are three irrational solutions and two imaginary solutions.
Graph questions 1-6 on your graphing calculator. Sketch the graph in an appropriate window. Then, find all the critical values, domain, range, and describe the end behavior.
1. $f(x)=2x^3+5x^2-4x-12$
2. $h(x)=-\frac{1}{4}x^4-2x^3-\frac{13}{4} x^2-8x-9$
3. $y=x^3-8$
4. $g(x)=-x^3-11x^2-14x+10$
5. $f(x)=2x^4+3x^3-26x^2-3x+54$
6. $y=x^4+2x^3-5x^2-12x-6$
7. What are the types of solutions in #2?
8. Find the two imaginary solutions in #3.
9. Find the exact values of the irrational roots in #5.
Determine if the following statements are SOMETIMES, ALWAYS, or NEVER true. Explain your reasoning.
10. The range of an even function is $(-\infty, max]$max is the maximum of the function.
11. The domain and range of all odd functions are all real numbers.
12. A function can have exactly three imaginary solutions.
13. An $n^{th}$$n$
14. The parent graph of any polynomial function has one zero.
15. Challenge The exact value for one of the zeros in #2 is $-4+\sqrt{7}$
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[SOLVED] Mutually independent
February 2nd 2009, 12:14 PM #1
Jul 2008
[SOLVED] Mutually independent
Let the three mutually independent events $C_{1},C_{2},C_{3},$ be such that $P(C_{1}) = P(C_{2}) = P(C_{3}) = \frac{1}{4}$.
Find $P[(C_{1}^c \cap C_{2}^c) \cup C_{3}]$.
By the definition of mutually independent events,
$P(C_{1} \cap C_{2} \cap C_{3}) = P(C_1) P(C_2) P(C_3)$.
Since they are mutually independent events, they are also pairwise e.g.
$P(C_1 \cap C_3) = P(C_1) P(C_3)$
So, by the distribution property i.e. $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$, then
$P[(C_{1}^c \cap C_{2}^c) \cup C_{3}] = P[(C_{1}^c \cup C_3 ) \cap (C_2^c \cup C_3)]$
Then by DeMorgan's Laws,
$= P [(C_1 \cap C_{3}^c)^c \cap (C_2 \cap C_{3}^c)^c)]$
Since C1, C2, and C3 are mut. ind., then
$= P((C_1 \cap C_{3}^c)^c P((C_2 \cap C_{c}^c)$
But from here I do not know how to continue.
Thank you for your time.
Last edited by Paperwings; February 2nd 2009 at 12:28 PM.
Recall the that complements of independent events are also independent.
$P\left[ {\left( {A^c \cap B^c } \right) \cup C} \right] = P\left( {A^c \cap B^c } \right)$$+ P(C) - P\left[ {\left( {A^c \cap B^c } \right) \cap C} \right] = P\left( {A^c } \right)P\left( {B^c }
\right) + P(C) - P\left( {A^c } \right)P\left( {B^c } \right)P\left( C \right)$
Ah, yes. I completely forgot about that. Thank you.
February 2nd 2009, 01:11 PM #2
February 2nd 2009, 01:34 PM #3
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Math Forum Discussions - Re: ZF - Powerset + Choice = ZFC ?
Date: Jan 12, 2013 11:36 AM
Author: ross.finlayson@gmail.com
Subject: Re: ZF - Powerset + Choice = ZFC ?
On Jan 8, 6:45 pm, "Ross A. Finlayson" <ross.finlay...@gmail.com>
> On Jan 7, 7:13 pm, "Ross A. Finlayson" <ross.finlay...@gmail.com>
> wrote:
> > On Jan 7, 2:30 pm, Dan Christensen <Dan_Christen...@sympatico.ca>
> > wrote:
> > > On Monday, January 7, 2013 4:51:47 PM UTC-5, david petry wrote:
> > > > Once again, I recommend that you read Nik Weaver's article.
> > > IIUC, he hopes to do mathematics (e.g. real analysis) without sets (or any equivalent notion). If he wants to be taken seriously, he should just go ahead and do so. I have tried to do so for a number of years myself to no avail. I don't much care for the ZF axioms of regularity and infinity myself. I haven't found any use for them in my own work, and haven't incorporated them (or any equivalent) in my own simplified set theory. But I really don't see how you can do foundational work without a powerset axiom.
> > > Dan
> > > Download my DC Proof 2.0 software athttp://www.dcproof.com
> > Doesn't it follow from pairing and union?
> > Basically we know that any element of a set is a set via union.
> > Then, each of those as elements as a singleton subset, is a set, as
> > necessary via inductive recursion and building back up the sets.
> > Via pairing, the two-elements subsets are sets, via pairing, the three
> > element sets are sets, ..., via induction, each of the subsets are
> > sets.
> > Then all the subsets are each sets, via pairing, that's a set.
> > It follows from pairing and union.
> Then, here it seems that there is a ready enough comprehension of the
> elements of the set, composed in each way, composing a set, that would
> be the set of subsets of a set, or its powerset, without axiomatic
> support, instead as a theorem of pairing and union, and into strata
> with countable or general choice.
> In a set theory, then what sets have powersets that don't exist via
> union and pairing? In a theory without well-foundedness perhaps those
> that are irregular, yet then the transitive closure would simply be
> irregular too and the powerset would be simply enough constructed (via
> induction and for the infinite, transfinite induction).
> So, what sets have powersets not constructible as the result of
> induction over union and pairing, in ZF - Powerset or ZFC - Powerset?
So, why axiomatize powerset if it's not an independent axiom?
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A Numerical Model of the Wave-Induced Currents in the Turbulent Coastal Zone
ISRN Civil Engineering
Volume 2013 (2013), Article ID 904180, 7 pages
Research Article
A Numerical Model of the Wave-Induced Currents in the Turbulent Coastal Zone
Faculty of Engineering, Cairo University, Giza 12613, Egypt
Received 16 February 2013; Accepted 24 March 2013
Academic Editors: I. Raftoyiannis and I. Smith
Copyright © 2013 O. Fahmy et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly cited.
A numerical model is developed, validated and applied to the turbulent coastal currents. The currents are driven by the sea surface slope and the radiation stresses of water waves. They are resisted
by friction due to turbulent eddies and sea bottom. The k-ε model is used to model the turbulent stresses. Five simultaneous nonlinear partial differential equations govern the depth-averaged
dynamics in the surf zone. An implicit finite-difference scheme is used to obtain an accurate numerical solution of the resulting initial-boundary value problem. It is tested against the case of
straight coast with uniform bottom slope and a protective jetty. To investigate the actual wave-induced currents, the model is applied to simulate the currents for three real case studies. Results
show that the model could be used to compute currents caused by the constructing coastal protection measures and could predict the locations of accretion and scouring.
1. Introduction
As surface waves progress onshore, through the nearshore zone, they transform due to the combined effect of ambient currents and bottom variations. As they approach the surfzone—between the shoreline
and the breaker line—the energy of the broken waves is transformed into radiation stresses, which drive turbulent currents. The effects of these currents on exciting bottom sediments and transporting
them offshore and longshore were studied by Martinez and Harbaugh [1].
Fahmy [2] showed that if the horizontal dimensions of the flow domain are much larger than the vertical dimension, the depth-averaged and wave period-averaged mass and momentum equations can be used
to represent the flow. As a result of large horizontal scales in the surf zone area, the flow can be considered turbulent even for very small velocity values. There are two approaches of mathematical
models of wave-induced current: the first is based on the radiation stress theory, while the second is based on the Boussinesq equation. Recently, many researchers dealt with this topic. Significant
publications include the one of Mengguo and Chongren [3]. They used a simplified model for the resisting force caused by eddy viscosity. Kim and An [4] presented a numerical model that takes the
wave-current interaction into consideration and quasi-3D equations were used in circulation analyses.
2. Model Theoretical Formulation
2.1. Model Assumptions
The depth-average shallow water model equations (SWE) are invoked for coastal regions where hydrostatic approximation is valid [5]. These equations are based on the following assumptions. (i) The
flow is incompressible. (ii) The vertical acceleration and Coriolis acceleration are neglected. We further assume that the two driving forces of wind stress and horizontal density gradient are
2.2. Model Governing Equations
Consider the turbulent flow in the surf zone between the shoreline and the breaker line (Figure 1). The currents are affected by four dynamical forces. These are categorized as follows:(a)two driving
forces of water wave radiation stress and surface slope ,(b)two resisting forces of bottom friction and friction due to turbulent eddies . The SWE take the form where is the depth-average speed, is
the turbulent eddies frictional stress tensor, is the water depth, is the water desist, is the water surface elevation, , is the time, is acceleration of gravity, are the offshore and longshore
coordinates, respectively, and is the radiation stress tensor defined by Dean and Dalrymble [6].
2.3. Model of Driving Forces
For a surface gravity wave incident with an angle , energy , and group velocity-celerity ration , The gradients of the radiation stress components provide driving forces in the nearshore zone. For
waves propagating obliquely into the surf zone, breaking will result in a reduction in wave energy and an associated decrease in , which is manifested as a wave thrust . The components of are
2.4. Model of Resisting Forces
2.4.1. The Bottom Stress
Consider the following equation: where Chezy coefficient, given by Handreson [7], and is the eddy viscosity
2.4.2. The Turbulent Stress: The Model
To evaluate the turbulent stresses, the depth averaged eddy viscosity is calculated from the transport equation of the depth average model, Rastogi and Rodi [8], where is the turbulent kinetic
energy, is the time rate of dissipation of , and is the production of due to vertical velocity gradient is the production of due to vertical velocity gradient and , , , , , , and are constants given
by Rodi [9]; see Table 1.
is the turbulent energy production due to mean motion is the shear velocity
These are five simultaneous nonlinear partial differential equations (1)–(8). They are continuity equation and momentum transport equations, with the turbulence model equations being the governing
equations that are used to calculate the velocity vector in the surf zone. The two driving forces that appear in the momentum equations are the gradients of the wave's radiation stress and the mean
slope of sea level. To feed the model with the radiation stress components, the authors developed a wave model to study wave propagation from deep water to the shore line; see Fassieh et al. [10].
2.5. Boundaries of the Study Area
The surf zone is the flow region that follows the breaking zone where water waves break and transform their energy into turbulent energy whose momentum flux derives coastal currents. The surf zone is
bounded by the shore line and three open boundaries, an offshore boundary and two lateral boundaries (see Figure 1). These open boundaries are artificial boundaries through which water flows and are
needed for reasons of computational economy. The off shore boundary is placed before waves breaking by about one wave length. The lateral boundaries are perpendicular to the shore line (too far from
the study area to affect calculations).
2.6. Boundary Conditions
At the shore line boundary, the normal velocity component vanishes:
At the offshore boundary, Dirichlet boundary conditions are imposed via the wave model. The water level is taken to be equal to the local mean sea level. The steady (depth and period-averaged)
velocity components due to mass flux defined from the wave model are At the two lateral boundaries, the velocity gradient in the normal direction (-direction) is taken to be zero to ensure their
perfect nonreflective character,
2.7. Initial Conditions
The water surface in the study area is assumed initially to be horizontal (zero surface displacement with no setup or setdown). Since the bottom topography is known, the local water depth is given
initially at each point. The initial velocity distribution is chosen such that the mass conservation equation is satisfied everywhere, that is, at each grid point; see Fahmy [11].
2.8. Numerical Solution
The discrete form of any variable in the () space will be , where in which , are the grid spacing in and directions, respectively, and are the grid point labels in and directions respectively, is
the time step, and is the time level.
2.9. The Computational Grid
The transport equations are discretized over a staggered uniform rectangular mesh; see Anderson et al. [12]. The staggered mesh is superimposed on the area under study as shown in Figure 2. In the
staggered grid technique, the domain is divided into control volumes. The scalar variables , , ε, , , , and are defined at the center of the cell, so that the transport equations of the scalar
variables and and the continuity equation are discretized at the point (), which is the center of the control volume. The velocity component u is defined at the two -constant boundaries of the
control volume, so that the -direction momentum equation is discretized at point (). The velocity component is defined at the two -constant boundaries of the control volume, so that the -direction
momentum equation is discretized at point (). The boundaries of the area under study lie at the control-volume boundaries to form a stair-step outer boundary.
An implicit scheme is used to discretize the five governing equations (1)–(8). Centered space difference is used for spatial derivatives, and backward difference is used for time derivatives. Upwind
difference is used to discretize the convective terms. The various terms of the governing equations are discretized at the time level . The produced system of algebraic equations has the following
form: The coefficients contain viscosity, convective velocities from both directions, and water depth. The terms are the source terms for each variable. Starting with the initial velocity and water
depth, the linear algebraic equations are solved by iteration to get values at the next time step using Gauss Seidel method. The calculated values from the previous time step are taken to be initial
guesses for the time next step.
3. Model Verification and Applications
Given the basin lateral dimensions and bottom topography, a Cartesian mesh is superimposed to define the computational domain. The water depth is assigned at all the grid nodes of the mesh covering
the domain. The wave model [10] simulates the manner by which the incident wave characteristics in deep water are transformed through wave refraction, shoaling, diffraction, and breaking as the wave
propagates over the water surface of a basin-varying bottom topography. The wave model is used to compute wave height and wave orthogonal direction at all the grid nodes. These results are used to
compute the components of the radiation stress tensor and the particle velocity components at each grid point. This surf zone model uses the radiation stress components as source terms that represent
wave forces, for the depth-averaged period-averaged momentum equations. The numerical solution of the coupled system of the five nonlinear partial differential equations of momentum, mass, turbulent
energy, and eddy viscosity produces the surf zone velocity field.
3.1. Nearshore Currents around a Protective Jetty
Consider a square basin of one dimensional uniform slope depth, whose formula is as follows: The basin dimensions are 6m 6m. A square computational mesh whose grid size is m, is imposed on the
basin. This grid size gives total of () grid points at which the bottom depth array, , is defined. A 1.5m long jetty is situated perpendicular to the shoreline, as shown in Figure 3. Regular surface
gravity waves are obliquely incident at 30° to the shoreline, with height 1.8cm and wave period 1.2 seconds. Upon breaking, part of their energy is dissipated and the reformed part is transformed
into an induced nearshore current that moves around the jetty. An infinitely long straight shoreline is imposed for the case study of the boundary condition utilizing a pure longshore current.
The same case study was studied by Nishimura et al. [13] and Watanabe and Maruyama [14]. They also built a physical model and took measurements for the longshore currents around the jetty. Figure 3
shows the computed longshore current obtained by Nishimura et al. [13]. The observed current velocity vectors (measured in the physical model) are plotted in Figure 4. The sets of parallel arrows in
the figure denote longshore components of local velocities measured by using small current meters. Figure 5 shows the current field computed by this model. The velocity vectors show that two
circulation zones are formed: one zone upstream of the jetty (the illuminated region) and another, much larger, circulation zone downstream the jetty (the shadow region). A fair agreement is found
between the predictions of this model and the physical model measurements. However, the present model results are more clear and accurate in simulating the current dynamics at the sharp edges of the
jetty and the narrow corners near the coast.
3.2. Application 1: Current System near the Nile Delta Coast
The coastal area extending from Gamasa to Ras EL-Bar (about 50km long) is covered by a mesh of grid size 500m 500m. The basin bathymetry is entered through digitization of the available contour
map for the Egyptian Nile Delta coast. Linear interpolation is used to determine water depth at the grid points lying between contour lines. The bathymetry of the study area is shown in Figure 6.
The random surface gravity waves propagating toward the Nile Delta coast have an average period of seconds and significant wave height that ranges from 1m. to 3m. coming from northwest direction
during summer and winter, while coming from north-northwest during fall season.
Figure 7 shows the velocity fields in the study area during winter season. The highest magnitude velocity vectors are observed around the head of Ras El-Bar.
3.3. Application 2: The Flow around Damietta Harbor
The flow around Damietta harbor is studied using a grid of 50.0m. 50.0m. The deep water wave height is 1 meter and coming from the northwest direction. The xx-component of the radiation stress
tensor is shown in Figure 8. The velocity vectors around the harbor are shown in Figure 9. The velocity vectors are shown to bend around the harbor walls with decreasing velocity in the front of the
main wall due to blocking which may cause accretion at the shore line near the main wall and increasing velocity vectors around the minor wall which may cause a scouring at the shore near the minor
3.4. Application 3: The Flow around Detached Break Waters
This case study is for a set of three detached break waters parallel to the shore at a distance of 200 meters from the shore line. The break waters were 200-meter long and 20.0-meter width, with 100
meters spacing between them. A rectangular mesh of 20.0 meters 5.0 meters is used to simulate the flow in the study area. The radiation stress component distribution is shown in Figure 10, while the
flow pattern is shown in Figure 11.
4. Conclusions
In the present study, a nonlinear model of the wave-induced unsteady currents in the turbulent coastal zone was presented. The main conclusions obtained from this study are as follows.(1)The present
model could be applied to accurately compute wave-induced unsteady currents within and outside of the surf zone.(2)The present model could be used to compute currents caused by the constructing
coastal protection measures like detached break waters parallel to the shore and protective jetties. (3)The highest magnitude velocities are observed around the head of Ras El-Bar which may cause a
scouring at this location.(4)The wave-induced current around Damietta harbor shows a decreasing velocity in the front of the main wall which may cause accretion at the shoreline near the main wall,
and increasing velocity around the minor wall which may cause scouring at the shore near the minor wall.
This work was performed as a part of a research project sponsored by the National Authority for Remote Sensing and Space Sciences (NARSS). The project title is “Assessment of the impact of
constructed coastal protection measures on the stability of the Nile Delta shoreline.”
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Wilson Geisler, Director SEA 4.328A, Mailcode A8000, Austin, TX 78712 • 512-471-5380
Brian Evans
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to the original. Lower resolution displays on cell phones and PDAs perform a similar operation. This operation to reduce resolution of gray levels or colors is called halftoning. Evans' group has
developed grayscale and color halftoning algorithms that are optimized for a model of visual quality. The halftoning algorithms are amenable for cost-effective implementation in printers and
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N. Damera-Venkata, B. L. Evans, and J. Tuqan, "Design of Optimum Multi-Dimensional Energy Compaction Filters", Proc. IEEE Asilomar Conf. on Signals, Systems, and Computers, Nov. 3-6, 2002, vol. 1,
pp. 350-354, Pacific Grove, CA USA.
