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# R squared always higher than 1

I am trying to implement an algorithm that solves a linear regression problem with the following objective function (LASSO):

$$\min_\beta \frac{1}{2}||y-X\beta||_2^2 + \lambda ||\beta||_1$$

for various values of $\lambda$ under several constraints which are added or change from time to time. $y$ and $X$ are my training data which have been standardized to have mean 0 and normalized to have unit $l_2$-norm. For all the regression problems that I solve (remember I add a few constraints from time to time), I want to calculate an out-of-sample $R^2$ on a validation set in order to compare the models. The validation set has also been standardized, yet I used the mean of the training set and the validation set was not normalized.

When I calculate the $R^2$ in the following way I receive values greater than 1:

$$R^2= \frac{\sum_{i=1}^n(\hat{y_i}-\bar{y_i})^2}{\sum_{i=1}^n (y_i - \bar{y_i})^2}$$

Since the training set was standardized to have 0 mean and the mean from the training set is used for the calculation of $R^2$, the above term simplifies to:

$$R^2= \frac{\sum_{i=1}^n(\hat{y_i})^2}{\sum_{i=1}^n (y_i )^2}$$

All my $R^2$ values are higher than 1 (about 1.5 to 1.6). Even if I use the same calculation on the training set, the value exceeds 1 (note that in the case of the training set the denominator equals 1 as the training set was normalized to have unit $l_2$-norm.

I sense that something is going utterly wrong here, yet I did not manage to find the mistake. I thought that maybe this standard calculation of $R^2$ does not work for my LASSO-objective function. If that is the case, which would be the correct way to calculate $R^2$ here?

• possible duplicate with this stats.stackexchange.com/questions/246347/… ? – utobi Nov 26 '16 at 18:50
• Apparently, the user there did have a similar question. Nevertheless there was no answer on how to calculate $R^2$ in the case for penalized models. – YukiJ Nov 27 '16 at 10:17

Your mistake doesn't come from putting the mean to zero, but from the general computation of $R^2$, which isn't the one you wrote. Using your notation we have several values:

• $SS_{tot} = \sum_i (y_i-\bar{y})^2$ total sum of squares
• $SS_{reg} = \sum_i (\hat{y}_i-\bar{y})^2$ explained sum of squares
• $SS_{res} = \sum_i (y_i-\hat{y}_i)^2$ residual sum of squares

Now the general formula is $R^2 = 1- \frac{SS_{res}}{SS_{tot}}$ depending on the ratio between the unexplained variance and the total variance of the data.

When $SS_{res} + SS_{reg} = SS_{tot}$ then the general formula is equivalent to the one you wrote: $R^2 = \frac{SS_{reg}}{SS_{tot}}$, which can be seen as the ratio between the explained variance to the total variance.

The condition $SS_{res} + SS_{reg} = SS_{tot}$ is true for instance in unregularized linear regression, but possibly not true with LASSO penalty.

• That's what I thought. Are there any suggestions on how to calculate $R^2$ in a regularized model? – YukiJ Nov 27 '16 at 7:57
• What's the problem with using the first (more general) formula? – etal Nov 27 '16 at 15:44
• The one you suggested cannot be used without an intercept as far as I am aware. So I wonder what I should do when my model does not allow a constant term (i.e. it always includes the origin). – YukiJ Nov 28 '16 at 11:12
• I don't understand why you cannot compute SSres and SStot and thus R even if your model includes the origin. You only need the original points and your predictions – etal Nov 28 '16 at 17:08
• Sure, but if it becomes negative that means you're better off always predicting the average (so 0) as your approximation. The formula implicitly assumes your model is at least as good as doing a constant prediction, which is the simplest thing you could do. – etal Nov 29 '16 at 14:00

This is an interesting question, see this and this for two related posts. As far as I understood from the literature, and judging from the answers/comments to the above cited posts, the calculation and interpretation of the coefficient of determination and the calculation of standard errors in penalised estimation approaches are currently open problems.

So my current answer to your question is: just give up using $R^2$ and adjusted $R^2$ in lasso-type problems. Perhaps, goodness-of-fit tests can be a viable alternative to $R^2$.


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# From binary to unary and EXPTIME

I read in [Razborov, "An equivalence between...", p.248] that

It is well recognized in Theoretical Computer Science that when we change the representation of integers from binary to unary, the class $$\mathsf{EXPTIME}$$ becomes not the class $$\mathsf{P}$$ but the class $$\mathsf{TIME}\left(2^{(\log n)^{O(1)}}\right)$$.

Unfortunatly there is no reference to this well recognized fact. I would like to get a proof of it or a reference. Thanks.

• Shouldn't it simply be replacing $n$ with $\log n$ in the definition of EXPTIME, which gives this result? Nov 30 '21 at 11:42
• I think you are right. If we denote by N the input length in unary and by n the input length in binary then it should be n = O(log N). Every algorithm that takes 2^(n^c) takes 2^(log N)^(O(1)) when expressed as a function of the unary input length.
– Aldo
Dec 1 '21 at 10:39

format: plain


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# Baratin and Freidel: a spin foam model of ordinary particle physics

by john baez
Tags: baratin, foam, freidel, model, ordinary, particle, physics, spin
 Share this thread:
 P: 360 I just read that new paper. As far as I can tell, it works out exactly as you would expect point particles to behave in MacDowell-Mansouri BeeF gravity. Judging from their introduction, they seem oddly excited about it though, so maybe I'm missing something. It could be the excitement stems from their description of these particles as field monopoles, but I'm not sure why that's so different than putting the point particle actions in by hand. Anyway, it's a decent treatment and I like the approach.
Astronomy
Sci Advisor
PF Gold
P: 23,101
 Quote by garrett I just read that new paper...
I am glad you had a look at it.
In a recent post, Baez mentioned that Freidel has 3 papers in the works with Starodubtsev and one in the work with Baratin. even if all don't come to fruition I'm inclined to expect at least a couple more in this same line of investigation.

It seems to me that you are especially well prepared to understand and comment, not only on this one but on the others when they come out.

The conclusions section speaks of a "forthcoming paper" in which they do a perturbation expansion in alpha

and thru that, they say, address the question of the flat limit of gravity and particles. I will get the quote

==quote==
First, since the alpha parameter is small, we can consider a perturbation theory of gravity coupled to particle(s) being the perturbation theory in alpha. The distinguished feature of this theory would be that it is, contrary to earlier approaches, manifestly diffeomorphism-invariant, so its framework it is possible to talk about weak gravitational field in the conceptual framework of full general relativity. These investigations, both in the case of beta = 0 and beta not = 0 will be presented in the forthcoming paper. The fuller control over the small alpha sector will presumably make it possible to address the outstanding question of what is the flat space limit of the theory of gravity, coupled to point particles. It has been claimed that such a theory will be not the special relativity, but some form of doubly special relativity
==endquote==

If they can show that the flatspace limit is not usual Lorentz but is, instead, some DSR, this would probably open up some possibilities to TEST. It would seem to me like considerable progress just to get a good flatspace limit of one sort or another.
 Astronomy Sci Advisor PF Gold P: 23,101 I am mulling over this parameter alpha that they want to do the perturbation expansion in. I think it came up in the earlier (Jan 2005?) Freidel Staro paper. You see it on page 3 of this paper, equation 2.3 if alpha and beta were both zero then S would be a usual BF action, but alpha perturbs it and makes it deviate from the usual BF action. Am I wrong? the nice thing is that we are now looking at a perturbation theory where we DO NOT HAVE A FIXED BACKGROUND GEOMETRY around which we perturb. I don't claim to have much grasp of this, but we seem to be contemplating the opportunity to "perturb around pure BeeF itself" so they hold out the attractive notion of a background independent perturbation theory or I guess what they said was a "manifestly diffeomorphism invariant" perturbation theory. that was what they said in conclusions on page 15. right now it looks to me as if they are proceeding with exactly what they promised in http://arxiv.org/hep-th/0501191 that they would do. rather than us getting new signals this time we are getting confirmation of progress along lines they said in january last year. Am I missing something?
 P: 360 Yep, that all sounds right. The $\alpha$ term is what makes BF into gravity. With $\alpha$ itself proportional to the gravitational constant. Rovelli wrote about this as well, in his propagator paper. And, urr, I do BF too -- although I came to it rather circuitously. "BF, it's what's for dinner."
Astronomy
Sci Advisor
PF Gold
P: 23,101
 Quote by garrett ... And, urr, I do BF too -- although I came to it rather circuitously.
Yes! and I am looking for you to surf this BF wave!

there is much truth in the saying
"BF, it's what's for dinner." I may adopt it as a signature.
Emeritus
PF Gold
P: 8,147
 Quote by Marcus there is much truth in the saying "BF, it's what's for dinner." I may adopt it as a signature.
Although Prof. Baez once remarked that it should be "EF" which spoils the pun. He said the so-called B part of BF theory was not in fact like magnetism (which B traditionally expresses) but like electricity, E.
Astronomy
Sci Advisor
PF Gold
P: 23,101
 Quote by selfAdjoint Although Prof. Baez once remarked that it should be "EF" which spoils the pun. He said the so-called B part of BF theory was not in fact like magnetism (which B traditionally expresses) but like electricity, E.
the equations look prettier with E and F instead of B and F
and the analogy is more correct, true, but still everybody says BF.
maybe we have to go with it.
Emeritus
PF Gold
P: 8,147
 Quote by marcus the equations look prettier with E and F instead of B and F and the analogy is more correct, true, but still everybody says BF. maybe we have to go with it.
Yeah, I agee. And who would want to give up that great Sig line? Even Baez seems to have bit the bullet.
Astronomy
Sci Advisor
PF Gold
P: 23,101
 Quote by selfAdjoint Yeah, I agee. And who would want to give up that great Sig line? Even Baez seems to have bit the bullet.
Garrett can have it back anytime he wants

On Thursday, two days hence, John Baez student Derek Wise will give a talk at Perimeter.

It is along the general lines Baez has been talking about but expecially about the papers of Baez, Wise, Crans and of Baez Perez.

I hope they put a video at the streamer site. here is the abstract:

Derek Wise
Exotic statistics and particle types in 3- and 4d BF theory
Thursday July 13, 2006, 1:30 PM
"Gravity in 2+1 dimensions has the remarkable property that momenta live most naturally not in Minkowski vector space but in the 3d Lorentz group SO(2,1) itself. Having group-valued momentum has interesting consequences for particles, including exotic statistics and a modified classification of elementary particle types. These results generalize immediately to 3d BF theory with arbitrary gauge group. Better yet, they generalize to 4d BF theory, where matter shows up as string-like defects. These 'strings' exhibit exotic statistics governed not by the usual braid group, but by its higher dimensional cousin: the 'loop braid group'. We discuss these statistics as well as the classification of elementary 'string types' in 4d BF theory."

http://perimeterinstitute.com/activi...&SeminarID=759
 Astronomy Sci Advisor PF Gold P: 23,101 I had the wrong post here earlier. Here is a question. if anyone wants to comment. In the first Freidel Starodubtsev paper they cited two "in preparation" papers one of them was something we know for sure has NOT appeared [13]Freidel Starodubtsev "perturbation gravity via spin foams" that would be the SPIN FOAM QUANTIZATION OF THE CLASSICAL WORK WE JUST SAW so if and when that paper comes out it will be kind of major. the other was [6] Freidel Kowalski-Glikman Starodubtsev "Background Independent Perturbation Theory for Gravity Coupled to Particles: Classical Analysis" Now my feeling is that Freidel has gotten cagey about saying "background independent" because that term is defined differently by string theorists and others and tends to provoke controversy. people feel threatened and start protesting that maybe string theory really IS "background independent" even though it might not be "manifestly" background independent, and then they go on to say "LQG" is not really background independent, and so on. The term irritates people---and has become associated with semantic conflict So my suspicion is that the paper that JUST CAME OUT REALLY IS THIS PAPER but RETITLED in a kind of inconspicuous ivy-league coat-and-tie way. the paper that just came out is titled "PARTICLES AS WILSON LINES OF GRAVITATIONAL FIELD" which is shocking if you think of it, but innocuous enough on the surface. the number is http://arxiv.org/gr-qc/0607014 (remember by Quatorze Juillet Bastille day) So I guess the question is, what do you think? Do you also think that the promised paper "Background Independent Perturbation Theory for Gravity Coupled to Particles: Classical Analysis" is actually the new one we have in hand called "Particles as Wilson Lines of Gravitational Field" but renamed? Notice if you look at "Particles as Wilson Lines" actually wilson lines is only a part of what they are doing and very much of what they are doing could be accurately described as a classical analysis of background independent (in the LQG sense) gravity-and-matter perturbation theory. and if so, any idea why they decided on the new name? ================ to repeat another point, that I think JB made, or various people have: to say "background independent perturbation theory" is a real kicker of a headline. Because perturbation theory is the customary predominant way to do fields and UP TILL THIS MOMENT all the perturbation field theory ever done has used a fixed BACKGROUND SPACETIME geometry. so when you hear that phrase you hear a slight breaking noise. (which among other things could motive people to deny that the paper could possibly be on the right track, causing the author a lot of bother answering them). I can understand how one might want the breaking noise to be inaudible.
P: 169
 Quote by marcus So I guess the question is, what do you think? Do you also think that the promised paper "Background Independent Perturbation Theory for Gravity Coupled to Particles: Classical Analysis" is actually the new one we have in hand called "Particles as Wilson Lines of Gravitational Field" but renamed?
I don't know. I remember Laurent saying they weren't even sure how many papers they were writing on this subject: two or three. They've done a lot of work, obviously, and for a big project like this one needs to keep rethinking the best way to slice the work into papers.

The paper they wrote doesn't actually do any "perturbation theory", apart from writing the MacDowell-Mansouri Lagrangian as the BF Lagrangian plus two extra terms, and analysing what this means... which they'd already done in a previous paper. The big new thing is to introduce particle worldlines as "defects" - curves removed from spacetime - much as had already been done in 3d gravity. So, it makes sense for their title to emphasize this.

In fact, their title is a bit more dramatic than what I might have chosen, because they don't really study these particle worldlines in the context of MacDowell-Mansouri gravity, except for one equation right near the end. Mostly they study these particles in the context of plain old 4d BF theory.

This nicely complements my own study, with Crans, Wise and Perez, of strings coupled to 4d BF theory. In fact the particles and strings fit hand in glove: particles like the A field, while strings like the B field - because particles have 1-dimensional worldlines and the A field is a 1-form, and strings have 2-dimensional worldsheets and the B field is a 2-form. I explain this a bit more in the latest issue of This Week's Finds, week235.

Unfortunately, Crans, Wise, Perez and I studied strings coupled to 4d BF theory for a general gauge group but didn't work out the details for the gauge group Freidel uses, namely SO(4,1). We focused on SO(3,1). It should be easy to do the SO(4,1) case now, though since Freidel & Company have worked out a lot of the necessary stuff.

After a talk I gave, Freidel guessed that the strings may be related to gravitons... or replace them, somehow. It's a big mystery: a nice structure is emerging, but it's not clear what it means! This is what makes physics fun.
Emeritus
PF Gold
P: 8,147
 Quote by john baez ...This is what makes physics fun.
Are you now finding it as much fun as math again? I know you were fed up for a while.

It is surely a wonderful gift you have to be able to work back and forth in the two areas; not only are you able to spot unnobvious connections, but there always seems to be something in one field or the other that really floats your boat.
 Astronomy Sci Advisor PF Gold P: 23,101 I think it's all one thing, basically. just the formalities of which department and which journal but if you see a glint in the eye of the universe or a little smile on the face of nature it probably doesnt matter much whether it is one or the other
Astronomy
Sci Advisor
PF Gold
P: 23,101
 Quote by john baez ... I explain this a bit more in the latest issue of This Week's Finds, week235.
Great news! Glad you found time!

 Unfortunately,...
UNFORTUNATELY? That's the way it's SUPPOSED to happen
general group case first, then specialize to SO(4,1)
couldnt be sweeter
certainly maximizes the pleasure and excitement for the sidelines observers like us anyway.
 ....Crans, Wise, Perez and I studied strings coupled to 4d BF theory for a general gauge group but didn't work out the details for the gauge group Freidel uses, namely SO(4,1). We focused on SO(3,1). It should be easy to do the SO(4,1) case now, though since Freidel & Company have worked out a lot of the necessary stuff. ...
hotdog

"After a talk I gave, Freidel guessed that the strings may be related to gravitons... or replace them, somehow. It's a big mystery: a nice structure is emerging, but it's not clear what it means! This is what makes physics fun."

Freidel: let's invent how spacetimematter works. My stuff can be the geometry and your stuff can be the gravitons that connect changes in the geometry, OK?

"but it's not clear what it means! This is what makes physics fun" at some point, this begins to sound like a memorable understatement

thanks for posting here, enjoy Shanghai, and don't forget to figure spacetime out for us
Emeritus
PF Gold
P: 8,147
 Quote by marcus I think it's all one thing, basically. just the formalities of which department and which journal but if you see a glint in the eye of the universe or a little smile on the face of nature it probably doesnt matter much whether it is one or the other
Sub specie aeternitatis, of course, you're right. But how few people are capable of working creatively in both! And you see so much (mostly tacit) dismissal of the concerns and interests of each field in so many practitioners of the other.
P: 1,667
 Quote by selfAdjoint Sub specie aeternitatis, of course, you're right. But how few people are capable of working creatively in both! And you see so much (mostly tacit) dismissal of the concerns and interests of each field in so many practitioners of the other.
Many smart people are creatively working both in mathematics and physics (see Ed Witten, Roger Penrose, Stephen Hawking, George Ellis, Yau, Paul Dirac, ... and many lesser Gods as well). When I read some threads here, there seem to be very strange opinions wandering around about the way physics is done''. Good physics inventions always *started* with a coherent intuitive picture of (a part of) nature suggested by experiment; a physicist has fun when he/she can find out a mathematical model incoorporating these intuitions and delivering the correct numbers. In some rare cases, he can get excited when some unexpected solutions come out which require *new* experiments to be done (or the inventor might even dismiss these as unphysical). History confirms this thesis over and over again - Einstein for example had the physical picture of GR already in his mind (at least) six years prior to writing down his field equations. His theory got experimental support by Eddington in 1919 and he was so surprised by the Schwarzschild solutions that he did not hesitate to refute them. Mathematicians on the other hand have fun exploring structures per se and in these days are not shy at all to sell some weak (possibly accidental) correspondences with some established theories as physics''. The fact that some physicists are interested in these merely expresses the lack of good ideas from their side. Once I heard from a mathematician that GR was the most beautiful theory one could imagine until mathematicians started formalising it - what distinguishes supreme physicists is their powerful intuition to recogize what to do (and what not), mathematical ability only serves as a very useful tool. A beautiful example of this is given by the mathematical genius Dirac (one of the very few to have done so much useful mathematical physics), who kept on insisting that QED was not a good physical theory and that its miracles could very well be accidental.