Z. Wang, S. Banerjee, B. L. Evans, and A. C. Bovik, "Generalized Bitplane-by-Bitplane Shift Method for JPEG2000 ROI Coding", Proc. IEEE Int. Conf. on Image Processing, Sep. 22-25, 2002, vol. III, pp.
81-84, Rochester, NY USA.
D. Arifler and B. L. Evans, "Modeling the Self-Similar Behavior of Packetized MPEG-4 Video Using Wavelet-Based Methods", Proc. IEEE Int. Conf. on Image Processing, Sep. 22-25, 2002, vol. 1, pp.
848-851, Rochester, NY USA.
N. Damera-Venkata, V. Monga, and B. L. Evans, "Clustered-dot FM Halftoning Via Block Error Diffusion", IEEE Transactions on Image Processing, submitted August 7, 2002.
M. Ding, A. J. Redfern, and B. L. Evans, "A Dual-path TEQ Structure For DMT-ADSL Systems",
Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Proc., May 13{17, 2002, vol. III, pp. 2573{
2576, Orlando, FL.
S. Banerjee and B. L. Evans, "Tuning JPEG2000 Image Compression for Graphics Regions", Proc.
IEEE Southwest Symposium on Image Analysis and Interpretation, April 7{9, 2002, pp. 67{71, Santa
Fe, NM.
K. C. Slatton, M. M. Crawford, and B. L. Evans, "Sensitivity Analysis of a Spatially-Adaptive Estimator for Data Fusion", Proc. IEEE Southwest Symposium on Image Analysis and Interpretation, April
7-9, 2002, pp. 72-76, Santa Fe, NM.
W. Schwartzkopf, B. L. Evans, and A. C. Bovik, "Entropy Estimation for Segmentation of Multi-Spectral Chromosome Images", Proc. IEEE Southwest Symposium on Image Analysis and Interpretation, April
7-9, 2002, pp. 234-238, Santa Fe, NM.
G. Arslan, B. L. Evans, and S. Kiaei, "Equalization for Discrete Multitone Receivers To Maximize Bit Rate", IEEE Transactions on Signal Processing, vol. 49, no. 12, pp. 3123-3135, Dec. 2001.
G. Arslan, B. L. Evans, and S. Kiaei, "Equalization for Discrete Multitone Receivers To Maximize Bit Rate", IEEE Transactions on Signal Processing, vol. 49, no. 12, pp. 3123-3135, Dec. 2001.
W. Schwartzkopf, B. L. Evans, and A. C. Bovik, ``Minimum Entropy Segmentation Applied to Multi-Spectral Chromosome Images'', Proc. IEEE Int. Conf. on Image Processing, Oct. 7-10, 2001, accepted for
N. Damera-Venkata and B. L. Evans, ``FM Halftoning Via Block Error Diffusion'', Proc. IEEE Int. Conf. on Image Processing, Oct. 7-10, 2001, accepted for publication.
N. Damera-Venkata and B. L. Evans, ``Color Error Diffusion with Generalized Optimum Noise Shaping'', Proc. IEEE Int. Conf. on Image Processing, Oct. 7-10, 2001, accepted for publication.
N. Damera-Venkata and B. L. Evans, "Design and Analysis of Vector Color Error Diffusion Halftoning Systems", IEEE Transactions on Image Processing, vol. 10, no. 10, pp. 1552-1565, Oct. 2001.
K. C. Slatton, M. M. Crawford, and B. L. Evans, ``Multiscale Adaptive Estimation for Fusing Interferometric Radar and Laser Altimeter Data'', Proc. IEEE Int. Geoscience and Remote Sensing Sym., Jul.
9-13, 2001, Sydney, Australia, accepted for publication.
N. Damera-Venkata and B. L. Evans, ``Matrix Gain Model for Vector Color Error Diffusion'', Proc. IEEE-EURASIP Workshop on Nonlinear Signal and Image Processing, June 3-5, 2001, Baltimore, MD, invited
H. R. Sheikh, S. Liu, B. L. Evans and A. C. Bovik, ``Real-Time Foveation Techniques for H.263 Video Encoding in Software'', Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Proc., May 7-11,
2001, Salt Lake City, UT, accepted for publication.
N. Damera-Venkata and B. L. Evans, "Adaptive Threshold Modulation for Error Diffusion Halftoning", IEEE Transactions on Image Processing, vol. 10, no. 1, pp. 104-116, Jan. 2001.
S. Banerjee, L. K. John, and B. L. Evans, ``The EASE Branch Predictor'', Proc. Int. Conf. on Communications, Computers, and Devices, Dec. 14-16, 2000, vol. 1, pp. 59-62, Kharagpur, India.
J. Wu, G. Arslan, and B. L. Evans, ``Efficient Matrix Multiplication Methods to Implement a Near-Optimum Channel Shortening Method for Discrete Multitone Transceivers'', Proc. IEEE Asilomar Conf. on
Signals, Systems, and Computers, Oct. 29-Nov. 1, 2000, vol. 1, pp. 152-157, Pacific Grove, CA.
S. Banerjee, H. R. Sheikh, L. K. John, B. L. Evans, and A. C. Bovik, "VLIW DSP vs. Superscalar Implementation of a Baseline H.263 Video Encoder'', Proc. IEEE Asilomar Conf. on Signals, Systems, and
Computers, Oct. 29-Nov. 1, 2000, vol. 2, pp. 1665-1669, Pacific Grove, CA.
Y. H. Cho, D. Brunke, G. E. Allen, and B. L. Evans, ``Optimization of Vertical and Horizontal Beamforming Kernels on the PowerPC G4 Processor with AltiVec Technology'', Proc. IEEE Asilomar Conf. on
Signals, Systems, and Computers, Oct. 29-Nov. 1, 2000, vol. 2, pp. 1670-1674, Pacific Grove, CA.
B. L. Evans and G. Arslan, ``A Signal Processing System-Level Design Course'', IEEE Signal Processing Education Workshop, Oct. 15-18, 2000, Hunt, Texas, invited paper.
B. Lu, L. D. Clark, G. Arslan, and B. L. Evans, "Fast Time-Domain Equalization for Discrete Multitone Modulation Systems'', IEEE Digital Signal Processing Workshop, Oct. 15-18, 2000, Hunt, Texas.
D. Talla, L. K. John, V. Lapinskii, and B. L. Evans, "Evaluating Signal Processing and Multimedia Applications on SIMD, VLIW and Superscalar Architectures'', Proc. IEEE Int. Conf. on Computer Design,
Sep. 17-20, 2000, pp. 163-172, Austin, TX.
Z. Wang, A. C. Bovik, and B. L. Evans, "Blind Measurement of Blocking Artifacts in Images'', Proc. IEEE Int. Conf. on Image Processing, Sep. 10-13, 2000, vol. III, pp. 981-984, Vancouver, Canada.
T. D. Kite, N. Damera-Venkata, B. L. Evans, and A. C. Bovik, "A Fast, High-Quality Inverse Halftoning Algorithm for Error Diffused Halftones", IEEE Transactions on Image Processing, vol. 9, no. 9,
pp. 1583-1592, Sep. 2000.
H. R. Sheikh, S. Banerjee, B. L. Evans, and A. C. Bovik, "Optimization of a Baseline H.263 Video Encoder on the TMS320C6x'', Proc. Texas Instruments DSP Educator's Conference, Aug. 2-4, 2000,
Houston, TX, 3 pages.
K. C. Slatton, M. M. Crawford, and B. L. Evans, "Combining Interferometric Radar and Laser Altimeter Data to Improve Topography Estimates'', Proc. IEEE Int. Geoscience and Remote Sensing Sym., July
24-28, 2000, vol. 3, pp. 960-962, Honolulu, Hawaii.
M. Torlak and B. L. Evans, "Self-Recovering RAKE Receiver for Asynchronous CDMA Systems'', Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Proc., June 5-9, 2000, vol. 5, pp. 2869-2872,
Istanbul, Turkey.
G. Arslan, B. L. Evans, and S. Kiaei, ``Optimum Channel Shortening for Discrete Multitone Transceivers'', Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Proc., June 5-9, 2000, vol. 5, pp.
2965-2968, Istanbul, Turkey.
N. Damera-Venkata and B. L. Evans, "Parallel Implementation of Multifilters'', Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Proc., June 5-9, 2000, vol. 6, pp. 3335-3338, Istanbul, Turkey.
M. Valliappan, B. L. Evans, M. Gzara, M. D. Lutovac, and D. V. Tosic, "Joint Optimization of Multiple Behavioral and Implementation Properties of Digital IIR Filter Designs'', Proc. IEEE Int. Sym. on
Circuits and Systems, May 28-31, 2000, vol. 4, pp. 77-80, Geneva, Switzerland.
J. I. Kim, A. C. Bovik, and B. L. Evans, "Generalized Predictive Binary Shape Coding Using Polygon Approximations", Signal Processing: Image Communication, vol. 15, no. 7-8, pp. 643-663, May 2000.
N. Damera-Venkata, B. L. Evans, and S. R. McCaslin, "Design of Optimal Minimum Phase Digital FIR Filters Using Discrete Hilbert Transforms", IEEE Transactions on Signal Processing, vol. 48, no. 5,
pp. 1491-1495, May 2000.
T. D. Kite, B. L. Evans, and A. C. Bovik, "Modeling and Quality Assessment of Halftoning by Error Diffusion", IEEE Transactions on Image Processing, vol. 9, no. 4, pp. 909-922, May 2000.
K. C. Slatton, M. M. Crawford, and B. L. Evans, "Improved Accuracy for Interferometric Radar Images Using Polarimetric Radar and Laser Altimetry Data'', Proc. IEEE Southwest Symposium on Image
Analysis and Interpretation, April 2-4, 2000, pp. 156-160, Austin, TX.
W. Schwartzkopf, J. Ghosh, T. E. Milner, B. L. Evans, and A. C. Bovik, "Two-Dimensional Phase Unwrapping Using Neural Networks'', Proc. IEEE Southwest Symposium on Image Analysis and Interpretation,
April 2-4, 2000, pp. 274-277, Austin, TX.
N. Damera-Venkata, T. D. Kite, W. S. Geisler, B. L. Evans, and A. C. Bovik, "Image Quality Assessment Based on a Degradation Mode", IEEE Transactions on Image Processing, vol. 9, no. 4, pp. 636-650,
Apr. 2000.
A. Deosthali, S. R. McCaslin, and B. L. Evans, "A Low-Complexity ITU-Compliant Dual Tone Multiple Frequency Detector", IEEE Transactions on Signal Processing, vol. 48, no. 3, pp. 911-916, Mar. 2000.
G. E. Allen and B. L. Evans, "Real-Time Sonar Beamforming on Workstations Using Process Networks and POSIX Threads", IEEE Transactions on Signal Processing, vol. 48, no. 3, pp. 921-926, Mar. 2000.
G. E. Allen, B. L. Evans, and L. K. John, "Real-Time High-Throughput Sonar Beamforming Kernels Using Native Signal Processing and Memory Latency Hiding Techniques'', Proc. IEEE Asilomar Conf. on
Signals, Systems, and Computers, Oct. 25-28, 1999, vol. I, pp. 137-141, Pacific Grove, CA.
G. Arslan, M. Valliappan, and B. L. Evans, "Quality Assessment of Compression Techniques for Synthetic Aperture Radar Images'', Proc. IEEE Int. Conf. on Image Processing, Oct. 25-28, 1999, vol. III,
pp. 857-861, Kobe, Japan.
T. D. Kite, B. L. Evans, and A. C. Bovik, "Fast Rehalftoning and Interpolated Halftoning Algorithms with Flat Low-Frequency Response'', Proc. IEEE Int. Conf. on Image Processing, Oct. 25-28, 1999,
vol. III, pp. 602-606, Kobe, Japan.
M. Milosevic, W. Schwartzkopf, T. E. Milner, B. L. Evans, and A. C. Bovik, ``Low-Complexity Velocity Estimation in High-Speed Optical Doppler Tomography Systems'', Proc. IEEE Int. Conf. on Image
Processing, Oct. 25-28, 1999, vol. II, pp. 658-662, Kobe, Japan.
M. Valliappan, B. L. Evans, D. A. D. Tompkins, and F. Kossentini, "Lossy Compression of Stochastic Halftones with JBIG2'', Proc. IEEE Int. Conf. on Image Processing, Oct. 25-28, 1999, vol. I, pp.
214-218, Kobe, Japan, invited paper.
D. V. Tosic, M. D. Lutovac, and B. L. Evans, "EMQF Filter Design in MATLAB'', Proc. IEEE Int. Conf. on Telecommunications in Modern Satellite Cable and Broadcasting Services, Oct. 13-15, 1999, Nis,
Yugoslavia, pp. 125-128.
S. Lee, A. C. Bovik, and B. L. Evans, "Efficient Implementation of Foveation Filtering'', Proc. Texas Instruments DSP Educator's Conference, Aug. 3-8, 1999, Houston, TX, 5 pages.
N. Damera-Venkata and B. L. Evans, "An Automated Framework for Multicriteria Optimization of Analog Filter Designs", IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing
, vol. 46, no. 8, pp. 981-990, Aug. 1999.
D. V. Tosic, M. D. Lutovac, and B. L. Evans, "Advanced Digital IIR Filter Design'', Proc. European Conference on Circuit Theory and Design, Aug. 1999, Stresa, Italy, pp. 1323-1326.
B. L. Evans and G. Arslan, "Raising the Level of Abstraction: A Signal Processing System Design Course'', Proc. IEEE-EURASIP Workshop on Nonlinear Signal and Image Processing, June 20-23, 1999,
Antalya, Turkey, invited paper, vol. II, pp. 569-573.
G. Arslan, F. A. Sakarya, and B. L. Evans, "Speaker Localization for Far-field and Near-field Wideband Sources Using Neural Networks'', Proc. IEEE-EURASIP Workshop on Nonlinear Signal and Image
Processing, June 20-23, 1999, Antalya, Turkey, vol. II, pp. 528-532.
B. Lu and B. L. Evans, "Channel Equalization by Feedforward Neural Networks'', Proc. IEEE Int. Sym. on Circuits and Systems, May 31-Jun. 2, 1999, Orlando, FL, vol. 5, pp. 587-590.
N. Damera-Venkata and B. L. Evans, "Optimal Design of Real and Complex Minimum Phase Digital FIR Filters'', Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Proc., Mar. 15-19, 1999, Phoenix,
AZ, vol. 3, pp. 1145-1148.
R. Bhargava, L. K. John, B. L. Evans, and R. Radhakrishnan, "Evaluating MMX Technology Using DSP and Multimedia Applications'', Proc. ACM/IEEE Int. Sym. on Microarchitecture, Nov. 30-Dec. 2, 1998,
Dallas, TX, pp. 37-46.
J.I. Kim and B. L. Evans, "System Modeling and Implementation of a Generic Video Codec'', Proc. IEEE Workshop on Multimedia Signal Processing, Dec. 7-9, 1998, Los Angeles, CA, pp. 311-316.
G. E. Allen, B. L. Evans, and D. C. Schanbacher, "Real-Time Sonar Beamforming on a Unix Workstation Using Process Networks and POSIX Threads'', Proc. IEEE Asilomar Conf. on Signals, Systems, and
Computers, Nov. 1-4, 1998, Pacific Grove, CA, invited paper, vol. 2, pp. 1725-1729.
S. Gummadi and B. L. Evans, "Cochannel Signal Separation in Fading Channels Using a Modified Constant Modulus Array'', Proc. IEEE Asilomar Conf. on Signals, Systems, and Computers, Nov. 1-4, 1998,
Pacific Grove, CA, invited paper, vol. 1, pp. 764-768.
B. Lu, D. Wei, B. L. Evans, and A. C. Bovik, "Improved Matrix Pencil Methods'', Proc. IEEE Asilomar Conf. on Signals, Systems, and Computers, Nov. 1-4, 1998, Pacific Grove, CA, vol. 2, pp. 1433-1437.
D. V. Tosic, M. D. Lutovac, B. L. Evans, and I. M. Markoski, "A Tool for Symbolic Analysis and Design of Analog Active Filters'', Proc. Int. Workshop on Symbolic Methods and Applications in Circuit
Design, Oct. 8-9, 1998, Kaiserslautern, Germany, pp. 71-74.
N. Damera-Venkata, T. D. Kite, M. Venkataraman, and B. L. Evans, "Fast Blind Inverse Halftoning'', Proc. IEEE Int. Conf. on Image Processing, Oct. 4-7, 1998, Chicago, IL, vol. 2, pp. 64-68.
T. D. Kite, N. Damera-Venkata, B. L. Evans, and A. C. Bovik, "A High Quality, Fast Inverse Halftoning Algorithm for Error Diffused Halftoned Images'', Proc. IEEE Int. Conf. on Image Processing, Oct.
4-7, 1998, Chicago, IL, vol. 2, pp. 59-63.
R. Bhargava, R. Radhakrishnan, B. L. Evans, and L. John, "Characterization of MMX-enhanced DSP and Multimedia Applications on a General Purpose Processor'', Digest of Workshop on Performance Analysis
and Its Impact on Design, July 27-28, 1998, Barcelona, Spain, pp. 16-23.
M. D. Felder, J. C. Mason, and B. L. Evans, ``Efficient Dual-Tone Multifrequency Detection Using the Nonuniform Discrete Fourier Transform'', IEEE Signal Processing Letters, vol. 5, no. 7, pp.
160-163, Jul. 1998.
N. Damera-Venkata, B. L. Evans, M. D. Lutovac, and D. V. Tosic, "Joint Optimization of Multiple Behavioral and Implementation Properties of Analog Filter Designs'', Proc. IEEE Int. Sym. on Circuits
and Systems, May 31-Jun. 3, 1998, Monterey, CA. vol. 6, pp. VI-286-VI-289.