Now, when some camp does not appreciate the worries of the other very well; it is usually so that the latter is not presenting a somewhat clear coherent picture of nature at all.

Careful
Emeritus
PF Gold
P: 8,147
 Quote by careful Many smart people are creatively working both in mathematics and physics (see Ed Witten, Roger Penrose, Stephen Hawking, George Ellis, Yau, Paul Dirac,
Umm, yes. Witten, Penrose, Hawking, are a world famous trio among hundreds of creative mathematicians and physicists. Yau is a mathematician whose work turned out to be significant to the physicists which has led him over; he certainly didn't prove Calabi's conjecture with string theory in mind. Elis I'm not familiar with. And Dirac, I believe, is Still Dead.
Astronomy
Sci Advisor
PF Gold
P: 23,101
 Quote by selfAdjoint Ellis I'm not familiar with...
when he was quite a bit younger George Ellis co-authored a book with Stephen Hawking (The Large Scale Structure of Space-Time). He is a distinguished cosmologist among other things.

I am afraid it is off topic of me to say so, but what excites me about Ellis recent contribution is his essay on Philosophical Issues in Cosmology. It is comprehensive and lays the issues out very clearly. The philosophical issues in cosmology are substantial and interesting, not just abstract hot air (as might be the case in some other areas of philosophy.)

he is not as well known as the others. Careful could have tossed the name in knowing that he would please a few Ellis-fans like me.

http://arxiv.org/abs/astro-ph/0602280
Issues in the Philosophy of Cosmology
George F. R. Ellis
To appear in the Handbook in Philosophy of Physics, Ed J Butterfield and J Earman (Elsevier, 2006).

"After a survey of the present state of cosmological theory and observations, this article discusses a series of major themes underlying the relation of philosophy to cosmology. These are: A: The uniqueness of the universe; B: The large scale of the universe in space and time; C: The unbound energies in the early universe; D: Explaining the universe -- the question of origins; E: The universe as the background for existence; F: The explicit philosophical basis; G: The Anthropic question: fine tuning for life; H: The possible existence of multiverses; I: The natures of existence. Each of these themes is explored and related to a series of Theses that set out the major issues confronting cosmology in relation to philosophy."

(probably far more than anyone wants to know about Ellis )

 Related Discussions Beyond the Standard Model 3 Beyond the Standard Model 0 General Physics 0 Beyond the Standard Model 18 General Physics 0


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# Subset of a partially ordered set: Least upper bound and greatest lower bound.

Let $(A, \leq)$ be a poset and $B \subseteq A$. I need to show

i) B may have at most one least upper bound in A

ii) B may have at most one greatest lower bound in A.

-

This is really a matter of unpacking and applying the definitions of what it means

• for $A$ to be a poset (poset: "partially ordered set")

• (See also the section of that entry subtitled Extrema)
• for $B$ to be a subset of $A$,
• for a set to have a least upper bound (lub) and
• for a set to have greatest lower bound (glb).

Recall the definition of a least upper bound as:

Let S be a poset:

1. $\alpha$ is an upper bound for S if $x \leq \alpha \space \forall x \in S$, (upper and lower bounds are NOT unique) and
2. $\beta$ is the least upper bound for S if $\beta$ is an upper bound, and $\beta \leq \alpha$ whenever $\alpha$ is an upper bound for S.

A symmetrical definition defines the greatest lower bound.

-
Sorry i m a newbie with this. I get the least upper bound but cant it it more than 1 least upper bound? –  Ahmed Apr 14 '13 at 17:01
The least upperbound and the greatest lower bound are unique. But there can be infinitely many "upper" or "lower" bounds. We're interested here in the least/greatest (respectively) bounds. –  amWhy Apr 14 '13 at 17:08
Also note that you are asked to show that $B$ has at most one least upper bound in A, at most one greatest lower bound in A. It may be the case that no such lub or glb for $B$ exists in $A$, but if either does exist in A, then it is the unique lub/glb of $B$ in A, respectively. Make sure you're clear about such bounds for $A$, given it's a poset. Then figure out what this means with respect to lub/glb bounds for $B\subset A$. –  amWhy Apr 14 '13 at 18:10
How are you coming on this question, Ahmed? –  amWhy Apr 14 '13 at 19:12
still confused i m reading book right now. But still trying. –  Ahmed Apr 14 '13 at 19:14
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# American Institute of Mathematical Sciences

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A kinetic theory description of liquid menisci at the microscale
June  2015, 8(2): 255-280. doi: 10.3934/krm.2015.8.255

## A Hamilton-Jacobi approach for front propagation in kinetic equations

 1 UMR CNRS 5669 UMPA' and INRIA project NUMED', École Normale Supérieure de Lyon, 46, allée d'Italie, F-69364 Lyon Cedex 07, France

Received  July 2014 Revised  November 2014 Published  March 2015

In this paper we use the theory of viscosity solutions for Hamilton-Jacobi equations to study propagation phenomena in kinetic equations. We perform the hydrodynamic limit of some kinetic models thanks to an adapted WKB ansatz. Our models describe particles moving according to a velocity-jump process, and proliferating thanks to a reaction term of monostable type. The scattering operator is supposed to satisfy a maximum principle. When the velocity space is bounded, we show, under suitable hypotheses, that the phase converges towards the viscosity solution of some constrained Hamilton-Jacobi equation which effective Hamiltonian is obtained solving a suitable eigenvalue problem in the velocity space. In the case of unbounded velocities, the non-solvability of the spectral problem can lead to different behavior. In particular, a front acceleration phenomena can occur. Nevertheless, we expect that when the spectral problem is solvable one can extend the convergence result.
Citation: Emeric Bouin. A Hamilton-Jacobi approach for front propagation in kinetic equations. Kinetic & Related Models, 2015, 8 (2) : 255-280. doi: 10.3934/krm.2015.8.255
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##### References:
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【online】Persistence of lower dimensional degenerate invariant tori ......

Moreover, it also depends on the existence of a connected path of real roots of approximating odd-degree real polynomials depending on parameter, which makes it possible to tackle the Melnikov's conditions.


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# Commensurability of finitely generated groups

Let’s begin with a definition.

Definition: Two groups $G,H$ are commensurable, which we’ll write $G\approx H$, if there exist subgroups $G_0 \leq G, H_0 \leq H$ such that $[G \colon G_0],[H \colon H_0]<\infty$ and $G_0 \cong H_0$.

This relation pops up in geometric group theory (among other places, I’m sure); for example the index in De La Harpe’s book lists about a dozen entries for “commensurable groups”.  Groups which are commensurable are easily seen to be quasi-isometric, and classifying groups up to quasi-isometry is a main focus in geometric group theory.

Commensurability of finitely generated groups is a countable Borel equivalence relation.  At some point, one begins to take the Borelness of these sorts of things for granted, so that part isn’t so surprising.  But the fact that the relation is countable is, I think.  It’s easy to see that a finitely generated group only has countably many finite-index subgroups; any finite-index subgroup is also finitely generated, and a finitely generated group only has countably many finitely generated subgroups.  But for this equivalence relation to be countable, one also needs to know that a given finitely generated group can only be a finite-index subgroup of countably many finitely generated subgroups.  This is not immediately obvious (at least, not to me).  As far as I can tell, the easiest way to see this is by appealing to the following theorem, due to Kaloujnine and Krasner:

Theorem (Kaloujnine-Krasner): Every extension of a group $N$ by a group $H$ is embeddable in the unrestricted wreath product $N \mathbin{Wr} H$.

So say that $N$ is a finite-index normal subgroup of a group $G$.  Then $G$ is the extension of $N$ by some finite group $H$, and so $G$ embeds into $N \mathbin{Wr} H = N \mathbin{wr} H$, since $H$ is finite.  Since $N \mathbin{wr} H$ is countable, it contains only countably many finitely generated subgroups, so there are only countably many possibilities for $G$ given $H$.  There are also only countably many possibilities for $H$, so there are only countably many groups which $N$ for which can be a finite-index normal subgroup.  What if $N$ is not a normal subgroup?  Then it contains a finite-index normal subgroup, so everything still works out.

Recall the terminology from the last post.  We’re going to look at the proof of the following result.

Theorem (Thomas): Commensurability of finitely generated groups is a universal countable Borel equivalence relation.

Note that it’s easy to see that commensurability is weakly universal, since it contains isomorphism of finitely generated groups as a subequivalence relation, and isomorphism is universal.  So this proof makes exactly the leap from weakly universal to universal that we’d like to make with the theorems I discussed last time.

The proof proceeds by creating a reduction from isomorphism to commensurability.  In fact, the reduction is easy to write down.  Let $S$ be your favorite infinite, finitely generated simple group, and let $A_5$ denote the alternating group on 5 elements.  Then the map

$G \mapsto W_G = (A_5 \mathbin{wr} G) \mathbin{wr} S$

is our reduction.  Clearly isomorphic groups will be sent to commensurable (in fact, isomorphic) groups.  The trick, of course, is to verify that this sends nonisomorphic groups to non-commensurable groups.

— Showing $W_A \approx W_B$ implies that particular subgroups are isomorphic —

Suppose that $W_A \approx W_B$.  We want to show that $A \cong B$.  It is enough to show that $(A_5 \mathbin{wr} A) \cong (A_5 \mathbin{wr} B)$, by a result of P.M. Neumann which states that $A \mathbin{wr} B \cong A' \mathbin{wr} B'$ iff $A\cong A'$ and $B\cong B'$.  The idea of the proof is to show that given a finite index subgroup of $W_A$ , one must be able to somehow recover $(A_5 \mathbin{wr} A)$.

A priori this could be quite difficult.  Suppose that you know two groups $G,H$ are isomorphic, say via $\pi$.  Say $K$ is a subgroup of $G$ that is somehow special or contains information you’ve coded into the construction of $G$, and $H$ also contains a special subgroup $L$.  There is no reason that $\pi(K)$ should have anything to do with $L$, and so there is no reason to expect the information you’ve coded in $K$ is the same as the information you’ve coded in $L$.

Let $K_A$ denote the base group of $W_A$.  Then other results of P.M. Neumann show that $W'_A=[S,K_A]S$.  This characterization, combined with a few other P.M. Neumann results (and here is where we use that $S$ is infinite and simple) tells us that if $H_A$ is a finite-index subgroup of $W_A$, then $W'_A=H'_A$.  So indeed, we can try and use the commutator subgroup of $W_A$ to recover $A$.  (All of these P.M. Neumann results are from the same paper, On the structure of standard wreath products of groups, which is a very thorough account of properties of wreath products.) This is great, because if $H_A \cong H_B$, then the isomorphism must send $H'_A$ to $H'_B$. So if we’ve coded information into $H'_A$ it must be the same as the information in $H'_B$. In this case, we’re hoping to be able to recover $A$ from $H'_A$.

— Finding $A$ in $H'_A$

So if $W_A \approx W_B$, we know that $[S,K_A]S\cong [S,K_B]S$.  We want to keep peeling away until we can see $A$ and $B$.  It seems intuitively obvious that we can get rid of the $S$‘s, but how?  Let $sf\in [S,K_A]S$ be such that $f\in[S,K_A]$, and $s\in S$, and let $\pi(sf)=s'f'\in[S,K_B]S$ with $f'\in[S,K_B]$ and $1\neq s'\in S$.  We’d like to show that $s\neq 1$ iff $s'\neq 1$, since this implies $[S,K_A]\cong [S,K_B]$.

We again look for certain subgroups that we know any isomorphism must preserve.  It would be nice to prove something like the following: $C_{W'_A}(sf)$ is cyclic iff $s\neq 1$.  Then since $C_{W'_B}(s'f')\cong C_{W'_A}(sf)$, we’d be done.  Unfortunately, things aren’t quite so easy.  Define $L_{sf}$ to be the normal closure of $sf$ in $W'_A$, and $R_{sf}=\langle (sf)^{L_{sf}} \rangle$ (i.e. $R_{sf}$ is the normal closure of $sf$ in $L_{sf}$).  Then one can show that $C_{L_{sf}}(R_{sf})=1$ iff $s=1$.  Since all of these groups are preserved under isomorphism, we now know $[S,K_A] \cong [S,K_B]$.

I don’t really have a good idea of how one decides to look at these groups beyond trial and error.  The one thing I’ll note is that centralizers are relatively easy to understand in wreath products $A \mathbin{wr} B$, since conjugating elements of the base group $K$ by elements of $B$ just shifts them around, and understanding conjugation by elements of $K$ is really only as difficult as understanding conjugation in $A$.  In particular, clearly $C_{A \mathbin{wr} B}(K)=\{e\}$, and so any subgroup containing $K$ will have trivial centralizer with respect to the whole group.  So it makes sense to look at centralizers.

For $X\subseteq S$, let $K^X_A=\{f\in K_A \mid \sigma(f)\subseteq X\}$, where $\sigma(f)$ denotes the support of $f$.  Now by a result of P.M. Neumann (again, I know), $[S,K_A]=W'_A\cap K_A$.  Obviously if $|X|=1$ we have $K^X_A\cong A_5 \mathbin{wr} A$, so if $K^X_A \leq W'_A \cap K_A$, we’d be well on our way to being done.  But in general this doesn’t happen.  We can do almost as well though.  Call $A_5 \mathbin{wr} A=\hat{A}$.  Using the P.M. Neumann result, it’s not too hard to check that if $s\neq t\in S$, $(W'_A\cap K_A^{\{s,t\}})/(W'_A\cap K_A^{\{s\}}) \cong \hat{A}$, and the analogous statements are true for $W_B$.  (Obviously $K_A^{\{s,t\}}/K_A^{\{s\}} \cong \hat{A}$, so this is just a matter of checking that enough of these groups are contained in $W'_A$ for such an isomorphism to still exist.)  So it is enough to show that there are some $a,b,c\in S$ with $a\neq b$ such that $\pi(W'_A \cap K_A^{\{c\}})=W'_B \cap K_B^{\{a\}}$ and $\pi(W'_A \cap K_A^{\{1,c\}})=W'_B \cap K_B^{\{a,b\}}$.

For $f\in [S,K_A]$, let $L_f=\langle f^{[S,K_A]} \rangle$, and define $R_f = C_{[S,K_A]}(L_f)$.  Note the similarity to the groups looked at couple of paragraphs ago.  A quick calculation shows that every element of $L_f$ has support contained in $\sigma(f)$, so $L_f \leq W'_A \cap K_A^{\sigma(f)}$.  We also note that since $A_5$ is simple and nonabelian, $\hat{A}$ has a unique nontrivial minimal normal subgroup, its base group $\hat{K}_A$.  Minimality is easy to check.  For uniqueness, since $C_{\hat{A}}(\hat{K}_A)=\{e\}$, we find that any normal subgroup of $\hat{A}$ must contain $\hat{K}_A$.  That second fact can be used to show that $L_f$ contains all the elements of $K_A$ with support contained in $\sigma(f)$ and range contained in $\hat{K}_A$.  It’s easy to see what happens to these functions under conjugation, and so after further calculation we find that $C_{[S,K_A]}(R_f)=W'_A \cap K_A^{\sigma(f)}$.  We now know that $|\sigma(f)|=1$ iff $C_{[S,K_A]}(R_f)$ is minimal among all groups of the form $C_{[S,K_A]}(R_h)$ with $h\in [S,K_A]$, and since similar results hold in $[S,K_B]$, we get our desired result.

To sum up, I think the main insight is that some subgroups don’t really change under isomorphism. (There is a little work to get from asking about commensurability to asking about isomorphism, but this is taken care of by P.M. Neumann’s results.) Derived subgroups map to derived subgroups; centralizers map to centralizers. This means that we can try and use those subgroups to encode the information we’re interested in preserving. Perhaps it’s an obvious idea, but until I actually wrote out what was happening, I hadn’t thought of things in those terms.


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Profit Per Student (in Dollars) at Dan’s Dojo,

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Profit Per Student (in Dollars) at Dan’s Dojo, [#permalink]  14 Jul 2018, 05:11
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Profit Per Student (in Dollars) at Dan’s Dojo, 2000–2004

 2000 60 2001 80 2002 80 2003 100 2004 162

At Dan’s Dojo, the percent increase from 2004 to 2005 (not shown) was the same as the percent increase from 2000 to 2001. What was the profit per student for 2005?

[Reveal] Spoiler: OA
216

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Re: Profit Per Student (in Dollars) at Dan’s Dojo, [#permalink]  14 Jul 2018, 05:30
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Explanation

The percent increase from 2000 to 2001 is:

Percent Change =$$\frac{Difference}{Original}\times 100 \%$$
Percent Change =$$\frac{20}{10}\times 100 \%=33.333333 \%$$

Now, apply a 33.33333%, or $$\frac{1}{3}$$, increase to 2004’s figure. The GRE calculator cannot accept a repeating decimal; instead, divide 162 by 3 to get the amount of increase, and then add 162 to get the new profit per student in 2005: 162 ÷ 3 + 162 = 216.
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Re: Profit Per Student (in Dollars) at Dan’s Dojo, [#permalink]  17 Sep 2019, 05:12
Previous percent increase is 33.3% so From 2004-2005; 162*33.3 = 5394.6 then divide it by 100 we get 53.946; adding this with 162 + 53.946 = 215.9460, this is the actual per-student profit whereas we take it as 216.
Now, my question is in the main/real GRE xm do I need to put the 215.94 or just by leveling 215.94 to 216 and then put 216 as an answer. ???
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Re: Profit Per Student (in Dollars) at Dan’s Dojo, [#permalink]  17 Sep 2019, 05:16
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The question will specify if the value must be rounded or you have to insert the exact value.

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Re: Profit Per Student (in Dollars) at Dan’s Dojo, [#permalink]  17 Sep 2019, 05:26
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huda wrote:
Previous percent increase is 33.3% so From 2004-2005; 162*33.3 = 5394.6 then divide it by 100 we get 53.946; adding this with 162 + 53.946 = 215.9460, this is the actual per-student profit whereas we take it as 216.
Now, my question is in the main/real GRE xm do I need to put the 215.94 or just by leveling 215.94 to 216 and then put 216 as an answer. ???