J.-I. Kim and B. L. Evans, "Predictive Shape Coding Using Generic Polygon Approximation'', Proc. IEEE Int. Sym. on Circuits and Systems, May 31-Jun. 3, 1998, Monterey, CA, invited paper, vol. 5, pp.
N. Damera-Venkata, S. Gummadi, and B. L. Evans, ``Comments on `Description of FIR Digital Filters in the form of Parallel Connection of Linear Phase FIRs' '', IEE Electronics Letters, vol. 34, no. 9,
pp. 866-867, Apr. 30, 1998.
M. Torlak, G. Xu, B. L. Evans, and H. Liu, ``Fast Estimation of Weight Vectors to Optimize Multi-Transmitter Broadcast Channel Capacity'', IEEE Transactions on Signal Processing, vol. 46, no. 1, pp.
243-246, Jan. 1998.
M. Torlak, B. L. Evans, and G. Xu, "Blind Estimation of FIR Channels in CDMA Systems with Aperiodic Spreading Sequences'', Proc. IEEE Asilomar Conf. on Signals, Systems, and Computers, Nov. 3-5,
1997, Pacific Grove, CA, vol. 1, pp. 495-499.
M. D. Lutovac, D. V. Tosic, and B. L. Evans, "Advanced Filter Design'', Proc. IEEE Asilomar Conf. on Signals, Systems, and Computers, Nov. 3-5, 1997, Pacific Grove, CA, vol. 1, pp. 710-715.
D. Wei, A. C. Bovik, and B. L. Evans, "Generalized Coiflets: A New Family of Orthonormal Wavelets'', Proc. IEEE Asilomar Conf. on Signals, Systems, and Computers, Nov. 3-5, 1997, Pacific Grove, CA,
vol. 2, pp. 1259-1263.
M. Ballan, F. A. Sakarya, and B. L. Evans, "A Fingerprint Classification Technique Using Directional Images'', Proc. IEEE Asilomar Conf. on Signals, Systems, and Computers, Nov. 3-5, 1997, Pacific
Grove, CA, vol. 1, pp. 101-104.
B. L. Evans, ``Designing Commutative Cascades of Multidimensional Upsamplers and Downsamplers'', IEEE Signal Processing Letters, vol. 4, no. 11, pp. 313-316, Nov. 1997.
T. D. Kite, B. L. Evans, A. C. Bovik, and T. L. Sculley, "Digital Halftoning As 2-D Delta-Sigma Modulation'', Proc. IEEE Int. Conf. on Image Processing, Oct. 26-29, 1997, Santa Barbara, CA, vol. I,
pp. 799-802.
D. Wei, B. L. Evans, and A. C. Bovik, ``Biorthogonal Quincunx Coifman Wavelets'', Proc. IEEE Int. Conf. on Image Processing, Oct. 26-29, 1997, Santa Barbara, CA, vol. II, pp. 246-249.
M. D. Lutovac, D. V. Tosic, and B. L. Evans, "Design Space Approach to Advanced Filter Design'', Proc. IEEE Int. Conf. on Telecommunications in Modern Satellite Cable and Broadcasting Services, Nis,
Serbia, Yugoslavia, Oct. 10-18, 1997, pp. 179-190.
D. Wei, B. L. Evans, and A. C. Bovik, ``Loss of Perfect Reconstruction in Multidimensional Filterbanks and Wavelets Designed via Extended McClellan Transformations'', IEEE Signal Processing Letters,
vol. 4, no. 10, pp. 295-297, Oct. 1997.
M. D. Lutovac, D. V. Tosic, and B. L. Evans, "Symbolic Analysis of Programmable Digital Filters'', Proc. IEEE Int. Conf. on Microelectronics, Sep. 14-17, 1997, Nis, Serbia, Yugoslavia, vol. 2, pp.
B. Lu, B. L. Evans, and D. V. Tosic, "Simulation and Synthesis of Artificial Neural Networks Using Dataflow Models in Ptolemy'', Proc. Seminar on Neural Network Applications in Electrical Engineering
, Sep. 8-9, 1997, Belgrade, Yugoslavia, invited paper, pp. 84-89.
M. Torlak, G. Xu, B. L. Evans, and H. Liu, "Estimation of Optimal Weight Vectors for Broadcast Channels'', Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Proc., Apr. 21-24, 1997, Munich,
Germany, vol. 5, pp. 4009-4012.
R. Ahmed and B. L. Evans, "Optimization of Signal Processing Algorithms'', Proc. IEEE Asilomar Conf. on Signals, Systems, and Computers, Nov. 3-6, 1996, Pacific Grove, CA, vol. II, pp. 1401-1406.
M. Torlak, G. Xu, B. L. Evans, and H. Liu, "Optimal Weight Vectors for Spatial Broadcast Channels'', Proc. IEEE Asilomar Conf. on Signals, Systems, and Computers, Nov. 3-6, 1996, Pacific Grove, CA,
vol. 1, pp. 65-69.
M. D. Lutovac, D. V. Tosic, and B. L. Evans, "An Algorithm for Symbolic Design of Elliptic Filters'', Proc. Int. Workshop on Symbolic Methods and Applications to Circuit Design, Oct. 10-11, 1996,
Leuven, Belgium, pp. 248-251. (paper is not available)
R. Mani, S. H. Nawab, J. M. Winograd, and B. L. Evans, "Integrated Numeric and Symbolic Signal Processing Using a Heterogeneous Design Environment'', Proc. SPIE Int. Sym. on Advanced Signals
Processing Algorithms, Aug. 6-8, 1996, Denver, CO, vol. 2846, pp. 445-456.
K. H. Chiang, B. L. Evans, W. T. Huang, F. Kovac, E. A. Lee, D. G. Messerschmitt, H. J. Reekie, and S. S. Sastry, "Real-Time DSP for Sophomores'', Proc. IEEE Int. Conf. on Acoustics, Speech, and
Signal Proc., May, 1996, Atlanta, GA, invited paper, vol. II, pp. 1097-1100.
G. Arslan, B. L. Evans, F. A. Sakarya, and J. L. Pino, "Performance Evaluation and Real-Time Implementation of Subspace, Adaptive, and DFT Algorithms for Multi-Tone Detection'', Proc. IEEE Int. Conf.
on Telecommunications, Apr. 14-17, 1996, Istanbul, Turkey, pp. 884-887.
B. L. Evans, D. R. Firth, K. D. White, and E. A. Lee, "Automatic Generation of Programs That Jointly Optimize Characteristics of Analog Filter Designs'', Proc. European Conference on Circuit Theory
and Design, Aug. 27-31, 1995, Istanbul, Turkey, pp. 1047-1050.
B. L. Evans, S. X. Gu, A. Kalavade, and E. A. Lee, "Symbolic Computation in System Simulation and Design'', Proc. SPIE Int. Sym. on Advanced Signals Processing Algorithms, Jul. 10-12, 1995, San
Diego, CA, invited paper, vol. 2563, pp. 396-407.
C. Schwarz, J. Teich, A. Vainshtein, E. Welzl, and B. L. Evans, "Minimal Enclosing Parallelogram with Application'', Proc. ACM Sym. on Computational Geometry, Jun. 5-7, 1995, Vancouver, Canada, pp.
R. H. Bamberger, B. L. Evans, E. A. Lee, J. H. McClellan, and M. A. Yoder, "Integrating Analysis, Simulation, and Implementation Tools in Electronic Courseware for Teaching Signal Processing'', Proc.
IEEE Int. Conf. on Acoustics, Speech, and Signal Proc., May 9-12, 1995, Detroit, MI, invited paper, vol. 5, pp. 2873-2876.
B. L. Evans and J. H. McClellan, "Algorithms for Symbolic Linear Convolution'', Proc. IEEE Asilomar Conf. on Signals, Systems, and Computers, Oct. 31-Nov. 2, 1994, Pacific Grove, CA, pp. 948-953.
B. L. Evans, S. X. Gu, and R. H. Bamberger, ``Interactive Solution Sets as Components of Fully Electronic Signals and Systems Courseware'', Proc. IEEE Asilomar Conf. on Signals, Systems, and
Computers, Oct. 31-Nov. 2, 1994, Pacific Grove, CA, pp. 1314-1319.
B. L. Evans, J. Teich, and C. Schwarz, "Automated Design of Two-Dimensional Rational Decimation Systems'', Proc. IEEE Asilomar Conf. on Signals, Systems, and Computers, Oct. 31-Nov. 2, 1994, Pacific
Grove, CA, pp. 498-502.
B. L. Evans, J. Teich, and T. A. Kalker, "Families of Smith Form Decompositions to Simplify Multidimensional Filter Bank Design'', Proc. IEEE Asilomar Conf. on Signals, Systems, and Computers, Oct.
31-Nov. 2, 1994, Pacific Grove, CA, pp. 363-367.
B. L. Evans and F. A. Sakarya, "Interactive Graphical Design of Two-Dimensional Compression Systems'', Proc. National Workshop on Signal Processing, pp. 173-178, Apr. 8-9, 1994, Marmaris, Turkey.
B. L. Evans, R. H. Bamberger, and J. H. McClellan, ``Rules for Multidimensional Multirate Structures'', IEEE Transactions on Signal Processing, vol. 42, no. 4, pp. 762-771, Apr. 1994.
B. L. Evans, T. R. Gardos, and J. H. McClellan, ``Imposing Structure on Smith-Form Decompositions of Rational Resampling Matrices'', IEEE Transactions on Signal Processing, vol. 42, no. 4, pp.
970-973, Apr. 1994.
B. L. Evans, J. H. McClellan, and H. J. Trussell, "Investigating Signal Processing Theory with Mathematica'', Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Proc., Apr. 27-30, 1993,
Minneapolis, MN, vol. I, pp. I-12-I-15.
B. L. Evans, L. J. Karam, K. A. West, and J. H. McClellan, ``Learning Signals and Systems with Mathematica'', IEEE Transactions on Education, vol. 36, no. 1, pp. 72-78, Feb. 1993.
B. L. Evans, ``General Changes in Version 2.0'', The Mathematica Journal, vol. 2, no. 1, pp. 19-20, Summer, 1992.
B. L. Evans, J. H. McClellan, and R. H. Bamberger, "A Symbolic Algebra for Linear Multidimensional Multirate Systems'', Proc. Conf. on Information Sciences and Systems, March, 1992, Princeton, NJ,
pp. 387-393.
B. L. Evans, J. H. McClellan, and K. A. West, "Mathematica as an Educational Tool for Signal Processing'', Proc. IEEE Southeastcon Conf., Apr. 8-10, 1991, Williamsburg, VA, pp. 1162-1166.
B. L. Evans, J. H. McClellan, and W. B. McClure, ``Symbolic Transforms with Applications to Signal Processing'', The Mathematica Journal, vol. 1, no. 2, pp. 70-80, Fall, 1990.
E. V. Garcia, M. D. Herbst, C. D. Cooke, N. F. Ezquerra, B. L. Evans, R. D. Folks, and E. G. DePuey, "Knowledge-Based Visualization of Myocardial Perfusion Tomographic Images'', Proc. IEEE Conf. on
Visualization in Biomedical Computing, May 1990, Atlanta, GA, pp. 157-161.
B. L. Evans, J. H. McClellan, and W. B. McClure, "Symbolic z-Transforms Using DSP Knowledge Bases'', Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Proc., Apr. 3-6, 1990, Albuquerque, NM,
vol. 3, pp. 1775-1778.
Brian Evans, Dept. of Electrical and Computer Engineering
Semester Course Unique No. Title
2014 Spr EE 445S 16895 Real-Time Digital Sig Proc Lab
2014 Spr EE 445S 16900 Real-Time Digital Sig Proc Lab
2014 Spr EE 445S 16905 Real-Time Digital Sig Proc Lab
2014 Spr EE 445S 16910 Real-Time Digital Sig Proc Lab
2013 Fall E E 445S 16840 Real-Time Digital Sig Proc Lab
2013 Fall E E 445S 16845 Real-Time Digital Sig Proc Lab
2013 Fall E E 445S 16850 Real-Time Digtial Sig Proc Lab
2013 Fall E E 445S 16855 Real-Time Digital Sig Proc Lab | {"url":"http://www.utexas.edu/cola/centers/cps/faculty/evansbl1","timestamp":"2014-04-19T10:17:05Z","content_type":null,"content_length":"56822","record_id":"<urn:uuid:d4360f5b-4628-4ce8-a222-b9c6a2d2cefd>","cc-path":"CC-MAIN-2014-15/segments/1397609537097.26/warc/CC-MAIN-20140416005217-00546-ip-10-147-4-33.ec2.internal.warc.gz"} |
Problematic quetion!
December 4th 2009, 03:31 PM #1
Problematic quetion!
Let {a_n} be a closed sequence and let {b_n} be a sequence, so that every b_k is a Partial Limit of sequnce {a_n}. Supose in addiotianly that lim(b_n)=L.
Prove that L is a partial limit of {a_n} also.
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Sequences and Series
This week’s topic is one usually studied in first year algebra: sequences and series. Let’s start with some definitions. A sequence is an ordered list of numbers, and a series is the sum of the terms
(the individual numbers) of a sequence. For more lessons, here are my weekly website picks.
• "A sequence is a list of numbers (or other things) that changes according to some sort of pattern." This ten lesson section introduces arithmetic sequences, series, sigma notation, geometric
sequences, mathematical induction and the binomial theorem. To move from one lesson to the next (each lesson is multiple pages), you need to return to this menu, which is linked at the bottom of
each page.
"To find a missing number, first find a Rule behind the Sequence. Sometimes you can just look at the numbers and see a pattern." After a few examples of how trial and error can help you discover
a rule, at the bottom of the page you'll find links to related topics, including Arithmetic Sequences, Geometric Sequences, Fibonacci Sequence and Triangular Sequence. Each of these topics also
include related links at the bottom of the page, so be sure to look for them.
• After defining arithmetic sequences, this Math Guide lesson explains how to calculate the nth term. "In order for us to know how to obtain terms that are far down these lists of numbers, we need
to develop a formula that can be used to calculate these terms. If we were to try and find the 20th term, or worse to 2000th term, it would take a long time if we were to simply add a number --
one at a time -- to find our terms." At the bottom of the page, you'll find four interactive quizzes on sequences and series.
• "While some sequences are simply random values, other sequences have a definite pattern that is used to arrive at the sequence's terms. Two such sequences are the arithmetic and geometric
sequences." This one-page lesson explains arithmetic sequences with lots of examples. At the bottom of the page is a link to a lesson about using a TI-83+/84+ graphing calculator for sequences
and series. Very cool.
• "Be careful that you don't think that every sequence that has a pattern in addition is arithmetic. It is arithmetic if you are always adding the SAME number each time." This one-page lesson with
practice problems is just one of three tutorials on the topic of sequences and series at the Virtual Math Lab. You'll find the others linked both in the introductory paragraph, and interspersed
in the lesson itself. The practice problems at the bottom of the page are meant to be worked out on your own before clicking through to the answer/discussion page.
Honorable Mentions
The following links are either new discoveries or sites that didn't make it into my newspaper column because of space constraints. Enjoy!
Cite This Page
• Feldman, Barbara. "Sequences and Series." Surfnetkids. Feldman Publishing. 2 Apr. 2013. Web. 20 Apr. 2014. <http://www.surfnetkids.com/resources/sequences-and-series/ >.
About This Page
• By Barbara J. Feldman. Originally published April 2, 2013. Last modified March 9, 2014. | {"url":"http://www.surfnetkids.com/resources/sequences-and-series/","timestamp":"2014-04-20T06:47:11Z","content_type":null,"content_length":"49072","record_id":"<urn:uuid:e7b6642b-6f16-4e3e-88d1-c50ef362fe5f>","cc-path":"CC-MAIN-2014-15/segments/1397609538022.19/warc/CC-MAIN-20140416005218-00421-ip-10-147-4-33.ec2.internal.warc.gz"} |
Prototype: Web-Friendly Visualizations in R
May 18, 2010
By Wojciech Gryc
Developing web-friendly data visualizations is not very difficult, though as far as I know, a package that allows one to do this directly in R does not exist (e-mail me if you know of one). As
someone who has been developing lots of data-oriented software tools, it's always nice to post visualizations online. To facilitate this task, I've been fooling around with creating a data
visualization prototype in R. While the package is very limited in what it does, I hope it'll generate a discussion on the types of visualization tools that could help R users post their work on the
At this stage, the package has three functions to illustrate scatter plots, line graphs, and social networks. Each function creates a new directory with all the necessary JavaScript and HTML files.
The HTML file could then be embedded using an inline frame (as done below) or used as a standalone website.
You can download the prototype here, and below are some examples of visualizations.
Scatter Plot
x = rnorm(25)
y = rnorm(25)
wv.scatterplot(x, y, "/wv-scatterplot", height=300, width=300, marginsize=0.1)
Line Graph
x = -100:100/10
y = sin(x)
wv.lineplot(x, y, "/wv-lineplot", height=300, width=300, marginsize=0.1)
Social Network
g <- erdos.renyi.game(15, 0.175)
wv.sna(g, "/wv-sna", rnorm(15, 2, 0.75), width=400, height=400)
Next Steps
I apologize in advance, as some of the code above may be buggy and it certainly isn't very customizable. The next step -- assuming there's interest -- is to abstract the graph drawing to individual
functions so one can then produce multiple graphs in one canvas or frame. Making more options for interactivity, labels, and so on is also a must. Again, comments and suggestions are very welcome.
for the author, please follow the link and comment on his blog:
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Among the most important applications of the circle to human progress has been circle-based (disk-shaped) food items. This is a family of foods that includes, among other things, cookies, bagels,
doughnuts, pies, and pizzas. (Some consider pizzas to be a subset of pies, but pepperoni pie doesn't sound nearly as delicious as pepperoni pizza.)
Critical to the development of circle-based foods was the concept of the central angle. Given ⊙O, an angle is a central angle of ⊙O if its vertex is at O. Simple as that.
Here, ∠1 is a central angle of ⊙O. So is ∠2. Every central angle has a buddy. The measure of a central angle and its buddy angle add up to 360°, the number of degrees in a full circle. In other
words, buddy angles complete each other. Aww.
With the discovery of the central angle, people could easily share circle-based foods as they saw fit. If you wanted to divide a pizza among five people, you could cut slices based on central angles
of 360° ÷ 5 = 72°.
Sample Problem
It's your birthday and you'll cry if you want to. There's no reason to cry though, since you got a massive chocolatey birthday cake. If there are 20 people total at your party, at what central angle
should you cut the cake?
Regardless of how big the cake is, it has a central angle of 360° because it's a circle. If there are 20 people, we should cut everyone a slice that is 360° ÷ 20 = 18° in measure. Time to bust out
the protractor.
Sample Problem
All circles are similar. True or false?
It's been a while since we talked about similarity, so here's a quick refresher: similarity exists when two figures are the same shape (all their angles are equal), but not the same size. This also
means they can be carried onto each other using similarity transformations (translation, reflection, rotation, and dilation).
Circles have 360° total. That won't change ever, so we took care of the angle requirement (as well as the rotation and reflection requirements). If a single point and a length defines a circle, we
can always translate the point to a different location and dilate the length so it matches another.
In other words, all circles are similar to each other because any similarity transformation can move one onto the other. (In fact, only dilation and translation are needed, so we leave reflection and
rotation at home.) | {"url":"http://www.shmoop.com/circles/central-angles-help.html","timestamp":"2014-04-21T09:50:46Z","content_type":null,"content_length":"32087","record_id":"<urn:uuid:7f0b9657-ada2-46c3-82b3-34b640f0c47d>","cc-path":"CC-MAIN-2014-15/segments/1397609539705.42/warc/CC-MAIN-20140416005219-00251-ip-10-147-4-33.ec2.internal.warc.gz"} |
Properties of Completions
First note that taking an inverse limit is a functor. I won’t need the functorial properties in the immediate future, but it would be good to state some of them. First off, the functor is not exact,
but it is left exact. So given an exact sequence of inverse systems $0\to \{A_n\}\to \{B_n\}\to \{C_n\}\to 0$ (it is exact at each level and all the diagrams commute) we get an exact sequence $\
displaystyle 0\to \lim_{\longleftarrow}A_n\to \lim_{\longleftarrow}B_n\to \lim_{\longleftarrow} C_n$.
It turns out that if the first system $\{A_n\}$ has the property that the homomorphisms $\theta_n$ are surjective, then the inverse limit is exact. So in our case with completions, this always
The properties I’d really like to prove are the ones listed in Hartshorne without proof. Suppose for the rest of this post that $(R, \frak{m})$ is a Noetherian local ring and $\hat{R}$ is its
completion with respect to the $\frak{m}$-adic topology. The numbers will refer to Hartshorne numbering:
5.4A(a) $\hat{R}$ is a local ring with maximal ideal $\hat{\frak{m}}=\frak{m}\hat{R}$ and there is a natural injective homomorphism $R\to \hat{R}$.
We already discussed the second part, since the kernel of the hom is just $\cap \frak{m}^n=0$. Using right exactness of tensoring and exactness of completions, we get that for any finitely generated
$R$-module $M$, $\hat{R}\otimes_R M\to \hat{M}$ is an iso (if we remove Noetherian on R, we only get surjective). This gives us that $\hat{R}\otimes_R \frak{m}\to \hat{\frak{m}}$ is an isomorphism
and since the image is $\frak{m}\hat{R}$ we get the first part of the statement.
Now we need that it is a unique maximal ideal. But applying the above result to any ideal (which is finitely generated since R is Noetherian) we get that $\widehat{\frak{a}^n}=\frak{a}^n\hat{R}=(\hat
{R}\frak{a})^n=(\hat{\frak{a}})^n$. Thus $R/\frak{a}^n\cong \hat{R}/\hat{\frak{a}}^n$. Taking inverse limits gives that $R/\frak{m}\cong \hat{R}/\hat{\frak{m}}$ and hence $\hat{\frak{m}}$ is a
maximal ideal since the quotient is a field. But for any $x\in\hat{m}$, we can define an inverse for $(1-x)$ formally by $(1-x)^{-1}=1+x+x^2+\cdots$. Since we are in the completion, this converges in
$\hat{R}$ and hence $x\in J(\hat{R})$. But a maximal ideal $\hat{\frak{m}}\subset J(\hat{R})$ means that it is the Jacobson radical and hence the unique maximal ideal.
5.4A (b) If $M$ is a finitely generated $R$-module, its completion $\hat{M}$ is isomorphic to $M\otimes_R \hat{R}$. Well, I needed this to prove (a) and briefly described how it would go, but since I
didn’t prove the exactness properties, it seems needlessly detailed to do a full proof using them. For more details, see posts at delta epsilons.
5.4A (c) $\dim R=\dim \hat{R}$.
Let’s use some of the machinery we developed. The dimensions are equal to the degree of the Hilbert polynomial, but $R/\frak{m}\cong \hat{R}/\hat{\frak{m}}$ says precisely that $\chi_m(n)=\chi_{\hat
{m}}(n)$. So the polynomials are actually the same.
5.4A (d) $R$ is regular if and only if $\hat{R}$ is regular.
We’ll hold off on this until I cover regularity (which will either be next time or the time after). | {"url":"http://hilbertthm90.wordpress.com/2009/11/15/properties-of-completions/","timestamp":"2014-04-21T14:40:17Z","content_type":null,"content_length":"86444","record_id":"<urn:uuid:bd6d0658-d18c-486c-9660-abe2d0b4ea13>","cc-path":"CC-MAIN-2014-15/segments/1397609540626.47/warc/CC-MAIN-20140416005220-00000-ip-10-147-4-33.ec2.internal.warc.gz"} |
Two-particle interference experiment with frequency-entangled photon pairs
We present experimental observations of a nonclassic and nonlocal interference effect with frequency-entangled photon pairs. We observed two-photon interference in a Hong–Ou–Mandel interferometer by
monitoring the coincidence-counting rate as we varied the path-length difference or the relative time delay between two photons. The quantum-mechanical probabilities of joint detection by two
detectors that respond to different frequency components exhibit differences that depend not only on the arrangement of the detector pairs but on the spectral filtering. The experimental results
provide more understanding of two-photon entanglement and of the interference effects that arise from superposition of indistinguishable probability amplitudes.
© 2003 Optical Society of America
OCIS Codes
(030.5260) Coherence and statistical optics : Photon counting
(270.0270) Quantum optics : Quantum optics
Heonoh Kim, Jeonghoon Ko, and Taesoo Kim, "Two-particle interference experiment with frequency-entangled photon pairs," J. Opt. Soc. Am. B 20, 760-763 (2003)
Sort: Year | Journal | Reset
1. A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777-780 (1935).
2. J. S. Bell, “On the Einstein Podolsky Rosen paradox,” Physics (Long Island City, N.Y.) 1, 195-200 (1964).
3. C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett.
70, 1895-1899 (1993).
4. D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeillinger, “Experimental quantum teleportation,” Nature 390, 575-579 (1997).
5. D. Boschi, S. Branca, F. De Martini, L. Hardy, and S. Popescu, “Experimental realization of teleporting an unknown pure quantum state via dual classical and Einstein-Podolsky-Rosen channels,”
Phys. Rev. Lett. 80, 1121-1125 (1998).
6. T. Jannewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729-4732 (2000).
7. D. S. Naik, C. G. Peterson, A. G. White, A. J. Berglund, and P. G. Kwiat, “Entangled state quantum cryptography: eavesdropping on the Ekert protocol,” Phys. Rev. Lett. 84, 4733-4736 (2000).
8. W. Tittle, J. Brendel, H. Zbinden, and N. Gisin, “Quantum cryptography using entangled photons in energy-time Bell states,” Phys. Rev. Lett. 84, 4737-4740 (2000).
9. S. F. Pereira, Z. Y. Ou, and H. J. Kimble, “Quantum communication with correlated nonclassical states,” Phys. Rev. A 62, 042311 (2000).
10. A. Gatti, E. Brambilla, L. A. Lugiato, and M. I. Kolobov, “Quantum entangled images,” Phys. Rev. Lett. 83, 1763-1766 (1999).
11. A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. 87, 123602 (2001).
12. L. Mandel, “Quantum effects in one-photon and two-photon interference,” Rev. Mod. Phys. 71, S274-S282 (1999).
13. A. Zeilinger, “Experiment and the foundations of quantum physics,” Rev. Mod. Phys. 71, S288-S297 (1999).
14. D. M. Greenberg, M. A. Horne, and A. Zeilinger, “Multiparticle interferometry and the superposition principle,” Phys. Today 46(8), 22-29 (1993).
15. C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044-2046 (1987).
16. Z. Y. Ou and L. Mandel, “Observation of spatial quantum beating with separated photodetectors,” Phys. Rev. Lett. 61, 54-57 (1988).
17. M. A. Horne, A. Shimony, and A. Zeilinger, “Two-particle interferometry,” Phys. Rev. Lett. 62, 2209-2212 (1989).
18. J. G. Rarity and P. R. Tapster, “Experimental violation of Bell’s inequality based on phase and momentum,” Phys. Rev. Lett. 64, 2495-2498 (1990). In this experiment, two tillable glass plates
were used to adjust the relative phases in a recombination of two color photons at the beam splitter.
19. J. G. Rarity and P. R. Tapster, “Two-color photons and nonlocality in fourth-order interference,” Phys. Rev. A 41, 5139-5146 (1990).
20. K. Mattle, M. Michler, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Non-classical statistics at multiport beam splitters,” Appl. Phys. B 60, S111-S117 (1995).
21. K. Mattle, H. Weinfurter, P. G. Kwiat, and A. Zeilinger, “Dense coding in experimental quantum communication,” Phys. Rev. Lett. 76, 4656-4659 (1996).
22. N. Lu¨tkenhaus, J. Calsamiglia, and K. A. Suominen, “Bell measurements for teleportation,” Phys. Rev. A 59, 3295–3300 (1999).
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Random subgraphs of the 2D Hamming graph: the supercritical phase
Hofstad, Remco and Luczak, Malwina J. (2010) Random subgraphs of the 2D Hamming graph: the supercritical phase. Probability Theory and Related Fields, 147 (1-2). pp. 1-41. ISSN 0178-8051
Full text not available from this repository.
We study random subgraphs of the 2-dimensional Hamming graph H(2, n), which is the Cartesian product of two complete graphs on n vertices. Let p be the edge probability, and write p = (1 + ε)/(2(n −
1)) for some ε ∈ R. In Borgs et al. (Random Struct Alg 27:137–184, 2005; Ann Probab 33:1886–1944, 2005), the size of the largest connected component was estimated precisely for a large class of
graphs including H(2, n) for ε ≤ �V−1/3, where � > 0 is a constant and V = n2 denotes the number of vertices in H(2, n). Until now, no matching lower bound on the size in the supercritical regime has
been obtained. In this paper we prove that, when ε � (log V)1/3V−1/3, then the largest connected component has size close to 2εV with high probability.We thus obtain a law of large numbers for the
largest connected component size, and show that the corresponding values of p are supercritical. Barring the factor (log V)1/3, this identifies the size of the largest connected component all the way
down to the critical p window.
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Lie brackets
The key to establishing whether a system is nonholonomic is to construct motions that combine the effects of two action variables, which may produce motions in a direction that seems impossible from
the system distribution. To motivate the coming ideas, consider the differential-drive model from (15.54). Apply the following piecewise-constant action trajectory over the interval :
The action trajectory is a sequence of four motion primitives: 1) translate forward, 2) rotate forward, 3) translate backward, and 4) rotate backward.
Figure 15.16: (a) The effect of the first two primitives. (b) The effect of the last two primitives.
The result of all four motion primitives in succession from is shown in Figure 15.16. It is fun to try this at home with an axle and two wheels (Tinkertoys work well, for example). The result is that
the differential drive moves sideways!^15.9 From the transition equation (15.54) such motions appear impossible. This is a beautiful property of nonlinear systems. The state may wiggle its way in
directions that do not seem possible. A more familiar example is parallel parking a car. It is known that a car cannot directly move sideways; however, some wiggling motions can be performed to move
it sideways into a tight parking space. The actions we perform while parking resemble the primitives in (15.71).
Algebraically, the motions of (15.71) appear to be checking for commutativity. Recall from Section 4.2.1 that a group is called commutative (or Abelian) if for any . A commutator is a group element
of the form . If the group is commutative, then (the identity element) for any . If a nonidentity element of is produced by the commutator, then the group is not commutative. Similarly, if the
trajectory arising from (15.71) does not form a loop (by returning to the starting point), then the motion primitives do not commute. Therefore, a sequence of motion primitives in (15.71) will be
referred to as the commutator motion. It will turn out that if the commutator motion cannot produce any velocities not allowed by the system distribution, then the system is completely integrable.
This means that if we are trapped on a surface, then it is impossible to leave the surface by using commutator motions.
Now generalize the differential drive to any driftless control-affine system that has two action variables:
Using the notation of (15.53), the vector fields would be and ; however, and are chosen here to allow subscripts to denote the components of the vector field in the coming explanation.
Figure 15.17: The velocity obtained by the Lie bracket can be approximated by a sequence of four motion primitives.
Suppose that the commutator motion (15.71) is applied to (15.72) as shown in Figure 15.17. Determining the resulting motion requires some general computations, as opposed to the simple geometric
arguments that could be made for the differential drive. If would be convenient to have an expression for the velocity obtained in the limit as approaches zero. This can be obtained by using Taylor
series arguments. These are simplified by the fact that the action history is piecewise constant.
The coming derivation will require an expression for under the application of a constant action. For each action, a vector field of the form is obtained. Upon differentiation, this yields
This follows from the chain rule because is a function of , which itself is a function of . The derivative is actually an Jacobian matrix, which is multiplied by the vector . To further clarify (
15.73), each component can be expressed as
Now the state trajectory under the application of (15.71) will be determined using the Taylor series, which was given in (14.17). The state trajectory that results from the first motion primitive can
be expressed as
which makes use of (15.73) in the second line. The Taylor series expansion for the second primitive is
An expression for can be obtained by using the Taylor series expansion in (15.75) to express . The first terms after substitution and simplification are
To simplify the expression, the evaluation at has been dropped from every occurrence of and and their derivatives.
The idea of substituting previous Taylor series expansions as they are needed can be repeated for the remaining two motion primitives. The Taylor series expansion for the result after the third
primitive is
Finally, the Taylor series expansion after all four primitives have been applied is
Taking the limit yields
which is called the Lie bracket of and and is denoted by . Similar to (15.74), the th component can be expressed as
The Lie bracket is an important operation in many subjects, and is related to the Poisson and Jacobi brackets that arise in physics and mathematics.
Example 15..9
(Lie Bracket for the Differential Drive)
The Lie bracket should indicate that sideways motions are possible for the differential drive. Consider taking the Lie bracket of the two vector fields used in (
). Let and . Rename and to and to allow subscripts to denote the components of a vector field.
By applying (15.81), the Lie bracket is
The resulting vector field is , which indicates the sideways motion, as desired. When evaluated at , the vector is obtained. This means that performing short commutator motions wiggles the
differential drive sideways in the direction, which we already knew from Figure
Example 15..10
(Lie Bracket of Linear Vector Fields)
Suppose that each vector field is a linear function of . The Jacobians and are constant.
As a simple example, recall the nonholonomic integrator (13.43). In the linear-algebra form, the system is
Let and . The Jacobian matrices are
Using (
This result can be verified using (
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solve equation
March 2nd 2010, 04:18 PM #1
Senior Member
Oct 2008
solve equation
solve the equation 2(a,3,c)+3(c,7,c)=(5,b+c,15) for a,b,c
i got to setting up 3 equation
i don't know where to go from here
Just isolate the variables then use substitution.
then put that in one of the other equations... then do it again until you only have 1 variable in the equation. Then repeat.
I just realized that the last equation only has variable 'c' in it... That would be the easiest one to use...
then put "c" in equation 1 and 2 to get values for "a" and "b"
March 2nd 2010, 04:27 PM #2
Jan 2010
March 2nd 2010, 04:29 PM #3
Jan 2010 | {"url":"http://mathhelpforum.com/algebra/131714-solve-equation.html","timestamp":"2014-04-21T06:09:04Z","content_type":null,"content_length":"32913","record_id":"<urn:uuid:c55a3c92-a3b6-4335-967f-e9796fe254e6>","cc-path":"CC-MAIN-2014-15/segments/1397609539493.17/warc/CC-MAIN-20140416005219-00497-ip-10-147-4-33.ec2.internal.warc.gz"} |
Independance probability question
May 25th 2009, 10:52 AM #1
Sep 2005
Independance probability question
I understand that for a event to have independence it follows that
P(A and B)= P(A) x P(B)
however i cannot understand this question
I have attached the question
any help is appreichated.
Maybe you can use another definition:
$P(B|A)=\frac{P(B \cap A)}{P(A)}$
i) Since the result of A will affect that of B, e.g. If A is 1 or 3, P(B)=2/5, If A is 5, P(B)=3/5, hence they are dependent.
Last edited by noob mathematician; May 25th 2009 at 07:02 PM.
May 25th 2009, 06:41 PM #2
Oct 2008 | {"url":"http://mathhelpforum.com/statistics/90441-independance-probability-question.html","timestamp":"2014-04-19T13:07:41Z","content_type":null,"content_length":"32229","record_id":"<urn:uuid:1d4c03c7-e796-47fb-9a45-aa785f94385a>","cc-path":"CC-MAIN-2014-15/segments/1397609537186.46/warc/CC-MAIN-20140416005217-00504-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Visual Calculus - Newton's Method
Objectives: In this tutorial, we examine Newton's Method and see the geometry underlying the derivation of the algorithm. We also look at a few examples that illustrate the method. After working
through these materials, the student should be able
• to understand graphically the derivation of Newton's Method,
• to apply Newton's Method to find the roots of a function.
Algorithm. Let f be a differentiable function. Choose a point x[1] near a root of f. Define recursively
If the point x[1] is chosen sufficiently close to the root then the x[n]'s are successively better approximations of the root.