Be careful, an increase from 60 to 80 is a 1/3 increase.
1/3 = 33.3333333.....%
In your solution, you rounded 33.3333333.....% down to 33.3%
So, your final answer is a bit off.

If we stick with the fraction increase of 1/3, we need only find 1/3 of 162, which is exactly 54

So, our final answer = 162 + 54 = 216

Cheers,
Brent
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Re: Profit Per Student (in Dollars) at Dan’s Dojo, [#permalink]  17 Sep 2019, 05:29
GreenlightTestPrep wrote:
huda wrote:
Previous percent increase is 33.3% so From 2004-2005; 162*33.3 = 5394.6 then divide it by 100 we get 53.946; adding this with 162 + 53.946 = 215.9460, this is the actual per-student profit whereas we take it as 216.
Now, my question is in the main/real GRE xm do I need to put the 215.94 or just by leveling 215.94 to 216 and then put 216 as an answer. ???

Be careful, an increase from 60 to 80 is a 1/3 increase.
1/3 = 33.3333333.....%
In your solution, you rounded 33.3333333.....% down to 33.3%
So, your final answer is a bit off.

If we stick with the fraction increase of 1/3, we need only find 1/3 of 162, which is exactly 54

So, our final answer = 162 + 54 = 216

Cheers,
Brent

Many Many thanks
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Re: Profit Per Student (in Dollars) at Dan’s Dojo, [#permalink]  30 Oct 2019, 20:49
he percent increase from 2000 to 2001 is:

Percent Change =DifferenceOriginal×100%DifferenceOriginal×100%
Percent Change =2010×100%=33.333333%2010×100%=33.333333%

Now, apply a 33.33333%, or 1313, increase to 2004’s figure. The GRE calculator cannot accept a repeating decimal; instead, divide 162 by 3 to get the amount of increase, and then add 162 to get the new profit per student in 2005: 162 ÷ 3 + 162 = 216.
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Re: Profit Per Student (in Dollars) at Dan’s Dojo,   [#permalink] 30 Oct 2019, 20:49
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# A New Exam Scheduling Algorithm Using Graph Coloring

Exam scheduling is a challenging task that universities and colleges face several times every year. The challenge is to schedule so many exams of courses in a limited, and usually short, period of time. An Exam schedule should avoid conflicts, in the sense that no two or more exams for the same student are scheduled at the same time. Part of the challenge is to achieve fairness for the students.

A fair schedule does not schedule more than two exams, for example for a student on one day. In the meantime, a fair schedule does not leave a big gap between exams for the students.

## Overview

An undirected graph $G$ is an ordered pair $(V, E)$ where $V$ is a set of nodes and $E$ is a set of non-directed edges between nodes. Two nodes are said to be adjacent if there is an edge between them. The graph coloring is a well-known problem. Node coloring assigns colors to the nodes of the graph such that no two adjacent nodes have the same color. Edge coloring assigns colors to the edges of the graph such that no two adjacent edges have the same color. Two edges are said to be adjacent if they both share a node in common. General graph coloring algorithms are well known and have been extensively studied by researchers.

Exam scheduling is a challenging task that universities and colleges face several times every year. The challenge is to schedule so many exams of courses in a limited, and usually short, period of time. An Exam schedule should avoid conflicts, in the sense that no two or more exams for the same student are scheduled at the same time. Part of the challenge is to achieve fairness for the students. A fair schedule does not schedule more than two exams, for example for a student on one day. In the meantime, a fair schedule does not leave a big gap between exams for the students. The exam scheduling problem is defined as follows: "We first represent the courses by nodes of a graph, where 2 nodes are adjacent if the 2 corresponding courses are registered by at least one student. Then, it is required to assign each course represented by a node a time slot, such that no two adjacent nodes have the same slot, in condition that a set of constraints imposed on the problem are also met." We solve this problem by using node graph coloring technique.

This study provides a mechanism for automatic exam-schedule generation that achieves fairness, and minimizes the exam period. As a result, this paper presents a graph-coloring-based algorithm for the exam scheduling application which achieves the objectives of fairness, accuracy, and optimal exam time period. Numerous studies have considered the problem of exam scheduling. The main difference between various studies is the set of assumptions and constraints taken into consideration. Burke, Elliman and Weare, for example, followed a similar approach using graph coloring. However, in their algorithm, they addressed only the conflicts without any constraints. Moreover, the algorithm presented in [9] does not eliminate conflicts, and only aims at minimizing conflicts. In this paper, we consider few but important assumptions and constraints, closely related to the general exam scheduling, and mainly driven from the real life requirements collected through the experience at various universities. Such assumptions and constraints are distinct from those present in more general graph coloring problems. We summarize the main assumptions and constraints as follows:

1. The number of exam periods per day (Time Slots ($TS$)) can be set by the user. TS depend on college/department specific constraints. For example, a university that uses a 2-hours exam period and begins the exam day at 8:00 am and finish at 8:00 pm, may set $TS$ to 5.
2. The number of concurrent exam sessions or concurrency level ($N_p$) depends on the number of available halls, and the availability of faculty to conduct the exams. Np is determined by the registrar's office. This paper assumes that $N_p$ is a system parameter and the scheduling algorithm has been examined with several $N_p$ values.
3. A student shall not have more than ($y$) exams per day (fairness requirement), and is treated as a system tunable parameter.
4. A student shall not have a gap of more than ($x$) days between two successive exams, and this factor is to be determined by the college or department (another fairness requirement).
5. The schedule shall be done in the minimal possible period of time, i.e., minimize the number of exam slots and/or number of exam days. The exam time period is an outcome of the scheduling algorithm.
6. Next, we give some more definitions that are relevant to the underlined problem. Let $C$ be a list of all courses to be scheduled. The length of this list is $n$. In other words, $n$ is the number of courses in the list. A course at position $i$ in the list $C$ is referred to using an index $c_i$. Let $G$ be the graph that represents the list $C$ of courses. We impose a weight $w_{ij}$ to each edge of $G$, where $w_{ij}$ is defined as the number of students present in both courses $c_i$ and $c_j$. An edge $e_{ij}$ exists between nodes $c_i$ and $c_j$ iff $w_{ij}$ is not $0$. We define a weight matrix $W$ to be an $nxn$ matrix, where $n$ is the number of courses to be scheduled for the exams, and $wij$ equals the weight of the edge $e_{ij}$ that joins the courses $c_i$ and $c_j$. Such a weight imposed on the edges of $G$ represents the exam conflict complexity present in courses $c_i$ and $c_j$. A multi-section course is considered as one course. However, the number of sections per course is taken into consideration in the process of hall assignment.

The degree $d_i$ of a node $c_i$ is defined as the number of edges connected to a node. A large degree of a node $c_i$ indicates that there is a large number of students registered in this course and $d_i$ other courses. The degree $d_i$ is also a measure of conflict complexity. An example of a weighted graph $G$ and the corresponding weight matrix $W$ is given in Figure 1 and , respectively. In Figure 1, $c_2$ and $c_5$ both have degree $3$. In Table 1, the weight of the edge $e_{15}$ is $4$.

$C1$$C2$$C3$$C4$$C5$
$C1$$0$$2$$0$$0$$4 C2$$2$$0$$1$$0$$3$
$C3$$0$$1$$0$$4$$0 C4$$0$$0$$4$$0$$3$

## Algorithm Color Schedule

The algorithm consists of two major steps. The first step builds the weight matrix and graph. The second step assigns colors to the nodes of the graph.

## Conclusion and Future Work

As discussed above, the number of concurrent exam sessions or concurrency level (Np) depends on the number of available halls, and the availability of faculty to conduct the exams. The value of Np is usually determined by the registrar's office, and the paper assumes that Np is a system parameter, and we will run the scheduling algorithm with several Np values. In a later work, the actual distribution of exam sessions to halls will be included. Also, the algorithm presented in this paper is claimed to achieve near optimal performance (close to minimal number of colors) in polynomial time. We are currently investigating a modification of the algorithm, which will achieve the absolute minimal for a certain set of graphs.

Love,
Ahmed Jazzar


text:
# The natural logarithmic function essay

The natural logarithmic function essay, The history of logarithms is the story of a correspondence around 1730, leonhard euler defined the exponential function and the natural logarithm by = →.

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Natural logarithms used with base e logarithmic function: definition & examples related study materials essay prompts. College essay financial the number e and the natural logarithm function of an exponential function natural logarithms are special types of. A-1 natural exponential function in lesson 21, we explored the world of logarithms in base 10 the natural logarithm has a base of “e” “e” is approximately. 51 the natural logarithmic function and differentiation and the definition of the natural logarithmic function: to this point later in this essay.

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We have all seen the natural logarithm appear in why does the natural logarithm appear so much in its inverse function, the natural logarithm will crop up a. Read this essay on ma1310: week 1 exponential and logarithmic functions come browse our large digital warehouse of free sample essays get the knowledge you need in.

From the early 1600's there have been two kinds of logarithms—natural logarithms and common logarithms common logarithms are base 10, denoted [math]\log_{10. Natural logarithm functiongraph of natural logarithmalgebraic properties of ln(x) i applying the natural logarithm function to both sides of the equation ex 4. Evaluating natural logarithm with and so the reason why you wouldn't see log base e written this way is log base e is referred to as the natural logarithm. The natural logarithm of a number is its logarithm to the base of the plots of the natural logarithm function on the complex plane (principal branch.