• Discussion [Using Flash]
• Example:
• Drill problems on using Newton's Method. | {"url":"http://archives.math.utk.edu/visual.calculus/3/newton.5/","timestamp":"2014-04-17T09:36:03Z","content_type":null,"content_length":"2110","record_id":"<urn:uuid:9991b6e0-c369-4799-a6dc-39a07d93b1e9>","cc-path":"CC-MAIN-2014-15/segments/1398223202548.14/warc/CC-MAIN-20140423032002-00413-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Applying interval arithmetic to real, integer and boolean constraints
Results 1 - 10 of 142
, 1994
"... The design and implementation of constraint logic programming (CLP) languages over intervals is revisited. Instead of decomposing complex constraints in terms of simple primitive constraints as
in CLP(BNR), complex constraints are manipulated as a whole, enabling more sophisticated narrowing procedu ..."
Cited by 121 (18 self)
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The design and implementation of constraint logic programming (CLP) languages over intervals is revisited. Instead of decomposing complex constraints in terms of simple primitive constraints as in
CLP(BNR), complex constraints are manipulated as a whole, enabling more sophisticated narrowing procedures to be applied in the solver. This idea is embodied in a new CLP language Newton whose
operational semantics is based on the notion of box-consistency, an approximation of arc-consistency, and whose implementation uses Newton interval method. Experimental results indicate that Newton
outperforms existing languages by an order of magnitude and is competitive with some state-of-the-art tools on some standard benchmarks. Limitations of our current implementation and directions for
further work are also identified.
- SIAM Journal on Numerical Analysis , 1997
"... This paper presents Newton, a branch & prune algorithm to find all isolated solutions of a system of polynomial constraints. Newton can be characterized as a global search method which uses
intervals for numerical correctness and for pruning the search space early. The pruning in Newton consists in ..."
Cited by 101 (7 self)
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This paper presents Newton, a branch & prune algorithm to find all isolated solutions of a system of polynomial constraints. Newton can be characterized as a global search method which uses intervals
for numerical correctness and for pruning the search space early. The pruning in Newton consists in enforcing at each node of the search tree a unique local consistency condition, called
box-consistency, which approximates the notion of arc-consistency well-known in artificial intelligence. Box-consistency is parametrized by an interval extension of the constraint and can be
instantiated to produce the Hansen-Segupta's narrowing operator (used in interval methods) as well as new operators which are more effective when the computation is far from a solution. Newton has
been evaluated on a variety of benchmarks from kinematics, chemistry, combustion, economics, and mechanics. On these benchmarks, it outperforms the interval methods we are aware of and compares well
with state-of-the-art continuation methods. Limitations of Newton (e.g., a sensitivity to the size of the initial intervals on some problems) are also discussed. Of particular interest is the
mathematical and programming simplicity of the method.
- CWI QUARTERLY VOLUME 11 (2&3) 1998, PP. 215 { 248 , 1998
"... We show that several constraint propagation algorithms (also called (local) consistency, consistency enforcing, Waltz, ltering or narrowing algorithms) are instances of algorithms that deal with
chaotic iteration. To this end we propose a simple abstract framework that allows us to classify and comp ..."
Cited by 89 (6 self)
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We show that several constraint propagation algorithms (also called (local) consistency, consistency enforcing, Waltz, ltering or narrowing algorithms) are instances of algorithms that deal with
chaotic iteration. To this end we propose a simple abstract framework that allows us to classify and compare these algorithms and to establish in a uniform way their basic properties.
- INT. CONF. ON LOGIC PROGRAMMING , 1999
"... Most interval-based solvers in the constraint logic programming framework are based on either hull consistency or box consistency (or a variation of these ones) to narrow domains of variables
involved in continuous constraint systems. This paper rst presents HC4, an algorithm to enforce hull consist ..."
Cited by 76 (13 self)
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Most interval-based solvers in the constraint logic programming framework are based on either hull consistency or box consistency (or a variation of these ones) to narrow domains of variables
involved in continuous constraint systems. This paper rst presents HC4, an algorithm to enforce hull consistency without decomposing complex constraints into primitives. Next, an extended denition
for box consistency is given and the resulting consistency is shown to subsume hull consistency. Finally, BC4, a new algorithm to eciently enforce box consistency is described, that replaces BC3the
original solely Newton-based algorithm to achieve box consistencyby an algorithm based on HC4 and BC3 taking care of the number of occurrences of each variable in a constraint. BC4 is then shown to
signicantly outperform both HC3 (the original algorithm enforcing hull consistency by decomposing constraints) and BC3. 1 Introduction Finite representation of numbers precludes computers from
exactly solv...
- J. ACM
"... We start with a mathematical definition of a real interval as a closed, connected set of reals. Interval arithmetic operations (addition, subtraction, multiplication and division) are likewise
defined mathematically and we provide algorithms for computing these operations assuming exact real arithme ..."
Cited by 76 (12 self)
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We start with a mathematical definition of a real interval as a closed, connected set of reals. Interval arithmetic operations (addition, subtraction, multiplication and division) are likewise
defined mathematically and we provide algorithms for computing these operations assuming exact real arithmetic. Next, we define interval arithmetic operations on intervals with IEEE 754 floating
point endpoints to be sound and optimal approximations of the real interval operations and we show that the IEEE standard’s specification of operations involving the signed infinities, signed zeros,
and the exact/inexact flag are such as to make a correct and optimal implementation more efficient. From the resulting theorems we derive data that are sufficiently detailed to convert directly to a
program for efficiently implementing the interval operations. Finally we extend these results to the case of general intervals, which are defined as connected sets of reals that are not necessarily
closed. 1
, 2004
"... A chapter for ..."
- Constraints , 1996
"... We consider constraint satisfaction problemswith variables in continuous,numerical domains. Contrary to most existing techniques, which focus on computing one single optimal solution, we address
the problem of computing a compact representation of the space of all solutions admitted by the constrai ..."
Cited by 56 (7 self)
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We consider constraint satisfaction problemswith variables in continuous,numerical domains. Contrary to most existing techniques, which focus on computing one single optimal solution, we address the
problem of computing a compact representation of the space of all solutions admitted by the constraints. In particular, we show how globally consistent (also called decomposable) labelings of a
constraint satisfaction problem can be computed.
- Handbook of Constraint Programming , 2006
"... Constraint propagation is a form of inference, not search, and as such is more ”satisfying”, both technically and aesthetically. —E.C. Freuder, 2005. Constraint reasoning involves various types
of techniques to tackle the inherent ..."
Cited by 51 (3 self)
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Constraint propagation is a form of inference, not search, and as such is more ”satisfying”, both technically and aesthetically. —E.C. Freuder, 2005. Constraint reasoning involves various types of
techniques to tackle the inherent
- CONSTRAINT PROGRAMMING: BASICS AND TRENDS, VOLUME 910 OF LNCS , 1995
"... Abstract. In this paper, we present anoverview on the use of interval arithmetic to process numerical constraints in Constraint Logic Program-ming. The main principle is to approximate n-ary
relations over IR with Cartesian products of intervals whose bounds are taken in a nite subset of I R.Variabl ..."
Cited by 47 (5 self)
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Abstract. In this paper, we present anoverview on the use of interval arithmetic to process numerical constraints in Constraint Logic Program-ming. The main principle is to approximate n-ary
relations over IR with Cartesian products of intervals whose bounds are taken in a nite subset of I R.Variables represent real values whose domains are intervals de ned in the same manner. Narrowing
operators are de ned from approximations. These operators compute, from an interval and a relation, aset included in the initial interval. Sets of constraints are then processed thanks to a local
consistency algorithm pruning at each stepvalues from initial intervals. This algorithm is shown to be correct and to terminate, on the basis of a certain number of properties of narrowing operators.
We focus here on the description of the general framework based on approximations, on its application to interval constraint solving over continuous and discrete quantities, we establish a strong
link between approximations and local consistency notions and show that arc-consistency is an instance of the approximation framework. We nally describe recentwork on di erent variants of the initial
algorithm proposed by John Cleary and developed by W. Older and A. Vellino which havebeen proposed in this context. These variants address four particular points: generalization of the constraint
language, improvement of domain reductions, e ciency of the computation and nally, cooperation with other solvers. Some open questions are also identi ed. 1
- PROCEEDINGS OF THE 6TH INTERNATIONAL CONFERENCE ON PRINCIPLES AND PRACTICE OF CONSTRAINT PROGRAMMING , 2000
"... Non-linear real constraint systems with universally and/or existentially quantified variables often need be solved in such contexts as control design or sensor planning. To date, these systems
are mostly handled by computing a quantifier-free equivalent form by means of Cylindrical Algebraic Decompo ..."
Cited by 46 (0 self)
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Non-linear real constraint systems with universally and/or existentially quantified variables often need be solved in such contexts as control design or sensor planning. To date, these systems are
mostly handled by computing a quantifier-free equivalent form by means of Cylindrical Algebraic Decomposition (CAD). However, CAD restricts its input to be conjunctions and disjunctions of polynomial
constraints with rational coefficients, while some applications such as camera control involve systems with arbitrary forms where time is the only universally quantified variable. In this paper, the
handling of universally quantified variables is first related to the computation of inner-approximation of real relations. | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=98303","timestamp":"2014-04-23T19:45:13Z","content_type":null,"content_length":"37569","record_id":"<urn:uuid:7d63b0fc-c5bf-478f-bcd2-6ed797fb7198>","cc-path":"CC-MAIN-2014-15/segments/1398223203422.8/warc/CC-MAIN-20140423032003-00656-ip-10-147-4-33.ec2.internal.warc.gz"} |
This Article
Bibliographic References
Add to:
PCBN: A High-Performance Partitionable Circular Bus Network for Distributed Systems
December 1993 (vol. 4 no. 12)
pp. 1298-1307
ASCII Text x
T.K. Woo, S.Y.W. Su, T.Y. Feng, "PCBN: A High-Performance Partitionable Circular Bus Network for Distributed Systems," IEEE Transactions on Parallel and Distributed Systems, vol. 4, no. 12, pp.
1298-1307, December, 1993.
BibTex x
@article{ 10.1109/71.250112,
author = {T.K. Woo and S.Y.W. Su and T.Y. Feng},
title = {PCBN: A High-Performance Partitionable Circular Bus Network for Distributed Systems},
journal ={IEEE Transactions on Parallel and Distributed Systems},
volume = {4},
number = {12},
issn = {1045-9219},
year = {1993},
pages = {1298-1307},
doi = {http://doi.ieeecomputersociety.org/10.1109/71.250112},
publisher = {IEEE Computer Society},
address = {Los Alamitos, CA, USA},
RefWorks Procite/RefMan/Endnote x
TY - JOUR
JO - IEEE Transactions on Parallel and Distributed Systems
TI - PCBN: A High-Performance Partitionable Circular Bus Network for Distributed Systems
IS - 12
SN - 1045-9219
EPD - 1298-1307
A1 - T.K. Woo,
A1 - S.Y.W. Su,
A1 - T.Y. Feng,
PY - 1993
KW - Index Termshigh-performance partitionable circular bus network; distributed systems; distributednetwork; graph traversal algorithm; maximal independent sets; synchronization; idlingproblem;
nonconflicting requests; distributed processing; graph colouring; synchronisation
VL - 4
JA - IEEE Transactions on Parallel and Distributed Systems
ER -
The authors present a dynamically partitionable circular bus network (PCBN) and efficient algorithms for maximizing its utilization. In their approach, a distributed network is transformed into a
graph, in which a vertex represents a communication request and an edge denotes the conflict between a pair of communication requests. A graph traversal algorithm is applied to the graph to identify
some maximal independent sets of vertices. The communication requests corresponding to the vertices of a maximum independent set call proceed in parallel. By computing the expected size of the
maximal independent sets of a graph, the improvement ratio of the network can be obtained. The network control and synchronization techniques of PCBN are described in detail. The idling problem in
the execution of nonconflicting requests is also discussed.
[1] B. W. Arden and R. Ginosar, "MP/C: A multiprocessor computer architecture,"IEEE Trans. Comput., vol. C-31, pp. 455-473, May 1982.
[2] C. K. Baru and S. Y. W. Su, "The architecture of SM3: A dynamically partitionable multicomputer system,"IEEE Trans. Comput., vol. C-35, pp. 790-801, Sept. 1986.
[3] S. K. Chang and A. Gill, "Algorithmic solution of the change-making problem,"J. Ass. Comput. Mach., vol. 17, pp. 113-122, Jan. 1970.
[4] M. R. Garey and D. S. Johnson,Computers and Intractability. San Francisco, CA: Freeman, 1979.
[5] R. M. Karp, "Reducibility among combinatorial problems," in R. E. Miller and J. W. Thatcher, Eds.,Complexity of Computer Computations. New York: Plenum, 1972.
[6] S. Kartashev and S. Kartashev, "Dynamic architecture: Problems and solutions,"Computer, vol. 11, pp. 7-15, July 1978.
[7] D. Matula, "On the complete subgraphs of a random graph," inProc. 2nd Chapel Hill Conf. Combinatorial Math. Appl., Univ. North Carolina, Chapel Hill, 1970, pp. 356-369.
[8] J. L. Mott, A. Kandel, and T. P. Baker,Discrete Mathematics for Computer Scientists. Reston, VA: Reston Publishing, a Prentice-Hall Co., 1983.
[9] S. Y. W. Su and C. K. Baru, "Dynamically partitionable multicomputers with switchable memory,"J. Parallel Distributed Comput., vol. 1, pp. 152-184, Nov. 1984.
[10] C. C. Wang, "An algorithm for the chromatic number of a graph,"J. Ass. Comput. Mach., vol. 21, pp. 385-391, July 1974.
[11] T. K. Woo, "Improving resource utilization in a partitionable bus network using graph coloring and coin-changing algorithms," Ph.D. dissertation, Dep. Comput. Inform. Sci., Univ. Florida,
Gainesville, May 1989.
[12] T. K. Woo, "Achieving parallel communication in distributed systems," inProc. Phoenix Conf. Comput. Commun., Scottsdale, AZ, Mar. 1991, pp. 160-166.
[13] T. K. Woo, S. Y. W. Su, and R. Newman-Wolfe, "Resource allocation in a partitionable bus network using a graph coloring algorithm,"IEEE Trans. Commun., vol. 39, pp. 1794-1801, Dec. 1991.
Index Terms:
Index Termshigh-performance partitionable circular bus network; distributed systems; distributednetwork; graph traversal algorithm; maximal independent sets; synchronization; idlingproblem;
nonconflicting requests; distributed processing; graph colouring; synchronisation
T.K. Woo, S.Y.W. Su, T.Y. Feng, "PCBN: A High-Performance Partitionable Circular Bus Network for Distributed Systems," IEEE Transactions on Parallel and Distributed Systems, vol. 4, no. 12, pp.
1298-1307, Dec. 1993, doi:10.1109/71.250112
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SD Toolbox - File Exchange - MATLAB Central
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Comments and Ratings (26)
08 Mar
08 Mar why i got the comment "fB" is undefined when i run this code?can anyone help me to solve this error?
15 Jul very helpful
31 May Very good, I am very appreciated. Thanks a lot!
12 Nov The SD ToolBox has only one kind of resonator which has two delays，I want to use the kind which has one delay，what should I do ？Thanks！！
Dear Prof.Simona Brigati
12 May I've read your paper, Behavioral Modeling of Switched-Capacitor Sigma–Delta Modulators. It is excellent. Both the paper and the toolbox help me a lot.
2010 Thanks!
14 Sep Thanks a lot! It's precious!
Since I had the same problem as Anna Kuncheva and Donghwi Kim, and no one has answered their question, I thought I would. When you set the path to the SDtoolbox folder, be sure to include
15 Apr all the subfolders. Otherwise, the block set will work, but the demo won't find its subblocks.
Very impressive demos! and convenient block set. Many thanks!
08 Mar
16 May
02 Mar
04 Dec good simualtion program
When I open the building block of sampling jitter,I don't know how to fill the space of "Random number spped" and what's "Random number spped" ?
25 Oct
2006 Would you do me a favor!
Hello, Anna Kuncheva,
While i am trying SDToolbox2 I also have a exactly same problem with yours.
When I try to open Demos in SDToolbox, error message arises with text: "Error evaluating 'OpenFcn' callback of SubSystem block (mask)'SDToolbox/Demos'.Undefined function or variable
05 Jun 'sdtoolboxdemo'".I have Matlab 7.
Did you guys solve this problem? Please help me if you can.
Thank you so much,
Donghwi Kim
23 Apr
17 May
Hi sir, i have looked at the example low pass SD2 and i encountered problem in the "function calSNR" line16:signal=(N/sum(w))*sinusx(vout(1:N).*w,f,N);
07 May where the matrix dimension is not matched.
2005 Could u give me some guide in solving the problem.
Dear Sir:
Thanks for your tool box.
27 Apr I have tried to use it but ome parameter Ts is always undefined in SD2.
2005 Could you tell me how to solve this problem.
18 Jan please send me .m programs on sigma-delta ADC
13 Nov VERY GOOD EXAMPLE.
dear sir
26 May please send me delta moduolaion box in simulink
2004 best regard
18 May It is very practical for the beginer to understand the nonideality.
15 Sep Very good starting point for SC SD-modulator
2003 design. The models are easy to expand.
27 Jun This is a very good tool box. It might be making some assumptions which are not true in all delta sigma designs but its very easy to expand these models to incoporate more non idealities.
2003 Simulink is much better then just using plain matlab.
23 Jan Very practical tools for Switched-capacitor SDM simulation!
21 Nov | {"url":"http://www.mathworks.com/matlabcentral/fileexchange/2460-sd-toolbox","timestamp":"2014-04-25T01:47:58Z","content_type":null,"content_length":"40915","record_id":"<urn:uuid:94759e66-191c-498c-bf04-f9a45841a3d7>","cc-path":"CC-MAIN-2014-15/segments/1398223207046.13/warc/CC-MAIN-20140423032007-00553-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Multiplication and Division as Inverse Operations
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March 15th 2010, 08:28 PM
Let $A$ be a commutative ring with unity.
If $M,N$ are distinct maximal ideals of $A$, then
(1) $M+N=A$.
(2) $M^a+N^b=A (a,b\ge1)$.
March 15th 2010, 10:45 PM
For (1), the sum of ideals is again an ideal (link). Thus, M+N is an ideal containing M. By hypthesis, M+N should properly contain an maximal ideal M. Thus M+N=A.
For (2), every proper ideal in A is contained in a maximal ideal in A and note that A contains the unity (link).
Assume $M^a+N^b , a,b\ge1$ is a proper ideal in A. Then, $M^a+N^b ,a,b\ge1$ should be contained in a maximal ideal. It follows that $M^a+N^b , a,b\ge1$ should be contained in either M or N (check
their intersection). Contradiction !
Thus, $M^a+N^b , a,b\ge1$ is an ideal in A which is not a proper ideal in A. We conclude that $M^a+N^b=A ,a,b\ge1$. | {"url":"http://mathhelpforum.com/advanced-algebra/134013-comaximal-coprime-print.html","timestamp":"2014-04-18T14:29:58Z","content_type":null,"content_length":"7352","record_id":"<urn:uuid:4017bc73-4fb5-4fb8-bb29-47647ed627d3>","cc-path":"CC-MAIN-2014-15/segments/1397609533689.29/warc/CC-MAIN-20140416005213-00420-ip-10-147-4-33.ec2.internal.warc.gz"} |
Family of curves on a surface (Differential geometry)
May 11th 2008, 04:09 PM #1
Mar 2008
Family of curves on a surface (Differential geometry)
Here is a question i am not sure how to tackle, I am not familiar with how to deal with family of curves and don't really have much time to look around for the definition as i am sitting the exam
in two days.
The question is divided into three parts: Here is my attempt any help appreciated.
1) I am have no idea, i think it is a case of knowing the definition and i don't.
2) It is simply constraining the local parametrization to the given function so:
$xz-hy=> v*sin(u)=h(1-cos(u))=> h= v*sin(u)/(1-cos(u))$ which is a constant.
3) $\psi(u,v)=const$ is like phi therefore the tangent vectors to the family defined by the psi are of the multiples of $\psi_{v}x_{u}-\psi_{u}x_{v}$
So for the families to be orthogonal their tangent must be orthogonal and so $(\psi_{v}x_{u}-\psi_{u}x_{v}).(\phi_{v}x_{u}-\phi_{u}x_{v})=0$
Using the fundamental forms E=1=G and F=0 we get $\psi_{v}\phi_{v}+\psi_{u}\phi_{u}=0$ which after differentiating gives $\psi_{v}\sin(u)-\psi_{u}v=0$
And after that i am stuck ... any help would be appreciated.
Ok, lets see...
1) Consider a curve in the domain. The gradient of $\phi$ is normal to the tangent $(du,dv)$ of the curve. Since all directions of such surface curves are given by $x_{u}du+x_vdv$, the functional
dependence of $x$ and $\phi$ means that the determinant of their Jacobian matrix is zero, whence the required result.
2) We have $<br /> \psi_{v}\sin(u)-\psi_{u}v=0,<br />$ so $\sin(u)du=vdv,$ from which $\psi(u,v)=\frac{v^2}{2}-\cos(u)=const.$
October 27th 2008, 12:18 PM #2 | {"url":"http://mathhelpforum.com/differential-geometry/38017-family-curves-surface-differential-geometry.html","timestamp":"2014-04-19T02:04:14Z","content_type":null,"content_length":"36045","record_id":"<urn:uuid:7463133e-d160-4858-ada5-5e48be602670>","cc-path":"CC-MAIN-2014-15/segments/1397609535745.0/warc/CC-MAIN-20140416005215-00001-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Summary: ISRAELJOURNALOF MATHEMATICS,Vol. 64, No. 2, 1988
School ofMathematical Sciences, Raymond and BeverlySackler Faculty of Exact Sciences,
Tel Aviv University, Tel Aviv 69978, Israel
We prove that algebras of sub-exponential growth and, more generally, rings
with a sub-exponential "growth structure" have the unique rank property. In
the opposite direction the proof shows that if the rank is not unique one gets
lowerbounds on the exponent ofgrowth. Fixing the growthexponent it shows
that an isomorphism between free modules of greatly differing ranks can only
be implemented by matrices with entries of logarithmically proportional high
It is well known that there exist rings over which the rank of a free left
module is not uniquely defined, i.e., they have modules that are free on two
bases of different cardinalities. A ring for which this distressing phenomenon
does not happen is said to have the (left) uniquerankproperty;we also say that
it has the "UR" property or, simply, that it "has UR". A commutative ring
always has the UR property since it has a non-trivial (i.e., with 1 going to 1) | {"url":"http://www.osti.gov/eprints/topicpages/documents/record/233/5350431.html","timestamp":"2014-04-19T00:22:32Z","content_type":null,"content_length":"8290","record_id":"<urn:uuid:7e124b1a-12b3-45c6-bd9a-4247649e97ac>","cc-path":"CC-MAIN-2014-15/segments/1397609535535.6/warc/CC-MAIN-20140416005215-00420-ip-10-147-4-33.ec2.internal.warc.gz"} |
kinematics in 2d problem
February 7th 2010, 11:09 AM #1
Junior Member
Dec 2009
kinematics in 2d problem
Hi, Im having problems with this question:
A boat heads north and 20kph
A wind is blowing SE with a constant speed of 2ms^1
a) Find the magnitude of the speed of the boat
b) Find the bearing it travels at
Im having problems with the triangle, drawing it out makes the 2 ms^1 the hypotenuse when, to find the speed shouldnt be the hypotenuse i.e the hypotenuse has to be found?
The using pythagoras I get the resultant being 5.2ms^1 which is complete rubbish.
Please could anyone give me some helpful pointers please?
Thankyou very much in advance!
The angle between the velocity vector of the boat (20 km/hr) and the velocity vector of the wind (7.2 km/hr) is 45 degrees.
The speed of the boat can be found by the law of cosines.
$c^{2} = 20^{2}+7.2^{2} - 2(20)(7.2) \cos (45°)$
c = 15.75 km/hr
The direction of the boat can be found by the law of sines.
$\frac{15.7542}{\sin(45°)} = \frac{7.2}{\sin B}$
$\sin B = \frac{(7.2)(\frac{\sqrt{2}}{2})}{15.7542}$
B = 18.9° east of north
How did you work out what the angle was though?
February 7th 2010, 11:37 AM #2
February 7th 2010, 01:05 PM #3
Junior Member
Dec 2009
February 7th 2010, 01:25 PM #4 | {"url":"http://mathhelpforum.com/math-topics/127636-kinematics-2d-problem.html","timestamp":"2014-04-17T13:53:06Z","content_type":null,"content_length":"39503","record_id":"<urn:uuid:88fbd435-6d1b-4f01-a3d1-c8707513c2ca>","cc-path":"CC-MAIN-2014-15/segments/1397609530131.27/warc/CC-MAIN-20140416005210-00525-ip-10-147-4-33.ec2.internal.warc.gz"} |
Alumni Dissertations
• Magnetic Resonance Studies of Energy Storage Materials
Rafael Vazquez Reina
Year of Dissertation:
Abstract Magnetic Resonance Studies of Energy Storage Materials by Rafael Vázquez Reina Adviser: Professor Steven G. Greenbaum In today's society there is high demand to have access to energy for
portable devices in different forms. Capacitors with high performance in small package to achieve high charge/discharge rates, and batteries with their ability to store electricity and make
energy mobile are part of this demand. The types of internal dielectric material strongly affect the characteristics of a capacitor, and its applications. In a battery, the choice of the
electrolyte plays an important role in the Solid Electrolyte Interphase (SEI) formation, and the cathode material for high output voltage. Electron Paramagnetic Resonance (EPR) and Nuclear
Magnetic Resonance (NMR) spectroscopy are research techniques that exploit the magnetic properties of the electron and certain atomic nuclei to determine physical and chemical properties of the
atoms or molecules in which they are contained. Both EPR and NMR spectroscopy technique can yield meaningful structural and dynamic information. Three different projects are discussed in this
dissertation. First, High energy density capacitors where EPR measurements described herein provide an insight into structural and chemical differences in the dielectric material of a capacitor.
Next, as the second project, Electrolyte solutions where an oxygen-17 NMR study has been employed to assess the degree of preferential solvation of Li+ ions in binary mixtures of EC (ethylene
carbonate) and DMC (dimethyl carbonate) containing LiPF6 (lithium hexafluo- rophosphate) which may be ultimately related to the SEI formation mechanism. The third project was to study Bismuth
fluoride as cathode material for rechargeable batteries. The objective was to study 19F and 7Li MAS NMR of some nanocomposite cathode materials as a conversion reaction occurring during
lithiation and delithation of the BiF3/C nanocomposite.
• An experimental investigation into the mechanisms of bacterial evolution
Year of Dissertation:
This thesis studies the two fundamental mechanisms of bacterial evolution — horizontal gene transfer and spontaneous mutation, in the bacterium Escherichia coli through novel experimental assays
and mathematical simulations. First, I will develop a growth assay utilizing the quantitative polymerase chain reaction (qPCR) to provide real-time enumeration of genetic marker abundance within
bacterial populations. Second, I will focus on horizontal gene transfer in E. coli occurring through a process called conjugation. By fitting the qPCR data to a resource limited, logistic growth
model, I will obtain estimated values of several key parameters governing the dynamics of DNA transfer through conjugation under two different conditions: i) in the absence of selection; ii) in
the presence of negative selection pressure — bacteriophage infection. Last, I will investigate spontaneous mutation through qPCR assay of competition between wild-type and mutator phenotype E.
coli. Mutator phenotype has an elevated mutation rate due to defects in DNA proofreading and repairing system. By introducing antibiotic selective pressure, I will examine the fixation
probability of mutators competing with wild-type in novel environment. I also will utilize simulations to study the impact of three parameters on the fixation probability.
Year of Dissertation:
The application of concepts from equilibrium statistical mechanics to out of equilibrium systems has a long history of describing diverse systems ranging from glasses to granular materials. These
systems are considered "complex" since equilibrium statistics is insufficient in its attempt to describe the system dynamics. An appealing approach for understanding these complex systems is to
study the properties of the system's "potential energy landscape" (PEL), described by the 3N-coordinates of all particles in the multi-dimensional configuration space, or landscape, of the
potential energy of the system (N is the number of particles). For dissipative jammed systems- granular materials or droplets- a key concept introduced by S. Edwards in 1989 is to replace the
energy ensemble describing conservative systems by the volume ensemble. However, this approach is no able to describe the jamming critical point (J-point) for deformable particles like emulsions,
whose geometric configurations are influenced by the applied external stress. Therefore, the volume ensemble requires augmentation by the ensemble of stresses. Just as volume fluctuations in the
Edwards ensemble can be described by compactivity, the stress fluctuations give rise to an angoricity, another analogue of temperature in equilibrium systems. In this Thesis, we test the combined
volume-stress ensemble for granular matter by comparing the statistical properties of jammed configurations obtained by dynamics with those averaged over the ensemble as a test of ergodicity.
Agreement between both methods suggests the idea of "thermalization" at a given angoricity and compactivity. These intensive variables elucidate the thermodynamic order of the jamming phase
transition by showing the absence of critical fluctuations above jamming in static observables like pressure and volume. Our results demonstrate the possibility of calculating important
observables such as the entropy, volume, pressure, coordination number and the distribution of interparticle forces to fully characterize the scaling laws near the jamming transition from a
statistical mechanics point of view. We also study the energy-landscape network. We find the stable basins and the first order saddles connecting them, and identify them with the network nodes
and links, respectively. We analyze the network properties and model the system's evolution.
• QUASI-NORMAL MODES IN RANDOM MEDIA
Year of Dissertation:
This thesis is an experimental study of microwave transmission through quasi-one-dimensional random samples via quasi-normal modes. We have analyzed spectra of localized microwave transmitted
through quasi-one-dimensional random samples to obtain the central frequency, linewidth and field speckle pattern of the modes for an ensemble of samples at three lengths. We find strong
correlation between modal field speckle patterns. This leads to destructive interference between modes which explain strong suppression of steady state transmission and of pulsed transmission at
early times. At longer times, the rate of decay of transmission slows down because of the increasing prominence of long-lived modes. We have also studied the statistics of mode spacings and
widths in localized samples. The distribution of mode spacings between adjacent modes is close to the Wigner surmise predicted for diffusive waves, which exhibit strong level repulsion. However,
a deviation from Wigner distribution can be seen in the distribution of spacings beyond the nearest ones. A weakening in the rigidity of the modal spectrum is also observed as the sample length
increases because of reduced level repulsion for more strongly localized waves. In contrast to residual diffusive behavior for level spacing statistics, the distribution of level widths are
log-normal as predicted for localized waves. But the residual diffusive behavior can be seen from the smaller variance of the normalized mode width as compared to predictions for strongly
localized waves. We also measured the steady state and dynamic fluctuations and correlation of localized microwave transmitted through random waveguides. We find the degree of intensity
correlation first increases, and then decays with time delay, before increasing dramatically. The variation in the spatial correlation of intensity with time delay is due to the changing
effective number of modes that contribute to transmission. A minimum in correlation is reached when the number of modes contributing appreciably to transmission peaks. At long times, the degree
of intensity correlation and the variance of total transmission increase dramatically. This reflects the reduced role of short-lived overlapping states and the growing weight of long-lived
spectrally isolated modes.
Year of Dissertation:
In a thermal system, the Brownian motion of the constituent particles implies that the system dynamically explores the available energy landscape, such that the notion of a statistical ensemble
applies. For densely packed systems of interest in this study, in which enduring contacts between particles are important, the potential energy barrier prohibits an equivalent random motion. At
first sight it seems that the thermal statistical mechanics do not apply to these systems as there is no mechanism for averaging over the configurational states. Hence, these systems are
inherently out of equilibrium. On the other hand, if the granular material is gently tapped such that the grains can slowly explore the available configurations, the situation becomes analogous
to the equilibrium case scenario. It has been shown that the volume of the system is dependent on the applied tapping regime, and that this dependence is reversible, implying ergodicity. This
result gives support to the proposed statistical ensemble valid for dense, static and slowly moving granular materials which was first introduced by Edwards and Oakeshott in 1989. Through this
approach, notions of macroscopic quantities such as entropy and compactivity were also introduced to granular matter.
• Dipolar interactions, long range order and random fields in a single molecule magnet, Mn12-acetate
Year of Dissertation:
In this thesis, I will present an experimental study of two single molecule magnets, Mn[12]–ac and Mn[12]–ac–MeOH. I will show that in both systems, the temperature dependence of the inverse
susceptibility yields a positive intercept on the temperature axis (a positive Weiss temperature), implying the existence of a ferromagnetic phase at low temperature. Applying a magnetic field in
the transverse direction moves the Weiss temperature downward towards zero. This implies that the transverse field triggers mechanisms in the system that compete with the dipolar interaction and
suppress the long–range ordering. I will then show that the suppression in Mn[12]–ac is considerably stronger than that expected for a pure TFIFM (Transverse Field Ising Ferromagnetic) model
system. By contrast, the behavior of Mn[12]–ac–MeOH is consistent with the model. We attribute the difference between the two systems to the presence of randomness in Mn[12]–ac associated with
isomer disorder. Thus, in addition to spin–canting and thermal fluctuations, which contribute to the suppression of long–range order in both materials in the same way, the random fields due to
isomer disorder that exist in Mn[12]–ac and not in Mn[12]–ac–MeOH causes further suppression of ferromagnetism in Mn[12]–ac. The behavior observed for Mn[12]–ac is consistent with a random field
model calculated for this system by Millis et al.
• Time Reversal Optical Tomography and Decomposition Methods for Detection and Localization of Targets in Highly Scattering Turbid Media
Year of Dissertation:
New near-infrared (NIR) diffuse optical tomography (DOT) approaches were developed to detect, locate, and image small targets embedded in highly scattering turbid media. The first approach,
referred to as time reversal optical tomography (TROT), is based on time reversal (TR) imaging and multiple signal classification (MUSIC). The second approach uses decomposition methods of
non-negative matrix factorization (NMF) and principal component analysis (PCA) commonly used in blind source separation (BSS) problems, and compare the outcomes with that of optical imaging using
independent component analysis (OPTICA). The goal is to develop a safe, affordable, noninvasive imaging modality for detection and characterization of breast tumors in early growth stages when
those are more amenable to treatment. The efficacy of the approaches was tested using simulated data, and experiments involving model media and absorptive, scattering, and fluorescent targets, as
well as, "realistic human breast model" composed of ex vivo breast tissues with embedded tumors. The experimental arrangements realized continuous wave (CW) multi-source probing of samples and
multi-detector acquisition of diffusely transmitted signal in rectangular slab geometry. A data matrix was generated using the perturbation in the transmitted light intensity distribution due to
the presence of absorptive or scattering targets. For fluorescent targets the data matrix was generated using the diffusely transmitted fluorescence signal distribution from the targets. The data
matrix was analyzed using different approaches to detect and characterize the targets. The salient features of the approaches include ability to: (a) detect small targets; (b) provide
three-dimensional location of the targets with high accuracy (~within a millimeter or 2); and (c) assess optical strength of the targets. The approaches are less computation intensive and
consequently are faster than other inverse image reconstruction methods that attempt to reconstruct the optical properties of every voxel of the sample volume. The location of a target was
estimated to be the weighted center of the optical property of the target. Consequently, the locations of small targets were better specified than those of the extended targets. It was more
difficult to retrieve the size and shape of a target. The fluorescent measurements seemed to provide better accuracy than the transillumination measurements. In the case of ex vivo detection of
tumors embedded in human breast tissue, measurements using multiple wavelengths provided more robust results, and helped suppress artifacts (false positives) than that from single wavelength
measurements. The ability to detect and locate small targets, speedier reconstruction, combined with fluorophore-specific multi-wavelength probing has the potential to make these approaches
suitable for breast cancer detection and diagnosis.