The natural logarithmic function essay
Rated 4/5 based on 39 review

2017. bstermpaperwcen.allstarorchestra.info


text:
## 9.3 The Difficulty of Bayesian Inference for Clustering

Two problems make it pretty much impossible to perform full Bayesian inference for clustering models, the lack of parameter identifiability and the extreme multimodality of the posteriors. There is additional discussion related to the non-identifiability due to label switching in the label switching section.

### Non-Identifiability

Cluster assignments are not identified—permuting the cluster mean vectors mu leads to a model with identical likelihoods. For instance, permuting the first two indexes in mu and the first two indexes in each soft_z[n] leads to an identical likelihood (and prior).

The lack of identifiability means that the cluster parameters cannot be compared across multiple Markov chains. In fact, the only parameter in soft $$K$$-means is not identified, leading to problems in monitoring convergence. Clusters can even fail to be identified within a single chain, with indices swapping if the chain is long enough or the data are not cleanly separated.

### Multimodality

The other problem with clustering models is that their posteriors are highly multimodal. One form of multimodality is the non-identifiability leading to index swapping. But even without the index problems the posteriors are highly multimodal.

Bayesian inference fails in cases of high multimodality because there is no way to visit all of the modes in the posterior in appropriate proportions and thus no way to evaluate integrals involved in posterior predictive inference.

In light of these two problems, the advice often given in fitting clustering models is to try many different initializations and select the sample with the highest overall probability. It is also popular to use optimization-based point estimators such as expectation maximization or variational Bayes, which can be much more efficient than sampling-based approaches.


text:
# Difference between Decimal, Float and Double in .NET?

What is the difference between Decimal, Float and Double in .NET?

When would someone use one of these?

-
IMO, Skeet's answer is not complete without cgreeno's answer –  Chibueze Opata Jul 15 '12 at 14:38
interesting article zetcode.com/lang/csharp/datatypes –  GibboK Mar 1 '14 at 14:20

float and double are floating binary point types. In other words, they represent a number like this:

10001.10010110011

The binary number and the location of the binary point are both encoded within the value.

decimal is a floating decimal point type. In other words, they represent a number like this:

12345.65789

Again, the number and the location of the decimal point are both encoded within the value – that's what makes decimal still a floating point type instead of a fixed point type.

The important thing to note is that humans are used to representing non-integers in a decimal form, and expect exact results in decimal representations; not all decimal numbers are exactly representable in binary floating point – 0.1, for example – so if you use a binary floating point value you'll actually get an approximation to 0.1. You'll still get approximations when using a floating decimal point as well – the result of dividing 1 by 3 can't be exactly represented, for example.

As for what to use when:

• For values which are "naturally exact decimals" it's good to use decimal. This is usually suitable for any concepts invented by humans: financial values are the most obvious example, but there are others too. Consider the score given to divers or ice skaters, for example.

• For values which are more artifacts of nature which can't really be measured exactly anyway, float/double are more appropriate. For example, scientific data would usually be represented in this form. Here, the original values won't be "decimally accurate" to start with, so it's not important for the expected results to maintain the "decimal accuracy". Floating binary point types are much faster to work with than decimals.

-
float/double usually do not represent numbers as 101.101110, normally it is represented as something like 1101010 * 2^(01010010) - an exponent –  Hazzard Aug 13 '14 at 21:50
@Hazzard: That's what the "and the location of the binary point" part of the answer means. –  Jon Skeet Aug 13 '14 at 21:57
I must have missed that, my bad –  Hazzard Aug 13 '14 at 21:59

Precision is the main difference.

Float - 7 digits (32 bit)

Double-15-16 digits (64 bit)

Decimal -28-29 significant digits (128 bit)

Decimals have much higher precision and are usually used within financial applications that require a high degree of accuracy. Decimals are much slower (up to 20X times in some tests) than a double/float.

Decimals and Floats/Doubles cannot be compared without a cast whereas Floats and Doubles can. Decimals also allow the encoding or trailing zeros.

float flt = 1F/3;
double dbl = 1D/3;
decimal dcm = 1M/3;
Console.WriteLine("float: {0} double: {1} decimal: {2}", flt, dbl, dcm);

Result :

float: 0.3333333
double: 0.333333333333333
decimal: 0.3333333333333333333333333333

-
@Thecrocodilehunter: sorry, but no. Decimal can represent all numbers that can be represented in decimal notation, but not 1/3 for example. 1.0m / 3.0m will evaluate to 0.33333333... with a large but finite number of 3s at the end. Multiplying it by 3 will not return an exact 1.0. –  Erik P. Nov 29 '11 at 21:14
@Thecrocodilehunter: I think you're confusing accuracy and precision. They are different things in this context. Precision is the number of digits available to represent a number. The more precision, the less you need to round. No data type has infinite precision. –  Igby Largeman Jan 6 '12 at 17:42
@Thecrocodilehunter: You're assuming that the value that is being measured is exactly 0.1 -- that is rarely the case in the real world! Any finite storage format will conflate an infinite number of possible values to a finite number of bit patterns. For example, float will conflate 0.1 and 0.1 + 1e-8, while decimal will conflate 0.1 and 0.1 + 1e-29. Sure, within a given range, certain values can be represented in any format with zero loss of accuracy (e.g. float can store any integer up to 1.6e7 with zero loss of accuracy) -- but that's still not infinite accuracy. –  Daniel Pryden Jan 10 '12 at 1:49
@Thecrocodilehunter: You missed my point. 0.1 is not a special value! The only thing that makes 0.1 "better" than 0.10000001 is because human beings like base 10. And even with a float value, if you initialize two values with 0.1 the same way, they will both be the same value. It's just that that value won't be exactly 0.1 -- it will be the closest value to 0.1 that can be exactly represented as a float. Sure, with binary floats, (1.0 / 10) * 10 != 1.0, but with decimal floats, (1.0 / 3) * 3 != 1.0 either. Neither is perfectly precise. –  Daniel Pryden Jan 10 '12 at 18:27
@Thecrocodilehunter: You still don't understand. I don't know how to say this any more plainly: In C, if you do double a = 0.1; double b = 0.1; then a == b will be true. It's just that a and b will both not exactly equal 0.1. In C#, if you do decimal a = 1.0m / 3.0m; decimal b = 1.0m / 3.0m; then a == b will also be true. But in that case, neither of a nor b will exactly equal 1/3 -- they will both equal 0.3333.... In both cases, some accuracy is lost due to representation. You stubbornly say that decimal has "infinite" precision, which is false. –  Daniel Pryden Jan 10 '12 at 19:29

The Decimal structure is strictly geared to financial calculations requiring accuracy, which are relatively intolerant of rounding. Decimals are not adequate for scientific applications, however, for several reasons:

• A certain loss of precision is acceptable in many scientific calculations because of the practical limits of the physical problem or artifact being measured. Loss of precision is not acceptable in finance.
• Decimal is much (much) slower than float and double for most operations, primarily because floating point operations are done in binary, whereas Decimal stuff is done in base 10 (i.e. floats and doubles are handled by the FPU hardware, such as MMX/SSE, whereas decimals are calculated in software).
• Decimal has an unacceptably smaller value range than double, despite the fact that it supports more digits of precision. Therefore, Decimal can't be used to represent many scientific values.
-

float 7 digits of precision

double has about 15 digits of precision

decimal has about 28 digits of precision

If you need better accuracy, use double instead of float. In modern CPUs both data types have almost the same performance. The only benifit of using float is they take up less space. Practically matters only if you have got many of them.

I found this is interesting. What Every Computer Scientist Should Know About Floating-Point Arithmetic

-
-1: Use decimal in accounting applications. –  Roger Lipscombe Jul 6 '13 at 18:10
@RogerLipscombe: I would consider double proper in accounting applications in those cases (and basically only those cases) where no integer type larger than 32 bits was available, and the double was being used as though it were a 53-bit integer type (e.g. to hold a whole number of pennies, or a whole number of hundredths of a cent). Not much use for such things nowadays, but many languages gained the ability to use double-precision floating-point values long before they gained 64-bit (or in some cases even 32-bit!) integer math. –  supercat May 29 '14 at 17:57
Your answer implies precision is the only difference between these data types. Given binary floating point arithmetic is typically implemented in hardware FPU, performance is a significant difference. This may be inconsequential for some applications, but is critical for others. –  saille Jan 15 at 3:16
1. Double and float can be divided by integer zero without an exception at both compilation and run time.
2. Decimal cannot be divided by integer zero. Compilation will always fail if you do that.
-
They sure can! They also also have a couple of "magic" values such as Infinity, Negative Infinity, and NaN (not a number) which make it very useful for detecting vertical lines while computing slopes... Further, if you need to decide between calling float.TryParse, double.TryParse, and decimal.TryParse (to detect if a string is a number, for example), I recommend using double or float, as they will parse "Infinity", "-Infinity", and "NaN" properly, whereas decimal will not. –  BrainSlugs83 Jun 23 '11 at 19:29

for more information you can go to source of this picture:

-
-1 for claiming that decimal is 96 bits. MSDN clearly says 128. –  Ben Voigt Oct 22 '14 at 1:09
And if you dig deeper, it's 102 bits used, 128 stored. No way to fit that in 12 bytes. –  Ben Voigt Oct 22 '14 at 1:24

Integers, as was mentioned, are whole numbers. They can't store the point something, like .7, .42, and .007. If you need to store numbers that are not whole numbers, you need a different type of variable. You can use the double type, or the float type. You set these types of variables up in exactly the same way: instead of using the word int, you type double, or float. Like this:

float myFloat;
double myDouble;

(Float is short for "floating point", and just means a number with a point something on the end.)