Year of Dissertation:
Under a longitudinal rescaling of coordinates x0,3 → λ x0,3 , λ << 1, the classical QCD action simplifies dramatically. This is the high-energy limit, as λ ∼ s-1/2 where s is the center-of-mass
energy squared of a hadronic collision. We find the quantum corrections to the rescaled action at one loop, in particular finding the anomalous powers of λ in this action, with λ < 1. The method
is an integration over high-momentum components of the gauge field. This is a Wilsonian renormalization procedure, and counterterms are needed to make the sharp-momentum cut-off gauge invariant.
Our result for the quantum action is found, assuming |lnλ| << 1, which is essential for the validity of perturbation theory. If λ is sufficiently small (so that |lnλ| >> 1), then the perturbative
renormalization group breaks down. This is due to uncontrollable fluctuations of the longitudinal chromomagnetic field.
• Crystal Growth and Neutron Scattering Studies of High Temperature Superconductors
Year of Dissertation:
Since the discovery of the first high temperature superconductor in the 1980's, there have been continuing efforts to understand the mechanism of high-T[C] superconductivity. Studies on the
cuprate systems seem to suggest that there is an intimate relationship between superconductivity and magnetism, and recently this has also shown to be the case for the newly discovered Fe-based
superconductors. Neutron scattering is a powerful tool for studying magnetism in superconductors, which can provide important information about the momentum and energy dependence of magnetic
correlations. The work presented in this thesis is divided into two main sections. Since high-quality large-size single crystals are necessary for the neutron scattering experiments. The first
section is about sample preparation, where I will introduce single crystal growths via the Floating-zone technique as well as unidirectional solidi¯cation methods. The second section is neutron
scattering experiments, which will show neutron scattering and transport measurements results in two high-temperature superconductor systems: La[2-x]Ba[x]CuO[4] (LBCO), and Fe[1+y]Te[1-x]Se[x]
(FeTeSe). In the LBCO system, we found that static magnetic order competes with bulk superconductivity. In addition, applying a magnetic field to or adding Zn impurities in the sample will
enhance the static magnetic order and suppress the superconductivity. In the FeTeSe system, we found that spin resonance is associated with superconductivity, while resonance and
superconductivity are simultaneously suppressed by an applied magnetic field or adding Fe impurities. Our results suggest that the magnetic correlations are important for the superconductivity,
and proper tuning of these correlations may be a key for superconductivity.
• A Classical And Quantum Noise Model
Year of Dissertation:
We develop detailed statistics of a noise model that consists of $N$ independent harmonic oscillators where the total force is given by the sum of the individual forces. This model was first
proposed in the paper by Ford, Kac, and Mazur that was aimed at deriving the Langevin equation from first principles. We extend the model and calculate relevant probability distributions and
other statistical quantities such as the autocorrelation function. In the usual model one assumes that the initial position and momentum values are stochastic variables that determine the
statistical features of the force by ensemble averaging over those quantities. We extend that by also treating the mass and frequency as statistical quantities. We consider both the equilibrium
case, that is the canonical distribution for the initial positions and momenta, for the but we also consider other initial distributions and show that this leads to non-stationary autocorrelation
functions. One of our basic aims is to also develop this model for the quantum case and compare the results with the classical case. The general approach we use for the calculation of the
statistical quantities is by way of the characteristic function. We use the characteristic function approach because the oscillators are independent of each other. However, the quantum
characteristic function present unique difficulties because the initial momentum and position operators do not commute. We use the Weyl correspondence to define the quantum characteristic
function and we derive explicit expressions for both the pure case and mixtures. We show that many of the statistical quantities can be expressed in terms of the Wigner distribution. In addition,
we consider the time-frequency Wigner spectrum of momentum governed by the Langevin equation when the random driving term is quantum noise. We obtaine an explicit equation. The equation is solved
exactly and includes both the transient and the stationary part. The time-dependent Wigner spectrum generalizes the result of Wang and Uhlenbeck wherein they showed that for the white noise
driving force the power spectrum in the stationary state regime is Lorenzian. We show that our solution reduces to the classical solution when the parameters of the quantum noise are such that
the white noise limit is approached and when the long time limit is taken. | {"url":"http://gc.cuny.edu/GC-Header/Alumni/Alumni-Dissertations.aspx?program=Physics&sortby=author&page=6","timestamp":"2014-04-18T23:17:44Z","content_type":null,"content_length":"133257","record_id":"<urn:uuid:797a3899-954b-4648-bb81-7bfe0418d77e>","cc-path":"CC-MAIN-2014-15/segments/1398223205375.6/warc/CC-MAIN-20140423032005-00650-ip-10-147-4-33.ec2.internal.warc.gz"} |
Time Value of Money
1. 118527
A. Starting with $10,000, how much will you have in 10 years if you can earn 15 percent on your money?
b. If you inherited $25,000 today and invested all of it in a security that paid a 10 percent rate of return, how much would you have in 25 years?
c. If the average new home costs $125,000 today, how much will it cost in 10 years if the price increases by 5 percent each year?
d. If you can earn 12 percent, how much will you have to save each year if you want to retire in 35 years with $1 million?
The solution has various questions relating to time value of money | {"url":"https://brainmass.com/business/the-time-value-of-money/118527","timestamp":"2014-04-19T04:48:41Z","content_type":null,"content_length":"25480","record_id":"<urn:uuid:6e9dccd9-031d-41f7-bfc3-a38ef5906071>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00484-ip-10-147-4-33.ec2.internal.warc.gz"} |
Open dots and a closed dots
Hi Cynthia,
Suppose that you want to plot the solution to the inequality
1< x ≤ 5
To do this you would draw a line segment from 1 to 5 but somehow, on the graph, you need to show that x = 5 satisfies the inequality but x = 1 does not. We use the open and closed dots to make this
distinction. I put a closed dot at the "5" end of the interval to indicate that x = 5 satisfies the inequality, and I put an open dot at the "1" end of the interval to indicate that x = 1 does not
satisfy the inequality.
And another example | {"url":"http://mathcentral.uregina.ca/QQ/database/QQ.09.05/cynthia1.html","timestamp":"2014-04-16T18:56:52Z","content_type":null,"content_length":"3758","record_id":"<urn:uuid:1a111478-d404-4ca1-a9f9-ffc0219f13ab>","cc-path":"CC-MAIN-2014-15/segments/1398223206147.1/warc/CC-MAIN-20140423032006-00612-ip-10-147-4-33.ec2.internal.warc.gz"} |
natural language processing blog
NIPS is over as of last night. Overall I thought the program was strong (though I think someone, somewhere, is trying to convince me I need to do deep learning -- or at least that was the topic
d'jour... or I guess d'an? this time). I wasn't as thrilled with the venue (details at the end) but that's life. Here were some of the highlights for me, of course excluding our own papers :P, (see
the full paper list here)... note that there will eventually be videos for everything!
• User-Friendly Tools for Studying Random Matrices
Joel A Tropp
This tutorial was awesome. Joel has apparently given it several times and so it's really well fine-tuned. The basic result is that if you love your Chernoff bounds and Bernstein inequalities for
(sums of) scalars, you can get almost exactly the same results for (sums of) matrices. Really great talk. If I ever end up summing random matrices, I'm sure I'll use this stuff!
• Emergence of Object-Selective Features in Unsupervised Feature Learning
Adam Coates, Andrej Karpathy, Andrew Y. Ng
They show that using only unlabeled data that is very heterogenous, some simple approaches can pull out faces. I imagine that some of what is going on is that faces are fairly consistent in
appearance whereas "other stuff" often is not. (Though I'm sure my face-recognition colleagues would argue with my "fairly consistent" claim.)
• Scalable nonconvex inexact proximal splitting
Suvrit Sra
I just have to give props to anyone who studies nonconvex optimization. I need to read this -- I only had a glance at the poster -- but I definitely think it's worth a look.
• A Bayesian Approach for Policy Learning from Trajectory Preference Queries
Aaron Wilson, Alan Fern, Prasad Tadepalli
The problem solved here is imitation learning where your interaction with an expert is showing them two trajectories (that begin at the same state) and asking them which is better. Something I've
been thinking about recently -- very happy to see it work!
• FastEx: Hash Clustering with Exponential Families
Amr Ahmed, Sujith Ravi, Shravan M. Narayanamurthy, Alexander J. Smola
The idea here is to replace the dot product between the parameters and sufficient statistics of an exp fam model with an approximate dot product achieved using locality sensitive hashing. Take a
bit to figure out exactly how to do this. Cool idea and nice speedups.
• Identifiability and Unmixing of Latent Parse Trees
Daniel Hsu, Sham M. Kakade, Percy Liang
Short version: spectral learning for unsupervised parsing; the challenge is to get around the fact that different sentences have different structures, and "unmixing" is the method they propose to
do this. Also some identifiability results.
• Tensor Decomposition for Fast Parsing with Latent-Variable PCFGs
Shay B. Cohen and Michael Collins
Another spectral learning paper, this time for doing exact latent variable learning for latent-variable PCFGs. Fast, and just slightly less good than EM.
• Multiple Choice Learning: Learning to Produce Multiple Structured Outputs
Abner Guzman-Rivera Dhruv Batra Pushmeet Kohli
Often we want our models to produce k-best outputs, but for some reason we only train them to produce one-best outputs and then just cross our fingers. This paper shows that you can train
directly to produce a good set of outputs (not necessarily diverse: just that it should contain the truth) and do better. It's not convex, but the standard training is a good initializer.
• [EDIT Dec 9, 11:12p PST -- FORGOT ONE!]
Query Complexity of Derivative-Free Optimization
Kevin G. Jamieson, Robert D. Nowak, Benjamin Recht
This paper considers derivative free optimization with two types of oracles. In one you can compute f(x) for any x with some noise (you're optimizing over x). In the other, you can only ask
whether f(x)>f(y) for two points x and y (again with noise). It seems that the first is more powerful, but the result of this paper is that you get the same rates with the second!
• I didn't see it, but Satoru Fujishige's talk Submodularity and Discrete Convexity in the Discrete Machine Learning workshop was supposedly wonderful. I can't wait for the video.
• Similarly, I heard that Bill Dolan's talk on Modeling Multilingual Grounded Language in the xLiTe workshop was very good.
• Ryan Adam's talk on Building Probabilistic Models Around Deterministic Optimization Procedures in the "Perturbations, Optimization and Statistics" workshop (yeah, I couldn't figure out what the
heck that meant either) was also very good. The Perturb-and-MAP stuff and the Randomized Optimum models are high on my reading list, but I haven't gotten to them quite yet.
• As always, Ryan McDonald and Ivan Titov gave very good talks in xLiTe, on Advances in Cross-Lingual Syntactic Transfer and Inducing Cross-Lingual Semantic Representations of Words, Phrases,
Sentences and Events, respectively.
I'm sure there was lots of other stuff that was great and that I missed because I was
working hard on
Really my only gripe about NIPS this year was the venue. I normally wouldn't take the time to say this, but since we'll be enjoying this place for the next few years, I figured I'd state what I saw
as the major problems, some of which are fixable. For those who didn't come, we're in Stateline, NV (on the border between CA and NV) in two casinos. Since we're in NV, there is a subtle note of old
cigarette on the nose fairly constantly. There is also basically nowhere good to eat (especially if you have dietary restrictions) -- I think there are a half dozen places on yelp with 3.5 stars or
greater. My favorite tweet during NIPS was Jacob Eisenstein who said: "stateline, nevada / there is nothing but starbucks / the saddest haiku". Those are the "unfixables" that make me think that I'll
think twice about going to NIPS next year, but of course I'll go.
The things that I think
fixable... there was no where to sit. Presumably this is because the casino wants you to sit only where they can take your money, but I had most of my long discussions either standing or sitting on
the ground. More chairs in hallways would be much appreciated. There was almost no power in rooms, which could be solved by some power strips. The way the rooms divided for tutorials was really
awkward, as the speaker was clear on one side of the room and the screen was on the other (and too high to point to) so it was basically like watching a video of slides online without ever seeing the
presenter. Not sure if that's fixable, but seems plausible. And the walls between the workshop rooms were so thin that often I could hear another workshop's speaker better than I could hear the
speaker in the workshop I was attending. And the internet in my hotel room was virtually unusably slow (though the NIPS specific internet was great).
No comments: | {"url":"http://nlpers.blogspot.com/2012/12/nips-stuff.html","timestamp":"2014-04-17T15:28:13Z","content_type":null,"content_length":"95963","record_id":"<urn:uuid:99b5e63c-edda-4252-898d-d7c2528dbdc8>","cc-path":"CC-MAIN-2014-15/segments/1397609530136.5/warc/CC-MAIN-20140416005210-00511-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Teaching an older child behind in math
The challenge to help an older child who struggles with math is to rebuild a math foundation and repair a negative self-concept. In this situation, if the child is required to learn from a textbook
that is several grade levels below his current grade, the rebuilding process is likely to be very discouraging. Math on the Level contains every math concept which must be mastered -- without
textbooks and with no grade levels assigned to concepts. During the first few weeks of using Math on the Level, the 5-A-Day review process will let you asses the child's retention of all concepts and
find out what needs to be re-taught.
With other math programs, children may be taught before being maturationally ready to understand the concept, and they often compensate by memorizing steps instead of truly understanding what was
taught. This flawed approach creates a foundation filled with holes. Typically, after years of memorizing steps without understanding, everything falls apart. This often happens between 6th and 8th
grades, which is when algebra is generally introduced.
With Math on the Level, you can take your child back to the basic concepts, re-teach and review them in a way that emphasizes comprehension, and rebuild a solid math foundation. Because there are no
grade levels in the curriculum, a forgotten concept can be re-taught without the stigma of having to use a far-below-grade-level textbook. You can target instruction exactly where it is needed, so
the re-learning process can be done efficiently (as opposed to having to wade through pages of textbooks).
The Math Adventures volume emphasizes real-life applications of math and includes many activities that are appropriate for older children. For example, math can be practiced through preparing a
budget, by analyzing the calories in a recipe, or in the context of starting a small business. When students successfully build math skills by using math, they are far more motivated than just doing
computations on a worksheet.
Another advantage of using Math on the Level with older students is the 5-A-Day review system, which is very effective in getting information into long term memory. The ongoing review process is
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Overall, Math on the Level is an excellent choice if your older student is behind in math skills. It will help build a solid foundation and rekindle a successful attitude toward learning and using | {"url":"https://www.mathonthelevel.com/index.php/component/content/article?id=165","timestamp":"2014-04-18T18:49:25Z","content_type":null,"content_length":"13155","record_id":"<urn:uuid:5970ee2f-7bc2-4dba-8c80-5d4fe9ff2c24>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.7/warc/CC-MAIN-20140416005215-00061-ip-10-147-4-33.ec2.internal.warc.gz"} |
Factoring Numbers
"Factors" are the numbers you multiply to get another number. For instance, the factors of 15 are 3 and 5, because 3×5 = 15. Some numbers have more than one factorization (more than one way of being
factored). For instance, 12 can be factored as 1×12, 2×6, or 3×4. A number that can only be factored as 1 times itself is called "prime". The first few primes are 2, 3, 5, 7, 11, and 13. The number 1
is not regarded as a prime, and is usually not included in factorizations, because 1 goes into everything. (The number 1 is a bit boring in this context, so it gets ignored.)
You most often want to find the "prime factorization" of a number: the list of all the prime-number factors of a given number. The prime factorization does not include 1, but does include every copy
of every prime factor. For instance, the prime factorization of 8 is 2×2×2, not just "2". Yes, 2 is the only factor, but you need three copies of it to multiply back to 8, so the prime factorization
includes all three copies.
On the other hand, the prime factorization includes ONLY the prime factors, not any products of those factors. For instance, even though 2×2 = 4, and even though 4 is a divisor of 8, 4 is NOT in the
PRIME factorization of 8. That is because 8 does NOT equal 2×2×2×4! This accidental over-duplication of factors is another reason why the prime factorization is often best: it avoids counting any
factor too many times. Suppose that you need to find the prime factorization of 24. Sometimes a student will just list all the divisors of 24: 1, 2, 3, 4, 6, 8, 12, and 24. Then the student will do
something like make the product of all these divisors: 1×2×3×4×6×8×12×24. But this equals 331776, not 24. So it's best to stick to the prime factorization, even if the problem doesn't require it, in
order to avoid either omitting a factor or else over-duplicating one.
In the case of 24, you can find the prime factorization by taking 24 and dividing it by the smallest prime number that goes into 24: 24 ÷ 2 = 12. (Actually, the "smallest" part is not as important as
the "prime" part; the "smallest" part is mostly to make your work easier, because dividing by smaller numbers is simpler.) Now divide out the smallest number that goes into 12: 12 ÷ 2 = 6. Now divide
out the smallest number that goes into 6: 6 ÷ 2 = 3. Since 3 is prime, you're done factoring, and the prime factorization is 2×2×2×3.
An easy way of keeping track of the factorization is to do upside-down division. It looks like this:
The nice thing about this upside-down division is that, when you're done, the prime factorization is the product of all the numbers around the outside. The factors are circled in red above. By the
way, this upside-down division is something that should probably be done on scratch-paper, and not handed in as part of your homework. Copyright © Elizabeth Stapel 2000-2011 All Rights Reserved
• Find the prime factorization of 1050.
I'll do the upside-down division:
Then my answer is: 1050 = 2 × 3 × 5 × 5 × 7
Some texts prefer that answers such as this be written using exponential notation, in which case the final answer would be written as 2×3×5^2×7.
You can do the repeated division "right-side up", too, if you prefer. The process works the same way, but the division is reversed in orientation. The above problem would be worked out like this:
I'll do the repeated division:
1092 = 2 × 2 × 3 × 7 × 13
This answer might also be written as 2^2×3×7×13.
By the way, there are some divisibility rules that can help you find the numbers to divide by. There are many divisibility rules, but the simplest to use are these:
• If the number is even, then it's divisible by 2.
• If the number's digits sum to a number that's divisible by 3, then the number itself is divisible by 3.
• If the number ends with a 0 or a 5, then it's divisible by 5.