The difference between the two is in the size of the numbers that they can hold. For float, you can have up to 7 digits in your number. For doubles, you can have up to 16 digits. To be more precise, here's the official size:

float: 1.5 × 10-45 to 3.4 × 1038
double: 5.0 × 10-324 to 1.7 × 10308

Float is a 32-bit number and double is a 64-bit number.

Double click your new button to get at the code. Add the following three lines to your button code:

double myDouble;
myDouble = 0.007;
MessageBox.Show(myDouble.ToString());

Halt your program and return to the coding window. Change this line:

myDouble = 0.007;
myDouble = 12345678.1234567;

Run your programme and click your double button. The message box correctly displays the number. Add another number on the end, though, and C# will again round up or down. The moral is, if you want accuracy, careful of rounding!

-
Upvoted for using .42 and .007 in your example. : D –  PruitIgoe Aug 12 '13 at 17:14

This has been an interesting thread of me, as today, we've just had a nasty little bug, concerning "decimal" having less precision than a "float".

In our C# code, we are reading numeric values from an Excel spreadsheet, converting them into a decimal, then sending this decimal back to a Service, to save into a SQL Server database.

Microsoft.Office.Interop.Excel.Range cell = ...
object cellValue = cell.Value2;
if (cellValue != null)
{
decimal value = 0;
Decimal.TryParse(cellValue.ToString(), out value);
}

Now, for almost all of our Excel values, this worked beautifully. But for some, very small Excel values, using "decimal.TryParse" lost the value completely. One such example:

• cellValue = 0.00006317592

• Decimal.TryParse(cellValue.ToString(), out value); would return 0

The solution, bizarrely, was to convert the Excel values into a double first, and then into a decimal.

Microsoft.Office.Interop.Excel.Range cell = ...
object cellValue = cell.Value2;
if (cellValue != null)
{
double valueDouble = 0;
double.TryParse(cellValue.ToString(), out valueDouble);
decimal value = (decimal)valueDouble;
...
}

Even though double has less precision than a decimal, this actually ensured small numbers would still be recognised. For some reason, "double.TryParse" was actually able to retrieve such small numbers, whereas "decimal.TryParse" would set them to zero.

Odd. Very odd.

-
Out of curiosity, what was the raw value of cellValue.ToString()? Decimal.TryParse("0.00006317592", out val) seems to work... –  micahtan Aug 27 '12 at 23:57
-1 Don't get me wrong, if true, it's very interesting but this is a separate question, it's certainly not an answer to this question. –  weston May 22 '13 at 14:19
Maybe because the Excel cell was returning a double and ToString() value was "6.31759E-05" therefore the decimal.Parse() didn't like the notation. I bet if you checked the return value of Decimal.TryParse() it would have been false. –  SergioL Oct 15 '14 at 20:44

float ~ ±1.5 x 10-45 to ±3.4 x 1038 --------7 figures
double ~ ±5.0 x 10-324 to ±1.7 x 10308 ------15 or 16 figures
decimal ~ ±1.0 x 10-28 to ±7.9 x 1028 --------28 or 29 figures

-

For applications such as games and embedded systems where memory and performance are both critical, float is usually the numeric type of choice as it is faster and half the size of a double. Integers used to be the weapon of choice, but floating point performance has overtaken integer in modern processors. Decimal is right out!

-
+1 for the most creative answer. –  DiwasP8 Jul 16 '14 at 4:54

No one has mentioned that

In default settings, Floats (System.Single) and doubles (System.Double) will never use overflow checking while Decimal (System.Decimal) will always use overflow checking.

I mean

decimal myNumber = decimal.MaxValue;
myNumber += 1;

throws OverflowException.

But these does not:

float myNumber = float.MaxValue;
myNumber += 1;

&

double myNumber = double.MaxValue;
myNumber += 1;

-
float.MaxValue+1 == float.MaxValue, just as decimal.MaxValue+0.1D == decimal.MaxValue. Perhaps you meant something like float.MaxValue*2? –  supercat Jan 14 at 0:21
@supercar But it is not true that decimal.MaxValue + 1 == decimal.MaxValue –  Grkmksk Jan 14 at 6:12
@supercar decimal.MaxValue + 0.1m == decimal.MaxValue ok –  Grkmksk Jan 14 at 6:19
The System.Decimal throws an exception just before it becomes unable to distinguish whole units, but if an application is supposed to be dealing with e.g. dollars and cents, that could be too late. –  supercat Jan 14 at 16:15

The Decimal, Double, and Float variable types are different in the way that they store the values. Precision is the main difference where float is a single precision (32 bit) floating point data type, double is a double precision (64 bit) floating point data type and decimal is a 128-bit floating point data type.

Float - 32 bit (7 digits)

Double - 64 bit (15-16 digits)

Decimal - 128 bit (28-29 significant digits)

More about...the difference between Decimal, Float and Double

-

the main difference between each of these is the precision.

float is a 32-bit number, double is a 64-bit number and decimal is a 128-bit number.

-


text:
# Einstein: Concerning the Aether

1. Sep 10, 2010

### Rasalhague

I first learnt of this paper in a FAQ by Benjamin Crowell which gave a link to Einstein's handwritten manuscript, but no English translation, so I had a go at making http://www.oe.eclipse.co.uk/nom/aether.htm [Broken]. Some of the physics is above my head, especially the stuff on radiation and magnetism towards the end and in the 2nd footnote, so all advice welcome, linguistic or physics-related. I've linked to two other sites where the original can be read more conveniently:

Digitised German text: Wikilivres

Scans of German text as published:
SEALS (Swiss Electronic Academic Library Service): Retrodigitized Journals

Last edited by a moderator: May 4, 2017
2. Sep 10, 2010

### turbo

The paper was previously translated by Saunders, and is Chapter One in "The Philosophy of Vacuum" by Saunders and Brown. The book is quite pricey, but you can find clean used copies pretty reasonably.

3. Sep 10, 2010

### bcrowell

Staff Emeritus
Cool! Thanks, Rasalhague, for making your translation publicly available!

Please consider dedicating it to the public domain or putting it under a license such as CC-BY or CC-BY-SA that would allow it to be saved permanently on wikisource ( http://en.wikisource.org/wiki/Author:Albert_Einstein ) and reused freely by others.

In case anyone is interested, here is the FAQ that Rasalhague referred to:

Didn't Einstein say that general relativity was an aether theory?

No. Einstein wrote a 1924 paper in which he made the philosophical point that although relativity killed off the luminiferous aether as the supposed medium of electromagnetic vibrations, it still imbued the vacuum with specific physical characteristics, such as curvature and energy. The basic point of the paper is that we can't decide, purely based on philophical ideas like Mach's principle, whether the vacuum has its own properties; we actually have to go through the usual scientific cycle of theory and experiment in order to find out the answer. Internet kooks love to misinterpret and overinterpret this paper, or to misrepresent it by saying that Einstein referred to GR in general, throughout his career, as an aether theory.

A. Einstein, "Über den Äther," Schweizerische naturforschende Gesellschaft 105 (1924) 85
original text - http://www.wikilivres.info/wiki/Über_den_Äther
English translation - http://www.oe.eclipse.co.uk/nom/aether.htm [Broken]
commentary by John Baez - http://web.archive.org/web/20070204022629/http://math.ucr.edu/home/baez/RelWWW/wrong.html

Last edited by a moderator: May 4, 2017
4. Sep 10, 2010

### Rasalhague

You're welcome! And thanks for all material you've put online that's made my world so much more intersting of late. I certainly have no objection to anyone copying, sharing or, if need be, improving on my translation. I've just added a note to that effect.

5. Sep 10, 2010

### Mentz114

Rasalhague, nice job. I look forward to reading it again.

6. Sep 10, 2010

### turbo

I just took the time to read through your translation and Saunders' translation side-by-side, and they are quite consistent, varying primarily in nomenclature as opposed to substance. Good job!

7. Sep 10, 2010

### Rasalhague

Thanks, turbo-1. That's a relief!

8. Sep 10, 2010

### Saw

I am also amazed by the quality of the job, Rasalhague. If there was another translation, better for us, thus we can compare, but for sure the fact that yours was uncontaminated by the previous one is an advantage of it, it endows your work with more added value.

On the other hand, as to the contents of the paper, I am even more amazed at the fact that some people can interpret it in the sense that Einstein declared that there was no aether. Here, like in the 1920 lecture at the university of Leyden (http://www.tu-harburg.de/rzt/rzt/it/Ether.html [Broken]), he makes it clear that, in his opinion, "some aether" that is "physically real" is a logical necessity to sustain the propagation of EM waves. Ok, he also claims that this aether is not mechanical, it is quite different from ponderable matter, as opposed to the air-sound analogy prevailing in the XIXth century; furthermore, he also claims that this aether, unlike Lorentz's, is deprived not also of mobility (it cannot be dragged) but also of immutability (since it is affected by the presence of ponderable matter). But existing..., of course it does exist, he says. Or not?

Last edited by a moderator: May 4, 2017
9. Sep 10, 2010

### turbo

Einstein said:
He specifically claimed that "empty space" had properties that were conditioned by the presence of mass and energy. Lately, Thanu Padmahnaban has toyed with an ether model, but his model is hampered, IMO, because he claims that the ether is an immutable entity, such as the Lorentz model that you cite. Such an ether might allow the development of a kinetic model of quantum gravity, but certainly not a dynamic one. Einstein's view was that the ether of GR was not a background against which gravitational and inertial effects play out passively, but is itself a player in the emergence of such effects.

10. Sep 10, 2010

### Rasalhague

How does Saunders treat the equation in note 2? Out of ignorance, I played safe and left it as it appears in the German, complete with fraktur letters:

$$d\mathfrak{h} = -C \, dm \, \frac{\left [ \mathfrak{v},\mathfrak{r} \right ]}{r^3}$$

But I was wondering whether it would be correct, and clearer, to replace $\left [ \mathfrak{v},\mathfrak{r} \right ]$ with $\textbf{v} \times \textbf{r}$ by analogy with

$$\textbf{B} = C q \frac{\textbf{v} \times \textbf{r}}{r^3}$$

11. Sep 13, 2010

### Saw

Yes, that is true. It is reciprocal: Einstein's ether is affected and it affects, too. But the point of my comment was not so much dwelling on the various features of Einstein's ether, but only remarking that it does exist, in his opinion!

Of course, you can call it "empty space" if you wish, as long as you admit it has properties. But if you admit it has properties, if you admit that it is "something" that interacts with mass and energy, what is the point of calling it with a name that suggests that it is "nothing"?

In any case, clearly that was not Einstein's intention. He never claimed that the ether is "nothing". Do you agree to that?

12. Sep 13, 2010

### turbo

I certainly agree that that was Einstein's intent. It is clear that he thought that the ether is a dynamic player in the emergence of gravitational and inertial effects.

13. Sep 13, 2010

### JesseM

It's really just a matter of semantics, but I'd say the major difference between Einstein's "ether" and the classical ether was that the classical version was an actual physical medium, so at any point in spacetime an object would either be at rest relative to the local ether or in motion relative to it, whereas there'd be no way to assign meaning to the idea of being locally at rest or in motion relative to Einstein's ether.

14. Sep 13, 2010

### granpa

Many people may not realize that the classical aether could not have been a simple elastic.
In fact it would have needed pretty miraculous properties to
explain the behavior of classical electric, magnetic, or gravitational fields

Last edited: Sep 13, 2010
15. Sep 13, 2010

### Saw

I thought so, thanks for confirming.

Well, as usual, I would not say it is “just” semantics if it has practical consequences. A better definition of a concept should enable us to make better predictions. For example, what you say about the difference between classical ether and Einstein’s ether is not irrelevant in practical terms. What Granpa says in the next post goes in the same line.

A different thing is which definition should we adhere to. What you point out is probably correct. It is impossible to measure a state of motion wrt the ether. It is no medium, like water or air, composed of parts of ponderable matter. But would you say it is not “physical”? Yes, thus we go into the ugly problem of defining what is physical, which is usually quite subjective. But if we choose to define “physical” as something that interacts with other physical things, would you agre that Einstein’s ether is physical?

16. Sep 13, 2010

### JesseM

Whether we use the word "ether" to describe the dynamical properties of spacetime in GR (as defined by the Einstein field equations) doesn't have any practical consequences that I can see.
Word definitions aren't really important in physics, it's all about the mathematical definitions. If one person uses the word "ether" to describe GR spacetime but the other doesn't, yet they both use the same equations to describe how mass/energy curves spacetime and what paths objects take through curved spacetime, they will make exactly the same predictions.
This verbal definition still seems too ambiguous--for example, you could see spacetime as just part of a quantitative description of how different bits of mass/energy interact with one another gravitationally (similarly, is a 'force' in Newtonian physics a 'something' that 'interacts with' objects, or just part of our mathematical description of the rules of how objects interact with one another?)

17. Sep 14, 2010

### Austin0

Hi saw Well I definitely agree with your interpretation.
His words seem clear and unambiguous in conceptual physical terms.
Spacetime may be purely emergent from the existence of matter yet still has properties , seemed to be the essence of his point.
I also saw no distinction being made between areas of relative concentration of matter.

The semantic ambiguity seems to arise through interpretation, whereby through suitable definition, the essential meaning somehow disappears entirely into the aether.

18. Sep 14, 2010

### Saw

Starting with the last point, I agree that in physics (and probably in all fields of knowledge) you do not really describe the objects themselves but only how they interact with one another. In this sense, you mentioned the concept of “force”, but you could have also referred to “mass”. Actually, the “mass” is not the object itself but the outcome of a competition between the same and another standard reference of similar kind, in terms of resistance to acceleration, given an interaction with a common challenge . The object itself is a piece of “matter”. Certainly, we don’t plug “matter” into the equations, only mass or other properties. However, we didn´t get rid of the concept of “matter”. Likewise, I do not see why we should dispense with the word “ether”. (Or maybe we should trascend both concepts, I don’t know…)

Yes, that is valid way to put it. But I still I find that it is advisable to distinguish, as a question of mental order, between the objects themselves and, as you mention, spacetime, i.e. a drawing representing the readings of clock and rulers placed along the path of an object.

Practical consequences? Better predictions? In my view, the problem of speaking only about spacetime is that then some people tend to think that this spacetime is “the object itself”, the ultimate answer to the why question and that leads to making absurd predictions or at least to accepting absurd possibilities, like time travel, but that is a subject for another thread. I do not want to high-jack this one, which was about the translation of Einstein’s paper and there was a new question by the OP:

19. Sep 16, 2010

### Rasalhague

Thanks for resurrecting my question, Saw. I found it on Google books:


text:
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 Trudy Inst. Mat. i Mekh. UrO RAN, 2014, Volume 20, Number 2, Pages 13–28 (Mi timm1055)

Optimal control with connected initial and terminal conditions

A. S. Antipina, E. V. Khoroshilovab

a Dorodnitsyn Computing Centre of the Russian Academy of Sciences
b M. V. Lomonosov Moscow State University, Faculty of Computational Mathematics and Cybernetics

Abstract: An optimal control problem with linear dynamics is considered on a fixed time interval. The ends of the interval correspond to terminal spaces, and a finite-dimensional optimization problem is formulated on the Cartesian product of these spaces. Two components of the solution of this problem define the initial and terminal conditions for the controlled dynamics. The dynamics in the optimal control problem is treated as an equality constraint. The controls are assumed to be bounded in the norm of $\mathrm L_2$. A saddle-point method is proposed to solve the problem. The method is based on finding saddle points of the Lagrangian. The weak convergence of the method in controls and its strong convergence in state trajectories, conjugate trajectories, and terminal variables are proved.

Keywords: terminal control, boundary value problems, convex programming, Lagrange function, solution methods, convergence.

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English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2015, 289, suppl. 1, S9–S25

Bibliographic databases:

UDC: 517.977

Citation: A. S. Antipin, E. V. Khoroshilova, “Optimal control with connected initial and terminal conditions”, Trudy Inst. Mat. i Mekh. UrO RAN, 20, no. 2, 2014, 13–28; Proc. Steklov Inst. Math. (Suppl.), 289, suppl. 1 (2015), S9–S25

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. S. Antipin, E. V. Khoroshilova, “O kraevoi zadache terminalnogo upravleniya s kvadratichnym kriteriem kachestva”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika, 8 (2014), 7–28
2. A. S. Antipin, O. O. Vasilieva, “Dynamic method of multipliers in terminal control”, Comput. Math. Math. Phys., 55:5 (2015), 766–787
3. A. S. Antipin, E. V. Khoroshilova, “Mnogokriterialnaya kraevaya zadacha v dinamike”, Tr. IMM UrO RAN, 21, no. 3, 2015, 20–29
4. A. Antipin, E. Khoroshilova, “Saddle point approach to solving problem of optimal control with fixed ends”, J. Glob. Optim., 65:1, SI (2016), 3–17
5. A. Antipin, E. Khoroshilova, “On methods of terminal control with boundary-value problems: Lagrange approach”, Optimization and Its Applications in Control and Data Sciences, In honor of Boris T. Polyak’s 80th birthday, Springer Optimization and Its Applications, 115, ed. B. Goldengorin, Springer, 2016, 17–49
6. A. S. Antipin, V. Jaćimović, M. Jaćimović, “Dynamics and variational inequalities”, Comput. Math. Math. Phys., 57:5 (2017), 784–801
7. A. Antipin, “Sufficient conditions and evidence-based solutions”, 2017 Constructive Nonsmooth Analysis and Related Topics (CNSA), Dedicated to the Memory of V.F. Demyanov, ed. L. Polyakova, IEEE, 2017, 11–13
8. E. Khoroshilova, “Minimizing a sensitivity function as boundary-value problem in terminal control”, 2017 Constructive Nonsmooth Analysis and Related Topics (CNSA), Dedicated to the Memory of V.F. Demyanov, ed. L. Polyakova, IEEE, 2017, 149–151
9. Antipin A. Khoroshilova E., “Controlled Dynamic Model With Boundary-Value Problem of Minimizing a Sensitivity Function”, Optim. Lett., 13:3, SI (2019), 451–473
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