Of course, if the number is divisible twice by 2, then it's divisible by 4; if it's divisible by 2 and by 3, then it's divisible by 6; and if it's divisible twice by 3 (or if the sum of the digits is
divisible by 9), then it's divisible by 9. But since you're finding the prime factorization, you don't really care about these non-prime divisibility rules. There is a rule for divisibility by 7, but
it's complicated enough that it's probably easier to just do the division on your calculator and see if it comes out even.
If you run out of small primes and you're not done factoring, then keep trying bigger and bigger primes (11, 13, 17, 19, 23, etc) until you find something that works — or until you reach primes whose
squares are bigger than what you're dividing into. Why? If your prime doesn't divide in, then the only potential divisors are bigger primes. Since the square of your prime is bigger than the number,
then a bigger prime must have as its remainder a smaller number than your prime. The only smaller number left, since all the smaller primes have been eliminated, is 1. So the number left must be
prime, and you're done.
You can use the Mathway widget below to practice finding the prime factorization. Try the entered exercise, or type in your own exercise. Then click "Answer" to compare your answer to Mathway's.
(Clicking on "View Steps" on the widget's answer screen will take you to the Mathway site, where you can register for a free seven-day trial of the software.)
Top | Return to Index
Cite this article as: Stapel, Elizabeth. "Factoring Numbers." Purplemath. Available from
http://www.purplemath.com/modules/factnumb.htm. Accessed
This lesson may be printed out for your personal use. | {"url":"http://www.purplemath.com/modules/factnumb.htm","timestamp":"2014-04-20T23:50:35Z","content_type":null,"content_length":"40253","record_id":"<urn:uuid:b7e1beb5-9962-4be9-a4a1-d39228acbf3f>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00507-ip-10-147-4-33.ec2.internal.warc.gz"} |
Physics Forums - View Single Post - How can Higgs field explain proton's inertial resistance to acceleration?
If most of the mass/energy of a proton is due to the kinetic energy of its quarks and gluons, rather than interaction with the Higgs field, then how can we explain its inertial mass, i.e. its
resistance to acceleration, as being due to a drag induced by the Higgs field?
You can't. The proton mass has little to do with the Higgs. Therefore its resistance to acceleration has little to do with the Higgs. No one has claimed so.
But the Higgs field is Lorentz invariant so that if the inertial resistance force is due to the Higgs field then it shouldn't be any harder to accelerate the body whether it is spinning or not.
A Lorentz transformation never turns a non-rotating body into a rotating body (though it can affect the linear velocity). So Lorentz invariance doesn't give you any connection between the two | {"url":"http://www.physicsforums.com/showpost.php?p=3676488&postcount=2","timestamp":"2014-04-16T22:16:05Z","content_type":null,"content_length":"9048","record_id":"<urn:uuid:e09cbcf0-40ae-457f-af0d-66099dbf752b>","cc-path":"CC-MAIN-2014-15/segments/1398223202457.0/warc/CC-MAIN-20140423032002-00022-ip-10-147-4-33.ec2.internal.warc.gz"} |
Translating XML Web Data into Ontologies.
Conference Proceeding
Translating XML Web Data into Ontologies.
01/2005; DOI:10.1007/11575863_118 In proceeding of: On the Move to Meaningful Internet Systems 2005: OTM 2005 Workshops, OTM Confederated International Workshops and Posters, AWeSOMe, CAMS, GADA,
MIOS+INTEROP, ORM, PhDS, SeBGIS, SWWS, and WOSE 2005, Agia Napa, Cyprus, October 31 - November 4, 2005, Proceedings
ABSTRACT Translating XML data into ontologies is the problem of flnd- ing an instance of an ontology, given an XML document and a specifl- cation of the relationship between the XML schema and the
ontology. Previous study (8) has investigated the ad hoc approach used in XML data integration. In this paper, we consider to translate an XML web document to an instance of an OWL-DL ontology in the
Semantic Web. We use the semantic mapping discovered by our prototype tool (1) for the relationship between the XML schema and the ontology. Particularly, we deflne the solution of the translation
problem and develop an algorithm for computing a canonical solution which enables the ontology to answer queries by using data in the XML document.
Data provided are for informational purposes only. Although carefully collected, accuracy cannot be guaranteed. The impact factor represents a rough estimation of the journal's impact factor and does
not reflect the actual current impact factor. Publisher conditions are provided by RoMEO. Differing provisions from the publisher's actual policy or licence agreement may be applicable.
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SAT Kaplan problem
August 27th 2011, 10:46 AM #1
Junior Member
Jul 2011
SAT Kaplan problem
in the figure above, how many points are half as far from k as from m?
i don't get the question
Re: SAT Kaplan problem
is it asking like how many points are half as far from k as k as from m?
Re: SAT Kaplan problem
It's asking you to find points that are some distance from k and also twice that distance from m.
edit: I could have phrased that much better. You don't have to find the points, just find how many there are.
August 27th 2011, 10:52 AM #2
Junior Member
Jul 2011
August 27th 2011, 11:35 AM #3
Junior Member
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Topic: HELP! basic algebra, anyone?
Replies: 4 Last Post: Nov 18, 1996 8:03 PM
Messages: [ Previous | Next ]
HELP! basic algebra, anyone?
Posted: Nov 16, 1996 1:16 AM
Hello, the problem is as following:
2^(2x) - 3(2^(x+1)) + 8 = 0
Find the two values of x that satisfy the above equation:
Thanks in advance.
Date Subject Author
11/16/96 HELP! basic algebra, anyone? dpmyang@lynx.bc.ca (> gUFman
11/17/96 Re: HELP! basic algebra, anyone? David D Buchanan
11/18/96 Re: HELP! basic algebra, anyone? Darren Matthew Glass
11/18/96 Re: HELP! basic algebra, anyone? J Robert Archer
11/18/96 Re: HELP! basic algebra, anyone? James A Mccann | {"url":"http://mathforum.org/kb/thread.jspa?threadID=179398","timestamp":"2014-04-17T05:58:19Z","content_type":null,"content_length":"20696","record_id":"<urn:uuid:ea0c98b6-5b83-4d68-b3ed-d64a1e25f329>","cc-path":"CC-MAIN-2014-15/segments/1397609526252.40/warc/CC-MAIN-20140416005206-00503-ip-10-147-4-33.ec2.internal.warc.gz"} |
Graphing Linear FunctionsAlgebraLAB: Lessons
This lesson will make use of slopes and y-intercepts on a graph. If you need to review these topics, click here for
(linear equations slope.doc) or click here for y-intercepts. (linear equations intercepts.doc)
Let's start with a
that has a
positive slope
Suppose we are given the function y = 2x + 5. Because this is in slope-intercept form of a line, we can see that the slope is 2 and the y-intercept is 5.
This is enough information to graph the function. Because we know the y-intercept is 5, we can start by placing a point at (0, 5) on the coordinate system.
The slope of 2 gives us the additional information we need to complete the graph. Remember that slope is “rise over run” or “the change in y over the change in x”. In any case, we always need to
think about slope as a fraction. Since we know the slope is 2, we should think of it in the fraction form of graph below that the new point is at (1, 7).
Once you know two points on a graph, you can connect those two points with a line and you then have the graph of the equation.
To find other points on the line, simply choose a starting point, and then move right one space and up two spaces to satisfy the slope of space and down two spaces and still find another point on
the line. Why would this work? Think about this. When you move to the left one you are changing the x-value by -1. When you move down two you are changing the y-value by -2. This will give a
slope of line indicates. Look at the graph below to verify moving left 1 and down 2 will give you the point (-1, 3) which is on the graph.
Let’s now consider what should happen if we have a
negative slope
Look at the function y-intercept for this graph will be at the point (0, -3). This will be our starting point for finding other points on the graph.
Since the slope is
We need to remember that when there is a negative fraction, as there is in this case, it can be written in one of three ways.
□ This indicates that if we move down 2, we should move right three because of
□ It also says that if we move up 2, we should move left 3 because of
Let’s find two more points.
□ Using point at (0+3, -3-2) = (3, -5)
□ Using point at (0-3, -3+2) = (-3, -1)
The graph of
What is your answer?
What is your answer?
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MathGroup Archive: April 1993 [00062]
[Date Index] [Thread Index] [Author Index]
Re: the size of frames
• To: mathgroup at yoda.physics.unc.edu
• Subject: Re: the size of frames
• From: Jonathan Rich <rich at earth.nwu.edu>
• Date: Mon, 26 Apr 1993 10:46:23 -0500
----- Begin Included Message -----
>From mathgroup-adm at yoda.physics.unc.edu Mon Apr 26 02:03:38 1993
Date: Wed, 21 Apr 93 10:12:59 EDT
From: matteo at cns.nyu.edu (Matteo Carandini)
To: mathgroup at yoda.physics.unc.edu
Subject: the size of frames
Content-Length: 858
Hi mathgroup,
I have a simple but very urgent problem:
I am making GraphicArrays of Framed plots, with FrameTicks and FrameLabels,
and I NEED the frames to have all exactly the same size.
Now, Mma has a very sensible behavior that I would like to inhibit:
it scales each frame depending on the amount of stuff that is has to write
next to the axes, in such a way to preserve the AspectRatio.
For example a frame with y-values like 10000 will be smaller than one
with y-values like 10, because "10000" takes up more space than "10".
Similarly, the frame will get smaller if one adds a label.
Again, this is very sensible, but I really need to be able to tell Mma
to just make all the frames of the same size, which I will choose small enough
to leave room for labels.
Matteo Carandini
Center for Neural Science
New York University
----- End Included Message -----
I would like to second this question since I too face the same problem.
An example follows:
I know what is happening, namely that Mathematica is filling the display area
which includes the space taken up by the axes tick numbers. However this is
not the behavior I want, since it is much more important for me that the y-axes
line up vertically even if this means not utilizing all of the display area
for one of the plots. How can this be enforced? Thanks in advance.
P.S. On page 422 the book has the following example:
Show[bp,Frame->True,FrameLabel->{"label 1","label 2","label 3","label 4"}]
When I try this example, the output I see on my screen has all of the labels
displayed horizontally, but when I actually print it, it comes out as the book
shows it and as it should. Is this a bug or is this something peculiar to my
setup? Also the FontForm examples on pages 468 and 470 not only look different
on the screen but also look different, when printed, from what is shown in the
book?! (The text in the book appears MUCH larger!)
pg468=Show[ Plot[Sin[Sin[x]],{x,0,Pi}], DefaultFont-> {"Times-Italic", 6},
PlotLabel-> FontForm["The label", {"Helvetica-Bold", 12}]]
pg470=Show[Graphics[ Text[FontForm["Some text", {"Courier-Bold", 14}], {2,2}, {-1, 0}, {0, 1}]], Frame->True ]
To give some idea of my setup, I show the following which is printed when
Mathematica is invoked:
Mathematica 2.1 for HP 9000 RISC
Copyright 1988-92 Wolfram Research, Inc.
-- Motif graphics initialized --
Jonathan Rich
rich at earth.nwu.edu | {"url":"http://forums.wolfram.com/mathgroup/archive/1993/Apr/msg00062.html","timestamp":"2014-04-18T11:17:10Z","content_type":null,"content_length":"37072","record_id":"<urn:uuid:8480bc80-0d2e-48f9-8ef2-67f2a9de269c>","cc-path":"CC-MAIN-2014-15/segments/1397609533308.11/warc/CC-MAIN-20140416005213-00093-ip-10-147-4-33.ec2.internal.warc.gz"} |
st: paired samples proportions
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st: paired samples proportions
From "Lachenbruch, Peter" <lachenbruch@cber.fda.gov>
To "statalist (E-mail)" <statalist@hsphsun2.harvard.edu>
Subject st: paired samples proportions
Date Wed, 27 Aug 2003 09:10:06 -0400
As Joe Coveney notes this is the McNemar test situation. Thus, one is
really computing the sample size to detect a difference from 0.5 for a
single sample using only the discordant parts of the 2x2 table.
+ -
+ a b
- c d
The test is given by X2=(b-c)2/(b+c) - it's easy to show this is the
binomial test. Where we get hung up is that we don't know how many
discordant pairs there will be. I wrote a paper on this in 1992 in
Statistics in Medicine (p. 1521) that isn't the last word on it. I just
observed that the numbers b and c were bounded by the number of discordant
pairs and 0 and you could do some playing with it. So Stata will compute
the sample size for a one sample test against p=0.5, but you then need to
figure out what proportion of discordant pairs you expect.
Peter A. Lachenbruch, Ph. D.
Director, Division of Biostatistics
Phone (301) 827-3320
FAX (301) 827-5218
* For searches and help try:
* http://www.stata.com/support/faqs/res/findit.html
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/ | {"url":"http://www.stata.com/statalist/archive/2003-08/msg00501.html","timestamp":"2014-04-20T19:23:52Z","content_type":null,"content_length":"5501","record_id":"<urn:uuid:78048e79-1f2d-478b-ae64-19d65c319496>","cc-path":"CC-MAIN-2014-15/segments/1398223206120.9/warc/CC-MAIN-20140423032006-00081-ip-10-147-4-33.ec2.internal.warc.gz"} |
A Symbolic Computational Approach to a Problem Involving Multivariate Poisson Distributions
By Eduardo Sontag and Doron Zeilberger .pdf LaTeX source Written: June 3, 2009.
[Appeared in Advances in Applied Mathematics v. 44 (2010), 359-377.]
How many Prussian officers per year do you expect to be kicked by their horses? How many Email messages will you get today? How many people will show up in your favorite restaurant? All these are
easy. But what if you know that twice the number of Prussian officers plus three times the letters plus twice the number of diners equals something, and you want to predict the individual numbers,
what would you do? Read this paper, to find the answers! [Of course, there are many other, more "serious" applications, to Biology and elsewhere (and phone calls!).]
Maple Package
Important: This article is accompanied by Maple package MVPoisson
[Version of March 15, 2010, incorporating helpful comments of Arthur Hipke]
Sample Input and Output for MVPoisson | {"url":"http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/mvp.html","timestamp":"2014-04-21T09:36:16Z","content_type":null,"content_length":"4830","record_id":"<urn:uuid:1b63a367-5ff1-4edf-bde6-1a4b4c713dc6>","cc-path":"CC-MAIN-2014-15/segments/1397609539705.42/warc/CC-MAIN-20140416005219-00241-ip-10-147-4-33.ec2.internal.warc.gz"} |
Find each derivative in simplest factored form
December 17th 2009, 09:24 AM #1
Sep 2009
Find each derivative in simplest factored form
Hi, I'm having some trouble with these problems.
1. $f(x) = 3sin(cos4x)$
2. $f(x) = 5cos(sin(sin(4x^2)))$
3. $f(x) = sin^2(3x^2)$
4. $f(x) = cos^4(sin(cos(3x+2)))$
I'm not really sure what these 2 problems are asking me to do:
5. Write the equation of the tangent line to the curve $f(x) = 3cos2x$ at x = 0.
6. Write the equation of the normal line to the curve $f(x) = tan3x$ at x = pi/12
Any help is appreciated, thanks in advance!
You aren't suppose to post lists of problems like this. It's againsts the rules of the forum. You should show some of your work so that we know what you're struggling with.
If you're in a calculus class, and you've been assigned these problems, then you should be familiar with the basic rules of differentiation. The functions you listed are all composite trig
functions. This means that you need to use the chain rule, and the derivatives of the trig functions in their most basic forms. I'll help you get started with a few of them.
Problem 1
Think of it like this. Let $u=cos(4x)$.
Then $y=3sin(u)$
The derivative can be found by the chain rule.
See if you can finish this.
Problem 2
For a function like $y=f(x)=5cos(sin(sin(4x^2)))$
Just apply an extended version of the chain rule.
Let $u=sin(4x^2)$
So you can write:
You still have a composite function here, so use another variable:
Let $v=sin(u)$
Now just apply the chain rule.
$\frac{dy}{dx}=\frac{dy}{dv}\frac{dv}{du}\frac{du}{ dx}$
See if you can finish.
Problems 5 and 6
For tangent lines, you should be familiar with the fact that the equation of the tangent line to the curve $y=f(x)$ at the point $(a, f(a))$ is $y-f(a) =f'(a)(x-a)$ . This is a basic fact, and
should be easy to apply to the functions you were given in problems 5 and 6.
Thanks so much! I wasn't aware that this wasn't allowed, I'll make sure to look over the rules before posting again. I have another question, on problem 6, I found the derivative to be
$f'(x) = 3sec^2(3x)$
And I'm trying to plug pi/12 into the equation to find the slope but I'm stuck at this step
Where can you go from here? Thanks!
Thanks so much! I wasn't aware that this wasn't allowed, I'll make sure to look over the rules before posting again. I have another question, on problem 6, I found the derivative to be
$f'(x) = 3sec^2(3x)$
And I'm trying to plug pi/12 into the equation to find the slope but I'm stuck at this step
Where can you go from here? Thanks!
Remember the reciprocal identities from trignometry? You can write the slope at $x=\frac{\pi}{12}$ as follows:
$f'\left(\frac{\pi}{12}\right)=3sec^2\left(\frac{\p i}{4}\right)$
It's easy to show that:
The square of this is just $\frac{1}{2}$, so you should find that the slope of the tangent line simplifies to $f'\left(\frac{\pi}{12}\right)=6$. However, they are asking for the NORMAL line at $x
=\frac{\pi}{12}$. This is the line perpendicular to the tangent line at that point. Perpendicular lines have slopes that are negative reciprocals of each other. So the equation will have the
You should be able to take it from here. Did that all make sense?
December 17th 2009, 11:19 AM #2
Super Member
Jun 2009
United States
December 17th 2009, 12:03 PM #3
Sep 2009
December 17th 2009, 12:58 PM #4
Super Member
Jun 2009
United States | {"url":"http://mathhelpforum.com/calculus/120960-find-each-derivative-simplest-factored-form.html","timestamp":"2014-04-21T00:42:39Z","content_type":null,"content_length":"45510","record_id":"<urn:uuid:c5f0f9c7-735f-4254-8173-81e99bb49727>","cc-path":"CC-MAIN-2014-15/segments/1397609539337.22/warc/CC-MAIN-20140416005219-00547-ip-10-147-4-33.ec2.internal.warc.gz"} |